magnetoresistance of nanoscale molecular devices based on aharonov

Magnetoresistance of nanoscale molecular devices based on Aharonov-Bohm interferometry

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IOP PUBLISHING JOURNAL OF PHYSICS CONDENSED MATTER

J Phys Condens Matter 20 (2008) 383201 (32pp) doi1010880953-89842038383201

TOPICAL REVIEW

Magnetoresistance of nanoscale moleculardevices based on AharonovndashBohminterferometryOded Hod1 Roi Baer2 and Eran Rabani3

1 Department of Chemistry Rice University Houston TX 77005-1892 USA2 Institute of Chemistry and the Fritz Haber Center for Molecular Dynamics The HebrewUniversity of Jerusalem Jerusalem 91904 Israel3 School of Chemistry The Sackler Faculty of Exact Sciences Tel Aviv UniversityTel Aviv 69978 Israel

E-mail odedhodriceedu roibaerhujiacil and rabanitauacil

Received 17 January 2008 in final form 11 July 2008Published 27 August 2008Online at stacksioporgJPhysCM20383201

AbstractControl of conductance in molecular junctions is of key importance in the growing field ofmolecular electronics The current in these junctions is often controlled by an electric gatedesigned to shift conductance peaks into the low bias regime Magnetic fields on the otherhand have rarely been used due to the small magnetic flux captured by molecular conductors(an exception is the Kondo effect in single-molecule transistors) This is in contrast to a relatedfield electronic transport through mesoscopic devices where considerable activity withmagnetic fields has led to a rich description of transport The scarcity of experimental activity isdue to the belief that significant magnetic response is obtained only when the magnetic flux isof the order of the quantum flux while attaining such a flux for molecular and nanoscaledevices requires unrealistic magnetic fields

Here we review recent theoretical work regarding the essential physical requirementsnecessary for the construction of nanometer-scale magnetoresistance devices based on anAharonovndashBohm molecular interferometer We show that control of the conductance propertiesusing small fractions of a magnetic flux can be achieved by carefully adjusting the lifetime ofthe conducting electrons through a pre-selected single state that is well separated from otherstates due to quantum confinement effects Using a simple analytical model and more elaborateatomistic calculations we demonstrate that magnetic fields which give rise to a magnetic fluxcomparable to 10minus3 of the quantum flux can be used to switch a class of different molecular andnanometer rings ranging from quantum corrals carbon nanotubes and even a molecular ringcomposed of polyconjugated aromatic materials

The unique characteristics of the magnetic field as a gate is further discussed anddemonstrated in two different directions First a three-terminal molecular router devices thatcan function as a parallel logic gate processing two logic operations simultaneously ispresented Second the role of inelastic effects arising from electronndashphonon couplings on themagnetoresistance properties is analyzed We show that a remarkable difference betweenelectric and magnetic gating is also revealed when inelastic effects become significant Theinelastic broadening of response curves to electric gates is replaced by a narrowing ofmagnetoconductance peaks thereby enhancing the sensitivity of the device

(Some figures in this article are in colour only in the electronic version)

0953-898408383201+32$3000 copy 2008 IOP Publishing Ltd Printed in the UK1

J Phys Condens Matter 20 (2008) 383201 Topical Review

Contents

1 Introduction 22 Defining open questions 3

21 AharonovndashBohm interferometry 322 Length scales 423 Open questions 5

3 Basic concepts 531 Two-terminal devices 532 Three-terminal devices 8

4 Atomistic calculations 1141 Conductance 1142 Retarded and advanced Greenrsquos functions and

self-energies 1243 An alternativemdashabsorbing imaginary potentials 1444 Electronic structuremdashtight binding magnetic

extended Huckel theory 1445 Results for two-terminal devices 1546 Results for three-terminal devices 20

5 Inelastic scattering effects in AB molecular interferom-eters 2151 Hamiltonian 2252 Conductance 2353 Results 23

6 Summary and prospects 25References 26

1 Introduction

In the past two decades molecular electronics has emerged asone of the most active and intriguing research fields [1ndash23]Scientifically this field offers insights into fundamental issuesregarding the physics of low-dimensional and nanometer-scalesystems as well as their response to external perturbationswhen embedded in complicated environments Froma technological point of view electronic functionalityat the molecular scale may revolutionize our currentconception of electronic devices and may lead to considerableminiaturization of their dimensions [1 13] Interestingly theparadigm of molecular electronics was set by a theoreticalstudy long before the first experimental realization waspresented [24] In their seminal work Aviram and Ratnersuggested that a donorndashbridgendashacceptor type of moleculecould in principle act as a molecular rectifier What designatesthis study is the fact that for the first time a single moleculewas considered to act as an electronic component rather thanjust being a charge transfer medium Therefore it could serveas a potential building block for future nanoscale electronicdevices [6 7 13 16 22]

In recent years a number of experimental techniqueshave been developed for the synthesis and fabrication ofjunctions that allow for the measurement of the electronictransport through molecular-scale systems These includemechanically controllable break junctions [25ndash28] electro-migration break junctions [29ndash32] electron beam lithogra-phy [33 34] feedback-controlled lithography [35] shadow

evaporation [36 37] electrochemical deposition [38ndash40] mer-cury droplet technologies [41ndash43] cross-wire tunnel junc-tions [44] STM [45 46] and conducting AFM [47] tip mea-surements and more [48ndash50] In typical molecular electronicsexperiments a molecule (or a group of molecules) is trappedwithin a gap between the electrodes while chemically or phys-ically attaching to the conducting leads Once the molecularjunction is obtained a bias voltage is applied and the cur-rentvoltage characteristics are recorded and analyzed Certaintechniques can also utilize an electrical or chemical gate elec-trode in order to control the response of the molecule to theapplied bias [51ndash58 34 59ndash68] A simplified picture of theeffect of the gate voltage can be viewed as a shift in the energyof the molecular states of the device with respect to the Fermienergy of the conducting leads allowing for fine tuning of theconductance through pre-selected molecular energy states

Unlike the common use of electrical gates as controlknobs for the transport characteristics of molecular junctionsmagnetic fields have rarely been used in conjunction withmolecular electronics This is in contrast to a relatedfield electronic transport through mesoscopic systems whereconsiderable activity with magnetic fields has led to thediscovery of the quantum Hall effect [69] and to a richdescription of AharonovndashBohm [70ndash87] magneto-transportphenomena in such conductors [88ndash100]

The scarcity of experimental activity involving magneticfields in molecular electronic set-ups is due to the beliefthat a significant magnetic response is obtained only whenthe magnetic flux is of the order of the quantum fluxφ0 = hqe (qe being the electron charge and h Planckrsquosconstant) and attaining such a flux for molecular andnanoscale devices requires unrealistically huge magnetic fieldsNevertheless several interesting experimental studies on theeffect of magnetic fields in nanoscale systems have beenpublished recently The major part of these regard Zeemansplitting of spin states in quantum dots [101] and in carbonnanotubes [48 102 103] the Kondo effect measured formesoscale quantum dots [104 105] fullerenes [106ndash109] andsingle molecules [30 34] and the quantum Hall effect ingraphene nanoribbons [110]

In this review we discuss recent theoretical work utilizingmagnetic fields as gates in molecular AharonovndashBohminterferometers [111ndash115] We show how the combination ofelectric and magnetic fields provide additional control over theelectrical current The unique symmetry breaking nature of themagnetic field allows for the design of novel logical devicessuch as multi-terminal electron routers [113] which are notfeasible when electrical gating is applied alone We also showthat when inelastic scattering effects are taken into account aremarkable difference between electric and magnetic gating isrevealed [114] The inelastic broadening of response curves toelectric gates is replaced by narrowing of magnetoconductancepeaks thereby enhancing the sensitivity of the device

This review starts with a brief introduction to theAharonovndashBohm (AB) effect and its feasibility in nanoscalesystems (section 2) The basic concepts are described insection 3 for two- and three-terminal devices The analysisof these AB interferometers is based on adopting a simple

2


continuum model Next in section 4 we described detailedatomistic calculations motivated by the results of the simplecontinuum approach We illustrate that molecular ABinterferometers based on a variety of systems can be used asswitching devices at relatively low magnetic fields We alsoillustrate a parallel molecular logic gate based on a three-terminal device Finally we describe the effect of inelasticscattering on the behavior of the molecular AB interferometersin section 5 Conclusions and a brief summary are given insection 6

2 Defining open questions

21 AharonovndashBohm interferometry

In classical mechanics the force exerted on a charged particletraversing a region in space which incorporates an electricandor a magnetic field is given by the Lorentz force lawF = q(E + v times B) where q is the charge of the particle Eis the electric field v is the particlersquos velocity and B is themagnetic field [116] It can be seen that the electric fieldoperates on a particle whether static or not and contributes aforce parallel to its direction and proportional to its magnitudewhile the magnetic field operates only on moving particles andcontributes a force acting perpendicular to its direction andto the direction of the particlersquos movement When solvingNewtonrsquos equation of motion F = dP

dt the resulting trajectoriesfor a classical charged particle entering a region of a uniformmagnetic field will thus be circular One can define scalar andvectorial potentials using the following definitions E(r t) =minusnablaV minus partA(rt)

part t and B(r t) = nabla times A(r t) respectively Whendefining the following Lagrangian L = 1

2 mv2minusqV (r)+qvmiddotAand deriving the canonical momentum using the relation Pi =partLpart ri

one can in principle solve the EulerndashLagrange or Hamiltonequations of motion This procedure even though easier tosolve for some physical problems is absolutely equivalent tosolving Newtonrsquos equations of motion and will produce theexact same trajectories

The influence of electric and magnetic fields on thedynamics of quantum charged particles was investigated byAharonov and Bohm in a seminal work from 1959 [71]revealing one of the fundamental differences between theclassical and the quantum description of nature Accordingto Aharonov and Bohm while in classical mechanics thetransition from using electric and magnetic fields to usingscalar and vectorial potentials is lsquocosmeticrsquo and may beregarded as a mathematical pathway for solving equivalentproblems in quantum mechanics the fundamental quantitiesare the potentials themselves

In order to demonstrate this principle consider a double-slit experiment applied to electrons The experimental set-up consists of a source emitting coherent electrons which arediffracted through two slits embedded in a screen The electronintensity is measured in a detector placed on the opposite sideof the screen In the absence of a magnetic field the intensitymeasured by the detector when placed directly opposite tothe source will be maximal due to the positive interferencebetween the two electron pathways which are of equal length

Figure 1 An AharonovndashBohm ring-shaped interferometerconnecting a coherent electron source and a detector

When applying a point magnetic field perpendicular to theinterference plane the interference intensity is altered Unlikethe classical prediction this change in the interference patternis expected even if the magnetic field is excluded from bothelectron pathways This can be traced back to the fact thateven though the magnetic field is zero along these pathwaysthe corresponding vector potential does not necessarily vanishat these regions

A more quantitative description of this phenomenon canbe obtained by considering an analogous model consisting ofa ring-shaped ballistic conductor forcing the bound electronsto move in a circular motion connecting the source and thedetector as shown in figure 1 The Hamiltonian of the electronsunder the influence of electric and magnetic potentials is givenby

H = 1

2m

[P minus qA(r)

]2 + V (r) (1)

Field quantization is disregarded in the entire treatment and weassume that the potentials are time-independent In the abovem is the mass of the particle P = minusihnabla is the canonicalmomentum operator and h is Planckrsquos constant divided by 2π In the absence of electrostatic interactions the Hamiltonianreduces to the free particle Hamiltonian with the appropriate

kinetic momentum operator ˜P = P minus qA(r)In the set-up shown in figure 1 the wavefunction splits into

two distinct parts one traveling through the upper arm and theother through the lower arm of the ring Since the magneticfield is excluded from these paths (nabla times A = 0) and both aresimply connected in space it is possible to write a solution tothe stationary Schrodinger equation with this Hamiltonian ateach path as follows

ud = eiint

ud (kminus qh A)middotdl

(2)

where k is the wavevector of the charged particle and theintegration is taken along its pathway with ud standing forthe upper (up) and lower (down) pathways respectively Dueto the circular symmetry of the system the spatial phase factorsaccumulated along the upper or the lower branches of the ringcan be easily calculated in the following manner

k =int 0

π

k middot dl = πRk (3)

3


Here k is taken to be along the ring and k = |k| is thewavenumber of the electron When a uniform4 magneticfield is applied perpendicular to the cross section of the ringB = (0 0 Bz) the vector potential may be written as A =minus 1

2 r times B = 12 Bz(minusy x 0) = RBz

2 (minus sin(θ) cos(θ) 0) Thusthe magnetic phase accumulated by the electron while travelingfrom the source to the detector in a clockwise manner throughthe upper path is given by

um = minusq

h

int 0

π

A middot dl = πφ

φ0 (4)

Here φ = Bz S is the magnetic flux threading the ring φ0 = hq

is the flux quantum and S = πR2 is the cross-sectional areaof the ring For an electron traveling through the lower path ina counterclockwise manner the magnetic phase has the samemagnitude with an opposite sign

lm = minusq

h

int 2π

π

A middot dl = minusπ φφ0 (5)

The electron intensity measured at the detector isproportional to the square absolute value of the sum of theupper and lower wavefunction contributions

I prop ∣∣u +d∣∣2 =

∣∣∣∣eiπ(

Rk+ φ

φ0

)+ e

iπ(

Rkminus φ

φ0

)∣∣∣∣2

= 2

[1 + cos

(2π

φ

φ0

)] (6)

The intensity measured at the detector is therefore a periodicfunction of the magnetic flux threading the ringrsquos cross sectionAs mentioned before this general and important result holdsalso when the magnetic field is not applied uniformly andmeasurable intensity changes may be observed at the detectoreven if the applied magnetic field is excluded from thecircumference of the ring to which the electrons are bound

22 Length scales

The model described above presents an idealized system forwhich the charge carrying particles travel from the source tothe detector without losing either momentum or phase Whenconsidering the issue of measuring the AB effect in a realisticsystem a delicate balance between three important lengthscales is needed (a) the Fermi wavelength (b) the mean freepath and (c) the coherence length scale In what follows weshall give a brief description of each of these length scales andtheir importance for transport

4 The calculation given here assumes a uniform magnetic field even thoughit was claimed that equation (2) holds only in regions in space where B = 0Nevertheless due to the fact that the electrons are confined to move on the one-dimensional ring there is no essential difference between the case of a singularmagnetic field and a homogeneous magnetic field Therefore the usage of thephases calculated here in equation (6) is valid Furthermore it should be notedthat due to Stokesrsquo law the line integration of A over a closed loop will alwaysgive the magnetic flux threading the ring whether the field is uniform singularor of any other form

221 Fermi wavelength As in the optical double-slit experiment the wavelength of the conducting electronsdetermines the interference intensity measured at the detectorAt low temperatures and bias voltages the net current is carriedby electrons in the vicinity of the Fermi energy and thusby controlling their wavelength one can determine whetherpositive or negative interference will be measured at thedetector in the absence of a magnetic field

In order to achieve positive interference an integernumber of Fermi wavelengths should fit into half thecircumference of the ring [117 115] (figure 1) L = nλFHere n is an integer L = πR is half of the circumferenceof the ring and λF is the Fermi electronrsquos wavelength given bythe well-known de Broglie relation λF = h

PF= 2π

kF where PF is

the momentum of the Fermi electrons and kF is the associatedwavenumber For such a condition to be experimentallyaccessible the Fermi wavelength should be approximatelyof the order of magnitude of the device dimensions Tooshort wavelengths will give rise to extreme sensitivity of theinterference pattern on the Fermi wavenumber while too longwavelengths will show very low sensitivity We shall return tothe importance of controlling the wavelength of the conductingelectrons in realistic systems measurements in sections 31and 45

222 Momentum relaxation lengthmdashstatic scatterers Anelectron traveling in a perfect crystal can be viewed as a freeparticle with a renormalized mass [118] This mass whichis usually referred to as the effective mass of the electrons inthe crystal incorporates the net effect of the periodic nucleiarray on the conducting electrons When impurities or defectsexist in the crystalline structure the electrons may scatteron them Such scattering implies a random change in theelectronrsquos momentum and thus destroys the ballistic nature ofthe conductance [119 120] The mean free path for momentumrelaxation Lm is the average distance an electron travelsbefore its momentum is randomized due to a collision withan impurity It is typically connected to the momentum ofthe conducting electrons (hkF) their effective mass mlowast and thecollision time (τ which is connected to the imaginary part ofthe self-energy of the quasiparticle) Lm = hkF

mlowast τ If the scattering taking place at the impurities sites

is elastic such that the energy of the electron and themagnitude of its momentum are conserved and only thedirection of the momentum is randomized then for a giventrajectory a stationary interference pattern will be observedHowever different electron trajectories will give rise todifferent interference intensities and the overall interferencepattern will not be stationary and thus will be averaged tozero [119]

223 Phase relaxation lengthmdashfluctuating scatterersInelastic scattering may occur when the impurities haveinternal degrees of freedom which may exchange energywith the scattered electrons Such scattering events alterthe phase of the conducting electron AB interferometryrequires that a constant phase difference exists between thetrajectories of electrons moving in the upper and the lower

4


arms of the interferometer ring Thus for stationary inelasticscatterers which shift the phase of the electrons in a persistentmanner a constant interference pattern will be measured at thedetector Although the intensity at zero magnetic field will notnecessarily be at its maximal value AB oscillations will beobserved [119] When the scatterers are not stationary andthe phase shifts they induce are not correlated between the twoarms of the ring the coherent nature of the conductance isdestroyed and the AB interference pattern is diminished [119]A length scale Lφ is assigned to this process and itsphysical meaning is that length electrons travel before they losetheir phase Apart from impurities inelastic scattering mayoccur due to electronndashphonon processes and electronndashelectroninteractions The latter conserve the total energy but allowenergy exchange and thus phase randomization

224 Regimes of transport There are several differentregimes for transport depending on the values of the abovelength scales If we denote the length of molecular device byL then the ballistic regime is achieved when L Lm Lφ When Lm L Lφ the transport is diffusive and isreduced relative to the ballistic regime Localization occurswhen Lm Lφ L and when the device is larger than bothLm and Lφ the system is considered to be classical with ohmicresistance

23 Open questions

The design of a realistic AB interferometer requires a carefulconsideration of the typical length scales of the conductingelectrons In order to measure a significant AB periodicityit is necessary to reduce the dimensions of the interferometerbelow the momentum and phase relaxation lengths and makeit comparable to the de Broglie wavelength

As an example consider a ring made of GaAsAlGaAs-based two-dimensional electron gas The typical de Brogliewavelength is of the order of sim30 nm the mean freepath (momentum relaxation length scale) is three orders ofmagnitude larger (sim30 μm) and the phase relaxation length issim1 μm [119] Thus one can expect to observe AB oscillationsin the conductivity in loops with dimensions that are smallerthan asymp1 μm with an AB period in the range of millitesla

hqπ R2 asymp 1 times 10minus3 T [117]

For molecular rings these scales are quite different Thetypical de Broglie wavelength is of the order of severalchemical bonds (sub-nanometer) [111] while the mean freepath of electrons and phase relaxation length are considerablylarger than the dimensions of the ring Thus molecularAB interferometers at the nanometer scale are expected toshow many AB oscillations as long as the temperature islow enough so that inelastic effects arising from electronndashphonon couplings are suppressed However in addition to theengineering challenge of fabricating such small and accuratedevices another substantial physical limitation becomes amajor obstacle When considering a nanometer-sized AB loopthe period of the magnetic interference oscillations increasesconsiderably and becomes comparable to h

qπ R2 asymp 1 times 103 TMagnetic fields of these orders of magnitude are by far not

accessible experimentally and thus AB interferometry is notexpected to be measured for systems of small dimensions atthe nanometer scale

In view of the above several open questions arise

(i) Can nanometric electronic devices based on an ABinterferometer be made despite the large magnetic fieldrequired to complete a full AB period

(ii) If such devices can be made what are their physicalproperties At what range of parameters (temperaturecontact coupling dimensions etc) will they be able tooperate

(iii) What are the advantagesdisadvantages of such magneti-cally gated devices over conventional electric gating

In the remainder of this review we attempt to provideanswers to the above problems Despite the fact that thetreatment described herein is theoretical we believe that futuretechnology will enable the development of realistic devicesbased on the concepts present below

3 Basic concepts

In this section we describe the basic concepts of AharonovndashBohm interferometry at the nanometer molecular scale Weconsider two- and three-terminal devices Transport ismodeled within a simple continuum model that has beenconsidered in mesoscopic physics for a two-terminal deviceonly [89 121 90ndash92 95 122 119 123] An exact solution forthe conduction in both cases and analysis of the role of differentmodel parameters is described The two-terminal device isused to establish the necessary condition for switching amolecular AharonovndashBohm interferometer at reasonably smallmagnetic fields while the three-terminal device is used todemonstrate one of the advantages of magnetic gating inmolecular AB interferometers

31 Two-terminal devices

Consider a simple continuum model of an AB interferometer[89 121 90ndash92 95 122 119 123 124] The model consists ofa one-dimensional (1D) single-mode conducting ring coupledto two 1D single-mode conducting leads as depicted in figure 2The transport is considered to be ballistic along the conductinglines Elastic scattering occurs at the two junctions only Eventhough this model disregards the detailed electronic structureof the molecular device and neglects important effects such aselectronndashelectron interactions and electronndashphonon couplingit succeeds in capturing the important physical features neededto control the profile of the AB period

The main feature differing the current arrangement fromthe one shown in figure 1 is that an lsquooutgoingrsquo lead replaces theabsorbing detector Hence the interference intensity measuredat the detector is replaced by a measurement of the conductancebetween the two leads through the ring Although it mightseem insignificant this difference is actually the heart of theapproach developed to control the shape of the AB periodIn the original setting an electron arriving at the detector isimmediately absorbed and therefore the intensity measured is

5


Figure 2 An illustration of a 1D coherent transport continuummodel of an AB interferometer

the outcome of the interference of the two distinct pathwaysthe electron can travel For the set-up considered in figure 2an electron approaching each of the junctions can be eithertransmitted into the corresponding lead or be reflected backinto one of the arms of the ring Consequently the conductancethrough the device is a result of the interference of an infiniteseries of pathways resulting from multiple scattering events ofthe electron at the junctions It is obvious that the behaviorpredicted by equation (6) has to be modified to account for theinterference between all pathways

A scattering matrix approach [121 92 122 119123 125 126] can now be used in order to give a quantitativedescription of the present model Within this approach onelabels each part of the wavefunction on every conductingwire with a different amplitude which designates its travelingdirection Here we use the following notation (as can beseen in the left panel of figure 3) L1 designates the right-going amplitude of the wavefunction on the left lead while L2

is the left-going amplitude on the same lead Similarly R1

and R2 stand for the right-and left-going wave amplitudes onthe right lead respectively For the upper arm of the ring U1

and U2 represent the clockwise and counterclockwise travelingamplitudes respectively whereas D2 and D1 are the clockwiseand counterclockwise traveling amplitudes on the lower arm

