Molecular Electronics

Molecular electronics: Some views on transport junctionsand beyondChristian Joachim* and Mark A. Ratner†‡

*Groupe NanoSciences, Centre d’Elaboration de Materiaux et d’Etudes Structurales, Centre National de la Recherche Scientifique, 29,Rue Jeanne Marvig, BP 94347, 31055 Toulouse Cedex 4, France; and †Department of Chemistry, Northwestern University, 2145Sheridan Road, Evanston, IL 60208-3113

Contributed by Mark A. Ratner, January 19, 2005

The field of molecular electronics comprises a fundamental set of issues concerning the electronic response of molecules as parts of amesoscopic structure and a technology-facing area of science. We will overview some important aspects of these subfields. The mostadvanced ideas in the field involve the use of molecules as individual logic or memory units and are broadly based on using thequantum state space of the molecule. Current work in molecular electronics usually addresses molecular junction transport, wherethe molecule acts as a barrier for incoming electrons: This is the fundamental Landauer idea of ‘‘conduction as scattering’’ general-ized to molecular junction structures. Another point of view in terms of superexchange as a guiding mechanism for coherent elec-tron transfer through the molecular bridge is discussed. Molecules generally exhibit relatively strong vibronic coupling. The last sec-tion of this overview focuses on vibronic effects, including inelastic electron tunneling spectroscopy, hysteresis in junction chargetransport, and negative differential resistance in molecular transport junctions.

How a single molecule or anordered network of molecules(1–7) can perform transport(8, 9) or a computation using

mechanical (10), magnetic (11), or elec-tronic (12) degrees of freedom is becom-ing an active field of research, after thefirst suggestive ideas of the 1970s (13, 14).Use of electronic and nuclear moleculardegrees of freedom to create a devicefunction embedded inside a unique mole-cule is a major challenge in the applica-tion of molecular electronics ideas (15,16). Its ultimate functionality would berealized by using the physical resourcesinside a single molecule to integrate a fullarithmetic and�or logic unit (17).

Molecular electronics entails a series ofconceptual, experimental, and modelingchallenges (1–7, 18–20). Those challengesconcern the architecture of molecules, theinterconnects and, more generally, thenanocommunication schemes within andamong molecules, the chemistry, and(from a more applied point of view) nano-fabrication and nanopackaging techniques.

In architecture, the main challenge is todetermine whether a molecule can pro-vide an interconnect, a switch, a transistor(13, 14, 21) or a more complex function,such as a logic gate or even a full arith-metic and logic unit (17, 92). If the ulti-mate miniaturization scale of an electronicdevice is one molecule per transistor, suchhybrid molecular electronic technologymight approach what is expected for fu-ture semiconductor nanoelectronics. Sub-stantial progress in this direction hasoccurred since 1974, as exemplified by thecontribution from Ebling et al. (22) in thisSpecial Feature dealing with molecularrectification. Going further, if there areenough quantum resources in a singlemolecule to integrate more than one de-vice function, then new computational

schemes might open. These new schemeswill require extensive modeling studies toselect among a semiclassical architecture,forcing the molecule to behave like atraditional electronic circuit (23), a quasi-quantum one employing electronic inter-ference effects (24) or a Hamiltonian orquantum computing-like architecture,where the intramolecular quantum dy-namics of the molecule defines a compu-tation scheme (25).

For interconnects permitting data,charge, and energy exchanges between asingle molecule and the external world(the so-called nanocommunication prob-lem), one major challenge is to create anatomically clean technology in which thefabrication precision and the positioningof a single atom, molecule, or atomic wireon a surface is better than 0.1 nm, as dis-cussed by the Gourdon (26), Hersam (27),and Ho (28) groups in this issue of PNAS.Atomic scale precision positioning is avery active new field of research accompa-nied by a progressive shift from metallicto semiconducting surfaces to intercon-nect very precisely a single molecule to amacroscopic lead (29, 30).

The main chemical challenge may notbe the synthesis of a conjugated molecular‘‘board’’ to perform electronic devicefunctions or even act as a digital logicgate. A major chemistry challenge re-quires equipping the board with lateralchemical groups, not contributing to thefunction directly but protecting the molec-ular electronic functionality and assem-bling and stabilizing the molecule on agiven substrate. One example involveschemistry to maintain the molecularboard away from the surface while facili-tating scanning tunneling microscope sin-gle molecule manipulation (31, 32). Otherchemical groups must provide a goodelectronic coupling between the molecular

device or logic and local leads (33, 34).Aside from simple end groups, such asthiols, adding all of these lateral groups isproblematic for the deposition procedure(solubility, sublimation, and self-assembly),requiring new deposition techniques. Themolecular weight of these molecular facili-tators may be larger than the board itself.

