Basic concepts of quantum interference andelectron transport in single-molecule electronics
C. J. Lambert
This tutorial outlines the basic theoretical concepts and tools which underpin the fundamentals of
phase-coherent electron transport through single molecules. The key quantity of interest is the
transmission coefficient T(E), which yields the electrical conductance, current–voltage relations, the
thermopower S and the thermoelectric figure of merit ZT of single-molecule devices. Since T(E) is
strongly affected by quantum interference (QI), three manifestations of QI in single-molecules are
discussed, namely Mach–Zehnder interferometry, Breit–Wigner resonances and Fano resonances. A sim-
ple MATLAB code is provided, which allows the novice reader to explore QI in multi-branched
structures described by a tight-binding (Huckel) Hamiltonian. More generally, the strengths and
limitations of materials-specific transport modelling based on density functional theory are discussed.
Key learning points(1) The central role of the transmission coefficient T(E) in determining electrical conductances and thermoelectric coefficients.(2) Three manifestations of QI in single-molecules, namely Mach–Zehnder interferometry, Breit–Wigner resonances and Fano resonances.(3) Strengths and weaknesses of density functional theory.(4) A simple tight-binding model of para-, meta- and ortho-linked rings.(5) An ability to model multi-branched structures.
A. Introduction
The aim of this brief tutorial is to provide an introduction tobasic theoretical concepts and tools, which pervade the field of
room-temperature single-molecule electronics. The technologygoal of the field is to enable the transformative development ofviable, molecular-based and molecularly-augmented electronicdevices. Such devices would enable new sensor technologiesfor pressure, acceleration and radiation, along with chemicalsensors capable of detection and analysis of a single molecule.They could also lead to new designs for logic gates and memorywith orders of magnitude reduction of both power requirementsand footprint area, and new thermoelectric devices with theability to scavenge energy with unprecedented efficiency. How-ever, realisation of these benefits of single-molecule electronicswill require the unprecedented assembly of bespoke moleculesin reproducible and scalable platforms with multiple electrodesseparated by molecular length scales. At present, the requiredlevel of control is far beyond current technologies and researchon single-molecule electronics is focussed mainly on the funda-mental science that will underpin future developments.
One example of such fundamental science is the room-temperature observation of quantum interference (QI) in singlemolecules connected to nano-gap electrodes. At a fundamentallevel, almost all anticipated performance enhancements associatedwith single-molecule electronics are derived from transportresonances associated with QI. Quantum interference controlsand is controlled by molecular conformation, charge distribution
C. J. Lambert
Colin Lambert joined LancasterUniversity in 1983 and wasawarded a Professorship in 1990.During the period 2005–2010 hewas Associate Dean of Research, inthe Faculty of Science and Techno-logy and in 2010 was awarded aResearch Professorship. He createdthe Lancaster Quantum TechnologyCentre and was the Founding Direc-tor until 2013. Over the past20 years, he has led five Europeanresearch collaborations, each invol-ving 10 EU partners. He has pub-
lished over 240 papers on the theory of quantum transport innanostructures and single-molecule junctions and in related areas.
Dept. of Physics, Lancaster University, Lancaster, LA1 4YB, UK.
and the energies of frontier orbitals. The symbiotic relationshipbetween quantum interference and physical, chemical and electronicstructure, which are sensitive to environmental factors and canbe manipulated through conformational control, polarisation orredox processes (electrical vs. electrochemical gating) leads tonew opportunities for controlling the electrical and thermoelectricalproperties of single molecules connected to nano-electrodes.Room-temperature QI in single molecules was demonstratedexperimentally only recently1–9 and it timely to identify newstrategies for exploiting QI in technologically-relevant platforms.
The tutorial below is intended to provide the backgroundconcepts and theoretical tools, which will inform the developmentof such strategies. The tutorial is based on notes and examplesgiven to early-stage researchers in my group and assumes only abasic background in mathematics.
B. The standard model of phase-coherent electron transport throughsingle-molecules
To begin, this tutorial I shall briefly outline a ‘standard model’of single-molecule electronics, both from a conceptual andcomputational viewpoint. The glossary below summarisessome of the key concepts and ingredients pervading currentresearch in room-temperature single-molecule electronics,including three types of resonances arising from QI, namelyBreit–Wigner, Fano and Mach–Zehnder resonances.
Not shown in the glossary are the electrodes, which makeelectrical contact to the anchor groups on the left and right ends ofthe molecules. Usually these are made from gold and contact ismade using a scanning – probe microscope of a mechanically-controlled break junction. However gold is rather mobile andforbidden from a CMOS lab. Consequently other electrode materialshave been explored, including Pt, Pd, carbon nanotubes andgraphene.9 Indeed recently, Si-based platforms for molecularelectronics have begun to be explored.10
To illustrate the standard approach to computing theelectrical conductance of single molecules, consider the mole-cule shown in Fig. 1, located between two gold electrodes.
The electrodes are connected to external reservoirs (not shown),which feed electrons of energy E into the gold electrodes. Theenergy distribution of electrons entering the left (right) gold leadfrom the left (right) reservoir is fleft(E) (fright(E)) and according to theLandauer formula, the current passing from left to right is
I ¼ 2e
h
� �ð1�1
dETðEÞ fleftðEÞ � frightðEÞ� �
(1)
where e = |e| is the electronic charge, h is Planck’s constant and T(E)is the transmission coefficient for electrons passing from one leadto the other via the molecule. Clearly I = 0 when fleft(E) = fright(E),because only differences in the distributions contribute to the netcurrent.
Close to equilibrium, fleftðEÞ ¼ eb E�EleftFð Þ þ 1
h i�1and
frightðEÞ ¼ eb E�ErightFð Þ þ 1
h i�1, where Eleft
F (ErightF ) is the Fermi
energy of the left (right) reservoir and b ¼ 1
kBT, where T is the
temperature. If V is the voltage difference between the left and
right reservoirs, then EleftF ¼ EF þ
eV
2and E
rightF ¼ EF �
eV
2. This
means that at zero temperature, but finite voltage
I ¼ 2e
h
� �ðEFþeV=2
EF�eV=2dETðEÞ (2)
Fig. 1 A molecule in contact with two gold electrodes.
Consequently, the conductance G = I/V is obtained by averagingT(E) over an energy window of width eV centred on the Fermienergy. On the other hand, if T(E) does not vary significantlyover an energy range of order eV, then the Fermi functions canbe Taylor expanded to yield the electrical conductance in thezero-voltage, finite temperature limit:
G ¼ I=V ¼ G0
ð1�1
dETðEÞ �df ðEÞdE
� �(3)
where G0 is the quantum of conductance,
G0 ¼2e2
h
� �
Since the quantity �df (E)/dE is a normalised probabilitydistribution of width approximately kBT, the above integralrepresents a thermal average of the transmission functionT(E) over an energy window of width kBT.
Finally, in the limit of zero voltage and zero temperature,one obtains
G = G0T(EF) (4)
The transmission coefficient T(E) is a property of the wholesystem comprising the leads, the molecule and the contactbetween the leads and the molecule. Nevertheless, if the contactto the electrodes is weak, then a graph of T(E) versus E will reflectthe energy-level structure of the isolated molecule. In particularif the isolated molecule has energy levels En, (where n representsthe quantum numbers labelling the energy levels), then T(E) willpossess a series of peaks (i.e. resonances) located at energies inthe vicinity of the levels En. These resonances will be discussed inmore detail below.
The above expressions assume that during the time it takesfor an electron to pass from one lead to the other, the system ofleads plus molecule can be described by a time-independentmean-field Hamiltonian H. This means that an electronremains phase coherent as it passes from one gold lead tothe next and does not undergo inelastic scattering. For shortmolecules, there is evidence that this is a reasonable assumption.11
However as the length of a molecule increases, the probability ofinelastic scattering (e.g. from phonons or other electrons) becomesnon-negligible12 and the above expressions require modification. Inwhat follows, we shall confine the discussion to systems where theabove expressions are valid.