Each junction is also assigned with appropriate scatteringamplitudes In what follows we assume that the junctionsare identical and are characterized by scattering amplitudesas shown in the right panel of figure 3 a is the probabilityamplitude for an electron approaching the junction from oneof the arms of the ring to be reflected back into the same arm

b is the probability amplitude to be transmitted from one armof the ring to the other upon scattering at the junction and cis the probability amplitude for an electron approaching thejunction from the lead to be reflected back into the lead

radicε is

the probability amplitude to be transmitted to (or out of) eitherarms of the ring

For each junction it is now possible to formulatea scattering matrix equation relating the outgoing waveamplitudes to the incoming wave amplitudes For the leftjunction one gets

( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)( L1

U2ei1

D2ei2

) (7)

Here 1 = k minusm is the phase accumulated by an electrontraveling from the right junction to the left junction through theupper arm of the ring and2 = k +m is the correspondingphase accumulated along the lower arm of the ring k andm equiv u

m are defined in equations (3) and (4) respectivelyAn analogous equation can be written down for the rightjunction

( R1

U2

D2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

U1ei2

D1ei1

) (8)

In order to ensure current conservation during each scatteringevent at the junctions one has to enforce the scattering matrixto be unitary This condition produces the following relationsbetween the junction scattering amplitudes a = 1

2 (1 minus c)b = minus 1

2 (1 + c) and c = radic1 minus 2ε It can be seen that the

entire effect of the elastic scattering occurring at the junctionscan be represented by a single parameter εmdashthe junctiontransmittance probability

Solving these equations and setting the right incomingwave amplitude R2 equal to zero one gets a relation betweenthe outgoing wave amplitude R1 and the incoming waveamplitude L1 Using this relation it is possible to calculatethe transmittance probability through the ring which is givenby [127 111 115]

T =∣∣∣∣

R1

L1

∣∣∣∣2

= A[1 + cos(2φm)]1 + P cos(2φm)+ Q cos2(2φm)

(9)

c

ε

a

b

Figure 3 An illustration of the amplitudes of the different parts of the wavefunction (left panel) and the junction scattering amplitudes (rightpanel) in the 1D continuum model

6


where the coefficients are functions of the spatial phase and thejunction transmittance probability and are given by [111]

A = 16ε2[1 minus cos(2k)]R

P = 2(c minus 1)2(c + 1)2 minus 4(c2 + 1)(c + 1)2 cos(2k)

R

Q = (c + 1)4

R

R = (c minus 1)4 + 4c4 + 4 minus 4(c2 + 1)(c minus 1)2 cos(2k)

+ 8c2 cos(4k)

(10)

The numerator of equation (9) resembles the result obtainedfor the interference of two distinct electron pathways (given byequation (6)) The correction for the case where the electronsare not absorbed at the detector is given by the denominatorexpression An alternative approach to obtain the result givenby equation (9) is based on a summation of all the paths ofthe electron entering the ring and completing n + 1

2 loopsbefore exiting where n is the winding number For n = 0 onerecovers the simplified result of equation (6) The contributionof all higher-order terms (n gt 0) is given by the denominatorof equation (9)

From equations (9) and (10) it can be seen that twoimportant independent parameters control the shape of themagneto-transmittance spectrum the junction transmittanceprobability ε and the conducting electron wavenumber kappearing in the spatial phase k = πRk In figure 4we present the transmittance probability (T ) through the ringas a function of the normalized magnetic flux threading itfor a given value of the spatial phase and several junctiontransmittance probabilities

It can be seen that for high values of the junctiontransmittance probability the magneto-transmittance behavioris similar to that predicted by equation (6) ie a cosinefunction As ε is reduced from its maximal value of 1

2 the widthof the transmittance peaks is narrowed At large values of εthe lifetime of the electron on the ring is short and thereforethe energy levels characterizing the ring are significantlybroadened Short lifetime also means that the electron travelingthrough the ring completes only very few cycles around itIn a path integral language this implies that the interferencepattern is governed by paths with low winding numbers Theapplication of a magnetic field will change the position of theseenergy levels however they will stay in partial resonancewith the energy of the incoming electrons for a wide range ofmagnetic fields Mathematically the transmission through thering when the electron exits the ring after it passes only alongone of the arms can be reduced to the case where the detectoris placed at the exit channel and thus one would expect acosine function describing the transmission as a function ofmagnetic field When reducing the coupling (low values of ε)the doubly degenerate energy levels of the ring sharpen If oneassumes that at zero magnetic flux two such energy levels arein resonance with the incoming electrons then upon applyinga finite magnetic field the degeneracy is removed such that onelevel has its energy raised and the other lowered This splitting

Figure 4 AB transmittance probability calculated usingequations (9) and (10) as a function of the magnetic flux for differentjunction transmittance probabilities and k R = 1

2 For high values ofε (dashedndashdotted line) the transmittance probability is similar to thatpredicted by equation (6) As ε is decreased the transmittance peaksnarrow (dashed line) For very small values of ε the peaks becomeextremely narrow (solid line)

causes both sharp energy levels to shift out of resonance andthus reduces the transmittance probability through the ringdramatically This however is not the case shown in figure 4where the narrow transmittance peaks are located around thecenter of the AB period Therefore one needs to find a wayto shift the transmittance resonances towards the low magneticfield region

To gain such control we realize that the situation describedabove for the low coupling regime is in fact resonanttunneling occurring through the slightly broadened (due tothe coupling to the leads) energy levels of a particle ona ring Here the free particle time-independent scatteringwavefunction on the wires ψk(z) = eikz has the same form ofthe wavefunction of a particle on a ring ψm(θ) = eimθ wherem = 0plusmn1plusmn2 Upon mounting the ring the electronrsquoswavenumber km is quantized according to the followingrelation Rkm = m The value of m for which resonanttunneling takes place is determined by the condition that thekinetic energy of the free electron on the wire equals a sharpenergy eigenvalue of the ring [117 128]

h2k2

2m= h2(m minus φ

φ0)2

2mR2 (11)

In order to achieve resonance one has to require thatk = (m minus φ

φ0)R A slight change in the magnetic flux

disrupts this resonance condition and reduces the transmittanceconsiderably For different values of the wavenumber on theleads the resonance condition in equation (11) will be obtainedat different values of the magnetic flux

This effect is depicted in figure 5 where the transmittanceprobability is plotted against the normalized magnetic fluxthreading the ring for a given value of ε and several spatialphase factors Changing the spatial phase results in a shift ofthe location of the transmittance peaks along the AB period

7


Figure 5 AB transmittance probability as a function of the magneticflux for different spatial phases and ε = 025 By changing the valueof k R from simn (dashedndashdotted line) to simn + 1

2 (solid line) where nis an integer it is possible to shift the transmittance peaks from thecenter of the AB period to its edges

This is analogous to the change in the position of the intensitypeaks of the interference pattern observed in the optical double-slit experiment when varying the wavelength of the photonsFor k R values of sim0 1 2 the peaks are located near thecenter of the AB period while for values of sim 1

2 1 12 2 1

2

the peaks are shifted toward the periodrsquos lower and higheredges In a realistic system such control can be achievedby the application of an electric gate field that serves toaccelerate (or decelerate) the electron as it mounts the ringThe gate potential Vg thus modifies the resonance conditionof equation (11) to

h2k2

2m= h2(m minus φ

φ0)2

2mR2+ Vg (12)

Equation (12) implies that a change in the gate potentialinfluences the magnetic flux at which resonance is attainedTherefore the transmittance resonancersquos position along the ABperiod can be varied as shown in figure 5

Considering the original goal of measuring a significantmagnetoresistance effect in nanometer scale AB interferome-ters it is clearly evident that even though the full AB periodis out of experimental reach a delicate combination of an ap-propriate wavenumber of the conducting electrons and weakleadsndashring coupling (so-called lsquobad contactrsquo) enables us to shiftthe transmittance peak toward the low magnetic fields regimewhile at the same time dramatically increasing the sensitivity tothe external magnetic field The combined effect is discussedin figure 6 where magnetic switching for a 1 nm radius ring isobtained at a magnetic field of sim1 T while the full AB period(see the inset of the figure) is achieved at magnetic fields ordersof magnitude larger

This result resembles the change in the interference in-tensity measured by the optical MachndashZehnder interferome-ter [129 130] upon altering the phase of the photons on oneof the interferometric paths The plausibility of applying these

Figure 6 Low field magnetoresistance switching of a 1 nm ringweakly coupled to two conducting wires as calculated using thecontinuum model The parameters chosen in this calculation areε = 0005 and k R asymp 1 Inset showing the full AB period ofasymp1300 T

principles to realistic molecular systems is the subject of sec-tion 4 But first an important question which considers theuniqueness of using magnetic fields has to be answered

32 Three-terminal devices

A legitimate question that may be raised at this point is whyuse magnetic fields to switch the conductance Switchingdevices based on other external perturbations such as fieldeffect transistors (FETs) already exist and operate even at themolecular scale [51 131 52 132 55 34 59 133 60ndash68]

One simple answer is that magnetic fields can provide anadditional control and perhaps for certain cases will enablecontrol at small length scales where electrical fields areextremely hard to manipulate However as will become clearbelow there are two additional motivations to employ magneticfields as gates One will be discussed shortly and the otherinvolves inelastic effects which are described at the end of thisreview

Consider an extension of the two-terminal continuummodel described in section 31 to the case of three terminalsIn a mesoscopic system or in the absence of magneticfields such devices have been considered by several groups(see [134ndash144]) An illustration of the three-terminal set-upis given in figure 7 The scattering matrix approach may beused in a similar manner to that described for the two-terminalset-up An illustration of the three-terminal set-up parametersdesignation is given in figure 8 We denote (panel (a)) by α βand γ = 2π minus α minus β the angles between the three conductingleads Similar to the two-terminal model we label the waveamplitudes on each ballistic conductor of the system as shownin panel (b) of figure 8

The scattering amplitudes characterizing the junctionsare given in figure 3 and obey the scattering matrix unitarycondition c = radic

1 minus 2ε a = 12 (1 minus c) and b = minus 1

2 (1 + c)For simplicity in what follows we assume that all the junctions

8


Figure 7 An illustration of a 1D coherent transport continuummodel of a three-terminal AB interferometer

are identical This assumption can be easily corrected to thecase of non-identical junctions

Using these notations it is again possible to writescattering matrix relations between the incoming and outgoingwave amplitudes at each junction

( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

L1

U2eiα1

D2eiβ2

⎞⎠

( R1

M1

U2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

M2eiγ1

U1eiα2

)

( I1

D2

M2

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

I2

D1eiβ1

M1eiγ2

⎞⎠

(13)

Here =αβγ1 = Rk minus

φ

φ0and =αβγ

2 = Rk +φ

φ0

Eliminating the equations for the wave amplitudes on thering (U12 D12M12) similar to the two-terminal continuummodel treatment and plugging the solution into the equationsfor the outgoing amplitudes I1 and R1 one finds a relation

between the incoming amplitude on the left lead L1 and bothoutgoing amplitudes The probability to transmit throughthe upper (lower) outgoing lead is given by T u = | R1

L1|2

(T l = | I1L1

|2) The exact expressions for these transmittanceprobabilities even when setting the incoming wave amplitudesR2 and I2 to zero are somewhat tedious to obtain [145] Thuswe provide the final results for completeness

The denominator of the transmittance probability for bothoutput channels is given by the following expression

Tdenominator = 116 (c

2 + 1)(19 minus 12c + 2c2 minus 12c3 + 19c4)

+ 32c3 cos(4πkr)+ 2(c minus 1)4ccos[4πkr(1 minus 2α)]+ cos[4πkr(1 minus 2β)] + cos[4πkr(1 minus 2γ )]minus 8(c minus 1)2c(c2 + 1)cos[4πkr(α minus 1)]+ cos[4πkr(β minus 1)] + cos[4πkr(γ minus 1)]minus 4(c minus 1)2(2 minus c + 2c2 minus c3 + 2c4)[cos(4πkrα)

+ cos(4πkrβ)+ cos(4πkrγ )]+ 2(c minus 1)4(c2 + 1)cos[4πkr(α minus β)]+ cos[4πkr(α minus γ )] + cos[4πkr(β minus γ )]minus 0125(c + 1)4 times minus4[1 + c(c minus 1)] cos(2πkr)

+ (c minus 1)2[cos[2πkr(1 minus 2α)] + cos[2πkr(1 minus 2β)]+ cos[2πkr(1 minus 2γ )]] cos

(2π

φ

φ0

)

+ 1

16(c + 1)6 cos2

(2π

φ

φ0

) (14)

The numerator of the transmittance probability through theupper output channel is given by

T unumerator = minus 1

2ε2

minus4(1 + c2)+ 2(c minus 1)2 cos(4πkrα)

+ (c + 1)2 cos(4πkrβ)+ 2(c minus 1)2 cos(4πkrγ )

+ 4c cos[4πkr(α+γ )] minus (cminus1)2 cos[4πkr(αminusγ )]

minus 2c(c + 1) cos

[2π

(φ

φ0minus kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0+ kr

)]

(a) (b)

Figure 8 Parameter designations in the three-terminal continuum model Panel (a) the angular separations between the three terminalsPanel (b) the amplitudes of different parts of the wavefunction

9


08

07

06

05

04

03

02

01

0

09

08

07

06

05

04

03

02

01

0

Figure 9 The transmittance of a three-terminal device as a function of the magnetic flux and the wavenumber of the conducting electron ascalculated using the continuum model Left panel high coupling (ε = 0495) right panel low coupling (ε = 0095) Color code deepbluemdashno transmittance (T = 0) deep redmdashfull transmittance (T = 1)

minus (c2 minus 1) cos

[2π

(φ

φ0+ (1 minus 2α)kr

)]

+ (c2 minus 1) cos

[2π

(φ

φ0minus (1 minus 2α)kr

)]

+ 2(c + 1) cos

[2π

(φ

φ0+ (1 minus 2β)kr

)]

+ 2c(c + 1) cos

[2π

(φ

φ0minus (1 minus 2β)kr

)]


[2π

(φ

φ0+ (1 minus 2γ )kr

)]

+ (c2 minus 1) cos

[2π

(φ

φ0minus (1 minus 2γ )kr

)] (15)

The numerator of the transmittance probability through thelower output channel is given by

T lnumerator = minus 1

2ε2

minus4(1 + c2)+ (c + 1)2 cos(4πkrα)

+ 2(c minus 1)2 cos(4πkrβ)+ 2(c minus 1)2 cos(4πkrγ )

+ 4c cos[4πkr(β+γ )] minus (cminus1)2 cos[4πkr(βminusγ )]

minus 2c(c + 1) cos

[2π

(φ

φ0+ kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0minus kr

)]

+ 2c(c + 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)] (16)

The back-scattering probability is the complementary part ofthe sum of the transmittance probability through both the upperand lower leads

The resulting transmittance probability through one of theoutgoing leads for the symmetric case where α = β =γ = 2π

3 is presented in figure 9 as a function of themagnetic flux threading the ring and the wavenumber of theconducting electron In the high coupling limit (left panelof figure 9) the system is characterized by a wide rangeof high transmittance which can be shifted along the ABperiod by changing the electronrsquos wavenumber Magneticswitching of the transmittance at a given wavenumber valuefor this coupling regime requires high magnetic fields sincethe transmittance peaks are quite broad

As the coupling is decreased from its maximal value ofε = 1

2 to very low values (right panel of figure 9) a resonanttunneling junction is formed and the transmittance probabilitybecomes very sensitive to the magnetic flux Similar to the two-terminal case this is translated into sharper peaks that developin the magneto-transmittance curve For the parameters shownin the right panel of figure 9 there is negligible transmittancefor all values of k at zero magnetic field Note that the positionof the first maximum in the transmittance depends linearly onthe value of φ and k Fine tuning the wavenumber to a valuethat satisfies k R = 1

2 results in the appearance of the sharptransmittance peak at finite relatively low magnetic flux andthus allows the switching of the device at feasible magneticfields

A careful examination of the transmittance probabilityspectrum for k R = 1

2 at the low magnetic flux regime revealsthat the appearance of the transmittance peak is sensitive tothe polarity of the magnetic flux While at a positive magneticflux a pronounced peak is observed at the negative magnetic

10


flux counterpart this peak is absent This magnetic rectificationphenomena is allegedly in contrast to the Onsager symmetryrelation [146ndash148 127] obeyed in the two-terminal case whichstates that g(φ) = g(minusφ) where g is the conductance

This contradiction is resolved by considering thetransmittance probability through the other output channelwhich can be obtained by applying a reflection transformationwith respect to a plain passing through the incoming lead andperpendicular to the cross section of the ring The Hamiltonianof the system is invariant to such a transformation only ifaccompanied by a reversal of the direction of the magneticfield It follows that the transmittance probability through oneoutgoing lead is the mirror image of the transmittance throughthe other Thus one finds that for the second output channel(not shown) the peak is observed at a negative magnetic fluxrather than at a positive one Onsagerrsquos condition is thereforeregained for the sum of the transmittance probabilities throughboth output channels

The above analysis implies that at zero magnetic field bothoutput channels are closed and the electron is totally reflectedThe application of a relatively small positive magnetic fieldopens only one output channel and forces the electrons totransverse the ring through this channel alone Reversing thepolarity of the magnetic field causes the output channels tointerchange roles and forces the electrons to pass through thering via the other lead

To summarize this section magnetic fields offer uniquecontrollability over the conductance of nanometer-scaleinterferometers For example their polarity can be used toselectively switch different conducting channels While non-uniform scalar potentials have been used in mesoscopic physicsto obtain a similar effect [149] such control cannot be obtainedvia the application of uniform scalar potentials which arecommonly used to control molecular-scale devices This isdue to the fact that such scalar potentials lack the symmetrybreaking nature of magnetic vector potentials

4 Atomistic calculations

The models presented in section 3 are based on a simplifieddescription where the AB interferometers were modeledwithin a continuum approach In order to capture themore complex nature of the electronic structure of realisticnanoscale AB interferometers we consider a model whichwas developed for the calculation of magnetoconductancethrough molecular set-ups [112] The approach is based onan extension of a tight binding extended Huckel approachwhich incorporates the influence of external magnetic fieldsThis approach is combined with a nonequilibrium Greenrsquosfunction (NEGF) [150 151] formalism to calculate theconduction [152 119 153ndash155] In the case of pureelastic scattering this reduces to the Landauer formalismwhich relates the conductance to the transmittance probabilitythrough the system [153] The resulting magnetoconductancespectrum can then be studied for different molecular set-upsand conditions as described below

41 Conductance

We are interested in calculating the conductance through amolecular device coupled to two macroscopic conductingleads in the presence of an external magnetic field Forcompleteness we provide the details of the approach adoptedfor the present study For more details on the theory ofmolecular conduction see [119 5 8 12 156]

The starting point is the current formula obtained withinthe NEGF framework [153]

IL(R) = 2e

h

intdE

2πTr[lt

L(R)(E)Ggtd (E)minusgt

L(R)Gltd (E)

]

(17)Here IL(R) is the net current measured at the left (right)moleculendashlead junction Glt

d (E) and Ggtd (E) are the lesser and

greater device Greenrsquos functions respectively and ltL(R) and

gtL(R) are the left (right) lesser and greater self-energy terms

respectively The first term in the trace in equation (17) canbe identified with the rate of in-scattering of electrons intothe device from the left (right) lead Similarly the secondterm represents the out-scattering rate of electrons into theleft (right) lead The difference between these two termsgives the net energy-dependent flow rate of electrons throughthe device When integrated over the energy and multipliedby twice (to account for spin states) the electronrsquos chargethis results in the net current flowing through the left (right)junction It can be shown [157] that IL = minusIR in analogy toKirchhoffrsquos law and therefore equation (17) represents the fullcurrent through the device

The lesser and greater GFs appearing in equation (17)are related to the retarded (Gr(E)) and advanced (Ga(E))GFs which will be discussed later through the Keldyshequation [150]

Gltd (E) = Gr

d(E)lt(E)Ga

d(E)

Ggtd (E) = Gr

d(E)gt(E)Ga

d(E)(18)

We now limit our discussion to the pure elastic case withina single-electron picture where as pointed out above theNEGF approach reduces to the Landauer formalism [153]We retain the description within the NEGF to provide acomplete description needed below when inelastic effects willbe considered In the pure elastic scattering case one assumesthat the lesser (greater) self-energies have contributions arisingfrom the coupling to the left and right leads only

lt = ltL + lt

R

gt = gtL +gt

R (19)

Furthermore the lesser and greater self-energy terms inequations (17) and (19) are related to the retarded (r

L(R)(E))and advanced (a

L(R)(E)) self-energies which will also bediscussed later in the following manner

ltL(R)(E) = minus fL(R)(E μL(R))

[r

L(R)(E)minusaL(R)(E)

]

gtL(R)(E) = [

1 minus fL(R)(E μL(R))] [r


]

(20)

11


where fL(R)(E μL(R)) = [1 + eβ(EminusμL(R))]minus1 is the FermindashDirac occupation distribution in the left (right) lead μL(R)

is the electrochemical potential of the left (right) leadand β = 1

kBT where kB is Boltzmannrsquos constant andT is the temperature Upon plugging equations (18)and (20) into (17) one obtains the following result for thecurrent

I = 2e

h

intdE

2π

[fL(E μL)minus fR(E μR)

]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (21)

where the level broadening matrix L(R) is given by

L(R)(E) equiv i[r


] (22)

It is now possible to identify T (E) = Tr[L(E)Grd(E)R(E)

Gad(E)] as the electron transmission probability through the

molecule and the difference in the FermindashDirac distributionfunctions as the net energy-dependent density of current car-rying particles from the left (right) lead to the right (left)lead due to the difference in their electrochemical poten-tials [120] The current is obtained by integration over the en-ergy

For calculating the conductance (g) one can define thedifferential conductance as the derivative of the current withrespect to the applied bias voltage (Vb)

g = part I

partVb= 2e

h

part

partVb

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (23)

If one assumes that the bias voltage applied does not alterconsiderably the energetic structure of the device the onlycontribution of the bias in equation (21) appears in the FermindashDirac distribution functions Therefore it is possible to directlyconduct the differentiation in equation (23) to get the followingrelation

g = part I

partVb= 2e

h

intdE

2π

part[

fL(E μL)minus fR(E μR)]

partVb

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (24)

It should be noted that for finite bias voltages the aboveequation is valid only in the so-called wide band limit (WBL)or in the single-electron limit [257]

If one further assumes that the bias potential drops sharplyand equally at both junctions [158] the electrochemicalpotentials are given by μL(R) = EL(R)

f +(minus) 12 eVb where EL(R)

fis the Fermi energy of the left (right) lead Thus the derivativewith respect to the bias voltage appearing in equation (24) isgiven by

part[


partVb

= eβ

2

eminusβ|EminusμL|

[1 + eminusβ|EminusμL|]2 + eminusβ|EminusμR|

[1 + eminusβ|EminusμR|]2

(25)