There is little current work on nanofab-rication and nanopackaging with atomicscale precision. Molecular self assemblyshould permit molecules to self-build suchstructures as interconnect junctions (35,36). It is fair to anticipate an explosion ofactivities aimed at such fabrication. Singlemolecular devices or logic gates will haveto be packaged to transition from the lab-oratory to the market.

More fundamentally, molecular elec-tronics work is contributing to the knowl-edge and capability base of molecular andcondensed matter science. In this contri-bution, we focus not on elastic transportin molecular junctions, which has beenrecently reviewed (7–9, 19). Rather, weexamine the fundamental nature of chargetransport in such systems wherein the mo-lecular entity acts as a guide�filter forcharge flow. In The Superexchange Mecha-nism, we show that new ways of under-standing the physics of signal exchangethrough a long molecular wire can enlargemolecular electronics by adding the lan-guage of the quantum information com-munity. This discussion can be generalizedfrom molecular wires interconnected ton � 2 electrodes to a larger molecule withN interfacial links.

Abbreviations: ET, electron transfer; IETS, inelastic electrontunneling spectroscopy; NDR, negative differential resis-tance.

‡To whom correspondence should be addressed. E-mail:[email protected].

© 2005 by The National Academy of Sciences of the USA

www.pnas.org�cgi�doi�10.1073�pnas.0500075102 PNAS � June 21, 2005 � vol. 102 � no. 25 � 8801–8808

SP

EC

IAL

FE

AT

UR

E:

PE

RS

PE

CT

IVE

The Superexchange MechanismConsider an isolated, zero-temperaturemolecule made of a longitudinal part (L)with a substituent group (M1) at one endof L and another group (M2) at the otherend of L. When initially prepared in aquantum nonstationary state M1�–L–M2,this molecule should ideally oscillate backand forth between the forms M1�–L–M2and M1–L–M2�. These time-dependentHeisenberg–Rabi oscillations are the sig-nature of a pure quantum superexchangeeffect (37, 38), which will then dephase,until the M1–L–M2 molecule has relaxedinto a (M1–L–M2)� state. This basicquantum phenomenon is at the origin ofnearly all of the proposed and�or experi-mented molecular electronics device orlogic functions.

The superexchange mechanism reflectsthe role of the molecular wire L in in-creasing the effective electronic couplingV12 between the two quantum statesM1�–M2 and M1–M2� when M1 and M2are maintained very far apart. In the co-herent regime, the effect of the bridge Lis not the creation of a classical materialchannel for the charge to be transferredfrom M1 to M2 by using stations along L(39). Rather, L facilitates tunneling byincreasing the size of quantum state space(40, 41) for the quantum trajectory d(t)describing the quantum evolution ofM1�–L–M2 to reach the vicinity of thetarget state representing M1–L–M2�, ascompared to the quantum trajectory toreach M1–M2� starting from M1�–M2.

Optimizing an average d(t) to pass inclose proximity to M1–L–M2� startingfrom M1�–L–M2 or minimizing the dis-tance between d(t) and M1–L–M2� for agiven value of t is a difficult quantum con-trol problem. Usually, it is reduced tooptimizing the Wtransfer electron transfer(ET) rate between M1�–L–M2 andM1–L–M2� (39). The electronic com-ponent of Wtransfer is simply the secularfrequency of the Heisenberg–Rabi nearlyperiodic oscillations between the M1�–L–M2 and the M1–L–M2� states. It de-pends (for situations with an injection gapfar exceeding thermal energy) exponen-tially on the length of the ligand L, onits conformation, and on the detailedproperties of the virtual electronic statesentering the ET process (39, 42). Thosevariables control the extension and qualityof d(t) on the corresponding quantumstate space. Reducing the d(t) characteris-tics to only Wtransfer is a drastic reductionof the quantum properties on the ligandL. For example, the time for an electronto oscillate between M1 and M2 througha hypothetical 5-nm-long polyene ligand ina binuclear metal complex is about 1 ps(40, 41). This time can be compared tothe 0.1 fs taken by a ballistic electron to

run over 5 nm of a mesoscopic metallicwire (estimated from the Landauer for-mula, under bias V of 1 V, or from theFermi velocity in copper). Using onlyWtransfer to optimize d(t) may not beenough to understand the large differencebetween the two processes.