On the other hand, just because a Hamiltonian is static onthe timescale required to pass through the molecule, does notmean that the system is completely stationary, because on alonger timescale, the molecule may change its shape, theelectrodes may distort or the environment surrounding themolecule may change. In this case, in the spirit of the Born–Oppenheimer approximation, one should construct an ensembleof Hamiltonians describing each realisation of the system (i.e.electrodes, molecule and environment) and compute T(E) foreach case. Since the environment and molecular conformationcan fluctuate many times on the timescale of a typical electricalmeasurement, an ensemble average of such transmission functionsis required. This ensemble average (i.e. average over snapshots)
does not include the even slower changes in H associated withthe pulling apart of electrodes in a break-junction experiment,or with the changes associated with the switching on of a dcbias or gate voltage. In the presence of such slow changes, anensemble average should be carried out at each electrodeseparation or gate voltage.
Starting from an arbitrary instantaneous mean-field HamiltonianH in a localised basis, the problem of computing the transmissioncoefficient T(E) was solved in13 and therefore the main problemis how to obtain H. Unfortunately there is no ‘‘text book ofHamiltonians’’ and theoreticians have to resort to a range ofapproximations and tricks to construct reasonable models. Inthe paper13 and in other papers of that decade, the strategy usedto obtain a Hamiltonian was to fit the matrix elements of a tight-binding Hamiltonian to band structures of the materials ofinterest. This is perfectly respectable, but it does not solve theproblem of modelling interfaces between different materials. Atthat time, a commonly-used approach was to approximate thematrix elements describing an interface between two materialsby a geometric average of the matrix elements of the individualmaterials. Clearly there is no microscopic justification for suchan approach. In the case of single-molecule electronics, anaccurate description of interfaces between the electrodes andthe molecule is vital and therefore an alternative approach isnecessary. To overcome this problem, the SMEAGOL code14,15
(Spin and Molecular Electronics in Atomically-Generated OrbitalLandscapes) was created to compute T(E) from the mean-fieldHamiltonian of the density functional theory (DFT) code SIESTAand indeed to modify the SIESTA Hamiltonian in the presence ofa finite bias. Similar strategies have been adopted by manygroups using a range of DFT codes, including TRANSIESTA16
and TURBOMOLE.17 Recently a next-generation code (GOLLUM)has been released,18 which is freely distributed and additionallydescribes the environmental effects due to a surrounding solventor gaseous atmosphere, the evolution of electrical propertiesduring the pulling apart of electrodes in break junction experi-ments and a range of other more exotic effects associated withthermoelectricity, superconductivity and magnetism.
The above DFT-based approach is widely adopted and can beconsidered a ‘standard model’ of phase-coherent electrontransport through single-molecules. The good news is thatDFT can predict ground state energies, binding energies,molecular conformations, trends in HOMO–LUMO gaps inhomologous series of molecules, symmetries of orbitals and arange of other properties (see below). The bad news is that DFTis not an exact theory and the underlying Hamiltonian cannotbe relied upon to reproduce essential quantities such as thepositions of the HOMO and LUMO levels relative to the Fermienergies of electrodes. There are many reasons for this lack ofreliability, ranging from approximations associated with theunderlying density functional, the fact that Kohn–Sham eigen-values are not the energy levels of a system and the presence ofmirror charges and screening due to the nearby electrodes.19
To illustrate this deficiency of DFT, consider the theoreticalpredictions for the HOMO–LUMO gaps of simple linear carbonchains (oligoynes) with various end caps, shown in Fig. 2. In the
limit that the number 2n of carbon atoms in the chain tends toinfinity (i.e. in the limit 1/n tends to zero), these should all agree,but clearly this is not the case. A SIESTA LDA implementation ofDFT yields a much-too-low value of less than 1 eV. For compar-ison, Fig. 2 shows the result of a GW many-body calculation,20
which for an infinite polyyne chain yields a HOMO–LUMO gap of2.2 eV, which is close to the literature consensus of experimentalvalues, shown in Fig. 3.
Despite the superior performance of GW, DFT remains a widely-used workhorse, because with currently-available computers, GWcan handle only small numbers of atoms and cannot describe manyof the molecules of interest for single-molecule electronics.
To accommodate the inadequacies of DFT, a DFT mean-fieldHamiltonian should be adjusted to accommodate knownexperimental facts or outputs from more accurate many-bodysimulations. In particular if the HOMO–LUMO (H–L) gap EG =EL � EH is known experimentally, then a scissor correction21,22
should be implemented so that the corrected Hamiltonianreproduces the known value. In addition if the locations ofthe HOMO and LUMO levels EH, EL relative to the Fermi energyEF of the electrodes are known, then the relative position of theFermi energy should be adjusted to correct value. Once the twoenergy differences EL � EF and EH � EF are fixed by experiment,then the resulting phenomenological Hamiltonian and associatedpredictions for transport properties are probably the most accurateobtainable. This approach mirrors that used widely by the
mesoscopics community to describe transport through quantumdots, where charging energies, electron affinities and ionisationpotentials are usually fixed to agree with experiment and rarelycomputed from first principles.
Within the related field of organic optoelectronics the questionof energy level alignment has also been investigated in some detail.Energy barriers for electron and hole injection are determinedby the offset from the HOMO and LUMO molecular levels of thecontact layer with respect to the Fermi level EF of the metalelectrode. Key parameters53 are the vacuum level of the metalEM
vac, the vacuum level of the organic EOvac, the work function of
the metal jM = EMvac � EF, the ionisation energy IE and electron
affinity EA of the organic material. If the interaction betweenthe organic and the metal is sufficiently weak, then one wouldexpect Schottky–Mott-like behaviour, in which EM
vac = EOvac. On
the other hand, if the interaction is stronger, then charge-transfer-induced surface dipoles and other chemical effectscauses the vacuum levels to differ such that EM
vac � EOvac = D.
In this case the hole and electron injection barriers are Eh =EH� EF = EF� IE + D and Ee = EF� EL = EF� EA + D respectively.Photoelectron spectroscopy (PES) clearly demonstrates theabsence of vacuum level alignment in many metal–organicinterfaces. In one such study54 Eh is measured as a functionof the work function jM of the metal. If the vacuum levels werealigned or if D is independent of jM, then eventually the holebarrier Eh would vanish. In practice, as EF approaches EHOMO,charge transfer causes the surface dipole to increase, theHOMO to be pinned to (just below) the Fermi level and the
Fig. 2 Theoretical values of the HOMO–LUMO gap (Eg) of oligoynes andas a function of 1/n (where n equals the number of carbon pairs) usingvarious methods. The last two items of data correspond to recent resultsfrom my own group using LDA in the SIESTA code and GW in VASP. Forreferences to original data, see the SI of ref. 20.
Fig. 3 Experimental values of the HOMO–LUMO gap of short oligoynesas a function of 1/n (where n equals the number of carbon pairs) fordifferent molecules terminated with symmetric and asymmetric arrange-ments of end groups. Linear extrapolation to the limit of infinite oligoynelength estimates the band gap of polyyne between 2.0 and 2.3 eV in themajority of these measurements, which is in excellent agreement with theGW results of Fig. 2. For references to original data, see the SI of ref. 20.
vacuum offset D to increase. These effects are not reliably predictedby density functional theory and one must rely on PES experimentsto reveal the values of the above parameters.