If both Fermi energies of the leads are aligned with theFermi energy of the device and the electrochemical potentialdifference arises from the bias voltage alone then at thelimit of zero temperature and zero bias conductance the

expression appearing in equation (25) approaches a sharp δfunction located at the Fermi energy of the molecule andequation (23) in turn reduces to the well established Landauerformula [119 159] which directly relates the conductanceto the transmittance probability of the current carrying entitythrough the relevant device

g = g0T (26)

where g0 = 2e2

h is the conductance quantumConsidering the full expression given by equation (24)

it can be seen that for the calculation of the conductanceor more specifically of the transmittance probability itis necessary to obtain the retarded and advanced GFs ofthe device and the retarded and advanced self-energies ofthe leads Obtaining these is the subject of the nextsection

42 Retarded and advanced Greenrsquos functions andself-energies

Following the lines of NEGF theory [153] the GF associatedwith a Hamiltonian H evaluated in a non-orthogonal basis setmust satisfy the following relation in energy space [160]

(E S minus H)G(E) = I (27)

where E is the energy S is the overlap matrix G(E) is the GFand I a unit matrix of the appropriate dimensions

As suggested above in molecular conductance cal-culations it is customary to divide the system into thelsquodevicersquo which is the conducting molecule5 and thelsquoleadsrsquo which are the macroscopic conducting contactsthrough which electrons are injected into and taken outof the lsquodevicersquo In a two-terminal set-up this is trans-lated into the following sub-matrices division of equa-tion (27) [160 161]⎛⎝ε SL minus HL ε SLd minus VLd 0

ε SdL minus VdL ε Sd minus Hd ε SdR minus VdR

0 ε SRd minus VRd ε SR minus HR

⎞⎠

times⎛⎜⎝

GrL(ε) Gr

Ld(ε) GrL R(ε)

GrdL(ε) Gr

d(ε) GrdR(ε)

GrRL(ε) Gr

Rd(ε) GrR(ε)

⎞⎟⎠ =

⎛⎝

IL 0 0

0 Id 0

0 0 IR

⎞⎠

(28)

Here HL(R) and Hd are the left (right) semi-infinite lead andthe device Hamiltonians respectively SL(R) and Sd are theleft (right) lead and the device overlap matrices respectivelyVL(R)d are the coupling matrices between the left (right) leadand the device and SL(R)d are the overlap matrices betweenthe left (right) lead and the device Since the Hamiltonianis Hermitian one finds that VdL(R) = V dagger

L(R)d and the same

requirement applies for the overlap matrices SdL(R) = SdaggerL(R)d

(see section 44) All the terms discussed above are calculated

5 Usually one defines an rsquoextended moleculersquo which is the molecule itselfaccompanied with a limited portion of the leads which is influenced by theproximity to the molecule

12


Figure 10 Division of the semi-infinite bulk lead into a set ofnearest-neighbors interacting layers

using a formalism which will be presented in section 44 Acomplex energy ε = limηrarr0E + iη has been introducedin order to shift the poles of the Greenrsquos function fromthe real axis and allow for the convergence of both ananalytical and a numerical evaluation of the integrals overthe GFs This softens the sharp singularities of the GF intoa smoother Lorentzian shape which in the limit of η rarr0 restores the δ function characteristics of the imaginarypart

Solving equation (28) for the middle column of the GFmatrix results in the following expression for the retardeddevice GF (which appears in equation (24)) Gr

d(ε) whencoupled to the two leads

Grd(ε) =

[(gr

d(ε))minus1 minus r

L(ε)minus rR(ε)

]minus1 (29)

where grd(ε) = [ε Sd minus Hd]minus1 is the retarded GF of the bare

device and rL(R)(ε) is the retarded self-energy of the left

(right) lead given by

rL(R)(ε) =

(ε SdL(R) minus HdL(R)

)gr

L(R)(ε)

times(ε SL(R)d minus HL(R)d

) (30)

Here grL(R)(ε) = [ε SL(R) minus HL(R)]minus1 is the retarded GF of the

bare left (right) lead The advanced device GF is the Hermitianconjugate of its retarded counterpart Ga

d(ε) = [Grd(ε)]dagger

The self-energy terms appearing in equations (29) and (30)represent the effect of the coupling to the leads on theGF of the device The calculation of these terms requiresthe GF of the bare lead gr

L(R)(ε) This is not a simpletask since the leads are macroscopic sources of electronswhose GFs are impossible to calculate directly A way toovercome this obstacle is to represent the leads as periodic bulkstructures of a semi-infinite nature Using common solid statephysics techniques the periodicity of the bulk can considerablyreduce the dimensionality of the problem and thus enablesthe calculation of the electronic structure of the leads Forthis one can use an efficient iterative procedure developed byLopez Sancho et al [162ndash164] Within this approach the semi-infinite bulk is divided into a set of identical principal layersas shown in figure 10 in such a manner that only adjacentlayers overlap and interact Using equation (27) the resultingGF can be represented as the inverse of a nearest-neighbors

block-tridiagonal matrix of the form⎛⎜⎜⎜⎜⎜⎜⎝

G00(ε) G01(ε) G02(ε) G03(ε) middot middot middotG10(ε) G11(ε) G12(ε) G13(ε) middot middot middotG20(ε) G21(ε) G22(ε) G23(ε) middot middot middotG30(ε) G31(ε) G32(ε) G33(ε) middot middot middot

⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝

ε S00 minus H00 ε S01 minus V01 0

ε S10 minus V10 ε S00 minus H00 ε S01 minus V01

0 ε S10 minus V10 ε S00 minus H00

0 0 ε S10 minus V10

0 middot middot middot0 middot middot middot

ε S01 minus V01 middot middot middotε S00 minus H00 middot middot middot

⎞⎟⎟⎟⎟⎠

minus1

(31)

Here SII = SIIII = SIIIIII = middot middot middot equiv S00 is the overlapmatrix of the principal layer with itself SIII = SIIIII =SIIIIV = middot middot middot equiv S01 is the overlap matrix between two adjacentlayers and as before S10 = [S01]dagger HII = HIIII = HIIIIII =middot middot middot equiv H00 is the Hamiltonian matrix of the principal layerVIII = VIIIII = VIIIIV = middot middot middot equiv V01 is the coupling matrixbetween two adjacent layers and V10 = [V01]dagger

Equation (31) is a matrix representation of a set ofequations for the semi-infinite bulk GF matrix elements Gi j(ε)

which can be solved iteratively to get a compact expression

G(ε) =[(ε S00 minus H00)+ (ε S01 minus H01)T (ε)

+ (ε Sdagger01 minus H dagger

01)T (ε)]minus1

(32)

The transfer matrices T and T are given by converging seriesof the form

T (ε) = t0 + t0t1 + t0 t1t2 + middot middot middot + t0 t1t2 middot middot middot tn

T (ε) = t0 + t0 t1 + t0t1 t2 + middot middot middot + t0t1t2 middot middot middot tn(33)

where ti and ti are defined by the recursion relations

ti = (I minus timinus1 timinus1 minus timinus1timinus1

)minus1t2iminus1


)minus1t2iminus1

(34)

with the following initial conditions

t0 =(ε S00 minus H00

)minus1V dagger

01


)minus1V01

(35)

After calculating the semi-infinite bulk GF given inequation (32) the left and right lead self-energies canbe determined using equation (30) and the L(R) matricesappearing in the conductance expression (equation (24)) can becalculated using equation (22) with a

L(R)(ε) = [rL(R)(ε)]dagger

13


43 An alternativemdashabsorbing imaginary potentials

A second methodology for the calculation of the transmit-tance probability involves the application of absorbing poten-tials [166 167] Within this approach the L(R) matrix appear-ing in equation (24) is a negative imaginary potential placeddeep within the left (right) lead The purpose of this poten-tial is to absorb an electron traveling in the lead away fromthe molecular device before it reaches the lsquoedgersquo of the leadThis ensures that the effect of reflections from the distant leadedge is suppressed to a desired accuracy and thus enables us totruncate the lead representation to a computable size

In this review we show results that are based on a Gaussianapproximation to the imaginary absorbing potentials of the

form VL(R) = V0eminus (zminuszL(R)0 )2

2σ2 Here V0 is the potential heightσ the potential width and zL(R)

0 its location along the left (right)lead assuming that the leads are located along the Z axisThe generalization of the above expression to the case of anarbitrary lead direction is straightforward The parametersof the potential are chosen such that electrons possessing akinetic energy in a wide band around the Fermi energy areeffectively absorbed [168] As a rule of thumb one can choosethe potential height to be the location of the Fermi energyEF above the bottom of the calculated valence band E0V0 asymp EFminusE0 The width of the potential should be sufficientlylarger than the Fermi wavelength For example the width for acarbon wire is taken to be σ asymp 20a0 where a0 is Bohrrsquos radiusThe origin of the potential zL(R)

0 is chosen such that the regionof the lead close to the device where the effect of the potentialis negligible contains at least a few Fermi wavelengths suchthat the metallic nature of the lead is appropriately capturedThis choice of parameters should serve as an initial try and aconvergence check should then be conducted The Gaussianabsorbing potential is of course not a unique choice and otherforms of absorbing potential expressions can be used [169]

Within the absorbing potentials methodology the systemis not divided into sub-units and the dimensionality of all thematrices appearing in equation (24) is the dimensionality ofthe full system (device+truncated leads) The L(R) matrixelements are calculated (usually numerically) as integrals ofthe atomic basis functions over the imaginary potential VL(R)It should be noted that similar to the WBL discussed above the matrices in the current methodology are energy-independentThis fact considerably reduces the computational effortsinvolved in the evaluation of the transmission probability

The device GF in equation (24) is now replaced by the GFof the whole system which similar to equation (29) is givenby

Gr(E) =[

E S minus H minus iL minus iR

]minus1 (36)

where S and H are the overlap and Hamiltonian matricesrespectively calculated for the whole system and E is thereal energy The advanced GF is as before the Hermitianconjugate of the retarded counterpart Ga(E) = [Gr(E)]dagger

The last remaining task for the calculation of theconductance is the calculation of the Hamiltonian overlap andcoupling matrices appearing in equation (28) This calculationis explained in the next section

44 Electronic structuremdashtight binding magnetic extendedHuckel theory

The final task before performing the calculations of theconductance is the appropriate representation of the electronicstructure of the system under the influence of the externalmagnetic field For this purpose a tight binding magneticextended Huckel theory (TB-MEHT) was developed Thisapproach provides a description of the Hamilton matrixelements as well as the corresponding basis set Thus itcan be used to include the effects of external perturbationssuch as magnetic fields by simply incorporating these into theHamiltonian and evaluating the perturbation matrix elementswith the given basis

Spin effects which have been considered mainlyfor mesoscopic AB interferometers utilizing spinndashorbitcouplings [170ndash181 142 182] are ignored here Sincemolecular rings lack a RashbaDresselhaus [183 184] fieldleading to spinndashorbit coupling and since one is interestedin small magnetic fields where the Zeeman spin splittingis negligible [145] it is safe to ignore the spin degree offreedom for the present application For a recent discussionof molecular spintronic devices utilizing an AB interferometersee [145]

Consider the one-body electronic Hamiltonian of the form

H = 1

2me

[P minus qA

]2 + V (r) (37)

The notation used here is similar to that used for theHamiltonian presented in equation (1) Within the TB-MEHTformalism the contribution at zero vector potential H(A =0) = P2

2m + V(r) is represented by the tight binding extended

Huckel (TB-EH) Hamiltonian H EH [185] which approximatesthe complex many-body problem using a mean-field single-particle semi-empirical approach The effect of applying avector potential is taken into account by simply adding theappropriate magnetic terms to the EH Hamiltonian

H = H EH minus q

2me

(P middot A + A middot P

)+ q2

2mA2 (38)

As before we assume that the magnetic field is uniform andconstant B = (Bx By Bz) and thus write down the relatedvector potential as A = minus 1

2 r times B = minus 12 (y Bz minus z By z Bx minus

x Bz x By minus y Bx) Putting this expression in equation (38) oneobtains the following electronic Hamiltonian [186]

H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me

is the Bohr magneton r2perp = r2 minus (rmiddotB)2

B2 is theprojection of r = (x y z) onto the plane perpendicular to Band L = r times P is the angular momentum operator

For the sake of simplicity and without limiting thegenerality of the solution we assume that the AB ring is placedin the Y ndashZ plane and that the magnetic field is applied parallelto the X axis With this the Hamiltonian in equation (39) isreduced to

H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14


A Slater-type orbital (STO) [187 188] basis set is used toevaluate the Hamiltonian and overlap matrices In the presenceof a magnetic field it is necessary to multiply each STO byan appropriate gauge factor which compensates for the finitesize of the set The resulting atomic orbitals are customarilyreferred to as gauge-invariant [189 190] Slater-type orbitals(GISTOs) and are given by

|n lm〉α = eiqh Aα middotr|n lm〉α = e

iq Bx2h (yαzminuszα y)|n lm〉α (41)

Here |n lm〉α is an STO characterized by the set of quantumnumbers (n lm) and centered on the atomic site α Aα

is the value of the vector potential A at the nuclear positionRα = (xα yα zα) and |n lm〉α is the GISTO situated at α

The generalized eigenvalue problem Hψn = En Sψis solved to get the electronic energy levels (En) and themolecular orbitals (ψn) characterizing the system Here

H and S are the Hamiltonian and overlap matrices with

elements given by S1α2β =α 〈n1 l1m1|n2 l2m2〉β and

H 1α2β =α 〈n1 l1m1|H |n2 l2m2〉β The Londonapproximation [189 190] is used to calculate the differentmatrix elements Within this approximation the gauge phaseappearing in equation (41) is taken outside the integralreplacing r by 1

2 (Rα + Rβ) The overlap integrals are thengiven by

S1α2β asymp S1α2βeiLαβ (42)

Here S1α2β =α 〈n1 l1m1|n2 l2m2〉β and Lαβ = q2h (Aβ minus

Aα) middot(Rα+Rβ) = q Bx

2h (zα yβ minus yαzβ) The Hamiltonian matrixelements are approximated as

H 1α2β asymp 12

[H1α2βeiLαβ + H1β2αeiLβα

] (43)

where H1α2β = α〈n1 l1m1|H |n2 l2m2〉β This form guaran-tees that the Hamiltonian matrix remains Hermitian under theapproximation

All matrix elements in the formulation presented above arecalculated analytically The EH Hamiltonian diagonal matrixelements are set to be equal to the ionization potentials of theappropriate atomic orbital H EH

1α1α = IP1α while the off-diagonal elements are given by an average of the corresponding

diagonal elements H EH1α2β = k

H EH1α1α+H EH

2β2β

2 S1α2β where k =175 is a parameter chosen to give the best fit to experimentaldata The overlap matrix is calculated analytically for every setof quantum numbers using a method developed by Guseinovet al [191ndash195] The matrix elements of the magnetic termsappearing in the Hamiltonian can then be expressed as a linearcombination of overlap integrals and are thus also calculatedwithout the need to perform numerical integration Theexpansion of the magnetic integrals in terms of correspondingoverlap integrals is presented elsewhere [196]

Before continuing it should be mentioned that while thecurrent model does take into account the explicit geometryof the system and also some of the details of the systemrsquoselectronic structure it remains an effective one-particle modeland neglects contributions from electronndashelectron correlationsand coupling to the vibrational degrees of freedom of the

molecular device Furthermore since the bias potential isassumed to drop sharply at the leadndashdevice junctions and itseffect on the energy levels of the device is neglected thecurrent model is valid only at the low bias regime

45 Results for two-terminal devices

In the present section we study the magnetoresistancebehavior of two-terminal devices as a function of the relevantphysical control parameters identified in section 31 Theatomistic calculations are compared to those obtained using thecontinuum model

451 Atomic corral First we consider a corral composedof monovalent atoms placed on a semi-conducting surface andcoupled to two atomic wires An experimental realizationof this set-up can be achieved using scanning tunnelingmicroscopy (STM) techniques [197ndash200] The whole set-upis then placed in a perpendicular homogeneous and constantmagnetic field and the conductance between the atomic wireleads through the corral is measured at the limit of zero biasvoltage as a function of the threading magnetic field Realizingthat the corral itself is also constructed of an atomic wire itis important to understand some basic properties of such one-dimensional structures

Since each atomic site contributes a monovalent electronthe number of atoms in the chain N must be even to geta closed shell wire The Fermi energy of the wire has aprincipal quantum number of N

2 which is also the number ofnodes the Fermi electrons wavefunctions have If the distancebetween the atomic sites d is constant then the separationbetween the nodes is given by Nd

N2 = 2d and thus the Fermiwavelength is λF = 2 times 2d = 4d Therefore one sees thatfor any regular atomic wire composed of monovalent atomicsites the conducting electronsrsquo wavelength is of the order offour interatomic distances This simple conjecture allows usto divide the symmetric atomic corrals into two prototypesthose having N = 4n and those having N = 4n + 2 atomson the circumference where n is an integer number For anN = 4n atomic corral the quantum number of the Fermiwavefunction of a particle on a ring is an integer mF =kFr = 2π

λF

Nd2π = Nd

4d = N4 = n and therefore the resonance

condition in equation (11) is obtained when the magnetic fieldis off However when N = 4n + 2 one obtains mF =n + 1

2 and resonance is achieved only when φ

φ0= l

2 wherel = plusmn1plusmn2

A similar picture arises when plotting the wavefunctionsof the Fermi electrons The complex normalized wavefunc-tions of a particle on a ring are given by plusmnm(θ) = 1radic

2πeplusmnimθ

A linear combination of each doubly degenerate functions hav-ing the same quantum number m results in real wavefunctionsof the form 1radic

πcos(mθ) and 1radic

πsin(mθ) The cosine functions

are plotted in panels (a) and (b) of figure 11 for the two corralprototypes having m = mF = N

4 As can be seen for the 4n prototype a standing wave is

obtained for which both input and output leads are located atnon-stationary points For the 4n + 2 corral at zero magneticflux the wavefunction is not stationary and the output lead

15


Figure 11 An illustration of the Fermi wavefunctions of a particle on a ring-shaped corral Panel (a) a cosine wavefunction for a 4n ring withn = 5 creates a stationary solution Panel (b) a cosine wavefunction of a 4n + 2 ring with the same n as before creates a destructiveinterference between the clockwise (blue) and counterclockwise (red) electron pathways Panel (c) a sine wave functions for N = 4n atomswith n = 5

is located exactly at a node of the interference between theclockwise and counterclockwise traveling electrons In factthe interference is destructive not only at the nodes but forevery point on the ring and thus this energy level does not existIt is interesting to note that the sine function which is shiftedby π

2 from the cosine counterpart has both leads located at itsnodes as can be seen in figure 11(c) Therefore at a givenleadsrsquo location and in the absence of a gate potential and amagnetic field only one of the two degenerate energy levelswill conduct and the maximum zero bias conductance will bethe quantum conductance g0

An alternative explanation for this class division can begiven based upon the energetic structure of the ring Due tothe symmetry of the ring apart from the lowest energy levelall energy levels of a particle on an uncoupled ring are doublydegenerate When considering a monovalent electron per sitecorral the occupation of the energy levels of the 4n prototypediffers from that of the 4n + 2 prototype

This can be seen in figure 12 where the highest occupiedmolecular orbital (HOMO) occupation is presented for bothprototypes The HOMO occupation of the 4n prototype (leftpanel) involves two vacancies and thus is suitable for electronconduction when brought to resonance with the leads whilethe HOMO occupation of the 4n + 2 prototype (right panel) isfull and therefore does not conduct electrons

After realizing these important features of one-dimensionalmonovalent atomic wires we turn to present atomistic calcu-lations results obtained using the imaginary potentials methodwithin the TB-MEHT formalism as discussed above In fig-ure 13 the conductance through atomic corrals composed of 40and 42 copper atoms is plotted [111] All atoms on the cor-ral are separated by a distance of 235 A The effect of a gatepotential was simulated by changing the corral atomic orbitalenergies by Vg

As discussed above at zero gate voltage the conductancepeak for the n = 40 corral is located at the low magneticfield region while for the n = 42 corral it is located near themiddle of the AB period Two common features are clearlyobserved for the two corral prototypes (a) a large magneticfield (sim500ndash600 T) is required to complete a full AB periodand (b) the conductance peaks (red spots) shift with the gatevoltage Vg The latter effect is analogous to the shift of peaksseen in figure 5 for the continuum model as the conductingelectron wavenumber is varied Therefore the application of

Figure 12 The occupation of the energy levels of the corral for the4n prototype (left panel) and for the 4n + 2 prototype (right panel)

a gate voltage allows control over the location of the peakconductance In particular it can be used to shift the maximalconductance to zero magnetic field for the N = 4n + 2 corralprototype and to fine tune the location of the N = 4n peak ifslightly shifted from the magnetic field axis origin

The next step is to control the width of the conductanceresonances as a function of the magnetic field In thecontinuum model this was done by reducing the transmissionamplitude ε In the molecular system this can be achieved byincreasing the distance between the ring and the edge lead atomclosest to the ring Alternatively one can introduce an impurityatom at the junctions between the lead and the ring Howeverfor quantum corrals the former approach seems more realistic

In figure 14 the conductance as a function of the magneticfield is depicted for several values of the leadndashring separationFor each generic corral prototype an appropriate gate voltageis applied to ensure maximal conductance at B = 0 TAs the leadndashring separation is increased the lifetime of theenergy levels on the ring is increased and therefore their widthdecreases This is translated into a sharpening of the switchingresponse to the magnetic field in the magnetoconductancespectrum At the highest separation studied one achieves aswitching capability of the order of a single tesla despite thefact the AB period is comparable to 500ndash600 T

Due to the dependence of the AB period on the magneticflux the effects discussed above are scalable with thedimension of the ring as long as the simple physics holdsFor a given leadndashcorral coupling strength doubling the crosssection of the corral will reduce the switching limit to half

16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)

Figure 13 Conductance as a function of the magnetic field and the gate voltage for a 40 (left panel) and a 42 (right panel) atomic corralcomposed of copper atoms at T = 1 K Color code redmdashg = g0 purplemdashg = 0

Figure 14 Conductance as a function of the magnetic field and the contact bond length (Rc) for a 40 (left panel) and a 42 (right panel) atomiccorral composed of copper atoms at T = 1 K The gate potential is 0 V (left) and minus0132 V (right)

its original value for a given leadndashdevice coupling at lowenough temperatures This can be clearly seen in figure 15where the magnetoconductance of several corrals with inter-site spacing of 1 A and at a temperature of 1 K is calculatedin the WBL with a coupling strength of 005 eV For a corralwith a diameter of sim13 nm switching occurs at sim4 T (solidcurve in the figure) When doubling the diameter of the corral(multiplying the cross section by 4) the switching thresholdreduces to sim1 T (dashed line in the figure) as expected Upon afurther increase in the dimensions of the corral (dashedndashdottedline) the switching threshold reduces respectively