By selecting a metallic atom and bond-ing pattern and extending the lateral sizeof M1 and M2, one can use the superex-change interaction to couple two metallicpads electronically (Fig. 1). In this case,the (M1)n–L–(M2)n electronic couplingwill depend also on a new size parametern, n3� for the metal–L–metal tunneljunction case. As for n � 1, adding oneelectron to (M1)n will trigger a superex-change, time-dependent process for theelectron to be transferred to (M2)n by themolecular ligand. The d(t) trajectory ofthis process is certainly more complexthan for n � 1 and will depend on theelectronic density of states of A � (M1)nand B � (M2)n. As a consequence, theelectronic coupling V12(n) between the(M1)n

�-L-(M2)n and the (M1)n–L–(M2)n�

states is a function of n, and the corre-sponding transfer rate Wreservoir differssubstantially from Wtransfer. MeasuringV12(n) for very large n is possible by usingmicrowave-type experiments, as alreadypracticed with mesoscopic superconductorqubits (43). The discussion in this sectionis dramatically modified by vibronic anddephasing processes as discussed in Deco-herence and Relaxation (Inelastic Effects)and in the contribution by Ness andFisher.

Source and Detection ofTunneling ElectronOne heuristic physical interpretation ofthe tunneling regime, based on ref. 44,involves a shot–noise-like approach. Sup-pose now that each time a single ETevent occurs between cluster A and clus-ter B in the A–L–B quantum system pre-sented in Fig. 1, the transferred electron isremoved from B and, immediately, an-other electron is provided to the clusterA, again preparing a nonstationary quan-tum state A�–L–B. Following this proce-

dure, the maximum electronic currentintensity measured in the external circuitis simply:

Itransf � ed�N� t��dt �4eW

–h

reservoir,

[1]

with N the number operator measuredon the right pad.

The corresponding gedanken experi-mental set up is presented in Fig. 2. As inFig. 1, the set up includes a detector ofsingle ET events able to follow in time theN(t) electron population of cluster B. Asource S is now acting in synchronizationwith the ET events. The current (I) is cal-culated by performing the time average ofthe N(t) time-dependent signal measuredat the detector. The ideal N(t) shape is asuccession in time of sinusoidal arcs. Therise of each arc corresponds to the growthof the electronic population of cluster Bin time, due to ET. When this populationapproaches its maximum, the electron ispumped out of cluster B. The maximumpossible current is obtained by supposingthat the source S is ideally synchronizedwith the ET events. This circumstanceleads to a periodic N(t) signal with nointerruption between each period. In thiscase, the current intensity depends onlyon the ET rate Wreservoir, which is calcu-lated by averaging in time the ideal N(t)signal shape given in Fig. 3.

Of course, an ideal synchronizedsource S as described in Fig. 2 is notknown. A source of voltage V is usuallyused to form a closed electrical circuitdelivering electrons to the A-L-B molec-ular junction. In this case, the maximumtunneling current is given by the Land-auer formula (45):

Iscat � �W reservoir2�X2�8he2V , [2]

where X is the lead electronic conductionbandwidth and V is the bias voltage of theA–L–B tunnel junction. This current isthe maximum that can be measured bythe macroscopic amperometer for a givenA–L–B junction. For example, a sufficient

Fig. 1. A molecular wire L adsorbed and providing electronic coupling between two metallic clusters Aand B. Injecting one electron on A � (M1)n will trigger an ET process guided by L between the state A�–L–Band A–L–B� with B � (M2)n. This process can be followed in time by using the electron detector N(t) beforeit stops because of decoherence and relaxation effects at the interface, on the molecule, and in the A andB clusters.

8802 � www.pnas.org�cgi�doi�10.1073�pnas.0500075102 Joachim and Ratner

increase of the lead ohmic resistance willeffectively transform the voltage source toa current source. If this lead resistance ismuch higher than the quantum of resis-tance h�2e2, no electrons will be providedto the A–L–B junction. The source will beblocked by the so-called Coulomb block-ade effect (46). Delivery of electrons tobe transferred through the molecular wireis a very delicate process whose time de-pendence will not follow the N(t) idealtime sequence (Fig. 3).

Whereas delivery of electrons to themolecular wire is a classical–quantumconversion process, the detection of theelectrons after its transfer through themolecular wire is a very peculiar quan-tum–classical measurement process, ascan be demonstrated by using the Ehren-fest theorem, indicating that the currentvalue given by (Eq. 2) results from themeasurement of a specific quantum ob-servable in the tunnel junction (47). It canbe shown that this intensity is given by thequantum average over junction scatteringeigenstates of the commutator [N(t)�2,H], where H is the single-electron Hamil-tonian of the A–L–B tunnel junction.Both the source of tunneling electronsand detection of the transferred electronsto obtain a measurable tunneling currentat the macroscopic amperometer havebeen studied very little, either experimen-tally or theoretically.