Within single molecule junctions, other useful informationfor locating the energy differences EL � EF and EH � EF includesthe shape of current–voltage plots, the sign of the thermopower(see below), spectroscopic data and the temperature dependence ofthe electrical conductance. Trends in measured properties ofhomologous series of molecules and conductance changes due toelectrostatic gating or the surrounding environment also providevaluable information for locating EF. For example, systematicexperimental and theoretical studies with thiol-terminatedaliphatic and aromatic molecular wires confirm that electronscan remain phase coherent when traversing single molecules,with a non-resonant tunneling process as the main transportmechanism. The tunneling decay parameter b in G = A�e�bL, withG and L as the molecular conductance and length, respectively,can be as large as (0.4 Å�1) for oligo-para-phenylenes23 and(0.2 Å�1) for oligophenyleneethynylenes.24 Smaller b values arefound for oligothiophenes (b = 0.1 Å�1),25 oligoynes (b =0.06 Å�1),26 and Zn-porphyrin-containing wires.27 Such valuesof b is can help to determine the position of the Fermi energyrelative to the LUMO or HOMO levels. In principle, such correc-tions account for the effect of image charges in the electrodes,which are known to renormalize EL � EF and EH � EF by up toseveral hundred meV,19 with the precise value depending on theelectrode spacing. More generally, DFT may also incorrectlypredict the order of energy levels, compared with more accuratemany-body calculations. In this case, a self-energy correctionshould be implemented to correct both the order and values ofenergy levels.28
C. Influence of quantum interferenceon electron transport throughsingle-molecules
Having dwelt upon the skeletons in the cupboard of DFT, I nowdiscuss some concepts which underpin descriptions of QI withinsingle molecules and which allow us to interpret the outcomes ofcomplex material-specific DFT-based calculations. Since QI takesmany forms, it is worth noting that QI in isolated molecules isdistinct from QI in molecules connected to conducting electrodes.In the former, the system is closed and constructive QI leads to theformation of molecular orbitals {|jni} and a discrete energyspectrum {En}, obtained by solving the Schrodinger equationH|jni = En|jni. In the latter, the system is open and the energiesE of electrons entering a molecule form a continuum. In this case,once the Hamiltonian is known, we are interested in calculatingthe transmission coefficient T(E) of the molecule, which describesthe transmission of de Broglie waves of energy E, which enter themolecule from the electrodes. The transmission coefficient T(E) canexhibit both constructive and destructive interference, whereas theorbitals {|jni} are formed from constructive interference only.
In an open system, QI should be described in terms of theinterference pattern of de Broglie waves due to electrons
traversing a molecule from one contact to another. The tech-nical term for such an interference pattern is a Greens function.As a simple analogy, imagine a singer at the entrance to anauditorium, emitting a note of constant frequency. If the note issustained for long enough this will set up a standing wavewithin the auditorium, with a complex pattern of nodes andantinodes. When the singer is located at position i, then theamplitude of the standing wave at position j is proportional tothe Greens function G( j,i). If a window is opened at position jand the location of the window coincides with an antinode ofthe standing wave G( j,i), a loud sound will exit from thewindow. This is an example of constructive interference. Onthe other hand, if the window is located at a node, no soundwill exit and destructive interference is said to occur. In thisanalogy, the auditorium and the sound wave are analogues of amolecule and a de Broglie wave respectively and in a modelbased on atomic orbitals, i and j label atomic orbitals at specificlocations within a molecule. In this case, G( j,i) is the amplitudeof a de Broglie wave at j, due to a source at i and is a property ofthe isolated molecule. If an electrode is attached to i = 1 and asecond electrode attached to j = 2, then the transmissioncoefficient T(E) will vanish when G(2,1) = 0; i.e. when atomicorbital 2 coincides with a node.
In what follows, four examples of QI in molecules aredescribed by analysing in detail a simple tight-binding model ofp orbitals. Since this is a tutorial introduction, many mathematicalsteps are included, so the reader should be able to reproduce allresults with relative ease.
Example 1. Mach–Zehnder interferometers
As a first example of QI and to further contrast QI in opensystems with QI in closed systems, Fig. 4a shows an example ofa tight-binding ring with 6 atomic orbitals (sometimes referredto as ‘atoms’ or ‘sites’ for brevity), labelled j = 1, 2,. . .6,described by a Schrodinger equation
e0jn1 � gjn
6 � gjn2 = Enj
nj1
e0jnj � gjn
j�1 � gjnj+1 = Enj
nj ( j = 2, 3, 4, 5)
e0jn6 � gjn
5 � gjn1 = Enj
nj1 (5)
In this simple tight-binding (Huckel) model, jnj is the
amplitude of the nth molecular orbital on atom j and En isthe energy of the nth molecular orbital. The parameters e0 and�g are ‘‘site energies’’ and ‘‘hopping elements’’ determinedby the type of atom in the rings and the coupling between the atoms.
Fig. 4 (a) Shows a closed system, (b) shows an open system with paracoupling to the leads and (c) shows an open system with meta coupling.
To solve this set of equations, it is illuminating to first considerthe Schrodinger equation of an infinite chain of such atoms,which takes the form
e0jj � gjj�1 � gjj+1 = Ejj (�N o j o N) (6)
The solution to this equation is jj = eikj, where �p o k o p.Substituting this into eqn (5b) yields (e0� ge�ik� geik)eikj = Eeikj,so after cancelling eikj on both sides, one obtains the dispersionrelation
E = e0 � 2g cos k
This means that the 1-d chain possess a continuous band ofenergies between E = e0 � 2g and E = e0 + 2g. More generally, thesolution to eqn (6) is jj = eikj + ae�ikj, where a is an arbitraryconstant. For such an open system, the problem is not tocompute E, since any value within the band is allowed. Insteadwe choose E and compute properties of interest as a function ofE. For example, the dimensionless wave vector k is given by
k(E) = cos�1[(e0 � E)/2g] (7)
and the group velocity is v(E)/�ha where a is the atomic spacing, �h
is Planck’s constant and
vðEÞ ¼ dE
dk¼ 2g sin kðEÞ: (8)
(The latter expression illustrates why it is convenient to use�g rather than g to denote hopping elements, because with thisnotation, if g is positive, then v(E) has the same sign as k(E).)
In contrast with the open system described by eqn (6), whenthe solution jn
j = eikj is substituted into eqn (5), there are only 6linearly independent solutions which also satisfy the first andlast equations are found to be jn
j = eiknj, where
kn ¼2pn6; n ¼ �2;�1; 0; 1; 2; 3, with energies En = e0� 2g cos kn.
Since energies corresponding to kn and �kn are degenerate, thecomplex exponentials eiknj and e�iknj can be added or subtracted toyield the following 6 real solutions: jn
j = sin kn j (where n = 1, 2)and jn
j = cos knj (where n = 0, 1, 2, 3). Clearly the levels E1 and E2
are doubly degenerate and the levels E0 and E3 are non-degenerateand in contrast with the open system of eqn (6), the allowedenergies are no longer continuous.
This example can be considered to be a simple model of pelectrons in a phenyl ring, with one electron per orbital, inwhich case the 6 electrons occupy the lowest three levels up tothe HOMO EH = E1 and the LUMO is EL = E2. Clearly the mid-point of the HOMO–LUMO gap is %E = (EH + EL)/2 = e0 and theparameters e0, g could be approximated by choosing E1 and �E2
to coincide with the ionisation potential and electron affinity ofthe molecule.
Now consider attaching a 1-dimensional lead to atom i = 1and a second 1-dimensional lead to a different atom j, to createan open system with two electrodes. If j = 4 (as shown inFig. 4b), then this is known as a para-coupled ring, whereasif j = 3 (or 5), as shown in Fig. 4c, it is a meta-coupled ring.
For j = 2, the coupling is ‘ortho’. In physics, the optical analogueof such a structure is known as a Mach–Zehnder interferometer.
Since the leads are infinitely long and connected to macro-scopic reservoirs (not shown), systems 4b and 4c are opensystems. In these cases, the transmission coefficient T(E) forelectrons of energy E incident from the first lead is obtained bynoting that the wave vector k(E) of an electron of energy ofenergy E traversing the ring is given by eqn (7). When Ecoincides with the mid-point of the HOMO–LUMO gap, i.e.when E = e0, this yields k(E) = p/2. Since T(E) is proportional to|1 + eikL|2, where L is the difference in path lengths between theupper and lower branches, one obtains constructive interfer-ence in the para case, (where eikL = ei0 = 1) and destructiveinterference in the meta case, (where eikL = ei2k = �1).
More precisely, the Greens function of the isolated ring of Natoms in Fig. 4a (see example 4 below for derivation) is
Gring( j,i) = A cos k(| j � i| � N/2) (9)
where N = 6 for the ring in Fig. 4a. In this expression, k is givenby eqn (7) and the amplitude A is
A ¼ 1
2g sin k sin kN=2(10)
The transmission in the case of para coupling is determinedby the Greens function Gring(4,1) = A. In the case of metacoupling, it is determined by Gring(3,1) = A cos k. In the caseof ortho coupling, it is determined by Gring(2,1) = A cos 2k.