452 Carbon nanotubes Another interesting (andperhaps somewhat more accessible experimentally) sys-tem is an AB interferometer based on a carbon nan-otube (CNT) [201] CNTs have been investigated inthe context of molecular electronics both experimen-tally [48 52 53 131 202ndash205 55 206 132 133] andtheoretically [207ndash209] The AB effect in single-walledand multi-walled carbon nanotubes (SWCNT and MWCNT

respectively) has been addressed experimentally and theoreti-cally as well [210ndash219] (for a recent review see [220]) Thesestudies focus on the AB effect in CNTs while in this review wedescribe the conditions for which switching can be obtained atmagnetic fields that are much smaller than those correspondingto a full AB period in CNTs [112]

Similar to the quantum corral set-up when a magneticfield is applied perpendicular to the cross section of the tube(along its main axis) electron pathways traversing the circularcircumference in a clockwise and a counterclockwise mannergain different magnetic phases and thus AB interferenceoccurs As the coupling between the CNT and the conductingleads is decreased a resonant tunneling junction forms Thisresults in an increase of the electronrsquos lifetime on the CNTand thus in a narrowing of the energy levelsrsquo width Usinga biasgate potential it is possible to tune the resonance suchthat the transmittance is high at zero magnetic field Theapplication of low magnetic fields shifts the narrow energylevel out of resonance Thus switching occurs at fields muchsmaller than those required to achieve a full AB cycle

17


Figure 15 Magnetoconductance through an atomic corral as afunction of the corral dimensions As the diameter of the corral isdoubled from sim13 nm (solid line) to sim26 nm (dashed line) theswitching threshold reduces by a factor of 4 For the largest corralconsidered (dashedndashdotted line) the switching threshold is sim05 Tfor a coupling strength of 005 eV

Two different experimental configurations are consideredfor this purpose [112] The first consists of an SWCNTplaced on an insulating substrate between two thin conductingcontacts (see figure 16(a)) and a bias potential is appliedbetween the contacts Similar set-ups have been recentlydemonstrated experimentally [30 221ndash223] In the secondconfiguration an SWCNT is placed on a conducting substratecoupled to a scanning tunneling microscope (STM) tip fromabove as described schematically in figure 16(b) The biaspotential is applied between the STM tip and the underlyingsurface For both configurations the resulting conductancebetween the leads can be calculated using the TB-MEHTapproach described above showing that high sensitivity to themagnetic field can be achieved

For configuration (a) (figure 16(a)) both leads are modeledby atomic conducting wires and the calculations are done usingthe imaginary potentials method (see section 43) while for

configuration (b) (figure 16(b)) the STM tip is modeled bya semi-infinite one-dimensional atomic conducting gold wireand the substrate is modeled by a semi-infinite slab of goldcrystal The iterative procedure discussed in section 42 wasapplied to obtain the semi-infinite bulk Greenrsquos function Thecalculations were conducted for a tube four unit cells in lengthusing minimum image periodic boundary conditions for thepassivation of the edge atoms Such short CNTs have beenrecently synthesized [224] Tests on longer tubes reveal thesame qualitative picture described below

The density of states on a 16 nm long (24 0) SWCNTsegment is plotted in figure 17 A sketch of the short segmentof the SWCNT is shown in the inset of the figure Twodoubly degenerate energy levels appear in near the Fermienergy which are separated from the rest of the spectrumby approximately 100 meV Resonant tunneling through thesestates accompanied by degeneracy lifting due to the AharonovndashBohm effect allows for delicate control over the conductancethrough the system

In figure 18 the conductance through the cross section ofthe (24 0) SWCNT segment as calculated for configuration(a) is plotted against the external axial magnetic field forseveral bias potentials The conductance at zero bias firstincreases as one switches on the magnetic field (negativemagnetoresistance) peaks near B = 10 T and subsequentlydecreases as the field grows vanishing at fields above 30 TThe maximum conductance observed gg0 = 2 is limitedby the number of open channels in the vicinity of the Fermienergy of the SWCNT In order to achieve switching capabilityat magnetic fields smaller than 1 T it is necessary to move theconductance peak to zero magnetic fields and at the same timereduce its width

When a small bias is applied to the sample theconductance peak splits into a doublet6 The position of thecorresponding peaks depends on the value of the bias and themaximum conductance is reduced by 25ndash50 As can be

6 This is due to the specific way one may chose the bias potential drop atthe two leadndashdevice junctions as described in section 41 For different biaspotential profiles the quantitative details of the calculations would changehowever the qualitative picture would remain the same

a b

Figure 16 An illustration of the experimental configurations suggested for measuring the cross-sectional magnetoresistance of a CNT Inconfiguration (a) the SWCNT is placed on an insulating surface between two narrow metallic contacts while in configuration (b) the SWCNTis placed on a conducting substrate and approached from above by an STM tip

18


Figure 17 Density of states in the vicinity of the Fermi energy ascalculated by the TB-MEHT for a (24 0) SWCNT segment Lightercoloured lines indicate the position of the energy levels Gaussianbroadening of 008 eV was introduced for clarity Inset axial andside views of the 16 nm long (24 0) SWCNT segment

Figure 18 Conductance versus the magnetic field for several biaspotentials as calculated by the TB-MEHT for a (24 0) SWCNTplaced as in configuration (a) The effect of the application of a biaspotential on the position of the conductance peaks is depicted at aconstant tubendashcontact separation of 24 A

seen in the figure by adjusting the bias potential it is possibleto shift one of the conductance peaks toward low values ofthe magnetic field such that the conductance is maximal atB = 0 T and positive magnetoresistance is achieved Theshift in the conductance peak can be attributed to the changein the energy level through which conductance occurs when asmall bias is applied As a result of this change the electronmomentum is altered resulting in the conductance peak shiftobserved in the calculation

In figure 19 the effect of changing the tubendashcontactseparation at constant bias potential is studied As oneincreases the separation between the tube and the contactstheir coupling decreases resulting in a reduction of thewidth of the energy resonances of the SWCNT Thus theconductance becomes very sensitive to an applied magneticfield and small variations in the field shift the relevantenergy level out of resonance In the magnetoresistance

Figure 19 Conductance versus the magnetic field for severaltubendashcontact separations as calculated by the TB-MEHT for a (24 0)SWCNT placed as in configuration (a) The effect of an increase inthe tubendashcontact separation is depicted at a constant bias potential of000679 V Inset the full AB period for a (24 0) SWCNT at zerobias potential and tubendashcontact separation of 24 A

spectrum this is translated to a narrowing of the transmittancepeaks similar to the case of the atomic corral discussed insection 451 For the smallest separation considered (24 A)the conductance seems to be constant on the logarithmicscale at the magnetic field range shown in the figure whileat the highest separation studied (32 A) the width of theconductance peak is comparable to 1 T At higher magneticfields (not shown) the conductance of the 24 A case reducesas well

Similar to the corral case the combined effect of the biaspotential and the tubendashcontact separation allows us to shiftthe position of the conductance peak to small magnetic fieldswhile at the same time reducing its width This is achievedby carefully selecting the values of the bias potential andtubendashcontact separation Under proper conditions one obtainspositive magnetoresistance with a sharp response occurring atmagnetic fields comparable to 1 T This result is significantsince it implies that despite the fact that the tube radius issmall (sim1 nm) and the corresponding full AB period requiresunrealistically large magnetic fields of the order of 1500 T(as shown in the inset of figure 19) it is possible to achievemagnetic switching at relatively small magnetic fields

A similar picture arises when considering configuration(b) In figure 20 we show the conductance as calculatedfor a (6 0) SWCNT placed between a sharp STM tip and aconducting surface for two bias voltages A smaller-diameterCNT was used in these calculations in order to be able toproperly describe the bulky nature of the conducting substratewith respect to the dimensions of the CNT

As can be seen when a bias voltage of sim0224 V isapplied the conductance peaks at a magnetic field of sim14 TBy changing the bias potential to sim0225 V the conductancepeak shifts toward zero magnetic field Under these conditionsswitching occurs at a magnetic field of sim10 T while the full ABperiod for this system is of the order of 2 times 104 T The CNTndashlead separation required to achieve high magnetoresistancesensitivity in this configuration is larger than the one needed

19


Figure 20 Conductance as a function of the magnetic field through a(6 0) CNT as calculated for configuration (b) at different biasvoltages The separation between the CNT and the conducting leadsused in this calculation is taken to be 41 A

for configuration (a) This is due to the difference in the CNTdiameters and the different lead geometries As the diameterof the tube becomes smaller the magnetic field needed togain a similar AB phase shift grows larger Thereforethe conductance peaks become wider so that larger CNTndashlead separations are required in order to narrow their widthFurthermore as the lead becomes more bulky its coupling tothe CNT needs to be decreased in order to achieve the samemagnetoresistance sensitivity

To conclude the CNT AB interferometry part it wasshown that SWCNTs can be used as magnetoresistanceswitching devices based on the AB effect As in the corralset-up the essential procedure involves the weak coupling ofthe SWCNT to the conducting leads in order to narrow theconducting resonances while at the same time controllingthe position of the resonances by the application of a biaspotential The control over the coupling between the SWCNTand the conducting leads in configuration (a) of figure 16 canbe achieved via a fabrication of a set of leads with proper gapsIn configuration (b) of the same figure one needs to controlthe distance between the STM tip and the CNT and betweenthe substrate and the CNT The former can be achieved bypiezoelectric control and the latter by covering the surface withmonolayers of an insulating material

453 Operative limitations The above calculations havebeen performed assuming a low temperature of 1 K (apartfor the (6 0) CNT where a temperature of 01 K was used)However the effect will hold even at higher temperaturesRecall that within the Landauer formalism the temperatureeffect is taken into account only through the difference of theleadsrsquo FermindashDirac population distributions fL(R)(E μL(R))This difference sets the (temperature-dependent) width of theenergy band through which conductance occurs Thus thiswidth should be narrow enough (the temperature should below enough) in order to resolve the magnetic field splittingof the originally degenerate energy levels of the ring

Figure 21 Magnetoconductance through an atomic corral composedof 40 single-electron sites with 1 A spacing at different temperaturesIt can be seen that as the temperature rises from 1 K (solid line) to10 K (dashed line) the device response becomes less sensitive to themagnetic field At 25 K (dashedndashdotted line) the conductivity isalmost insensitive to the magnetic field at the low magnetic fieldregion

Using equation (11) one finds that this splitting is givenby h2

2mR2 [(minus|mF| minus φ

φ0)2 minus (|mF| minus φ

φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ

φ0 Therefore the temperature should fulfill the following

condition in order for AB switching to be observed kBT lt2h2kFmR

φ

φ0

Considering an atomic corral with a diameter of sim13 nmand interatomic distances of 1 A (40 atomic sites) thenormalized magnetic flux at 5 T is φ

φ0asymp 15 times 10minus3 and

the Fermi wavelength is λF asymp 4 A resulting in a Fermiwavenumber of kF = 2π

λFasymp π

2 Aminus1

Assuming an effectivemass of m = 1 au one gets an upper limit of approximately7 K

This can be seen in figure 21 where the magnetoconduc-tance through such a corral is calculated using the WBL at dif-ferent temperatures The coupling to both leads is taken to be005 eV such that switching occurs at sim5 T As the temperaturerises from 1 K (solid curve in the figure) to 10 K (dashed linein the figure) the device response becomes less sensitive to themagnetic field even though an effect is still observable as pre-dicted At a higher temperature of 25 K (dashedndashdotted line)the conductivity becomes almost insensitive to the magneticfield at the magnetic field region considered

46 Results for three-terminal devices

As discussed in section 32 an interesting case in whichmagnetic fields provide unique control over the conductanceis based on the three-terminal set-up Here we describean example based on a polycyclic aromatic hydrocarbon(PAH) [113] A sketch of such a ring composed of48 conjugated benzene units with a diameter of sim3 nmis shown in figure 22 Similar PAH molecules havebeen synthesized by chemical means [225ndash227] and studiedtheoretically [228ndash231] This structure can also be viewed as

20


Figure 22 A realization of a three-terminal molecular ABinterferometer based on a polycyclic aromatic hydrocarbon moleculecoupled to three atomic gold wires

a closed form of a graphene nanoribbon [61 64 66 110] aclass of materials that has attracted a great deal of theoreticalactivity recently [232ndash237]

In figure 23 we show the zero bias conductance of themolecular switch as a function of the magnetic field intensityfor both output channels We focus on the region of realisticmagnetic fields (much smaller than the field required tocomplete a full AB period which is sim470 T for a ring with across-sectional area of about 875 nm2) A relatively large gatevoltage Vg = 185 V is needed in order to bring the systeminto resonance and the leadndashmolecule separation is taken to besim3 A

For zero magnetic field both channels are semi-openedand the conductance assumes a value of 04g0 at the selectedgate voltage When a magnetic field of sim25 T is appliedone finds that one output channel fully opens while theother closes shut As the polarity of the field changes signthe two output channels interchange their role Exactly thesame characteristics are captured by the continuum model(eg dashed lines in figure 23) Therefore it can be seen thatthe magnetic field polarity serves as a distributor that streamsthe electrons to the desired output channel

Based on this magnetic rectification behavior it is possibleto design a molecular logic gate which processes two differentlogic operations simultaneously This can be achieved bychoosing one input signal as the bias voltage (Vb) and the otherinput signal as the magnetic field (B) For the bias input signalVb we mark as lsquo0rsquo the case where Vb = 0 V and as lsquo1rsquo thecase where a small bias is applied For the magnetic fieldinput signal we mark as lsquo0rsquo the case where B asymp minus3 T andas lsquo1rsquo the case where B asymp 3 T The output signals are thecurrents measured at the two outgoing leads marked as I1 andI2 Using these definitions the truth table presented in table 1can be constructed

One sees that the output I1 gives the logic operationVb ampamp B while the output O2 gives the logic operation

Figure 23 Magnetoconductance of the three-terminal moleculardevice shown in figure 22 Solid lines correspond to the conductanceof the upper and lower outgoing terminals calculated by TB-MEHTformalism Dashed lines correspond to the conductance of the upperand lower outgoing terminals calculated by the continuum model

Table 1 Truth table for the parallel molecular logic gate

Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0

Vb ampamp B where the overbar stands for NOT Even though theconductance peaks are sharp relative to the full AB period thistruth table holds for a wide range of the threading magneticfield intensity plusmn(2ndash4) T and is thus suitable for robust logicgate operations Shifting the conductance peaks via the changeof the gate potential will give rise to different logic operationsof the same set-up

To summarize the current section it can be seen that singlecyclic molecules are promising candidates for the fabricationof magnetoresistance parallel switching and gating devices atfeasible magnetic fields As in the previous discussion carefulfine tuning of experimentally controllable physical parametersallows the selective switching of a single pre-selected outputchannel thus allowing for the design of a molecular logicgate processing two logic operations in parallel This is madepossible due to the unique symmetry breaking nature of themagnetic gate

5 Inelastic scattering effects in AB molecularinterferometers

Our discussion above neglects completely inelastic effects aris-ing from electronndashphonon couplings (EPCs) Such processeshave been considered in molecular junctions by several au-thors utilizing different theoretical approaches [238ndash243 165244ndash248] In AB interferometers EPC were considered inmesoscopic rings where EPC leads to a decay in the amplitudeof the AB oscillations [249] and more recently in the context of

21


electronndashelectron interaction [250] In this section we addressthe issue of inelastic effects arising from EPCs for molecularAB interferometers where we focus on the low magnetic fluxregime as appropriate for such systems [114] We show asexpected that the EPC broadens the conduction peak whenan electric gate is applied [244] In contrast dephasing pro-cesses arising from EPC narrow the magnetoconductance re-sponse and thus increase the sensitivity to the threading mag-netic flux This surprising result can be rationalized in terms ofa rapid loss of the phase of electrons at the exit channel aris-ing from the coupling to the phonons thus destroying the co-herence which is essential for AB interferometry as discussedbelow This important result emphasizes yet another major dif-ference between electric and magnetic gauges

51 Hamiltonian

We consider a tight binding model of an atomic-corral-basedinterferometer We describe the case of a single valenceelectron per atomic site where the interaction between theelectrons and the vibrations occurs on site namely an electronlocated at site i will couple to the local vibration of the i thatom The Hamiltonian of the system is described by

H = Hel + Hvib + Hint (44)

where Hel is the electronic part of the Hamiltonian Hvib is thepart of the Hamiltonian describing the vibrational modes of themolecule and Hint represents the interaction of the electronswith the vibrations of the molecule

The electronic part of the Hamiltonian may be written asa combination of two terms Hel = H 0

el + H intel where H 0

el =H device

el + H leadsel H device

el being the electronic Hamiltonian ofthe isolated corral (or any other device) H leads

el the electronicHamiltonian of the bare leads and H int

el is the leadndashdeviceelectronic interaction Hamiltonian One may write these termsusing second quantization operators For the non-interactingelectronic Hamiltonian

H 0el =

sumi j

ti j (B)cdaggeri c j +

sumlmisinL R

εlm(B)ddaggerl dm (45)

where cdaggeri and ci are creation and annihilation operators of

an electron on site i on the device and ddaggerl and dl are the

creation and annihilation operators of an electron on site l onthe relevant lead ti j (B) is the hopping integral between sitei and site j on the device and εlm(B) is the hopping integralbetween site l and site m on the relevant lead As before L andR stand for left and right leads

For the devicendashlead interaction part

H intel =

sumli

Vli (B)ddaggerl ci + Hc (46)

where Hc stands for Hermitian conjugate The matrixelements of all these terms are evaluated using the TB-MEHTapproach presented in section 44 Thus ti j (B) εlm(B) andVli (B) depend on the magnetic field B

When considering the vibrational part of the HamiltonianHvib the device atoms are allowed to vibrate around their

equilibrium position such that one can describe their vibrationsusing the harmonic approximation For the sake of simplicitythe atoms are assumed to be confined to move along thecircumference such that one can map the problem onto aone-dimensional (1D) chain of coupled harmonic oscillatorswith periodic boundary conditions For the case of a corralcomposed of identical atoms one has to define only the localvibrational frequency in order to obtain all normal vibrationsof the ring In local modes the vibrational Hamiltonian of thesystem is described by a simple form given by

Hvib = Tvib + Vvib =Nminus1sumk=0

p2k

2+ 1

22

Nminus1sumk=0

(xk+1 minus xk)2 (47)

Here N is the number of atoms and pk and xk denote themass-weighted momentum operator and the mass-weighteddeviation from the equilibrium position of atom k along thechain respectively Periodic boundary conditions are imposedon the system such that xN = x0 It is possible to define aunitary transformation (given by a matrix U ) that diagonalizesthe Hamiltonian (cf equation (47)) to obtain

Hvib = Tvib + Vvib = 12

Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)

This Hamiltonian represents a set of N uncoupled harmonicoscillators each with frequency ωk of the collective normalmode of vibration qk where all frequencies are defined in termsof and the unitary transformation U In terms of raising andlowering boson operators one finally arrives at

Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)

The last term in the full Hamiltonian (equation (44)) isthe electronndashvibration coupling term To lowest order in theelectronndashphonon coupling this term is modeled by

Hint = MNminus1sumi=0

cdaggeri ci(a

daggeri + ai ) (50)

where M is the electronndashphonon coupling strength which istaken to be identical to all atomic sites Using this form theelectron on site i represented by the number operator Ni equivcdagger

i ci is coupled to the local vibration of atom i represented byadagger

i + ai Here adaggeri and ai are the raising and lowering operators

of the local vibrational modes defined in terms of xk and pk

such that (adaggerk + ak) =

radic2h xk In terms of the normal modes

of the ring the electronndashphonon coupling Hamiltonian is givenby


cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger

i ci(bdaggerk + bk) (51)

where Mki equiv M

radicωk

Uki and Uki are the matrix elements of the

transformation matrix U

22


Collecting all the terms appearing in equations (45) (46)(49) and (51) the full Hamiltonian of the system is given bythe following expression

H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li

Vli (B)ddaggerl ci + Hc

)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger


As pointed out above the only two free parameters in thismodel are Mmdashthe electronndashphonon coupling strengthmdashandmdashthe local vibrational frequency The remaining hoppingterms are calculated from the TB-MEHT

52 Conductance

The calculation of the conductance is described as beforewithin the framework of the nonequilibrium Greenrsquos function(NEGF) method [119] In brief the total current I = Iel +Iinel is recast as a sum of elastic (Iel) and inelastic (Iinel)contributions given by [165]

Iel = 2e

h

intdε

2π

[f (ε μR)minus f (ε μL)

]

times Tr[ΓL(ε)Gr(ε)ΓR(ε)Ga(ε)

](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)

respectively The retarded (advanced) GFs satisfy the Dysonequation

Gra(ε) =[gra(ε)]minus1 minus Σra

L (ε)minus ΣraR (ε)minus Σra

ph (ε)minus1

(55)where gra(ε) is the uncoupled retarded (advanced) electronicGF of the ring In the above equations the retarded(advanced) self-energy arising from the coupling to the right(left) lead is Σra

LR(ε) = (ε minus Vdagger(B))graLR(ε)(ε minus V(B)) the

corresponding greater (lesser) self-energy is Σ≶LR(ε) = (δ≶ minus

f (ε μLR))[ΣrLR(ε)minusΣa

LR(ε)] where δ≶ equals 0 for lt and1 otherwise f (ε μ) = 1

1+eβ(εminusμ) and ΓLR(ε) = i[ΣrLR(ε) minus

ΣaLR(ε)] To obtain these self-energies one requires as input

the leadndashring hopping matrix V(B) with elements Vmj (B)and the retarded (advanced) uncoupled GF of the left or rightlead gra

LR(ε) The calculation of the current requires also theself-energy arising from the interactions with the phononswhich in the present study is calculated using the first Bornapproximation (FBA) and is given by [165]

Σrph(ε) = i

sumk

intdω

2πMk

Dlt

k (ω)gr(ε minus ω)

+ Drk(ω)g

lt(ε minus ω)+ Drk(ω)g

r(ε minus ω)

Mk (56)

Figure 24 Conductance versus gate voltage for a single energy levelcoupled to two electronic reservoirs in the absence (solid line) andthe presence (dashed line) of an interaction with a single vibrationalmode

where the Hartree term has been omitted [244] The lesser andgreater self-energies arising from the coupling to the phononsare given by

Σ≶ph(ε) = i

sumk

intdω

2πMk D≶

k (ω)g≶(ε minus ω)Mk (57)

In the above equations Drak and D≶

k are the uncoupledequilibrium retarded (advanced) and lesser (greater) GFs ofphonon mode k respectively g≶(ε) is the lesser (greater)uncoupled electronic GF of the ring and Mk is the EPC matrixof mode k (diagonal in the c j basis)