One indication of the poor perfor-mance of the actual voltage generator-interconnection lead-stray capacitancetunnel sources is the difference betweenthe intensity given by Eqs. 1 and 2. For ashort conjugated molecular wire with aneffective electronic coupling V12(n) be-tween the two electrodes of 10 meV (1eV � 1.602 � 10�19 J), the maximumcurrent is 1.5 �A in Eq. 1 and the scatter-ing current is 0.2 nA in Eq. 2, where thescattering current is calculated for a biasvoltage of V � 0.1 V and a lead band-width of 4 eV. This difference of fourorders of magnitude between the two cur-rents is an indication that the time inter-val between two ET events in a A-L-Btunnel junction is very large. It followsthat there is no phase relation betweenthe elementary ET events, nor a regulardistribution of electron delivery to thetunnel junction over time. This Schottkynoise indicates that electron deliveryto the junction is a Poissonian processin time (48). Each ET event can beconsidered as quantum coherent [underconditions discussed in Decoherence andRelaxation (Inelastic Effects)]. The overallmeasured tunneling is not coherent, butnoisy. There is room for extensive im-provement of the source to take advan-tage of the fast ET rate compared withthe very small rate of electron delivery tothe junction.

This discussion of the source and detec-tion of tunneling electrons may hold forany tunneling junction in the linear part ofits I–V characteristic. Application to ametal�molecule�metal tunnel junction canclarify interpretation and nomenclature.First, the measured conductance of themetal–molecule–metal junction is not the‘‘conductance of the molecule itself’’ northe intrinsic ET rate through the molecu-lar wire. It is simply the rate of delivery ofelectrons to the junction. Of course, thisdelivery rate must be lower than ET ratethrough the molecule to avoid any chargepile-up in the junction. Second, there islittle electron flux at low-bias voltage,which (as discussed in the next section)

implies that for a virtual resonance tunnelprocess where the bias voltage V is lowerthan any electronic resonance of the mo-lecular wire, the appellation ‘‘electrontransport through a molecular wire’’ maynot describe the real process occurring inthe molecular junction. Third, one candefine the contact conductance as thepart of the A–L–B junction conductanceindependent of the length of the molecu-lar wire (49). This conductance is a mea-sure of the ability of the molecule�metalatomic scale junction to ease both theclassical–quantum conversion and thequantum measurement occurring at thejunction microscopic level. Both processesare summarized in one constant, andwork remains to be done to disentanglethe characterization of the two processes.

A Transport Phenomenon or a Guide forExchange InteractionsAt low-bias V (as discussed in the preced-ing section), very few electrons are re-placed on the molecular wire in thetunneling regime. Each transferred elec-tron is localized mainly to the electrodesand not on the molecular wire: The mo-lecular wire is not reduced or oxidized.This process is purely quantum: The mo-lecular wire quantum states merely in-crease quantum state space of the A–L–Bsystem to open a quantum trajectory rep-resenting the ET event between A and B.Without this state space expansion, theET between A and B will be too slow.Even with it, elastic scattering at the elec-trode�molecule interface will reducethe value of the conductance, which is the‘‘conductance is scattering’’ view of theLandauer formula (50).

When the electronic transparency in-creases toward unity, the electronic statesof the electrodes begin to delocalize alongthe molecule, and a regime of ballistictransport can be reached (in the absenceof vibronic coupling or resonance), whereat least one excess electron can be (onaverage) delocalized along the molecularwire. At low transparency, a molecularwire can be better seen as a partial guidefor the electronic exchange interactionsbetween the two metallic electrodes of theA–L–B junction than as a material sup-porting a permanent flux of electrons(51). The large difference between the ETrate and the tunneling rate is a good pieceof evidence that there is no permanentelectron flux in a molecular wire junction,even if there is a permanent electronicinteraction introduced by the molecularwire in A–L–B. Of course, an ET eventcan be viewed as an information exchangebetween A and B via L. The informationexchanged is classical, the presence of anelectron prepared in a nonstationary stateand located on A. There is no known wayyet to control in detail the preparation of

Fig. 3. The ideal N(t) time-dependent electronpopulation on cluster B when the source S in Fig. 2is synchronized with each ET event. As indicated,each arc is supposed to be normalized to effectivelyone ET at a time. � is the full duration of the eventbefore its fast pumping out of cluster B, � �h�4V12(n).