In the centre of the HOMO–LUMO (H–L) gap, where k = p/2,these reduce to A, 0 and �A for the para, meta and ortho casesrespectively, which demonstrates that para and ortho corre-spond to constructive QI whereas meta corresponds todestructive QI.
As well as illustrating the effects of constructive and destruc-tive QI, the above expressions also illustrate the non-classicalmanner in which conductances add in parallel. For example,since Fig. 4b has two branches in parallel, (i.e. one branchformed from atoms 2 and 3 and the other from atoms 6 and 5)the classical conductance of Fig. 4b should be twice that of asingle branch. However for a linear chain of N0 atoms, theGreens function connecting atoms 1 and N0 at opposite ends ofthe chain is29
GchainðN 0; 1Þ ¼� sin k
g sin kðN 0 þ 1Þ (11)
which for N0 = 2 reduces to 1/g at the H–L gap centre. Hence atthe gap centre,
Gringð2; 1ÞGchainðN 0; 1Þ
¼ 1
2
which means that when the leads are only weakly coupled tothe ring, the transmission of the para-ring at the gap centre is(1/2)2 = 1/4 of the transmission of a single linear chain. Thisdemonstrates that QI must be respected when combiningconductances at a molecular level.
Finally, for completeness it is worth noting that the aboveGreens functions are those of isolated rings and chains, not
connected to leads. As shown in example 4 below, when theleads are attached, the Greens functions become
Gringð j; iÞ ¼cos k j j � ij �N=2ð Þ
2g sin k sinkN
2þ sring
(12)
and
GchainðN 0; 1Þ ¼� sin k
g sin kðN 0 þ 1Þ þ schain(13)
where sring and schain are self energies due to the attachment ofthe leads. (Both sring and schain are equal to zero in the limitthat the coupling to the leads vanishes.)
As real life examples of constructive and destructive interferencein polycyclic aromatics, Arroyo et al.7 presented experiments onphenyl rings connected by either meta or para coupling toanchor groups. And Aradhya et al.4 presented an experimentalstudy of stilbene derivatives comprising a para-terminated4,4-di-(methylthio)stilbene, moderately conducting 1,2-bis(4-(methylthio)phenyl)ethane and a lower-conductance meta-terminated 3,30-di(methylthio)stilbene.
Example 2. The Breit–Wigner formula
The above example shows that QI between multiple paths canlead to resonances or anti-resonances in the transmissionfunction T(E) controlling charge transport and therefore ifthese peaks or dips in T(E) are rather narrow, large values ofthe slope dT(E)/dE can be obtained. If this occurs near theelectrode Fermi energy, then the electrical conductance of themolecule can be extremely sensitive to changes in the environ-ment, including changes in the surrounding atmosphere, thetemperature, a gate voltage or changes in molecular conforma-tion or electrode spacing. If such sharp features in T(E) can becontrolled, then the system may behave as a sensitive detector,transistor or an efficient thermoelectric device.
The ability to tune molecular orbitals (e.g. by varying theparameters e0 and �g in eqn (5)) is an example of howconstructive interference can be exploited to control transport,since molecular orbitals are a direct result of constructiveinterference within the isolated molecule. For electrons ofenergy E passing through a single molecular orbital jn
j , thesimplest description of constructive interference is provided bythe Breit–Wigner formula30
T(E) = 4G1G2/[(E � en)2 + (G1 + G2)2], (14)
where T(E) is the transmission coefficient of the electrons, G1
and G2 describe the coupling of the molecular orbital to theelectrodes (labeled 1 and 2) and en = En � S is the eigenenergyEn of the molecular orbital shifted slightly by an amount S dueto the coupling of the orbital to the electrodes. This formulashows that when the electron resonates with the molecularorbital (i.e. when E = en), electron transmission is a maximum.The formula is valid when the energy E of the electron is closeto an eigenenergy En of the isolated molecule, and if the levelspacing of the isolated molecule is larger than (G1 + G2). In thecase of a symmetric molecule attached symmetrically to
identical leads, G1 = G2 and therefore on resonance, whenE = en, T(E) = 1. On the other hand in the case of an asymmetricjunction, where for example G1 c G2, the on-resonance value ofT(E) (when E = en) is approximately T(en) = 4G2/G1, which ismuch less than unity.
Although the resonance condition (E = en) described by theBreit–Wigner formula is a consequence of constructive inter-ference, it is worth noting that the formula also containsinformation about destructive interference. As an example,for the case of 1-dimensional leads, G1 is proportional to(jn
i )2 and G2 is proportional to (jnj )2, where jn
i and jnj are the
amplitudes of the wavefunction (i.e. molecular orbital) evalu-ated at the contacts and if either jn
i or jnj coincide with nodes
then G1 or G2 will vanish. For this simple example, G( j,i) =jn
i jnj /(E � en) and clearly vanishes when jn
i or jnj coincide with
nodes. More generally, G1 and G2 are determined by matrixelements at the contacts,30 which may vanish in the presence ofdestructive interference at the molecule–electrode interface.
Example 3. Fano resonances
In this third example of QI, we examine how destructivequantum interference can be used to control molecular-scaletransport. The control of electron transport by manipulatingdestructive interference was discussed in ref. 31–34, where itwas shown that Fano resonances can control electrical trans-port and lead to giant thermopowers and figures of merit insingle-molecule devices.
Fano resonances occur when a bound state is coupled to acontinuum of states. For a single molecule connected tometallic electrodes, the continuum of states is supplied bythe metal and therefore, not surprisingly, this type of destruc-tive interference is usually not present in the isolated molecule.In ref. 32, it was shown that when a ‘pendant’ orbital of energyep is coupled to the above Breit–Wigner resonance via acoupling matrix element a, eqn (14) is replaced by
T(E) = 4G1G2/[(E � e)2 + (G1 + G2)2] (15)
where e = en + a2/(E � ep). This formula shows that when theelectron anti-resonates with the pendant orbital (i.e. whenE = ep), e diverges and electron transmission is destroyed. Italso shows that a resonance occurs when E � e = 0. i.e. when(E � en)(E � ep) + a2 = 0. There are two solutions to thisquadratic equation. The solution close to E = en is the Breit–Wigner resonance, whereas the solution close to E = ep is aresonance close to the anti-resonance. This means that a Fanoresonance is composed of both an anti-resonance and a nearbyresonance (Fig. 5).
The vanishing of electron transmission due to a Fanoresonance is particularly dramatic and can be distinguishedfrom more general forms of destructive interference, such asmultiple-path quantum interference, because the latter is sen-sitive to the changes in the positions of the electrodes, whereasthe former is not. In the latter case, if for a specific location ofelectrode, G2 = 0, then G2 can be made non-zero by moving theelectrode to a nearby location. On the other hand, when E = ep,eqn (15) shows that T(E) vanishes no matter how large or small
the values of G1 or G2 and provided eqn (15) remains valid, thiscannot be remedied by adjusting the values of G1 or G2.
Finally, it should be noted that since a Fano resonance consistsof a ‘dip’ at E = ep, and a ‘peak’ at E = e, then if the Fano resonanceappears near the Fermi energy and if its position fluctuates rapidlyon the time scale of a measurement, possibly due to environmentalfluctuations, then the Fano ‘peak’ will dominate the averageconductance, rather than the ‘dip’, because the average of a largeand a small positive number is dominated by the larger number.[For example (1 + 10�6)/2 is approximately 1/2.]
The possibility of using Fano resonances to control electrontransport through single molecules was first highlighted inref. 31 where Fano resonances due to cross-conjugated oxygensand pendant bipyridyl units were discussed. This was followedby a joint experimental and theoretical study32 of the three 4 nm-long molecules shown in Fig. 6, which possess different pendantsubstituents on the central fluorene unit, namely the fluorenone 1,which possess a cross-conjugated oxygen, 2 a 1,3-dithiol-2-ylidene(electron donor) and 3 a di(4-pyridyl)methylene (electron acceptor).Clearly the LUMOs of 1 and 3 are localised on the pendant unitsand as shown in ref. 32 possess Fano resonances in their trans-mission coefficients near E = EF. In contrast, 2 possesses no suchFano resonance. Fano resonances were also discussed in ref. 33and their application to thermoelectricity discussed in ref. 34.