One should keep in mind that the first Born approximationis a truncation of the self-consistent Born procedure which byitself is an approximation to the solution of the real many-body problem Therefore only the low electronndashphononcoupling regime can be considered within the framework ofthis model [244 156]

53 Results

First we discuss the influence of electronndashvibration couplingon the gate voltage dependence of the zero bias zerotemperature conductance in the absence of a magnetic fieldFor simplicity we start by considering a single energy levelcoupled to a single vibrational mode The results are shownin figure 24 The vibrational frequency is taken to giveh = 001 eV the coupling to the leads is considered tobe symmetric such that L = R = 025h and =L + R = 05hω0 and the width of the vibrational energylevels due to their relaxation to an external thermal bath is takenas η = 2 times 10minus4 eV We plot the conductance as a functionof the gate potential for the case of no vibrational coupling( M

= 0) and for the case where the coupling of the electron tothe nuclear vibrations of the device equals the coupling of thedevice to the leads ( M

= 1) where M is the coupling strength

to the vibrational mode The latter condition defines the upper

23


limit for the relevance of the perturbative treatment discussedin the last section [244]

The two most significant observations are the expectedbroadening of the conduction when the EPC is turned on andthe value of the zero bias conduction (gg0 = 1 whereg0 = 2e2h is the quantum conductance) in the presence ofEPC To better understand these results we rewrite the currentfor the case that ΓL(ε) = ΓR(ε) equiv Γ(ε)2 in the followingway [251] I = 2e

h

intdε( f (ε minusμL)minus f (εminusμR))T (ε) where

T (ε) = i4 Tr Γ(ε)(Gr(ε) minus Ga(ε)) Note that T (ε) is the

transmission coefficient only when M = 0 In the wide bandlimit for the single resonant level model T (ε) can be reduced

to 4

minus2rphim(ε)

(εminusε0minusrphre(ε))

2+(2minusrphim(ε))

2 As a result of the fact that

rphim(0) = 0 at zero bias the only inelastic contribution to

the conduction comes from the real part of the phonon self-energy [244] From this it follows that even in the presence ofEPC the maximal conduction is gmaxg0 = 1 as can clearlybe seen in figure 24 It also immediately implies that the maincontribution to the broadening of the resonant conduction peakcomes from the energy-dependent real part of the phonon self-energy

Next we consider a similar situation in a molecularAB interferometer composed of N = 40 sites Thesites are identical and contribute a single electron which isdescribed by a single Slater s-like orbital The couplingbetween the ring and the leads is limited to the contactregion For simplicity the electronic self-energies arising fromthis coupling are approximated within the wide band limitSpecifically we approximate ΓLR(ε) with matrices that areindependent of energy where the only non-vanishing elementsare the diagonal elements (LR) corresponding to the two sitescoupled to the left or right lead The local phonon frequency = 00125 eV is characteristic of a low frequency opticalphonon in molecular devices [252 253] Since our modeldoes not include a secondary phonon bath required to relaxthe energy from the optical phonons we include a phononenergy level broadening η = 0016 The coupling toeach of the leads is taken to be L = R = 4 such thatthe magnetoconductance switching in the absence of EPC isobtained at sim5 T for an inter-site separation of 15

ξ where

ξ is the inverse correlation length of the Slater s-like orbital(ξ = 13 in atomic units)

In figure 25 a similar plot is presented for the twodegenerate levels of the atomic corral As can be seen thepicture that arises in the corral case (figure 25) resembles thatobtained for a single level (figure 24) The difference in thewidth of the lineshapes between the two cases results from thefact that the coupling to the electronic reservoirs is differentand that the coupling strength of the molecular levels to thevibrations in the corral case is not given by M itself but bya set of transformations proportional to M as described insection 51

We now turn to discuss the role of EPC on themagnetoconduction when the gate voltage is fixed to zerosuch that at zero magnetic field the corral fully conducts Infigure 26 we plot the magnetoconductance of the AB ring forseveral values of the EPC (M) and for different temperatures(T ) We focus on the regime of low magnetic flux φ = AB

Figure 25 Conductance versus gate voltage for a 40-site atomiccorral coupled to two electronic reservoirs in the absence (solid line)and the presence (dashed line) of electronndashvibration interactions

where A is the area of the ring and B is taken perpendicular tothe plane of the ring

The case M = 0 was discussed in detail in sections 3and 4 The solid curves in figure 26 present the coherentmagnetoconductance switching for different temperatures Asexpected we find that upon increasing the temperature themaximal value of gg0 is decreased and the width of themagnetoconductance peaks is increased linearly with T Thisincrease in the width with temperature is a result of the resonanttunneling conditions and the broadening of the leadsrsquo Fermidistribution functions as T is varied

Turning to discuss the case of M = 0 one of the majorquestions is related to the effects of EPC on the switchingcapability of small AB rings Based on the discussion ofthe results when the gate voltage was varied one mightexpect that an increase in M will lead to a broadening of themagnetoconductance peaks thereby increasing the value of themagnetic field required to switch a nanometer AB ring andperhaps leads to unphysical values of B required to reducedthe conduction significantly As can be seen in figure 26the numerical solution of the NEGF for M = 0 leads to areduction of the width of the magnetoconductance peaks andthe switching of the AB ring is achieved at lower values of themagnetic flux compared to the case where M = 0

This surprising observation can be explained in simpleterms [114] As discussed above even in the presence of EPCthe maximal conduction at zero bias and zero temperature isgmaxg0 = 1 as is clearly the case for the results shown in theupper left panel of figure 26 for φφ0 = 0 For the symmetricring of N = 4n the resonance condition at φφ0 = 0 isequivalent to the condition that electrons entering the ring fromthe left interfere constructively when they exit the ring to theright [254 255] This picture also holds when M = 0 andthe conduction takes a maximal value at φφ0 = 0 Theapplication of a magnetic field leads to destructive interferenceand increases the backscattering of electrons This loss ofphase is even more pronounced when inelastic effects arisingfrom EPC are included In the magnetoconductance this is

24


Figure 26 Conductance as a function of magnetic flux for several values of the EPC strength M and for different temperatures The resultsare for a ring of N = 40 sites Similar results were obtained for largersmaller rings signifying that the scaling behavior [254 255] for thephonon-free case also holds in the case of weak EPC

translated to a more rapid loss of conduction as a function ofthe magnetic field when M is increased In the weak devicendashlead coupling limit the same physics also describes otherclasses of rings [254 255] like the N = 4n + 1 N = 4n + 2and N = 4n + 3 However a gate voltage must be appliedto tune the resonance such that conduction is maximal at zeromagnetic field [111]

Mathematically the rapid decay of the conduction withthe magnetic field as the EPC is increased can be explainedby analyzing the dependence of T (ε) In figure 27 we plotTel(ε) = Tr[ΓL(ε)Gr(ε)ΓR(ε)Ga(ε)] which is the elastic (anddominant) contribution to T (ε) as a function of energy forseveral values of φφ0 for M = 0 or M = 1

4 In thelower panel we also plot the corresponding Fermi distributionintegration window At φφ0 = 0 Tel(ε) asymp 1 near the Fermienergy (εf) independent of M and the conduction is gg0 asymp 1The application of a small magnetic field results in a splitof Tel(ε) where each peak corresponds to a different circularstate [256] The separation between the two peaks in the elasticlimit = (ε2 minus ε1) prop φφ0 is proportional to the magneticflux where ε12 are the corresponding energies of the twocircular states When inelastic terms are included due to thefact that the imaginary part of Σr

ph(ε) is negligibly small therenormalized positions of the two peaks can be approximatedby εlowast

12 = ε12 +rphre(ε12) = ε12 plusmnr

phre(ε2) which impliesthat the renormalized separation between the two peaks canbe approximated by lowast = (εlowast

2 minus εlowast1 ) = + 2r

phre(ε2)Therefore as M is increased lowast is also increased consistentwith the numerical results shown in figure 27

To summarize this section as expected the gate voltagedependence of the conductance broadens upon switchingon the electronndashvibration coupling Nevertheless themagnetoconductance peaks reduce both in size and in widthwhen the vibrational coupling is taken into account This

implies a higher sensitivity of the device to the applicationof an external magnetic field than that predicted for the purecoherent case

6 Summary and prospects

Affecting the current through molecular or nanoscale junctionsis usually done by a combination of bias and gate voltagesMagnetic fields are less studied because nanodevices cancapture only small fractions of a magnetic flux In particularexploiting AharonovndashBohm interferometers at the nanometerscale was regarded as impossible due to the large magnetic fieldrequired to complete a full AharonovndashBohm period

The present review summarizes an attempt to overcomesome of the limitations associated with utilizing magneticfields as gates in molecular junctions We describe the physicalconditions at which magnetic fields significantly affect theconductance in molecular AharonovndashBohm interferometersSeveral examples were given to emphasize the majordifferences between electric and magnetic gates

Using a simple continuum model of an AharonovndashBohminterferometer ring that captures the essential physical featuresof the problem we have identified and isolated the importantparameters that allow control over the conductance througha nanometric ring using fractions of a magnetic flux (interms of the quantum flux) In contrast to the strive to gainbetter coupling between the leads and the molecule within amolecular junction magnetic control requires weak devicendashlead couplings (bad contacts) This results in a resonanttunneling junction in which the weakly coupled (and thusextremely narrow) energy levels of the ring allow the transportonly at well-defined energy values By the application of a gatevoltage it is possible to tune these resonances such that in theabsence of a magnetic field the device will conduct Turning

25


Figure 27 Plots of Tel(ε) as a function of energy for M = 0(solid curves) and M = 1

4 (dashed curves) for different values ofthe magnetic flux The dotted curve at the lower panel showspart

partμ f (ε minus μ) at T = 0007 where f (ε minus μ) is the difference

between the Fermi distributions of the left and right leads

on a magnetic field will shift the narrow doubly degenerateenergy levels of the ring out of resonance with the leads andthus will lower the conductance considerably Considering thecombined effect of increasing the lifetime of the electron on thering (through the coupling) and controlling its energy (using agate voltage) it is possible to narrow the magnetoconductancepeaks in the AharonovndashBohm spectrum while at the sametime shifting them toward the low magnetic field regime suchthat the device becomes extremely sensitive to small magneticfields This is despite the fact that the full AharonovndashBohmperiod involves magnetic fields orders of magnitude higher

In order to validate the basic principles revealed by thecontinuum model we have presented results based on a tightbinding magnetic extended Huckel theory which allows foratomistic calculation of the conductance through molecularset-ups under the influence of an external magnetic fieldSimilar to the conclusions drawn from the continuum modelthe results obtained from atomistic calculation indicate thatnanometric circular set-ups such as atomic corrals the crosssection of carbon nanotubes or polycyclic aromatic rings canbe used for AharonovndashBohm interferometry The conductancethrough the device becomes very sensitive to the magnetic field

when a delicate balance of coupling and gatebias voltages isreached Such devices can be used as electromagnetic switchesor as miniature magnetic sensors

Similar to the electric gate the magnetic field providesmeans to externally control the conductance of ring-shapedmolecular junctions However there are striking differencesin the properties of these two gauges This was illustrated ina multi-terminal device where the polarity of the magneticfield which couples to the electronic angular momentumplayed a key role In this device the unique symmetrybreaking nature of the magnetic vector potential allows fora selective control over the outgoing route of the electronThe magnetic field polarity determines through which ofthe two outgoing leads the electrons will transverse thering Another fundamental difference between electricaland magnetic gauges was observed with respect to inelasticeffects While the conductance as a function of the gatevoltage broadens upon coupling to phonons it actually narrowsconsiderably in response to a magnetic field This wasillustrated for a model system of an atomic corral whereimproved switching capability was observed as the electronndashphonon coupling strength was increased

The design of molecular junctions is an extremely difficulttask Moreover introducing external electrical gates is quitelimited and the fabrication and application of gate electrodes atthe nanometer scale is still not feasible In this context thedevelopment of molecular AharonovndashBohm interferometerssensitive to uniform magnetic fields provides yet anotheradvantage over electrical gating Devices can be designedutilizing different sizesshapes of rings that function at differentmagnetic fields thus generating a multi-component structure

Several issues still remain open Most importantlyis the experimental realization of the proposed theoreticalscheme We have suggested several realistic systems wherethe effects discussed in this review can be measured Onthe theoretical side several directions are not coveredhere First a treatment of high electronndashphonon couplingis needed [247] In this regime it is reasonable toexpect that the coherent nature of the transport would bedestroyed diminishing the interference pattern Thereforeit is important to identify the physical conditions at whichsuch effects are expected to be found Furthermore thetreatment of strongly correlated systems where electronndashelectron interactions become important is also an open subjectnot addressed in the context of molecular AharonovndashBohminterferometers Another interesting direction involves therole of the spin of the current carrying particles Sinceopposite spins respond differently when entering a magneticfield region one would expect that their AharonovndashBohminterference patterns would be different [145] Thus such set-ups can provide the grounds for the development of spintronicdevices based on molecular AharonovndashBohm interferometers

References

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[2] Roth C and Joachim C 1997 Atomic and Molecular Wires(Dordrecht Kluwer)

26


[3] Emberly E G and Kirczenow G 1998 Theory of electricalconduction through a molecule Mol Electron Sci Technol852 54ndash67

[4] Dekker C 1999 Carbon nanotubes as molecular quantum wiresPhys Today 52 22ndash8

[5] Ratner M 2000 Molecular electronicsmdashpushing electronsaround Nature 404 137ndash8

[6] Joachim C Gimzewski J K and Aviram A 2000 Electronicsusing hybrid-molecular and mono-molecular devicesNature 408 541ndash8

[7] Tour J M 2000 Molecular electronics synthesis and testing ofcomponents Acc Chem Res 33 791ndash804

[8] Nitzan A 2001 Electron transmission through molecules andmolecular interfaces Annu Rev Phys Chem 52 681ndash750

[9] Pantelides S T Di Ventra M and Lang N D 2001 Molecularelectronics by the numbers Physica B 296 72ndash7

[10] Bourgoin J-P 2001 Molecular electronics a review ofmetalndashmoleculendashmetal junctions Lect Notes Phys 579 105

[11] Avouris P 2002 Molecular electronics with carbon nanotubesAcc Chem Res 35 1026ndash34

[12] Nitzan A and Ratner M A 2003 Electron transport inmolecular wire junctions Science 300 1384ndash9

[13] Heath J R and Ratner M A 2003 Molecular electronic PhysToday 56 43ndash9

[14] Hush N S 2003 An overview of the first half-century ofmolecular electronics Ann New York Acad Sci 1006 1ndash20

[15] Wassel R A and Gorman C B 2004 Establishing the molecularbasis for molecular electronics Angew Chem Int EdnEngl 43 5120ndash3

[16] Seminario J M 2005 Molecular electronics approachingreality Nat Mater 4 111ndash3

[17] Joachim C and Ratner M A 2005 Molecular electronics someviews on transport junctions and beyond Proc Natl AcadSci 102 8801ndash8

[18] Tao N J 2006 Electron transport in molecular junctions NatNanotechnol 1 173ndash81

[19] Prasanna de Silva A and Uchiyama S 2007 Molecular logicand computing Nat Nanotechnol 2 399ndash10

[20] Park H 2007 Molecular electronics charges feel the heat NatMater 6 330ndash1

[21] Reddy P Jang S-Y Segalman R A and Majumdar A 2007Thermoelectricity in molecular junctions Science315 1568ndash71

[22] Weibel N Grunder S and Mayor M 2007 Functionalmolecules in electronic circuits Org Biomol Chem5 2343ndash53

[23] Franzon P Nackashi D Amsinck C DiSpigna N andSonkusale S 2007 Molecular electronicsmdashdevices andcircuits technology IFIP Int Federation Inf Process240 1ndash10

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[25] Reed M A Zhou C Muller C J Burgin T P and Tour J M 1997Conductance of a molecular junction Science 278 252ndash4

[26] Kergueris C Bourgoin J P Palacin S Esteve D Urbina CMagoga M and Joachim C 1999 Electron transport througha metalndashmoleculendashmetal junction Phys Rev B59 12505ndash13

[27] Weber H B Reichert J Weigend F Ochs R Beckmann DMayor M Ahlrichs R and von Lohneysen H 2002Electronic transport through single conjugated moleculesChem Phys 281 113ndash25

[28] Smit R H M Noat Y Untiedt C Lang N D van Hemert M Cand van Ruitenbeek J M 2002 Measurement of theconductance of a hydrogen molecule Nature 419 906ndash9

[29] Park H Lim A K L Alivisatos A P Park J and McEuen P L1999 Fabrication of metallic electrodes with nanometerseparation by electromigration Appl Phys Lett75 301ndash3

[30] Liang W J Shores M P Bockrath M Long J R andPark H 2002 Kondo resonance in a single-moleculetransistor Nature 417 725

[31] Selzer Y Cai L T Cabassi M A Yao Y X Tour J MMayer T S and Allara D L 2005 Effect of localenvironment on molecular conduction isolated moleculeversus self-assembled monolayer Nano Lett 5 61ndash5

[32] Selzer Y and Allara D L 2006 Single-molecule electricaljunctions Annu Rev Phys Chem 57 593ndash623

[33] Guillorn M A Carr D W Tiberio R C Greenbaum E andSimpson M L 2000 Fabrication of dissimilar metalelectrodes with nanometer interelectrode distance formolecular electronic device characterization J Vac SciTechnol B 18 1177ndash81

[34] Park J et al 2002 Coulomb blockade and the Kondo effect insingle atom transistors Nature 417 722ndash5

[35] Hersam M C Guisinger N P and Lyding J W 2000Silicon-based molecular nanotechnology Nanotechnology11 70ndash6

[36] Klein D L McEuen P L Bowen Katari J E Roth R andAlivisatos A P 1996 An approach to electrical studies ofsingle nanocrystals Appl Phys Lett 68 2574ndash6

[37] Davidovic D and Tinkham M 1998 Coulomb blockade anddiscrete energy levels in Au nanoparticles Appl Phys Lett73 3959ndash61

[38] Morpurgo A F Marcus C M and Robinson D B 1999Controlled fabrication of metallic electrodes with atomicseparation Appl Phys Lett 74 2084ndash6

[39] Li C Z He H X and Tao N J 2000 Quantized tunneling currentin the metallic N formed by electrodeposition and etchingAppl Phys Lett 77 3995ndash7

[40] Kervennic Y V Van der Zant H S J Morpurgo A FGurevich L and Kouwenhoven L P 2002 Nanometer-spacedelectrodes with calibrated separation Appl Phys Lett80 321ndash3

[41] Frank S Poncharal P Wang Z L and de Heer W A 1998Carbon nanotube quantum resistors Science 280 1744 ett

[42] Slowinski K Fong H K Y and Majda M 1999Mercuryndashmercury tunneling junctions 1 Electrontunneling across symmetric and asymmetric alkanethiolatebilayers J Am Chem Soc 121 7257ndash61

[43] Holmlin R E Haag R Chabinyc M L Ismagilov R FCohen A E Terfort A Rampi M A and Whitesides G M2001 Electron transport through thin organic films inmetalndashinsulatorndashmetal junctions based on self-assembledmonolayers J Am Chem Soc 123 5075ndash85

[44] Kushmerick J G Holt D B Yang J C Naciri J Moore M Hand Shashidhar R 2002 Metalndashmolecule contacts andcharge transport across monomolecular layersmeasurement and theory Phys Rev Lett 89 086802

[45] Dorogi M Gomez J Osifchin R Andres R P andReifenberger R 1995 Room-temperature Coulombblockade from a self-assembled molecular nanostructurePhys Rev B 52 9071ndash7

[46] Gittins D I Bethell D Schiffrin D J and Nichol R J 2000 Ananometre-scale electronic switch consisting of a metalcluster and redox-addressable groups Nature 408 67

[47] Cui X D Primak A Zarate X Tomfohr J Sankey O FMoore A L Moore T A Gust D Harris G andLindsay S M 2001 Reproducible measurement ofsingle-molecule conductivity Science 294 571ndash4

[48] Tans S J Devoret M H Dai H Thess A Smalley R EGeerligs L J and Dekker C 1997 Individual single-wallcarbon nanotubes as quantum wires Nature 386 474

[49] Porath D Bezryadin A de Vries S and Dekker C 2000 Directmeasurement of electrical transport through DNAmolecules Nature 403 635ndash8

[50] Dadoshand T Gordin Y Krahne R Khivrich I Mahalu DFrydman V Sperling J Yacoby A and Bar-Joseph I 2005

27


Measurement of the conductance of single conjugatedmolecules Nature 436 677ndash80

[51] Klein D L Roth R Lim A K L Alivisatos A P andMcEuen P L 1997 A single-electron transistor made from acadmium selenide nanocrystal Nature 389 699ndash701

[52] Tans S J Verschueren A R M and Dekker C 1998Room-temperature transistor based on a single carbonnanotube Nature 393 49ndash52

[53] Yao Z Postma H W Ch Balents L and Dekker C 1999 Carbonnanotube intramolecular junctions Nature 402 273

[54] Chen J Reed M A Rawlett A M and Tour J M 1999 Largeonndashoff ratios and negative differential resistance in amolecular electronic device Science 286 1550ndash2

[55] Postma H W Ch Teepen T Yao Z Grifoni M andDekker C 2001 Carbon nanotube single-electron transistorsat room temperature Science 293 76

[56] Cui Y and Lieber C M 2001 Functional nanoscale electronicdevices assembled using silicon nanowire building blocksScience 291 851ndash3

[57] Zhitenev N B Meng H and Bao Z 2002 Conductance of smallmolecular junctions Phys Rev Lett 88 226801

[58] Luo Y et al 2002 Two-dimensional molecular electronicscircuits ChemPhysChem 3 519ndash25

[59] Liang W Shores M P Bockrath M Long J R and Park H 2002Kondo resonance in a single-molecule transistor Nature417 725ndash9

[60] Keren K Berman R S Buchstab E Sivan U and Braun E 2003DNA-templated carbon nanotube field-effect transistorScience 302 1380ndash2

[61] Novoselov K S Geim A K Morozov S V Jiang D Zhang YDubonos S V Grigorieva I V and Firsov A A 2004 Electricfield effect in atomically thin carbon films Science306 666ndash9

[62] Piva P G DiLabio G A Pitters J L Zikovsky J Rezeq MDogel S Hofer W A and Wolkow R A 2005 Fieldregulation of single-molecule conductivity by a chargedsurface atom Nature 435 658ndash61

[63] Champagne A R Pasupathy A N and Ralph D C 2005Mechanically adjustable and electrically gatedsingle-molecule transistors Nano Lett 5 305ndash8

[64] Novoselov K S Geim A K Morozov S V Jiang DKatsnelson M I Grigorieva I V Dubonos S V andFirsov A A 2005 Two-dimensional gas of massless diracfermions in graphene Nature 438 197ndash200