Interesting theoretical discussions were also presented inref. 35–38. Experimental and theoretical evidence of destruc-tive interference associated with cross-conjugated pendantoxygens in anthraquinones (such as that shown in Fig. 1)was present in ref. 1, followed by related experiments inref. 3, 5 and 8.
Example 4. A multi-branched ‘molecule’
As an example of a more complex structure exhibiting QI, whichcan be solved analytically and contains the above three forms ofQI as special cases, consider the multi-branched structureshown in Fig. 7, which is composed of (generally different) leftand right leads connected to a structure containing M (generallydifferent) branches.
An analytic formula for the transmission coefficient of theabove structure is presented in ref. 29, where it is shown thatthe transmission coefficient is given by
TðEÞ ¼ vLaLgL
� �2GRLj j2 bR
gR
� �2vR (16)
In this expression, vL(vR) is the electron group velocity in theleft (right) lead, gL(gR) is the hopping element in the left (right)lead, aL(bR) are the coupling between the left (L) and right (R)nodal atom to the left (right) lead and GRL is the Greensfunction of the whole structure describing a wave propagatingfrom nodal atom L to nodal atom R. Clearly, this formulacaptures the intuitively-obvious property that for a wave topropagate from the left to the right lead, the electron velocitiesin the leads, their coupling to the nodal atoms and GRL must allbe non-zero.
Appendix 1 (below) provides a simple MATLAB code, whichevaluates this formula and provides plots of the transmissioncoefficient versus energy for user-selected, many-branchedstructures. It also evaluates the individual currents in each ofthe branches. Interestingly, although these currents must addup to the total current through the device, the individualcurrents in some branches can be greater than the total current,because the currents in other branches can be in the oppositedirection. By modifying this few lines of MATLAB, the readercan have a little fun and learn about quantum interference by
Fig. 5 A plot of eqn (14), for two different choices of e1. For the blackcurve ep = 0.9. For the brown curve ep = �1.5. The BW resonance occurs atE = en = 1.5.
Fig. 6 Three molecules with central fluorene units possessing eithera pendant oxygen 1, a pendant 1,3-dithiol-2-ylidene 2 or a pendantdi(4-pyridyl)methylene 3.
Fig. 7 A multi-branch structure with nodal sites L and R (on the left andright) connecting external current-carrying leads, by hopping matrix ele-ments �aL (on the left) and �bR (on the right), and to internal branches (l),by hopping matrix elements �al and �bl, respectively. The energies of thenodal sites are e0
L and e0R. The site energy and hopping matrix element of
evaluating the transmission coefficient of a variety of differentmodel systems.
To utilise eqn (16), the hopping elements gL, gR, gl andorbital energies eL, eR, el defining the left (L) and right (R) leadsand each branch l should be chosen. For a given energy E, thewavevectors in L, R and l are then given by kL(E) = cos�1(eL � E)/2gL, kR(E) = cos�1(eR � E)/2gR and kl(E) = cos�1(el � E)/2gl. Thesigns of the wave vectors are chosen such that the corres-ponding group velocities vL = 2gL sin kL(E), vR = 2gR sin kR(E)and vl = 2gl sin kl(E) are positive, or if the wavevector is complex,such that the imaginary part is positive. Next the orbitalenergies e0
L, e0R of the nodal sites L and R and their respective
couplings �aL, �al and �bR, �bl to the leads and branchesshould be chosen.
The final step in evaluating eqn (16) is to compute theGreens function GRL connecting the left nodal site L to theright nodal site R via the expression:
GRL = y/D (17)
In this equation, the numerator y contains informationabout the branches and their couplings to the nodal sites only(i.e. it contains no information about the left and right leads)and describes one contribution to quantum interference due tothe multiple paths within the structure of Fig. 7. It is given bythe following superposition of contributions from each of the Mbranches:
y ¼XMl¼1
yl (18)
where yl describes the ability of a wave to propagate from theleft to the right nodal site and is given by
yl ¼albl sin kl
gl sin kl Nl þ 1ð Þ (19)
where Nl is the number of atoms in branch l. (For the specialcase Nl = 1, one should choose al = bl = gl.)
This expression immediately reveals an interesting rule forcombining conductors in parallel, which is valid in the limitthat D does not depend on the number or nature of thebranches. (This approximation will be discussed further below.)In this case, for a structure containing a single branch l,eqn (16) yields for the transmission coefficient Tl(E)
Tl(E) = B( yl)2
where B ¼ vLaLgL
� �2jDj�2 bR
gR
� �2vR
On the other hand in the presence of two branches (l = 1 andl = 2), the transmission coefficient is
corresponding conductances (G, G1 and G2) are obtained by
multiplying the transmission coefficients by the quantum ofconductances, this means that the conductances satisfy
G ¼ G1 þ G2 þ 2ffiffiffiffiffiffiffiffiffiffiffiG1G2
p: (20)
If the branches are identical, so that G1 = G2, then one findsG = 4G1, instead of the classical result obtained from Kirchoff’slaw for two conductors in parallel of G = 2G1. This expressionwas derived in ref. 39. For three branches in parallel, thecorresponding expression is clearly
G ¼ffiffiffiffiffiffiG1
pþ
ffiffiffiffiffiffiG2
pþ
ffiffiffiffiffiffiG3
p� �2(21)
However, it should be emphasised that this expression isnot always valid, because the condition of ‘‘constant D’’ is notuniversally satisfied and in general the quantities yl do not havethe same sign. (For example the Mach–Zehnder ring consideredin example 1 has the property that the conductance of twobranches in parallel is less than the conductance of a singlebranch, by a factor of 1/4.)
To understand more precisely how parallel branches com-bine to yield the overall transmission coefficient, we need toevaluate the denominator D of eqn (17), which is given by29
D = y2 � (aL � xL)(aR � xR) (22)
In this expression, the quantities xL and xR describe how awave from the left or right nodal sites is reflected back to thosesites and are given by
xL ¼XMl¼1
xLl (23)
xR ¼XMl¼1
xRl (24)
where
xLl ¼al2 sin kl Nlð Þ
gl sin kl Nl þ 1ð Þ (25)
and
xRl ¼bl
2 sin klðNlÞgl sin klðNl þ 1Þ (26)
Finally, the quantities aL and aR contain information aboutthe nodal site energies and their coupling to the left and rightleads and are given by
aL ¼ e0L � E
� aL2
gLeikL (27)
and
aR ¼ e0R � E
� bR2
gReikR (28)
Eqn (22), reveals that the condition of ‘‘constant D’’ isapproximately satisfied if y, xL and xR are small compared withaL and aR, which means that the branches should be coupledonly weakly to the nodal sites.
Eqn (16) contains a great deal of information about mani-festations of quantum interference in a variety of contexts. Forexample in ref. 29, it is shown how the Breit–Wigner formula (14) isobtained from this equation as special case and how odd–eveneffects arise in linear chains of atoms. As another example, I nowderive eqn (12) for the Greens function GRL of the rings of atomsshown in Fig. 4b and c. In this case, there are two identicalbranches with N1 atoms in branch 1 and N2 atoms in branch 2.For a para-connected phenyl ring, N1 = N2 = 2, while for a metaconnect ring, N1 = 1 and N2 = 3. Since all atoms are identical, all siteenergies are chosen to be equal to a constant e0 and all couplings(except aR and aL) equal to g, i.e. a1 = b1 = g1 = g. This means that allwave vectors are equal to k(E) = cos�1(e0 � E)/2g and xL = xR.