[65] Li X Xu B Xiao X Yang X Zang L and Tao N 2006Controlling charge transport in single molecules usingelectrochemical gate Faraday Discuss 131 111ndash20

[66] Berger C et al 2006 Electronic confinement and coherence inpatterned epitaxial graphene Science 312 1191ndash6

[67] Han M Y Ozyilmaz B Zhang Y and Kim P 2007 Energyband-gap engineering of graphene nanoribbons Phys RevLett 98 206805

[68] Lemme M C Echtermeyer T J Baus M and Kurz H 2007 Agraphene field effect device IEEE Electron Device Lett28 282ndash4

[69] Klitzing K V Dorda G and Pepper M 1980 New method forhigh-accuracy determination of the fine-structure constantbased on quantized Hall resistance Phys Rev Lett45 494ndash7

[70] Eherenberg W and Siday R E 1949 The refractive index inelectron optics and the principles of dynamics Proc PhysSoc (London) B 62 8

[71] Aharonov Y and Bohm D 1959 Significance ofelectromagnetic potentials in the quantum theory Phys Rev115 485

[72] Chambers R G 1960 Shift of an electron interference patternby enclosed magnetic flux Phys Rev Lett 5 3ndash5

[73] Parks R D and Little W A 1964 Fluxoid quantization in amultiply-connected superconductor Phys Rev133 000A97

[74] Sharvin D Yu and Sharvin Yu V 1981 Quantization of amagnetic flux in a cylindrical film from normal metalJETP Lett 34 272

[75] Alrsquotshuler B L Aronov A G Spivak B Z Sharvin D Yu andSharvin Yu V 1982 Observation of the AharonovndashBohmeffect in hollow metal cylinders JETP Lett 35 588ndash90

[76] Mollensdedt G Schmid H and Lichte H 1982 Proc 10th IntCongress on Electron Microscopy (HamburgDeutschgesellschaft fur Elektronen Mikroskopie)

[77] Tonomura A Matsuda T Suzuki R Fukuhara A Osakabe NUmezaki H Endo J Shinagawa K Sugita Y andFujiwara H 1982 Observation of AharonovndashBohm effect byelectron holography Phys Rev Lett 48 1443ndash6

[78] Brandt N B Bogachek E N Gitsu D V Gogadze G AKulik I O Nikolaeva A A and Ponomarev Ya G 1982 Fluxquantization effects in metallic microcylinders in tiltedmagnetic field Fiz Nisk Temp 8 718

[79] Ladan F R and Maurer J 1983 Magnetic flux quantization in athin non-superconducting metal cylinder C R Acad Sci297 227ndash9

[80] Schmid H 1984 Proc 8th European Congress on ElectronMicroscopy ed A Csanady P Rohlich andD Szabo (Budapest Programme Committee of the 8th EurCongr on Electron Microsc)

[81] Gijs M van Haesandonck C and Bruynseraede Y 1984Resistance oscillations and electron localization incylindrical Mg films Phys Rev Lett 52 2069

[82] Gijs M van Haesandonck C and Bruynseraede Y 1984Quantum oscillations in the superconducting fluctuationregime of cylindrical Al films Phys Rev B 30 2964ndash7

[83] Pannetier B Chaussy J Rammal R and Gandit P 1984Magnetic flux quantization in the weak-localization regimeof a nonsuperconducting metal Phys Rev Lett 53 718ndash21

[84] Sharvin Yu V 1984 Weak localization and oscillatorymagnetoresistance of cylindrical metal films Physica B+C126 288ndash93

[85] Gordon J M 1984 Quantum phase sensitivity to macroscopicboundaries Al cylinders and wires Phys Rev B 30 6770ndash3

[86] Pannetier B Chaussy J Rammal R and Gandit P 1985 Firstobservation of AltshulerndashAronovndashSpivak effect in gold andcopper Phys Rev B 31 3209ndash11

[87] Dolan G J 1985 Bull Am Phys Soc 30 395[88] Alrsquotshuler B L Aronov A G and Spivak B Z 1981 The

AharonovndashBohm effect in disordered conductorsJETP Lett 33 94

[89] Buttiker M Imry Y and Landauer R 1983 Josephson behaviorin small normal one-dimensional rings Phys Lett A96 365

[90] Gefen Y Imry Y and Azbel M Ya 1984 Quantum oscillationsin small rings at low temperatures Surf Sci 142 203

[91] Gefen Y Imry Y and Azbel M Ya 1984 Quantum oscillationsand the AharonovndashBohm effect for parallel resistorsPhys Rev Lett 52 129ndash32

[92] Buttiker M Imry Y and Azbel M Ya 1984 Quantumoscillations in one-dimensional normal-metal ringsPhys Rev A 30 1982ndash9

[93] Carini J P Muttalib K A and Nagel S R 1984 Origin of theBohmndashAharonov effect with half flux quanta Phys RevLett 53 102ndash5

[94] Browne D A Carini J P Muttalib K A and Nagel S R 1984Periodicity of transport coefficients with half flux quanta inthe AharonovndashBohm effect Phys Rev B 30 6798ndash800

[95] Buttiker M Imry Y Landauer R and Pinhas S 1985Generalized many-channel conductance formula withapplication to small rings Phys Rev B 31 6207ndash15

[96] Webb R A Washburn S Umbach C P and Laibowitz R B1985 Observation of he AharonovndashBohm oscillations innormal-metal rings Phys Rev Lett 54 2696ndash9

[97] Timp G Chang A M Cunningham J E Chang T YMankiewich P Behringer R and Howard R E 1987

28


Observation of the AharonovndashBohm effect for ωcτ gt 1Phys Rev Lett 58 2814ndash7

[98] Yacoby A Heiblum M Mahalu D and Shtrikman H 1995Coherence and phase sensitive measurements in a quantumdot Phys Rev Lett 74 4047

[99] van Oudenaarden A Devoret M H Nazarov Yu V andMooij J E 1998 Magneto-electric AharonovndashBohm effectin metal rings Nature 391 768ndash70

[100] Auslaender O M Yacoby A de Picciotto R Baldwin K WPfeiffer L N and West K W 2002 Tunneling spectroscopyof the elementary excitations in a one-dimensional wireScience 295 825ndash8

[101] Ralph D C Black C T and Tinkham M 1997 Gate-voltagestudies of discrete electronic states in aluminumnanoparticles Phys Rev Lett 78 4087ndash90

[102] Cobden D H Bockrath M McEuen P L Rinzler A G andSmalley R E 1998 Spin splitting and evenndashodd effects incarbon nanotubes Phys Rev Lett 81 681ndash4

[103] Cobden D H Bockrath M Chopra N G Zettl A McEuen P LRinzler A Thess A and Smalley R E 1998 Transportspectroscopy of single-walled carbon nanotubes Physica B251 132ndash5

[104] Cronenwett S M Oosterkamp T H and Kouwenhoven L P1998 A tunable Kondo effect in quantum dots Science281 540ndash4

[105] Goldhaber-Gordon D Shtrikman H Mahalu DAbusch-Magder D Meirav U and Kastner M A 1998Kondo effect in a single-electron transistor Nature391 156ndash9

[106] Nyga ringrd J Cobden D H and Lindelof P E 2000 Kondophysics in carbon nanotubes Nature 408 342ndash6

[107] Buitelaar M R Nussbaumer T and Schonenberger C 2002Quantum dot in the Kondo regime coupled tosuperconductors Phys Rev Lett 89 256801

[108] Pasupathy A N Bialczak R C Martinek J Grose J EDonev L A K McEuen P L and Ralph D C 2004 TheKondo effect in the presence of ferromagnetism Science306 86ndash9

[109] Yu L H and Natelson D 2004 The Kondo effect in C60single-molecule transistors Nano Lett 4 79ndash83

[110] Geim A K and Novoselov K S 2007 The rise of grapheneNat Mater 6 183ndash91

[111] Hod O Baer R and Rabani E 2004 Feasible nanometricmagnetoresistance devices J Phys Chem B 108 14807

[112] Hod O Rabani E and Baer R 2005 Magnetoresistance devicesbased on single-walled carbon nanotubes J Chem Phys123 051103

[113] Hod O Baer R and Rabani E 2005 A parallel electromagneticmolecular logic gate J Am Chem Soc 127 1648

[114] Hod O Baer R and Rabani E 2006 Inelastic effects inAharonovndashBohm molecular interferometers Phys RevLett 97 266803

[115] Hod O Rabani E and Baer R 2006 Magneto-resistance ofnanoscale molecular devices Acc Chem Res 39 109ndash117

[116] Jackson J D 1999 Classical Electrodynamics(New York Wiley)

[117] Fuhrer A Luescher S Ihn T Heinzel T Ensslin KWegscheider W and Bichler M 2001 Energy spectra ofquantum rings Nature 413 822

[118] Kittel C 1996 Introduction to Solid State Physics(New York Wiley)

[119] Datta S 1995 Electronic Transport in Mesoscopic Systems(Cambridge Cambridge University Press)

[120] Datta S 2004 Electrical resistance an atomistic viewNanotechnology 15 S433ndash51

[121] Gefen Y Imry Y and Azbel N Y 1984 Quantum oscillationsand the AharonovndashBohm effect for parallel resistorsPhys Rev Lett 52 129

[122] Cahay M Bandyopadhyay S and Grubin H L 1989 Two typesof conductance minima in electrostatic AharonovndashBohmconductance oscillations Phys Rev B 39 12989

[123] Imry Y 2002 Introduction to Mesoscopic Physics 2nd edn(Oxford Oxford University Press)

[124] Aharony A Entin-Wohlman O and Imry Y 2003 Measuringthe transmission phase of a quantum dot in a closedinterferometer Phys Rev Lett 90 156802

[125] Benjamin C Bandopadhyay S and Jayannavar A M 2002Survival of φ02 periodicity in presence of incoherence inasymmetric AharonovndashBohm rings Solid State Commun124 331ndash4

[126] Jayannavar A M and Benjamin C 2002 Wave attenuationmodel for dephasing and measurement of conditional timesPramana J Phys 59 385ndash95

[127] Aharony A Entin-Wohlman O Halperin B I and Imry Y 2002Phase measurement in the mesoscopic AharonovndashBohminterferometer Phys Rev B 66 115311

[128] Vugalter G A Das A K and Sorokin V A 2004 A chargedparticle on a ring in a magnetic field quantum revivalsEur J Phys 25 157ndash70

[129] Zehnder L 1891 Ein neuer interferenzrefraktorZ Instrumentenkunde 11 275ndash85

[130] Mach L 1892 Uber einen interferenzrefraktorZ Instrumentenkunde 12 89ndash93

[131] Martel R Schmidt T Shea H R Hertel T and Avouris Ph 1998Single- and multi-wall carbon nanotube field-effecttransistors Appl Phys Lett 73 2447

[132] Bachtold A Hadley P Nakanishi T and Dekker C 2001 Logiccircuits with carbon nanotube transistors Science 294 1317

[133] Javey A Guo J Wang Q Lundstrom M and Dai H 2003Ballistic carbon nanotube field-effect transistorsNature 424 654

[134] Wu C H and Ramamurthy D 2002 Logic functions ofthree-terminal quantum resistor networks for electron wavecomputing Phys Rev B 65 075313

[135] Ramamurthy D and Wu C H 2002 Four-terminal quantumresistor network for electron-wave computing Phys Rev B66 115307

[136] Ionicioiu R and DrsquoAmico I 2003 Mesoscopic SternndashGerlachdevice to polarize spin currents Phys Rev B 67 041307

[137] Yi J and Cuniberti G 2003 A three terminal ring interferometerlogic gate Ann New York Acad Sci 1006 306

[138] Margulis V A and Pyataev M 2004 Fano resonances in athree-terminal nanodevice J Phys Condens Matter16 4315

[139] Shelykh I A Galkin N G and Bagraev N T 2005 Quantumsplitter controlled by rasha spinndashorbit coupling Phys RevB 72 235316

[140] Szafran B and Peeters F M 2005 Lorentz-force-inducedasymmetry in the AharonovndashBohm effect in athree-terminal semiconductor quantum ring Europhys Lett70 810ndash6

[141] Leturcq R Schmid L Ensslin K Meir Y Driscoll D C andGossard A C 2005 Probing the Kondo density of states in athree-terminal quantum ring Phys Rev Lett 95 126603

[142] Foldi P Kalman O Benedict M G and Peeters F M 2006Quantum rings as electron spin beam splitters Phys Rev B73 155325

[143] Leturcq R Schmid L Ihn T Ensslin K Driscoll D C andGossard A C 2006 Asymmetries of the conductance matrixin a three-terminal quantum ring in the coulomb blockaderegime Physica E 34 445ndash8

[144] Joe Y S Hedin E R and Satanin A M 2007 Manipulation ofresonances in an open three-terminal interferometer with anembedded quantum dot Phys Rev B 76 085419

[145] Cohen G Hod O and Rabani E 2007 Constructing spininterference devices from nanometric rings Phys Rev B76 235120

[146] Onsager L 1931 Reciprocal relations in irreversible processesII Phys Rev 38 2265ndash79

[147] Buttiker M 1986 Four-terminal phase-coherent conductancePhys Rev Lett 57 1761ndash4

29


[148] Entin-Wohlman O Aharony A Imry Y Levinson Y andSchiller A 2002 Broken unitarity and phase measurementsin AharonovndashBohm interferometers Phys Rev Lett88 166801

[149] Palm T and Thylen L 1996 Designing logic functions using anelectron waveguide y-branch switch J Appl Phys 79 8076

[150] Keldysh L V 1965 Diagram technique for nonequilibriumprocesses Sov PhysmdashJETP 20 1018

[151] Kadanoff L P and Baym G 1962 Quantum StatisticalMechanics Greenrsquos Function Methods in Equilibrium andNonequilibrium Problems (Reading MA Benjamin)

[152] Wagner M 1991 Expansions of nonequilibrium Greenrsquosfunctions Phys Rev B 44 6104ndash17

[153] Damle P S Ghosh A W and Datta S 2001 Unified descriptionof molecular conduction from molecules to metallic wiresPhys Rev B 64 201403

[154] Galperin M and Nitzan A 2003 NEGF-HF method inmolecular junction property calculations Mol Electron IIIAnn New York Acad Sci 1006 48ndash67

[155] Ghosh A W Damle P Datta S and Nitzan A 2004 Molecularelectronics theory and device prospects MRS Bull 29391ndash5

[156] Galperin M Ratner M A and Nitzan A 2007 Moleculartransport junctions vibrational effects J Phys CondensMatter 19 103201

[157] Todorov T N Briggs G A D and Sutton A P 1993 Elasticquantum transport through small structures J PhysCondens Matter 5 2389ndash406

[158] Pleutin S Grabert H Ingold G L and Nitzan A 2003 Theelectrostatic potential profile along a biased molecularwire a model quantum-mechanical calculation J ChemPhys 118 3756ndash63

[159] Landauer R 1957 Spatial variation of currents and fields dueto localized scatterers in metallic conduction IBM J ResDev 1 223ndash31

[160] Xue Y Datta S and Ratner M A 2002 First-principles basedmatrix Greenrsquos function approach to molecular electronicdevices general formalism Chem Phys 281 151ndash70

[161] Paulsson M 2002 Non equilibrium Greenrsquos functions fordummies introduction to the one particle NEGF equationsPreprint cond-mat0210519 v1

[162] Lopez Sancho M P Lopez Sancho J M and Rubio J 1984Quick iterative scheme for the calculation of transfermatrices application to mo (100) J Phys F Met Phys14 1205ndash15

[163] Lopez Sancho M P Lopez Sancho J M and Rubio J 1985Highly convergent schemes for the calculation of bulk andsurface Green functions J Phys F Met Phys 15 851ndash8

[164] Nardelli M B 1999 Electronic transport in extended systemsapplication to carbon nanotubes Phys Rev B 60 7828ndash33

[165] Galperin M Ratner M A and Nitzan A 2004 Inelastic electrontunneling spectroscopy in molecular junctions peaks anddips J Chem Phys 121 11965ndash79

[166] Neuhauser D and Baer M 1989 The time-dependentSchrodinger equation application of absorbing boundaryconditions J Chem Phys 90 4351ndash5

[167] Seideman T and Miller W H 1992 Quantum mechanicalreaction probabilities via a discrete variablerepresentationndashabsorbing boundary condition Greenrsquosfunction J Chem Phys 97 2499ndash514

[168] Baer R and Neuhauser D 2002 Phase coherent electronics amolecular switch based on quantum interference J AmChem Soc 124 4200

[169] Riss U V and Meyer H D 1996 Investigation on the reflectionand transmission properties of complex absorbingpotentials J Chem Phys 105 1409ndash19

[170] Nitta J Meijer F E and Takayanagi H 1999 Spin-interferencedevice Appl Phys Lett 75 695ndash7

[171] Frustaglia D Hentschel M and Richter K 2001 Quantumtransport in nonuniform magnetic fields AharonovndashBohmring as a spin switch Phys Rev Lett 87 256602

[172] Ionicioiu R and DrsquoAmico I 2003 SternndashGerlach device topolarize spin currents Phys Rev B 67 041307

[173] Popp M Frustaglia D and Richter K 2003 Spin filter effects inmesoscopic ring structures Nanotechnology 14 347ndash51

[174] Molnar B Peeters F M and Vasilopoulos P 2004Spin-dependent magnetotransport through a ring due tospinndashorbit interaction Phys Rev B 69 155335

[175] Hentschel M Schomerus H Frustaglia D and Richter K 2004AharonovndashBohm physics with spin I Geometric phases inone-dimensional ballistic rings Phys Rev B 69 155326

[176] Frustaglia D Hentschel M and Richter K 2004AharonovndashBohm physics with spin II Spin-flip effects intwo dimensional ballistic systems Phys Rev B 69 155327

[177] Frustaglia D and Richter K 2004 Spin interference effects inring conductor subject to Rashba coupling Phys Rev B69 235310

[178] Aeberhard U Wakabayashi K and Sigrist M 2005 Effect ofspinndashorbit coupling on zero-conductance resonances inasymmetrically coupled one-dimensional rings Phys RevB 72 075328

[179] Wang X F and Vasilopoulos P 2005 Spin-dependentmagnetotransport through a mesoscopic ring in thepresence of spinndashorbit interaction Phys Rev B 72 165336

[180] Foldi P Molnar B Benedict M G and Peeters F M 2005Spintronic single-qubit gate based on a quantum ring withspinndashorbit interaction Phys Rev B 71 033309

[181] Molnar B Vasilopoulos P and Peeters F M 2005Magnetoconductance through a chain of rings with orwithout periodically modulated spinndashorbit interactionstrength and magnetic field Phys Rev B 72 075330

[182] Ramaglia V M Cataudella V De Filippis G and Perroni CA 2006 Ballistic transport in one-dimensional loops withRashba and Dresselhaus spinndashorbit coupling Phys Rev B73 155328

[183] Rashba E I 1960 Properties of semiconductors with anextremum loop1 cyclotron and combinational resonancein a magnetic field perpendicular to the plane of the loopSov PhysmdashSolid State 2 1109ndash22

[184] Dresselhaus G 1955 Spinndashorbit coupling effects in zinc blendestructures Phys Rev 100 580ndash6

[185] Hoffmann R 1963 An extended Huckel theory IHydrocarbonsJ Chem Phys 39 1397

[186] Cohen-Tannoudji C Diu B and Laloe F 1977 QuantumMechanics (New York Wiley)

[187] Slater J C 1930 Atomic shielding constants Phys Rev36 57ndash64

[188] Pople J A and Beveridge D L 1970 Approximate MolecularOrbital Theory (New York McGraw-Hill)

[189] London F 1937 Theorie quantique des courantsinteratomiques dans les combinaisons aromatique J PhysRadium 8 397ndash409

[190] Pople J A 1962 Molecular-orbital theory of diamagnetism IAn approximate LCAO scheme J Chem Phys 37 53ndash9

[191] Guseinov I I 1970 Analytical evaluation of two-centreCoulomb hybrid and one-electron integrals for Slater-typeorbitals J Phys B At Mol Phys 3 1399ndash412

[192] Guseinov I I 1985 Evaluation of two-center overlap andnuclear-attraction integrals for Slater-type orbitals PhysRev A 32 1864ndash6

[193] Guseinov I I Ozmen A Atav U and Yuksel H 1998Computation of overlap integrals over Slater-type orbitalsusing auxiliary functions Int J Quantum Chem67 199ndash204

[194] Guseinov I I and Mamedov B A 1999 Computation ofmolecular integrals over Slater type orbitals I Calculationsof overlap integrals using recurrence relations J MolStruct (Theochem) 465 1ndash6

[195] Guseinov I I 2003 Response to the comment on thecomputation of molecular auxiliary functions an and bnPramana J Phys 61 C781

30


[196] Hod O 2005 Molecular nano-electronic devices based onAharonovndashBohm interferometry PhD Thesis Tel AvivUniversity

[197] Collins G P 1993 STM rounds up electron waves at the QMcorral Phys Today 46 17ndash9

[198] Crommie M F Lutz C P and Eigler D M 1993 Confinement ofelectrons to quantum corrals on a metal surfaceScience 262 218ndash20

[199] Manoharan H C Lutz C P and Eigler D M 2000 Quantummirages formed by coherent projection of electronicstructure Nature 403 512ndash5

[200] Nazin G V Qiu X H and Ho W 2003 Visualization andspectroscopy of a metalndashmoleculendashmetal bridgeScience 302 77

[201] Iijima S 1991 Helical microtubules of graphitic carbonNature 354 56

[202] Collins P G and Avouris P 2000 Nanotubes for electronicsSci Am 283 (December) 62

[203] Fuhrer M S et al 2000 Crossed nanotube junctionsScience 288 494

[204] Zhou C Kong J Yenilmez E and Dai H 2000 Modulatedchemical doping of individual carbon nanotubesScience 290 1552

[205] Rueckes T Kim K Joselevich E Tseng G Y Cheung C-L andLieber C M 2000 Carbon nanotube-based nonvolatilerandom access memory for molecular computingScience 289 94

[206] Bockrath M Liang W Bozovic D Hafner J H Lieber C MTinkham M and Park H 2001 Resonant electron scatteringby defects in single-walled carbon nanotubesScience 291 283

[207] Treboux G Lapstun P and Silverbrook K 1999 Conductancein nanotube y-junctions Chem Phys Lett 306 402

[208] Leonard F and Tersoff J 2000 Negative differential resistancein nanotube devices Phys Rev Lett 85 4767

[209] Andriotis A N Menon M Srivastava D andChernozatonskii L 2001 Rectification properties of carbonnanotube lsquoy-junctionsrsquo Phys Rev Lett 87 66802

[210] Ajiki H and Ando T 1994 AharonovndashBohm effect in carbonnanotubes Physica B 201 349ndash52

[211] Lin M F 1998 Magnetic properties of toroidal carbonnanotubes J Phys Soc Japan 67 1094ndash7