First consider the case of an isolated ring (Fig. 4a) for which
aL = bR = 0, in which case aL = aR = 2g cos k, xl ¼g sin kl Nlð Þ
sin kl Nl þ 1ð Þ,
yl ¼g sin kl
sin kl Nl þ 1ð Þ. Since g cos k� xl ¼ g sin kCl
Sl- where Sl = sin
kl(Nl + 1) and Cl = cos kl(Nl + 1), one obtains
aL � x ¼ g sin kC1
S1þ C2
S2
� �, y = g sin k(S1 + S2)/S1S2 and
D ¼ 4g2 sin2 kS1S2
sin2 kN=2, where N = N1 + N2 + 2. These combine
to yield
GRL ¼y
D¼
cos kN1 �N2
2
� �
2g sin k sin kN=2(29)
which is identical to eqn (8), because with the notation ofFig. 4a, for any choice of i and j in eqn (8) |i� j| = N1 + 1 and N =N1 + N2 + 2. More generally, when the coupling to the left andright leads (aL and bR) are not zero, aL = 2g cos k + sL where
sL ¼ e0L � e0
� aL2
gLeikL and similarly for aR. In this case, we
obtain eqn (12)
GRL ¼cos k
N1 �N2
2
� �
2g sin k sinkN
2þ sring
(30)
where
sring ¼2g sin k sin kN sL þ sRð Þ � S1S2sLsR
2g sin k sinkN
2
(31)
Furthermore, the calculation can easily be repeated for asingle branch to yield
GRL ¼� sin k
g sin k N1 þ 3ð Þ þ schain(32)
where
schain = �2 sin k(N1 + 2)(sL + sR) � sin k(N1 + 1)sLsR/g (33)
(Note that in the notation of eqn (13), N0 = N1 + 2).
As an example, for N = 6, k = p/2, eqn (31) for the a ring yields
GRL ¼�2g cos k N1 �N2
2
� �
4g2 � sin k N1 þ 1ð Þ sin k N2 þ 1ð ÞsLsR(34)
For the para case, where N1 = N2 = 2, this yields
GRL ¼�2g
4g2 � sLsR(35)
For the meta case, where N1 = 1, N2 = 3, it yields GRL = 0 andfor the ortho case, where N1 = 0, N2 = 4, it yields
GRL ¼2g
4g2 � sLsR(36)
These expressions demonstrate that at the centre of theHOMO–LUMO gap, ortho and para couplings lead to the sameelectrical conductance.
As a second example, of this odd–even conductance varia-tion as a function of N1, consider the Greens function of alinear chain at k = p/2. In this case eqn (32) yields
GRL ¼ ð�1ÞN1þ1
21
2 sL þ sRð Þ for N1 odd
and
GRL ¼ ð�1ÞN12
1
gþ sLsR=gfor N1 even
which shows that the conductance of such a chain also exhibitsan odd–even oscillation as a function of the chain length. Thisshows that well-known conductance oscillations in atomicchains40 are closely related to meta versus para conductancevariations found in aromatic rings.
The formula (16) is rather versatile, because the precisemeaning of the symbols depends on context. For examplesthe ‘sites’ within the scatterer could refer to molecular orbitalsor orbitals localised on groups of atoms within a molecule. Asan example of this versatility, consider the case of a singlebranch (M = 1) containing no sites (N1 = 0) and choose a1 = b1 =g1, bR = gR, e0
R = eR, e0L = En. In this case, the nodal site L plays the
role of a molecular orbital weakly coupled to the leads andFig. 7 reduces to Fig. 8a below, for which y = a1, x = 0, aR = eR �E � gR exp(ikR) = 2gR cos (kR) � gR exp(ikR) = gR exp(�ikR) and
aLR ¼ En � E � aL2
gLexp ikLð Þ. Substitution into eqn (16) and (17)
yields the Breit–Wigner formula (14), withP¼ aL2
gLcos kLð Þ þ
a12
gLcos kRð Þ, G1 ¼
vLaL2
2gL2¼ aL2
gLsin kLð Þ and
G2 ¼a12
gRsin kRð Þ. As a final example, the choice M = 2, N1 = 0,
a1 = b1 = g1, bR = gR, e0R = eR, e0
L = En, N2 = 1, b2 = 0, a2 = a yields thestructure shown in Fig. 7b, in which a pendant orbital of energyep is weakly coupled to the orbital En. In this case, eqn (16) and(17) combine to yield eqn (15) describing a Fano resonance.
D. Thermoelectricity and the role ofinter-scatterer quantum phase averaging
The transmission coefficient controls not only the electricalconductance of single molecules, but also a range of othertransport properties. In this section, we discuss issues asso-ciated with thermoelectricity. The thermopower or Seebeckcoefficient (S) and thermoelectric figure of merit (ZT) of amaterial or of a nanojunction are defined as S = �DV/DT andZT = S2GT/k where DV is the voltage difference created betweenthe two ends of the junction when a temperature difference DTis established between them, G is the electrical conductance, Tis the ambient temperature and k is the thermal conductance.Despite several decades of development, the best inorganicthermoelectric materials, such as bismuth telluride (Bi2Te3)-based alloys, possess a figure of merit ZT close to unity only. Asan alternative for achieving higher values, organic thermo-electric materials are now being investigated and are alreadyshowing promising values of both ZT and S.
A key strategy for improving the thermoelectric properties ofinorganic materials has been to take advantage of nanostruc-turing,41,42 which leads to quantum confinement of electronsand enhanced thermoelectric performance. The single-molecule building blocks of organic materials offer the ulti-mate limit of electronic confinement, with quantised energylevel spacings, which are orders of magnitude greater thanroom temperature. Therefore it is natural to examine thethermoelectric performance of single-molecule junctions as astepping stone towards the design of new materials. The abilityto measure thermopower in single-molecule junctions is rela-tively new43–49 and the thermoelectric properties of only a fewmolecules have been measured.
To calculate thermoelectric quantities34,50,51 it is useful tointroduce the non-normalised probability distribution P(E)defined by
PðEÞ ¼ �TðEÞ@f ðEÞ@E
;
where f (E) is the Fermi–Dirac function, whose moments aredenoted
Li ¼ð1�1
dEPðEÞ E � EFð Þi; (37)
where EF is the Fermi energy. With this notation, the low-biasconductance, G is proportional to the first moment:
GðTÞ ¼ 2e2
hL0; (38)
The thermopower S is proportional to the first moment(i.e. to the asymmetry of P(E) with respect to EF)
SðTÞ ¼ � 1
eT
L1
L0; (39)
the electronic contribution to the thermal conductance kel isproportional to the width of the distribution P(E):
kel ¼2
h
1
TL2 �
L12
L0
� �; (40)
and the electronic contribution to the figure of merit ZelT is
ZelT ¼1
L0L2
L12� 1
: (41)
As an aside, for E close to EF, if T(E) varies only slowly withE on the scale of kBT then these expressions take the well-known forms:
GðTÞ � 2e2
h
� �T EFð Þ; (42)
SðTÞ � �aeT d lnTðEÞdE
� �E¼EF
; (43)
k E L0TG, where a ¼ kB
e
� �2p2=3 is the Lorenz number.
The above approximate formulae suggest a number of strategiesfor enhancing the thermoelectric properties of single molecules. Forexample eqn (43) demonstrates the ‘‘rule of thumb’’ that S isenhanced by increasing the slope of ln T(E) near E = EF and thereforethe presence of sharp resonances near E = EF are of interest. For thepurpose of increasing the sharpness of such resonances, Fanoresonances appear to be more attractive than Breit–Wigner reso-nances, because the width of the latter is governed by the parametersG1 and G2, whereas the width of the former is determined by theparameter a in eqn (15). To achieve a stable junction, it is desirablethat there is strong coupling to the electrodes, so the parameters G1
and G2 should as large as possible and therefore in practice, Breit–Wigner resonances are likely to be rather broad. On the other handthe parameter a is an intra-molecular coupling, which is indepen-dent of the coupling to the electrodes and can be tuned withoutaffecting junction stability.
The above formulae provide guidelines for designing newthermoelectric materials from the single molecule upwards. Toillustrate such a strategy, consider two quantum scatterers(labelled 1 and 2) in series, whose transmission and reflectioncoefficients are T1, T2 and R1, R2 respectively. It can be shownthat the total transmission coefficient for the scatterers in series
is T ¼ T1T2
1� 2ffiffiffiffiffiffiffiffiffiffiffiR1R2
pcosjþ R1R2
, where j is a quantum phase
due to QI between the scatterers.51,52 Experimentally-quoted
Fig. 8 (a) A special case of Fig. 7, obtained by setting M = 1, N1 = 0, a1 = b1 = g1,bR = gR, e0R = eR, e0L = En. (b) A special case of Fig. 7, obtained by setting M = 2,N1 = 0, a1 = b1 = g1, bR = gR, e0R = eR, e0L = En, N2 = 1, b2 = 0, a2 = a.