[212] Bachtold A Strunk C Salvetat J-P Bonard J-M Forro LNussbaumer T and Schonenberger C 1999AharonovndashBohm oscillations in carbon nanotubesNature 397 673ndash5

[213] Fujiwara A Tomiyama K Suematsu H Yumura M andUchida K 1999 Quantum interference of electrons inmultiwall carbon nanotubes Phys Rev B 60 13492ndash6

[214] Ando T 2000 Theory of transport in carbon nanotubesSemicond Sci Technol 15 R13ndash27

[215] Roche S Dresselhaus G Dresselhaus M S and Saito R 2000AharonovndashBohm spectral features and coherence lengths incarbon nanotubes Phys Rev B 62 16092ndash9

[216] Zaric S Ostojic G N Kono J Shavey J Moore V CStrano M S Hauge R H Smalley R E and Wei X 2004Optical signatures of the AharonovndashBohm effect insingle-walled carbon nanotubes Science 304 1129

[217] Coskun U C Wei T-C Vishveshwara S Goldbart P M andBezryadin A 2004 he magnetic flux modulation of theenergy gap in nanotube quantum dots Science 304 1132

[218] Cao J Wang Q Rolandi M and Dai H 2004 AharonovndashBohminterference and beating in single-walled carbon-nanotubeinterferometers Phys Rev Lett 93 216803

[219] Cao J Wang Q and Dai H 2005 Electron transport in veryclean as-grown suspended carbon nanotubes Nat Mater4 745ndash9

[220] Charlier J C Blase X and Roche S 2007 Electronic andtransport properties of nanotubes Rev Mod Phys79 677ndash732

[221] Deshmukh M M Prieto A L Gu Q and Park H 2003Fabrication of asymmetric electrode pairs with nanometerseparation made of two distinct metals Nano Lett3 1383ndash5

[222] Xie F Q Nittler L Obermair C and Schimmel T 2004Gate-controlled atomic quantum switch Phys Rev Lett93 128303

[223] Biercuk M J Mason N Martin J Yacoby A and Marcus C M2005 Anomalous conductance quantization in carbonnanotubes Phys Rev Lett 94 026801

[224] Javey A Qi P Wang Q and Dai H 2004 Ten-to 50-nm-longquasi-ballistic carbon nanotube devices obtained withoutcomplex lithography Proc Natl Acad Sci 101 13408ndash10

[225] Staab H A and Diederich F 1978 Benzenoid versusannulenoid aromaticity synthesis and properties ofkekulene Angew Chem Int Edn Engl 17 372ndash4

[226] Staab H A and Diederich F 1983 Cycloarenes a new class ofaromatic compounds I Synthesis of kekulene Chem Ber116 3487ndash503

[227] Staab H A Diederich F Krieger C and Schweitzer D 1983Cycloarenes a new class of aromatic compounds IIMolecular structure and spectroscopic properties ofkekulene Chem Ber 116 3504ndash12

[228] Cioslowski J Orsquoconnor P B and Fleischmann E D 1991 Issuperbenzene superaromatic J Am Chem Soc113 1086ndash9

[229] Aihara J 1992 Is superaromaticity a fact or an artifact thekekulene problem J Am Chem Soc 114 865ndash8

[230] Steiner E Fowler P W Jenneskens L W and Acocella A 2001Visualisation of counter-rotating ring currents in kekuleneChem Commun 659ndash60

[231] Hajgato B and Ohno K 2004 Novel series of giant polycyclicaromatic hydrocarbons electronic structure andaromaticity Chem Phys Lett 385 512ndash8

[232] Ezawa M 2006 Peculiar width dependence of the electronicproperties of carbon nanoribbons Phys Rev B 73 045432

[233] Son Y-W Cohen M L and Louie S G 2006 Half-metallicgraphene nanoribbons Nature 444 347ndash9

[234] Son Y-W Cohen M L and Louie S G 2006 Energy gaps ingraphene nanoribbons Phys Rev Lett 97 216803

[235] Barone V Hod O and Scuseria G E 2006 Electronic structureand stability of semiconducting graphene nanoribbonsNano Lett 6 2748

[236] Hod O Barone V Peralta J E and Scuseria G E 2007Enhanced half-metallicity in edge-oxidized zigzaggraphene nanoribbons Nano Lett 7 2295ndash9

[237] Castro-Neto A H Guinea F Peres N M R Novoselov K S andGeim A K 2007 The electronic properties of graphenePreprint 07091163

[238] Ness H and Fisher A J 1999 Quantum inelastic conductancethrough molecular wires Phys Rev Lett 83 452ndash5

[239] Segal D and Nitzan A 2002 Conduction in molecularjunctions inelastic effects Chem Phys 281 235ndash56

[240] Montgomery M J Todorov T N and Sutton A P 2002 Powerdissipation in nanoscale conductors J Phys CondensMatter 14 5377ndash90

[241] Flensberg K 2003 Tunneling broadening of vibrationalsidebands in molecular transistors Phys Rev B 68 205323

[242] Petrov E G May V and Hanggi P 2004 Spin-bosondescription of electron transmission through a molecularwire Chem Phys 296 251ndash66

[243] Mii T Tikhodeev S G and Ueba H 2003 Spectral features ofinelastic electron transport via a localized state Phys RevB 68 205406

[244] Mitra A Aleiner I and Millis A J 2004 Phonon effects inmolecular transistors quantal and classical treatmentPhys Rev B 69 245302

[245] Paulsson M Frederiksen T and Brandybge M 2005 Modelinginelastic phonon scattering in atomic- and molecular-wirejunctions Phys Rev B 72 201101(R)

31


[246] Galperin M Nitzan A and Ratner M A 2006 Inelastictunneling effects on noise properties of molecular junctionsPhys Rev B 74 075326

[247] Muhlbacher L and Rabani E 2008 Real-time path integralapproach to nonequilibrium many-body quantum systemsPhys Rev Lett 100 176403

[248] Wang H and Thoss M unpublished[249] Guinea F 2002 AharonovndashBohm oscillations of a particle

coupled to dissipative environments Phys Rev B65 205317

[250] Ludwig T and Mirlin A D 2004 Interaction-induced dephasingof AharonovndashBohm oscillations Phys Rev B 69 193306

[251] Jauho A P Wingreen N S and Meir Y 1994 Time-dependenttransport in interacting and noninteractingresonant-tunneling systems Phys Rev B 50 5528

[252] Stipe B C Rezaei M A and Ho W 1998 Single-moleculevibrational spectroscopy and microscopy Science280 1732ndash5

[253] Thijssen W H A Djukic D Otte A F Bremmer R H andvan Ruitenbeek J M 2006 Vibrationally induced two-levelsystems in single-molecule junctions Phys Rev Lett97 226806

[254] Wu C H and Mahler G 1991 Quantum network theory oftransport with application to the generalizedAharonovndashBohm effect in metals and semiconductorsPhys Rev B 43 5012ndash23


[256] Silva A Oreg Y and Gefen Y 2002 Sign of quantum dotndashleadmatrix elements the effect of transport versus spectralproperties Phys Rev B 66 195316

[257] Ryndyk D A Gutierrez R Song B and Cuniberti G 2008Green function techniques in the treatment of quantumtransport at the molecular scale Preprint 08050628v1

32

1 Introduction


21 Aharonov--Bohm interferometry

22 Length scales

221 Fermi wavelength

222 Momentum relaxation length---static scatterers

223 Phase relaxation length---fluctuating scatterers

224 Regimes of transport

23 Open questions

3 Basic concepts




41 Conductance

42 Retarded and advanced Greens functions and self-energies

43 An alternative---absorbing imaginary potentials

44 Electronic structure---tight binding magnetic extended Huckel theory


451 Atomic corral

452 Carbon nanotubes

453 Operative limitations


5 Inelastic scattering effects in AB molecular interferometers

51 Hamiltonian

52 Conductance

53 Results


References

IOP PUBLISHING JOURNAL OF PHYSICS CONDENSED MATTER

J Phys Condens Matter 20 (2008) 383201 (32pp) doi1010880953-89842038383201

TOPICAL REVIEW

Magnetoresistance of nanoscale moleculardevices based on AharonovndashBohminterferometryOded Hod1 Roi Baer2 and Eran Rabani3

1 Department of Chemistry Rice University Houston TX 77005-1892 USA2 Institute of Chemistry and the Fritz Haber Center for Molecular Dynamics The HebrewUniversity of Jerusalem Jerusalem 91904 Israel3 School of Chemistry The Sackler Faculty of Exact Sciences Tel Aviv UniversityTel Aviv 69978 Israel

E-mail odedhodriceedu roibaerhujiacil and rabanitauacil

Received 17 January 2008 in final form 11 July 2008Published 27 August 2008Online at stacksioporgJPhysCM20383201

AbstractControl of conductance in molecular junctions is of key importance in the growing field ofmolecular electronics The current in these junctions is often controlled by an electric gatedesigned to shift conductance peaks into the low bias regime Magnetic fields on the otherhand have rarely been used due to the small magnetic flux captured by molecular conductors(an exception is the Kondo effect in single-molecule transistors) This is in contrast to a relatedfield electronic transport through mesoscopic devices where considerable activity withmagnetic fields has led to a rich description of transport The scarcity of experimental activity isdue to the belief that significant magnetic response is obtained only when the magnetic flux isof the order of the quantum flux while attaining such a flux for molecular and nanoscaledevices requires unrealistic magnetic fields

Here we review recent theoretical work regarding the essential physical requirementsnecessary for the construction of nanometer-scale magnetoresistance devices based on anAharonovndashBohm molecular interferometer We show that control of the conductance propertiesusing small fractions of a magnetic flux can be achieved by carefully adjusting the lifetime ofthe conducting electrons through a pre-selected single state that is well separated from otherstates due to quantum confinement effects Using a simple analytical model and more elaborateatomistic calculations we demonstrate that magnetic fields which give rise to a magnetic fluxcomparable to 10minus3 of the quantum flux can be used to switch a class of different molecular andnanometer rings ranging from quantum corrals carbon nanotubes and even a molecular ringcomposed of polyconjugated aromatic materials

The unique characteristics of the magnetic field as a gate is further discussed anddemonstrated in two different directions First a three-terminal molecular router devices thatcan function as a parallel logic gate processing two logic operations simultaneously ispresented Second the role of inelastic effects arising from electronndashphonon couplings on themagnetoresistance properties is analyzed We show that a remarkable difference betweenelectric and magnetic gating is also revealed when inelastic effects become significant Theinelastic broadening of response curves to electric gates is replaced by a narrowing ofmagnetoconductance peaks thereby enhancing the sensitivity of the device

(Some figures in this article are in colour only in the electronic version)

0953-898408383201+32$3000 copy 2008 IOP Publishing Ltd Printed in the UK1


Contents









1 Introduction








2















H = 1

2m

[P minus qA(r)

]2 + V (r) (1)




ud = eiint


(2)


k =int 0

π


3





um = minusq

h

int 0

π

A middot dl = πφ

φ0 (4)



lm = minusq

h

int 2π

π



I prop ∣∣u +d∣∣2 =

∣∣∣∣eiπ(

Rk+ φ

φ0

)+ e

iπ(

Rkminus φ

φ0

)∣∣∣∣2

= 2

[1 + cos

(2π

φ

φ0

)] (6)


22 Length scales





PF= 2π

kF where PF is






4




23 Open questions












3 Basic concepts





5










radicε is



( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)( L1

U2ei1

D2ei2

) (7)



( R1

U2

D2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

U1ei2

D1ei1

) (8)






T =∣∣∣∣

R1

L1

∣∣∣∣2


(9)

c

ε

a

b


6





R

Q = (c + 1)4

R


+ 8c2 cos(4k)

(10)










h2k2

2m= h2(m minus φ

φ0)2

2mR2 (11)





7





2 1 12 2 1

2


h2k2

2m= h2(m minus φ

φ0)2

2mR2+ Vg (12)













8





( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

L1

U2eiα1

D2eiβ2

⎞⎠

( R1

M1

U2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

M2eiγ1

U1eiα2

)

( I1

D2

M2

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

I2

D1eiβ1

M1eiγ2

⎞⎠

(13)


φ

φ0and =αβγ

2 = Rk +φ

φ0



L1|2

(T l = | I1L1




2 + 1)(19 minus 12c + 2c2 minus 12c3 + 19c4)




(2π

φ

φ0

)

+ 1

16(c + 1)6 cos2

(2π

φ

φ0

) (14)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0minus kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0+ kr

)]

(a) (b)


9


08

07

06

05

04

03

02

01

0

09

08

07

06

05

04

03

02

01

0



[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ 2c(c + 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)] (15)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0+ kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0minus kr

)]

+ 2c(c + 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)] (16)









10








41 Conductance



IL(R) = 2e

h

intdE

2πTr[lt


L(R)Gltd (E)

]







Gltd (E) = Gr

d(E)lt(E)Ga

d(E)

Ggtd (E) = Gr

d(E)gt(E)Ga

d(E)(18)


lt = ltL + lt

R

gt = gtL +gt

R (19)





[r


]

gtL(R)(E) = [



]

(20)

11





I = 2e

h

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (21)


L(R)(E) equiv i[r


] (22)





g = part I

partVb= 2e

h

part

partVb

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (23)


g = part I

partVb= 2e

h

intdE

2π

part[


partVb

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (24)





part[


partVb

= eβ

2

eminusβ|EminusμL|



(25)



g = g0T (26)

where g0 = 2e2










⎞⎠

times⎛⎜⎝

GrL(ε) Gr

Ld(ε) GrL R(ε)

GrdL(ε) Gr

d(ε) GrdR(ε)

GrRL(ε) Gr

Rd(ε) GrR(ε)

⎞⎟⎠ =

⎛⎝

IL 0 0

0 Id 0

0 0 IR

⎞⎠

(28)


L(R)d and the same




12






Grd(ε) =

[(gr


L(ε)minus rR(ε)

]minus1 (29)




rL(R)(ε) =


)gr

L(R)(ε)


) (30)








⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝







⎞⎟⎟⎟⎟⎠

minus1

(31)






01)T (ε)]minus1

(32)






)minus1t2iminus1


)minus1t2iminus1

(34)



)minus1V dagger

01


)minus1V01

(35)



13











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References


Contents









1 Introduction








2















H = 1

2m

[P minus qA(r)

]2 + V (r) (1)




ud = eiint


(2)


k =int 0

π


3





um = minusq

h

int 0

π

A middot dl = πφ

φ0 (4)



lm = minusq

h

int 2π

π



I prop ∣∣u +d∣∣2 =

∣∣∣∣eiπ(

Rk+ φ

φ0

)+ e

iπ(

Rkminus φ

φ0

)∣∣∣∣2

= 2

[1 + cos

(2π

φ

φ0

)] (6)


22 Length scales





PF= 2π

kF where PF is






4




23 Open questions












3 Basic concepts





5










radicε is



( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)( L1

U2ei1

D2ei2

) (7)



( R1

U2

D2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

U1ei2

D1ei1

) (8)






T =∣∣∣∣

R1

L1

∣∣∣∣2


(9)

c

ε

a

b


6





R

Q = (c + 1)4

R


+ 8c2 cos(4k)

(10)










h2k2

2m= h2(m minus φ

φ0)2

2mR2 (11)





7





2 1 12 2 1

2


h2k2

2m= h2(m minus φ

φ0)2

2mR2+ Vg (12)













8





( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

L1

U2eiα1

D2eiβ2

⎞⎠

( R1

M1

U2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

M2eiγ1

U1eiα2

)

( I1

D2

M2

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

I2

D1eiβ1

M1eiγ2

⎞⎠

(13)


φ

φ0and =αβγ

2 = Rk +φ

φ0



L1|2

(T l = | I1L1




2 + 1)(19 minus 12c + 2c2 minus 12c3 + 19c4)




(2π

φ

φ0

)

+ 1

16(c + 1)6 cos2

(2π

φ

φ0

) (14)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0minus kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0+ kr

)]

(a) (b)


9


08

07

06

05

04

03

02

01

0

09

08

07

06

05

04

03

02

01

0



[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ 2c(c + 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)] (15)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0+ kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0minus kr

)]

+ 2c(c + 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)] (16)









10








41 Conductance



IL(R) = 2e

h

intdE

2πTr[lt


L(R)Gltd (E)

]







Gltd (E) = Gr

d(E)lt(E)Ga

d(E)

Ggtd (E) = Gr

d(E)gt(E)Ga

d(E)(18)


lt = ltL + lt

R

gt = gtL +gt

R (19)





[r


]

gtL(R)(E) = [



]

(20)

11





I = 2e

h

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (21)


L(R)(E) equiv i[r


] (22)





g = part I

partVb= 2e

h

part

partVb

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (23)


g = part I

partVb= 2e

h

intdE

2π

part[


partVb

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (24)





part[


partVb

= eβ

2

eminusβ|EminusμL|



(25)



g = g0T (26)

where g0 = 2e2










⎞⎠

times⎛⎜⎝

GrL(ε) Gr

Ld(ε) GrL R(ε)

GrdL(ε) Gr

d(ε) GrdR(ε)

GrRL(ε) Gr

Rd(ε) GrR(ε)

⎞⎟⎠ =

⎛⎝

IL 0 0

0 Id 0

0 0 IR

⎞⎠

(28)


L(R)d and the same




12






Grd(ε) =

[(gr


L(ε)minus rR(ε)

]minus1 (29)




rL(R)(ε) =


)gr

L(R)(ε)


) (30)








⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝







⎞⎟⎟⎟⎟⎠

minus1

(31)






01)T (ε)]minus1

(32)






)minus1t2iminus1


)minus1t2iminus1

(34)



)minus1V dagger

01


)minus1V01

(35)



13











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References















H = 1

2m

[P minus qA(r)

]2 + V (r) (1)




ud = eiint


(2)


k =int 0

π


3





um = minusq

h

int 0

π

A middot dl = πφ

φ0 (4)



lm = minusq

h

int 2π

π



I prop ∣∣u +d∣∣2 =

∣∣∣∣eiπ(

Rk+ φ

φ0

)+ e

iπ(

Rkminus φ

φ0

)∣∣∣∣2

= 2

[1 + cos

(2π

φ

φ0

)] (6)


22 Length scales





PF= 2π

kF where PF is






4




23 Open questions












3 Basic concepts





5










radicε is



( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)( L1

U2ei1

D2ei2

) (7)



( R1

U2

D2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

U1ei2

D1ei1

) (8)






T =∣∣∣∣

R1

L1

∣∣∣∣2


(9)

c

ε

a

b


6





R

Q = (c + 1)4

R


+ 8c2 cos(4k)

(10)










h2k2

2m= h2(m minus φ

φ0)2

2mR2 (11)





7





2 1 12 2 1

2


h2k2

2m= h2(m minus φ

φ0)2

2mR2+ Vg (12)













8





( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

L1

U2eiα1

D2eiβ2

⎞⎠

( R1

M1

U2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

M2eiγ1

U1eiα2

)

( I1

D2

M2

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

I2

D1eiβ1

M1eiγ2

⎞⎠

(13)


φ

φ0and =αβγ

2 = Rk +φ

φ0



L1|2

(T l = | I1L1




2 + 1)(19 minus 12c + 2c2 minus 12c3 + 19c4)




(2π

φ

φ0

)

+ 1

16(c + 1)6 cos2

(2π

φ

φ0

) (14)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0minus kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0+ kr

)]

(a) (b)


9


08

07

06

05

04

03

02

01

0

09

08

07

06

05

04

03

02

01

0



[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ 2c(c + 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)] (15)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0+ kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0minus kr

)]

+ 2c(c + 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)] (16)









10








41 Conductance



IL(R) = 2e

h

intdE

2πTr[lt


L(R)Gltd (E)

]







Gltd (E) = Gr

d(E)lt(E)Ga

d(E)

Ggtd (E) = Gr

d(E)gt(E)Ga

d(E)(18)


lt = ltL + lt

R

gt = gtL +gt

R (19)





[r


]

gtL(R)(E) = [



]

(20)

11





I = 2e

h

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (21)


L(R)(E) equiv i[r


] (22)





g = part I

partVb= 2e

h

part

partVb

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (23)


g = part I

partVb= 2e

h

intdE

2π

part[


partVb

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (24)





part[


partVb

= eβ

2

eminusβ|EminusμL|



(25)



g = g0T (26)

where g0 = 2e2










⎞⎠

times⎛⎜⎝

GrL(ε) Gr

Ld(ε) GrL R(ε)

GrdL(ε) Gr

d(ε) GrdR(ε)

GrRL(ε) Gr

Rd(ε) GrR(ε)

⎞⎟⎠ =

⎛⎝

IL 0 0

0 Id 0

0 0 IR

⎞⎠

(28)


L(R)d and the same




12






Grd(ε) =

[(gr


L(ε)minus rR(ε)

]minus1 (29)




rL(R)(ε) =


)gr

L(R)(ε)


) (30)








⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝







⎞⎟⎟⎟⎟⎠

minus1

(31)






01)T (ε)]minus1

(32)






)minus1t2iminus1


)minus1t2iminus1

(34)



)minus1V dagger

01


)minus1V01

(35)



13











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References





um = minusq

h

int 0

π

A middot dl = πφ

φ0 (4)



lm = minusq

h

int 2π

π



I prop ∣∣u +d∣∣2 =

∣∣∣∣eiπ(

Rk+ φ

φ0

)+ e

iπ(

Rkminus φ

φ0

)∣∣∣∣2

= 2

[1 + cos

(2π

φ

φ0

)] (6)


22 Length scales





PF= 2π

kF where PF is






4




23 Open questions












3 Basic concepts





5










radicε is



( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)( L1

U2ei1

D2ei2

) (7)



( R1

U2

D2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

U1ei2

D1ei1

) (8)






T =∣∣∣∣

R1

L1

∣∣∣∣2


(9)

c

ε

a

b


6





R

Q = (c + 1)4

R


+ 8c2 cos(4k)

(10)










h2k2

2m= h2(m minus φ

φ0)2

2mR2 (11)





7





2 1 12 2 1

2


h2k2

2m= h2(m minus φ

φ0)2

2mR2+ Vg (12)













8





( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

L1

U2eiα1

D2eiβ2

⎞⎠

( R1

M1

U2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

M2eiγ1

U1eiα2

)

( I1

D2

M2

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

I2

D1eiβ1

M1eiγ2

⎞⎠

(13)


φ

φ0and =αβγ

2 = Rk +φ

φ0



L1|2

(T l = | I1L1




2 + 1)(19 minus 12c + 2c2 minus 12c3 + 19c4)




(2π

φ

φ0

)

+ 1

16(c + 1)6 cos2

(2π

φ

φ0

) (14)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0minus kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0+ kr

)]

(a) (b)


9


08

07

06

05

04

03

02

01

0

09

08

07

06

05

04

03

02

01

0



[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ 2c(c + 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)] (15)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0+ kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0minus kr

)]

+ 2c(c + 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)] (16)









10








41 Conductance



IL(R) = 2e

h

intdE

2πTr[lt


L(R)Gltd (E)

]







Gltd (E) = Gr

d(E)lt(E)Ga

d(E)

Ggtd (E) = Gr

d(E)gt(E)Ga

d(E)(18)


lt = ltL + lt

R

gt = gtL +gt

R (19)





[r


]

gtL(R)(E) = [



]

(20)

11





I = 2e

h

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (21)


L(R)(E) equiv i[r


] (22)





g = part I

partVb= 2e

h

part

partVb

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (23)


g = part I

partVb= 2e

h

intdE

2π

part[


partVb

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (24)





part[


partVb

= eβ

2

eminusβ|EminusμL|



(25)



g = g0T (26)

where g0 = 2e2










⎞⎠

times⎛⎜⎝

GrL(ε) Gr

Ld(ε) GrL R(ε)

GrdL(ε) Gr

d(ε) GrdR(ε)

GrRL(ε) Gr

Rd(ε) GrR(ε)

⎞⎟⎠ =

⎛⎝

IL 0 0

0 Id 0

0 0 IR

⎞⎠

(28)


L(R)d and the same




12






Grd(ε) =

[(gr


L(ε)minus rR(ε)

]minus1 (29)




rL(R)(ε) =


)gr

L(R)(ε)


) (30)








⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝







⎞⎟⎟⎟⎟⎠

minus1

(31)






01)T (ε)]minus1

(32)






)minus1t2iminus1


)minus1t2iminus1

(34)



)minus1V dagger

01


)minus1V01

(35)



13











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References




23 Open questions












3 Basic concepts





5










radicε is



( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)( L1

U2ei1

D2ei2

) (7)



( R1

U2

D2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

U1ei2

D1ei1

) (8)






T =∣∣∣∣

R1

L1

∣∣∣∣2


(9)

c

ε

a

b


6





R

Q = (c + 1)4

R


+ 8c2 cos(4k)

(10)










h2k2

2m= h2(m minus φ

φ0)2

2mR2 (11)





7





2 1 12 2 1

2


h2k2

2m= h2(m minus φ

φ0)2

2mR2+ Vg (12)













8





( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

L1

U2eiα1

D2eiβ2

⎞⎠

( R1

M1

U2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

M2eiγ1

U1eiα2

)

( I1

D2

M2

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

I2

D1eiβ1

M1eiγ2

⎞⎠

(13)


φ

φ0and =αβγ

2 = Rk +φ

φ0



L1|2

(T l = | I1L1




2 + 1)(19 minus 12c + 2c2 minus 12c3 + 19c4)




(2π

φ

φ0

)

+ 1

16(c + 1)6 cos2

(2π

φ

φ0

) (14)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0minus kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0+ kr

)]

(a) (b)


9


08

07

06

05

04

03

02

01

0

09

08

07

06

05

04

03

02

01

0



[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ 2c(c + 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)] (15)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0+ kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0minus kr

)]

+ 2c(c + 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)] (16)









10








41 Conductance



IL(R) = 2e

h

intdE

2πTr[lt


L(R)Gltd (E)

]







Gltd (E) = Gr

d(E)lt(E)Ga

d(E)

Ggtd (E) = Gr

d(E)gt(E)Ga

d(E)(18)


lt = ltL + lt

R

gt = gtL +gt

R (19)





[r


]

gtL(R)(E) = [



]

(20)

11





I = 2e

h

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (21)


L(R)(E) equiv i[r


] (22)





g = part I

partVb= 2e

h

part

partVb

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (23)


g = part I

partVb= 2e

h

intdE

2π

part[


partVb

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (24)





part[


partVb

= eβ

2

eminusβ|EminusμL|



(25)



g = g0T (26)

where g0 = 2e2










⎞⎠

times⎛⎜⎝

GrL(ε) Gr

Ld(ε) GrL R(ε)

GrdL(ε) Gr

d(ε) GrdR(ε)

GrRL(ε) Gr

Rd(ε) GrR(ε)

⎞⎟⎠ =

⎛⎝

IL 0 0

0 Id 0

0 0 IR

⎞⎠

(28)


L(R)d and the same




12






Grd(ε) =

[(gr


L(ε)minus rR(ε)

]minus1 (29)




rL(R)(ε) =


)gr

L(R)(ε)


) (30)








⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝







⎞⎟⎟⎟⎟⎠

minus1

(31)






01)T (ε)]minus1

(32)






)minus1t2iminus1


)minus1t2iminus1

(34)



)minus1V dagger

01


)minus1V01

(35)



13











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References










radicε is



( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)( L1

U2ei1

D2ei2

) (7)



( R1

U2

D2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

U1ei2

D1ei1

) (8)






T =∣∣∣∣

R1

L1

∣∣∣∣2


(9)

c

ε

a

b


6





R

Q = (c + 1)4

R


+ 8c2 cos(4k)

(10)










h2k2

2m= h2(m minus φ

φ0)2

2mR2 (11)





7





2 1 12 2 1

2


h2k2

2m= h2(m minus φ

φ0)2

2mR2+ Vg (12)













8





( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

L1

U2eiα1

D2eiβ2

⎞⎠

( R1

M1

U2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

M2eiγ1

U1eiα2

)

( I1

D2

M2

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

I2

D1eiβ1

M1eiγ2

⎞⎠

(13)


φ

φ0and =αβγ

2 = Rk +φ

φ0



L1|2

(T l = | I1L1




2 + 1)(19 minus 12c + 2c2 minus 12c3 + 19c4)




(2π

φ

φ0

)

+ 1

16(c + 1)6 cos2

(2π

φ

φ0

) (14)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0minus kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0+ kr

)]

(a) (b)


9


08

07

06

05

04

03

02

01

0

09

08

07

06

05

04

03

02

01

0



[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ 2c(c + 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)] (15)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0+ kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0minus kr

)]

+ 2c(c + 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)] (16)









10








41 Conductance



IL(R) = 2e

h

intdE

2πTr[lt


L(R)Gltd (E)

]







Gltd (E) = Gr

d(E)lt(E)Ga

d(E)

Ggtd (E) = Gr

d(E)gt(E)Ga

d(E)(18)


lt = ltL + lt

R

gt = gtL +gt

R (19)





[r


]

gtL(R)(E) = [



]

(20)

11





I = 2e

h

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (21)


L(R)(E) equiv i[r


] (22)





g = part I

partVb= 2e

h

part

partVb

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (23)


g = part I

partVb= 2e

h

intdE

2π

part[


partVb

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (24)





part[


partVb

= eβ

2

eminusβ|EminusμL|



(25)



g = g0T (26)

where g0 = 2e2










⎞⎠

times⎛⎜⎝

GrL(ε) Gr

Ld(ε) GrL R(ε)

GrdL(ε) Gr

d(ε) GrdR(ε)

GrRL(ε) Gr

Rd(ε) GrR(ε)

⎞⎟⎠ =

⎛⎝

IL 0 0

0 Id 0

0 0 IR

⎞⎠

(28)


L(R)d and the same




12






Grd(ε) =

[(gr


L(ε)minus rR(ε)

]minus1 (29)




rL(R)(ε) =


)gr

L(R)(ε)


) (30)








⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝







⎞⎟⎟⎟⎟⎠

minus1

(31)






01)T (ε)]minus1

(32)






)minus1t2iminus1


)minus1t2iminus1

(34)



)minus1V dagger

01


)minus1V01

(35)



13











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References





R

Q = (c + 1)4

R


+ 8c2 cos(4k)

(10)










h2k2

2m= h2(m minus φ

φ0)2

2mR2 (11)





7





2 1 12 2 1

2


h2k2

2m= h2(m minus φ

φ0)2

2mR2+ Vg (12)













8





( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

L1

U2eiα1

D2eiβ2

⎞⎠

( R1

M1

U2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

M2eiγ1

U1eiα2

)

( I1

D2

M2

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

I2

D1eiβ1

M1eiγ2

⎞⎠

(13)


φ

φ0and =αβγ

2 = Rk +φ

φ0



L1|2

(T l = | I1L1




2 + 1)(19 minus 12c + 2c2 minus 12c3 + 19c4)




(2π

φ

φ0

)

+ 1

16(c + 1)6 cos2

(2π

φ

φ0

) (14)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0minus kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0+ kr

)]

(a) (b)


9


08

07

06

05

04

03

02

01

0

09

08

07

06

05

04

03

02

01

0



[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ 2c(c + 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)] (15)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0+ kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0minus kr

)]

+ 2c(c + 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)] (16)









10








41 Conductance



IL(R) = 2e

h

intdE

2πTr[lt


L(R)Gltd (E)

]







Gltd (E) = Gr

d(E)lt(E)Ga

d(E)

Ggtd (E) = Gr

d(E)gt(E)Ga

d(E)(18)


lt = ltL + lt

R

gt = gtL +gt

R (19)





[r


]

gtL(R)(E) = [



]

(20)

11





I = 2e

h

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (21)


L(R)(E) equiv i[r


] (22)





g = part I

partVb= 2e

h

part

partVb

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (23)


g = part I

partVb= 2e

h

intdE

2π

part[


partVb

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (24)





part[


partVb

= eβ

2

eminusβ|EminusμL|



(25)



g = g0T (26)

where g0 = 2e2










⎞⎠

times⎛⎜⎝

GrL(ε) Gr

Ld(ε) GrL R(ε)

GrdL(ε) Gr

d(ε) GrdR(ε)

GrRL(ε) Gr

Rd(ε) GrR(ε)

⎞⎟⎠ =

⎛⎝

IL 0 0

0 Id 0

0 0 IR

⎞⎠

(28)


L(R)d and the same




12






Grd(ε) =

[(gr


L(ε)minus rR(ε)

]minus1 (29)




rL(R)(ε) =


)gr

L(R)(ε)


) (30)








⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝







⎞⎟⎟⎟⎟⎠

minus1

(31)






01)T (ε)]minus1

(32)






)minus1t2iminus1


)minus1t2iminus1

(34)



)minus1V dagger

01


)minus1V01

(35)



13











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References





2 1 12 2 1

2


h2k2

2m= h2(m minus φ

φ0)2

2mR2+ Vg (12)













8





( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

L1

U2eiα1

D2eiβ2

⎞⎠

( R1

M1

U2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

M2eiγ1

U1eiα2

)

( I1

D2

M2

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

I2

D1eiβ1

M1eiγ2

⎞⎠

(13)


φ

φ0and =αβγ

2 = Rk +φ

φ0



L1|2

(T l = | I1L1




2 + 1)(19 minus 12c + 2c2 minus 12c3 + 19c4)




(2π

φ

φ0

)

+ 1

16(c + 1)6 cos2

(2π

φ

φ0

) (14)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0minus kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0+ kr

)]

(a) (b)


9


08

07

06

05

04

03

02

01

0

09

08

07

06

05

04

03

02

01

0



[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ 2c(c + 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)] (15)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0+ kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0minus kr

)]

+ 2c(c + 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)] (16)









10








41 Conductance



IL(R) = 2e

h

intdE

2πTr[lt


L(R)Gltd (E)

]







Gltd (E) = Gr

d(E)lt(E)Ga

d(E)

Ggtd (E) = Gr

d(E)gt(E)Ga

d(E)(18)


lt = ltL + lt

R

gt = gtL +gt

R (19)





[r


]

gtL(R)(E) = [



]

(20)

11





I = 2e

h

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (21)


L(R)(E) equiv i[r


] (22)





g = part I

partVb= 2e

h

part

partVb

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (23)


g = part I

partVb= 2e

h

intdE

2π

part[


partVb

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (24)





part[


partVb

= eβ

2

eminusβ|EminusμL|



(25)



g = g0T (26)

where g0 = 2e2










⎞⎠

times⎛⎜⎝

GrL(ε) Gr

Ld(ε) GrL R(ε)

GrdL(ε) Gr

d(ε) GrdR(ε)

GrRL(ε) Gr

Rd(ε) GrR(ε)

⎞⎟⎠ =

⎛⎝

IL 0 0

0 Id 0

0 0 IR

⎞⎠

(28)


L(R)d and the same




12






Grd(ε) =

[(gr


L(ε)minus rR(ε)

]minus1 (29)




rL(R)(ε) =


)gr

L(R)(ε)


) (30)








⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝







⎞⎟⎟⎟⎟⎠

minus1

(31)






01)T (ε)]minus1

(32)






)minus1t2iminus1


)minus1t2iminus1

(34)



)minus1V dagger

01


)minus1V01

(35)



13











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References





( L2

U1

D1

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

L1

U2eiα1

D2eiβ2

⎞⎠

( R1

M1

U2

)=( c

radicε

radicεradic

ε a bradicε b a

)( R2

M2eiγ1

U1eiα2

)

( I1

D2

M2

)=( c

radicε

radicεradic

ε a bradicε b a

)⎛⎝

I2

D1eiβ1

M1eiγ2

⎞⎠

(13)


φ

φ0and =αβγ

2 = Rk +φ

φ0



L1|2

(T l = | I1L1




2 + 1)(19 minus 12c + 2c2 minus 12c3 + 19c4)




(2π

φ

φ0

)

+ 1

16(c + 1)6 cos2

(2π

φ

φ0

) (14)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0minus kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0+ kr

)]

(a) (b)


9


08

07

06

05

04

03

02

01

0

09

08

07

06

05

04

03

02

01

0



[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ 2c(c + 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)] (15)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0+ kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0minus kr

)]

+ 2c(c + 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)] (16)









10








41 Conductance



IL(R) = 2e

h

intdE

2πTr[lt


L(R)Gltd (E)

]







Gltd (E) = Gr

d(E)lt(E)Ga

d(E)

Ggtd (E) = Gr

d(E)gt(E)Ga

d(E)(18)


lt = ltL + lt

R

gt = gtL +gt

R (19)





[r


]

gtL(R)(E) = [



]

(20)

11





I = 2e

h

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (21)


L(R)(E) equiv i[r


] (22)





g = part I

partVb= 2e

h

part

partVb

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (23)


g = part I

partVb= 2e

h

intdE

2π

part[


partVb

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (24)





part[


partVb

= eβ

2

eminusβ|EminusμL|



(25)



g = g0T (26)

where g0 = 2e2










⎞⎠

times⎛⎜⎝

GrL(ε) Gr

Ld(ε) GrL R(ε)

GrdL(ε) Gr

d(ε) GrdR(ε)

GrRL(ε) Gr

Rd(ε) GrR(ε)

⎞⎟⎠ =

⎛⎝

IL 0 0

0 Id 0

0 0 IR

⎞⎠

(28)


L(R)d and the same




12






Grd(ε) =

[(gr


L(ε)minus rR(ε)

]minus1 (29)




rL(R)(ε) =


)gr

L(R)(ε)


) (30)








⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝







⎞⎟⎟⎟⎟⎠

minus1

(31)






01)T (ε)]minus1

(32)






)minus1t2iminus1


)minus1t2iminus1

(34)



)minus1V dagger

01


)minus1V01

(35)



13











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References


08

07

06

05

04

03

02

01

0

09

08

07

06

05

04

03

02

01

0



[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ 2c(c + 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)] (15)



2ε2




minus 2c(c + 1) cos

[2π

(φ

φ0+ kr

)]

minus 2(c + 1) cos

[2π

(φ

φ0minus kr

)]

+ 2c(c + 1) cos

[2π

(φ


)]

+ 2(c + 1) cos

[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)]

+ (c2 minus 1) cos

[2π

(φ


)]


[2π

(φ


)] (16)









10








41 Conductance



IL(R) = 2e

h

intdE

2πTr[lt


L(R)Gltd (E)

]







Gltd (E) = Gr

d(E)lt(E)Ga

d(E)

Ggtd (E) = Gr

d(E)gt(E)Ga

d(E)(18)


lt = ltL + lt

R

gt = gtL +gt

R (19)





[r


]

gtL(R)(E) = [



]

(20)

11





I = 2e

h

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (21)


L(R)(E) equiv i[r


] (22)





g = part I

partVb= 2e

h

part

partVb

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (23)


g = part I

partVb= 2e

h

intdE

2π

part[


partVb

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (24)





part[


partVb

= eβ

2

eminusβ|EminusμL|



(25)



g = g0T (26)

where g0 = 2e2










⎞⎠

times⎛⎜⎝

GrL(ε) Gr

Ld(ε) GrL R(ε)

GrdL(ε) Gr

d(ε) GrdR(ε)

GrRL(ε) Gr

Rd(ε) GrR(ε)

⎞⎟⎠ =

⎛⎝

IL 0 0

0 Id 0

0 0 IR

⎞⎠

(28)


L(R)d and the same




12






Grd(ε) =

[(gr


L(ε)minus rR(ε)

]minus1 (29)




rL(R)(ε) =


)gr

L(R)(ε)


) (30)








⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝







⎞⎟⎟⎟⎟⎠

minus1

(31)






01)T (ε)]minus1

(32)






)minus1t2iminus1


)minus1t2iminus1

(34)



)minus1V dagger

01


)minus1V01

(35)



13











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References








41 Conductance



IL(R) = 2e

h

intdE

2πTr[lt


L(R)Gltd (E)

]







Gltd (E) = Gr

d(E)lt(E)Ga

d(E)

Ggtd (E) = Gr

d(E)gt(E)Ga

d(E)(18)


lt = ltL + lt

R

gt = gtL +gt

R (19)





[r


]

gtL(R)(E) = [



]

(20)

11





I = 2e

h

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (21)


L(R)(E) equiv i[r


] (22)





g = part I

partVb= 2e

h

part

partVb

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (23)


g = part I

partVb= 2e

h

intdE

2π

part[


partVb

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (24)





part[


partVb

= eβ

2

eminusβ|EminusμL|



(25)



g = g0T (26)

where g0 = 2e2










⎞⎠

times⎛⎜⎝

GrL(ε) Gr

Ld(ε) GrL R(ε)

GrdL(ε) Gr

d(ε) GrdR(ε)

GrRL(ε) Gr

Rd(ε) GrR(ε)

⎞⎟⎠ =

⎛⎝

IL 0 0

0 Id 0

0 0 IR

⎞⎠

(28)


L(R)d and the same




12






Grd(ε) =

[(gr


L(ε)minus rR(ε)

]minus1 (29)




rL(R)(ε) =


)gr

L(R)(ε)


) (30)








⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝







⎞⎟⎟⎟⎟⎠

minus1

(31)






01)T (ε)]minus1

(32)






)minus1t2iminus1


)minus1t2iminus1

(34)



)minus1V dagger

01


)minus1V01

(35)



13











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References





I = 2e

h

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (21)


L(R)(E) equiv i[r


] (22)





g = part I

partVb= 2e

h

part

partVb

intdE

2π


]

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (23)


g = part I

partVb= 2e

h

intdE

2π

part[


partVb

times Tr[L(E)G

rd(E)R(E)G

ad(E)

] (24)





part[


partVb

= eβ

2

eminusβ|EminusμL|



(25)



g = g0T (26)

where g0 = 2e2










⎞⎠

times⎛⎜⎝

GrL(ε) Gr

Ld(ε) GrL R(ε)

GrdL(ε) Gr

d(ε) GrdR(ε)

GrRL(ε) Gr

Rd(ε) GrR(ε)

⎞⎟⎠ =

⎛⎝

IL 0 0

0 Id 0

0 0 IR

⎞⎠

(28)


L(R)d and the same




12






Grd(ε) =

[(gr


L(ε)minus rR(ε)

]minus1 (29)




rL(R)(ε) =


)gr

L(R)(ε)


) (30)








⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝







⎞⎟⎟⎟⎟⎠

minus1

(31)






01)T (ε)]minus1

(32)






)minus1t2iminus1


)minus1t2iminus1

(34)



)minus1V dagger

01


)minus1V01

(35)



13











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References






Grd(ε) =

[(gr


L(ε)minus rR(ε)

]minus1 (29)




rL(R)(ε) =


)gr

L(R)(ε)


) (30)








⎞⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎝







⎞⎟⎟⎟⎟⎠

minus1

(31)






01)T (ε)]minus1

(32)






)minus1t2iminus1


)minus1t2iminus1

(34)



)minus1V dagger

01


)minus1V01

(35)



13











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References











Gr(E) =[


]minus1 (36)







H = 1

2me

[P minus qA

]2 + V (r) (37)




H = H EH minus q

2me


)+ q2

2mA2 (38)




H = H EH minus μB

hL middot B + q2B2

8mer2perp (39)

Here μB = qh2me




H = H EH + iμB Bx

(ypart

partzminus z

part

party

)+ q2 B2

x

8me(y2 + z2) (40)

14
















H 1α2β asymp 12


] (43)





H EH1α1α+H EH

2β2β










λF

Nd2π = Nd




φ0= l



2πeplusmnimθ







15














16


B (Tesla)B (Tesla)

Vg

(eV

)

Vg

(eV

)







17










a b


18











19










φ0)2] = 2h2mF

mR2φ

φ0=

2h2kFmR

φ



φ

φ0




λFasymp π

2 Aminus1





20










Vb B I1 I2

0 0 0 00 1 0 01 0 0 11 1 1 0





21



51 Hamiltonian






el =H device





H 0el =

sumi j


sumlmisinL R






H intel =

sumli






p2k

2+ 1

22

Nminus1sumk=0




Nminus1sumk=0

p(q)k

2 + 12

Nminus1sumk=0

ω2k q2

k (48)


Hvib =Nminus1sumk=0

hωk

(bdagger

kbk + 12

) (49)



cdaggeri ci(a

daggeri + ai ) (50)









cdaggeri ci(a

daggeri + ai) =

Nminus1sumik=0

Mki cdagger


where Mki equiv M

radicωk



22



H =sumi j


sumlmisinL R

εlm(B)ddaggerl dm

+(sum

li


)+

Nminus1sumk=0

hωk

(bdagger

k bk + 12

)

+Nminus1sumik=0

Mki cdagger



52 Conductance


Iel = 2e

h

intdε

2π


]


](53)

and

Iinel = 2e

h

intdε

2πTr[Σlt

L (ε)Gr(ε)Σgt

ph(ε)Ga(ε)

minus ΣgtL (ε)G

r(ε)Σltph(ε)G

a(ε)] (54)




ph (ε)minus1










Σrph(ε) = i

sumk

intdω

2πMk

Dlt


+ Drk(ω)g


r(ε minus ω)

Mk (56)



Σ≶ph(ε) = i

sumk

intdω

2πMk D≶





53 Results





23




h




to 4

minus2rphim(ε)


2+(2minusrphim(ε))





ξ where









24

















25












References



26


















































27


















































28





















































29


















































30




















































31














32

1 Introduction



22 Length scales





23 Open questions

3 Basic concepts




41 Conductance





451 Atomic corral





51 Hamiltonian

52 Conductance

53 Results


References

magnetoresistance of nanoscale molecular devices based on aharonov

Documents