If the phase j is uniformly distributed between 0 and 2p,then the third term on the right hand side averages tozero, because of the mathematical identityÐ 2p0 djlog10 1� 2
ffiffiffiffiffiffiffiffiffiffiffiR1R2
pcosjþ R1R2
¼ 0. Hence all informa-
tion about the inter-scatterer quantum phase is lost and
log10T ¼ log10T1 þ log10T2, or equivalentlyGTotal
G0¼ G1
G0
G2
G0. More
importantly, if S1 and S2 are the thermopowers of 1 and 2respectively, then according to eqn (43) the combined thermo-power of the new molecule is S = S1 + S2.
This strategy for designing new materials was demonstratedrecently in ref. 47, where it was shown that the thermopower oftwo C60s coupled in series is approximately twice that of asingle C60.
E. Conclusions
Rather than provide a review of the literature, the aim of thistutorial has been to gather together some basic concepts andmathematical tools, which underpin the fundamentals ofphase-coherent electron transport through single molecules.The key quantity of interest has been the transmission coeffi-cient T(E), which yields the electrical conductance and variousthermoelectric coefficients. Since T(E) is strongly affected by QI,control of QI provides us with new strategies for increasing theperformance of molecular-scale devices. The MATLAB code inthe appendix provides a route to exploring QI in multi-branched structures described by a tight-binding Hamiltonian.The GOLLUM, TRANSIESTA and TURBOMOLE codes providemethods of exploring such effects at the level of density func-tional theory. These codes can be used to explore differentrealisations of the concepts discussed in this tutorial. Single-molecule electronics is a field very much in its infancy. Theoptimal combinations of electrodes, anchor groups and func-tional units for delivering enhanced performance are stilllargely unknown. The concepts discussed in this tutorial, whencombined with synthesis of new molecules and experimentalexploration of single-molecule devices structures should help toaddress these challenges.
Appendix: a simple MATLAB programto evaluate eqn (5) and (20) of Sparks,Garcia-Suarez, Manrique and Lambert(SGML)19
Below is a simple MATLAB formula for evaluating the SGMLformula (20) and also eqn (20) of ref. 19 for the currents within
the individual branches of the multi-branch structure shown inFig. 6.
Enjoy exploring QI in a variety of systems by varying theinput parameters!
%%%Start of program%%%%%Choose the parameters defining the systemgammaL=1.1; %gamma of left leadepsiL=0; %epsi of left leadalphaL=0.5; %coupling to left leadepsi0L=0; %site energy of left nodegammaR=1.1; %gamma of right leadepsiR=0; %epsi of right leadbetaR=0.5; %coupling to right leadepsi0R=0; %site energy of right nodem=2; %number of branchesalpha=ones(1,m); %array of left coupling to branchesbeta=ones(1,m); %array of right coupling to branchesgamma=ones(1,m); %array of gammas of branchesepsi=zeros(1,m); %array of site energies of branchesn=2*ones(1,m); %array containing the number of atoms in
each branch%In the above array ‘n’ both branches have 2 atoms so this is a%para-connected ring with 6 atoms%For a meta-connected ring with 6 atoms, add the following
two %lines (ie remove the %s)%n(1,1)=1%n(1,2)=3%%%%%Choose a set of energies for graph plottingnenergies=10000;energies=ones(1,nenergies);trans=ones(1,nenergies);current=ones(m,nenergies);emin=-2*max(gammaL,gammaR);emax=2*max(gammaL,gammaR);deltae=(emax-emin)/nenergies;%%%%%Loop over all energies
for count=1: nenergies;E=emin+deltae*count;energies(1,count)=E;
kL=acos((epsiL-E)/(2*gammaL)); %wavevector of left leadif abs((epsiL-E)/(2*gammaL)) o=1
vL=2*gammaL*sin(kL); %group velocity of left leadelse
vL=0;disp(‘no open channel in left lead’)
endkR=acos((epsiR-E)/(2*gammaR)); %wavevector of left leadif abs((epsiR-E)/(2*gammaR)) o=1
vR=2*gammaR*sin(kR); %group velocity of left leadelse
(gammaL*gammaR))^2*abs(GRL)^2;trans(1,count)=transmission;current(:,count)=transmission*yl’./y;end%%%%%Plot the graphssubplot(2,2,1)plot(energies,trans)title([‘T (‘,‘m=’,num2str(m),’)’])subplot(2,2,2)plot(energies,current(1,:),‘k’)title([‘current in lead 1 ’,‘(‘,‘ n1=’,num2str(n(1,1)),’)’])if m 4=2
subplot(2,2,3)plot(energies,current(2,:),‘r’)title([‘current in lead 2 ’,‘(‘,‘ n2=’,num2str(n(1,2)),’)’])endif m 4=3
subplot(2,2,4)plot(energies,current(3,:),‘g’)title([‘current in lead 3 ’,‘(‘,‘ n3=’,num2str(n(1,3)),’)’])end
Acknowledgements
This article is based on many fruitful interactions with experi-mental colleagues in the FUNMOLS and MOLESCO ITNs andtheoretical colleagues in a 20-years series of EU networks, ofwhich the latest is the NanoCTM ITN. In particular I would liketo thank Thomas Wandlowskii, Martin Bryce, Richard Nichols,Simon Higgins, Nicholas Agrait, Harry Anderson and AndrewBriggs for many fruitful discussions about experimental design,measurement and synthesis and Jaime Ferrer and VictorGarcia-Suarez for theoretical discussions and co-developmentof GOLLUM. I would also like to thanks PhD students andpostdocs in my group for their helpful comments. This work issupported by the UK EPSRC grants EP/K001507/1, EP/J014753/1,EP/H035818/1 and by the EU ITN MOLESCO 606728.
Notes and references
1 W. Hong, H. Valkenier, G. Meszaros, D. Z. Manrique,A. Mishchenko, A. Putz, P. Moreno Garcıa, C. J. Lambert,J. C. Hummelen and T. Wandlowski, Beilstein J. Nanotechnol.,2011, 2, 699–713.
2 H. Vazquez, R. Skouta, S. Schneebeli, M. Kamenetska,R. Breslow, L. Venkataraman and M. S. Hybertsen, Nat.Nanotechnol., 2012, 7, 663.
3 S. Ballmann, R. Hartle, P. B. Coto, M. Elbing, M. Mayor,M. R. Bryce, M. Thoss and H. B. Weber, Phys. Rev. Lett.,2012, 109, 056801.
4 S. V. Aradhya, J. S. Meisner, M. Krikorian, S. Ahn,R. Parameswaran, M. L. Steigerwald, C. Nuckolls andL. Venkataraman, Nano Lett., 2012, 12, 1643–1647.
5 V. Kaliginedi, P. Moreno-Garcıa, H. Valkenier, W. Hong,V. M. Garcıa-Suarez, P. Buiter, J. L. H. Otten, J. C. Hummelen,C. J. Lambert and T. Wandlowski, J. Am. Chem. Soc., 2012,134(11), 5262.
6 S. V. Aradhya and L. Venkataraman, Nat. Nanotechnol., 2013,8, 399.
7 C. R. Arroyo, S. Tarkuc, R. Frisenda, J. S. Seldenthuis, C. H. M.Woerde, R. Eelkema, F. C. Grozema and H. S. J. van der Zant,Angew. Chem., Int. Ed., 2013, 52, 3152–3155.
8 C. M. Guedon, H. Valkenier, T. Markussen, K. S. Thygesen,J. C. Hummelen and S. J. van der Molen, Nat. Nanotechnol.,2012, 7, 304.
9 F. Prins, A. Barreiro, J. W. Ruitenberg, J. S. Seldenthuis,N. Aliaga-Alcalde, L. M. K. Vandersypen and H. S. J. van derZant, Nano Lett., 2011, 11, 4607.
10 G. J. Ashwell, L. J. Phillips, B. J. Robinson, S. A. Barnes,A. T. Williams, B. Urasinska-Wojcik, C. J. Lambert, I. M. Grace,T. I. Cox and I. C. Sage, Angew. Chem., 2011, 50, 8722.
11 G. Sedghi, L. J. Esdaile, H. L. Anderson, V. M. Garcıa-Suarez,C. J. Lambert, S. Martin, D. Bethell, S. J. Higgins andR. J. Nichols, Nat. Nanotechnol., 2011, 6, 517.
12 X. Zhao, C. Huang, M. Gulcar, A. S. Batsanov, M. Baghernejad,W. Hong, M. R. Bryce and T. Wandlowski, Chem. Mater., 2013,25, 4340.
13 S. Sanvito, C. J. Lambert, J. H. Jefferson and A. M. Bratkovsky,Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 11936.
14 A. R. Rocha, V. M. Garcia-Suarez, S. W. Bailey, C. J. Lambert,J. Ferrer and S. Sanvito, Nat. Mater., 2005, 4, 335.
15 A. R. Rocha, V. M. Garcia-Suarez, S. W. Bailey, C. J. Lambert,J. Ferrer and S. Sanvito, Phys. Rev. B: Condens. Matter Mater.Phys., 2006, 73, 085414.
16 M. Brandbyge, J.-L. Mozos, P. Ordejon, J. Taylor andK. Stokbro, Phys. Rev. B: Condens. Matter Mater. Phys.,2002, 65, 165401.
17 F. Furche, R. Ahlrichs, C. Hattig, W. Klopper, M. Sierka andF. Weigend, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2014,4, 91.
18 J. Ferrer, C. J. Lambert, V. M. Garcıa-Suarez, D. Zs.Manrique, D. Visontai, L. Oroszlani, R. Rodrıguez-Ferradas, I. Grace, S. W. D. Bailey, K. Gillemot, H. Sadeghiand L. A. Algharagholy, New J. Phys., 2014.
19 M. L. Perrin, C. J. O. Verzijl, C. A. Martin, A. J. Shaikh,R. Eelkema, J. H. van Esch, J. M. van Ruitenbeek,J. M. Thijssen, H. S. van der Zant and D. Dulic, Nat.Nanotechnol., 2013, 8(4), 282–287.
20 A. Al-Backri, V. Zolyomi and C. J. Lambert, J. Chem. Phys.,2014, 140(10), 104306.
21 D. J. Mowbray, G. Jones and K. S. Thygesen, J. Chem. Phys.,2008, 128, 111103.
22 V. M. Garcıa-Suarez and C. J. Lambert, New J. Phys., 2011,13, 053026.
23 L. Venkataraman, J. E. Klare, C. Nuckolls, M. S. Hybertsenand M. L. Steigerwald, Nature, 2006, 442, 904.
24 Q. Lu, K. Liu, H. Zhang, Z. Du, X. Wang and F. Wang, ACSNano, 2009, 3, 381.
25 R. Yamada, H. Kumazawa, T. Noutoshi, S. Tanaka andH. Tada, Nano Lett., 2008, 8, 1237.
26 C. Wang, A. S. Batsanov, M. R. Bryce, S. Martın, R. J. Nichols,S. J. Higgins, V. M. Garcıa-Suarez and C. J. Lambert, J. Am.Chem. Soc., 2009, 131, 15647.
27 G. Sedghi, V. M. Garcıa-Suarez, L. J. Esdaile, H. L. Anderson,C. J. Lambert, S. Martin, D. Bethell, S. J. Higgins, M. Elliott,N. Bennett, J. E. Macdonald and R. J. Nichols, Nat. Nano-technol., 2011, 6, 517.
28 Z. F. Liu, S. Wei, H. Yoon, O. Adak, I. Ponce, Y. Jiang,W.-D. Jang, L. M. Campos, L. Venkataraman and J. B. Neaton,Nano Lett., 2014, 14, 5365–5370.
29 R. E. Sparks, V. M. Garcıa-Suarez and C. J. Lambert, Phys.Rev. B: Condens. Matter Mater. Phys., 2011, 83, 075437.
30 N. R Claughton, M. Leadbeater and C. J. Lambert, J. Phys.:Condens. Matter, 1995, 7(46), 8757.
31 T. A. Papadopoulos, I. M. Grace and C. J. Lambert, Phys. Rev.B: Condens. Matter Mater. Phys., 2006, 74, 193306.
32 C. Wang, M. R. Bryce, J. Gigon, G. J. Ashwell, I. Grace andC. J. Lambert, J. Org. Chem., 2008, 73, 4810.
33 S.-H. Ke, W. Yang and H. U. Baranger, Nano Lett., 2008, 8, 3257.34 C. M. Finch, V. M. Garcıa-Suarez and C. J. Lambert, Phys.
Rev. B: Condens. Matter Mater. Phys., 2009, 79, 033405.35 J. P. Bergfield, M. Solis and C. A. Stafford, ACS Nano, 2010,
4, 5314.36 A. B. Ricks, G. C. Solomon, M. T. Colvin, A. M. Scott,
K. Chen, M. A. Ratner and M. R. Wasielewski, J. Am. Chem.Soc., 2010, 132, 5427.
37 T. Markussen, J. Schiotz and K. S. Thygesen, J. Chem. Phys.,2010, 132, 224104.
38 G. C. Solomon, J. P. Bergfield, C. A. Stafford andM. A. Ratner, Beilstein J. Nanotechnol., 2011, 2, 862.
39 M. Magoga and C. Joachim, Phys. Rev. B: Condens. MatterMater. Phys., 1999, 59, 1601.
40 P. Major, V. M. Garcia-Suarez, S. Sirichantaropass, J. Cserti,C. J. Lambert, J. Ferrer and G. Tichy, Phys. Rev. B: Condens.Matter Mater. Phys., 2006, 73, 045421.
41 R. Venkatasubramanian, E. Siivola, T. Colpitts andB. O’Quinn, Nature, 2001, 413, 597.
42 T. C. Harman, P. J. Taylor, M. P. Walsh and B. E. LaForge,Science, 2002, 297, 2229.
43 J. A. Malen, S. K. Yee, A. Majumdar and R. A. Segalman,Chem. Phys. Lett., 2010, 491, 109.
44 P. Reddy, S.-Y. Jang, R. A. Segalman and A. Majumdar,Science, 2007, 315, 1568.
45 K. Baheti, J. A. Malen, P. Doak, P. Reddy, S. Y. Jang,T. D. Tilley, A. Majumdar and R. A. Segalman, Nano Lett.,2008, 8, 715.
46 S. K. Yee, J. A. Malen, A. Majumdar and R. Segalman, NanoLett., 2011, 11, 4089.
47 C. Evangeli, K. Gillemot, E. Leary, M. T. Gonzalez, G. Rubio-Bollinger, C. J. Lambert and N. Agrait, Nano Lett., 2013,13(5), 2141–2145.
48 J. A. Malen, P. Doak, K. Baheti, T. D. Tilley, A. Majumdarand R. A. Segalman, Nano Lett., 2009, 9, 3406.
49 M. Buerkle, L. A. Zotti, J. K. Viljas, D. Vonlanthen,A. Mishchenko, T. Wandlowski, M. Mayor, G. Schoen andF. Pauly, Phys. Rev. B: Condens. Matter Mater. Phys., 2012,86, 115304.
50 V. M. Garcıa-Suarez, C. J. Lambert, D. Z. Manrique andT. Wandlowski, Nanotechnology, 2014, 25(20), 205402.
51 C. J. Lambert and M. F. Thorpe, Phys. Rev. B: Condens.Matter Mater. Phys., 1982, 26, 4742–4744.
52 C. J. Lambert, J. Phys. Chem., 1984, 17, 2401–2414.53 See e.g. A. El-Sayed, et al., ACS Nano, 2013, 7, 6914 and
references therein.54 S. Braun, W. Osikowicz, Y. Wang and W. R. Salaneck, Org.