Fig. 2. A source S is added to Fig. 1, ideally synchronized with each ET event from A to B through L. Anideal S is supposed to provide an electron at cluster A each time the previous one had been pumped outof cluster B. A macroscopic amperometer is added to measure the tunnel current density in the circuit.

Joachim and Ratner PNAS � June 21, 2005 � vol. 102 � no. 25 � 8803

this state, for example, by preparing a pe-culiar phase in the mixing of the molecu-lar orbital involved in the definition of theA�–L–B initial state. We do not haveenough control yet of the quantum stateof the electrons delivered by the elec-trodes to the molecular wire to be able toexchange quantum information betweenA and B.

Many parameters characterize an ex-change interaction guide: the strength ofthe interaction, the level of dissipationand importance of the dispersion effects,the bandwidth, the noise level, and thecommunication capacity of a channelformed by the molecular wire in the tun-nel regime. Aside from the interactionstrength (spectral density) and some vi-bronic components of the dissipation,none of the other parameters are accessi-ble to simple I–V experiments because thecurrent intensity does not provide anydirect information about them.

Decoherence and Relaxation(Inelastic Effects)Two phenomena, decoherence and relax-ation, limit the quantum capability (dis-cussed above) of a molecular wire toexchange information between metallicelectrodes. Both phenomena increase withincreasing temperature and length of thewire (39), which implies that the ET timebetween electrodes is becoming long rela-tive to some internal characteristic timeof the molecular medium defining theexchange guide.

Decoherence. Preparation of the ET pro-cess by the tunnel source on the left ofthe A-L-B junction creates a wave packet,which is supposed to reach the right elec-trode and be detected. Wave packet prop-agation is dispersive, so it is very difficultto optimize the chemical structure of amolecular wire to ensure perfect (quan-tum yield of unity) wave packet propaga-tion. Some component of the wave packetwill be phase-shifted in time, leading to adeformation of the wave packet. As aconsequence, in the coherent tunnelingregime with a gap, exponential decay withlength of the electronic coupling V12(n) �Vo(n) exp(��L) between the electrodesthrough the molecular wire will occur,where � is an inverse decay length of theprocess (39, 52). A fascinating challengeinvolves finding a peculiar molecular wirestructure or wave packet quantum trajec-tory control to approach dispersionlessmotion. The second order character of theSchrodinger equation is responsible for �being bound from below (53). But an-other second-order equation, the Maxwellequation, produces remarkable propaga-tion behavior with material of negativerefractive index (54). Might such strangebehavior be induced into a quantum wave

packet moving along a molecular wire? Insearch for a supertunnel effect, this behav-ior would provide a way to minimize �and to couple the A and B electrodesover large distances. A first step in thisdirection was to recognize that the spec-tral second moment describing the molec-ular wire electronic level distribution inenergy is the second parameter control-ling � after the highest occupied molecu-lar orbital–lowest occupied molecularorbital gap of the molecular wire (39, 55).

Finding a molecular wire architectureto optimize this second-order distributionwill lead to very small �. Unfortunatelyfor this optimization of V12, a simple tun-neling junction does not access all virtualelectronic excitations. This incompletenessis one explanation of the difference be-tween Eqs. 1 and 2, as can be clearly ob-served in electronic I–V characteristicsrecorded on a single molecule with ascanning tunneling microscope metal sur-face–molecule–metal tip apex tunnel junc-tion. The recorded spectra can usually beinterpreted as tunneling through a quan-tum system showing only very simplemonoelectronic excitations (56). How atunneling junction is filtering (selecting)the relevant electronic virtual excitationsto create a given channel is essentiallyan open question. Deviations from theexpected simple (57) I–V curves are some-times seen in scanning tunneling micro-scope I–V spectra on single molecules (29)and are not yet completely understood.

Inelastic Effects (Relaxation). Molecularelectronics (1–9) differs from more tradi-tional semiconductor electronics in severalcrucial ways. An important one firstpointed out by Yablanovich (58) is thatmolecules nearly always exhibit vibroniccoupling: Molecular spectra show Stokesshifts and the excited state minimum ge-ometry of molecules is hardly ever thesame as the ground state geometry, just asthe behavior of molecular ions almostalways differs from that of the stable mol-ecules. The absence of such vibronic cou-pling in silicon is one of the importantaspects of silicon semiconductors, givingstable, unchanging geometries as the elec-tronic population changes. Because thisabsence is not true in molecules, molecu-lar electronics will show several importantdifferences from silicon electronics.

For example, in traditional semiconduc-tors the so-called ‘‘Coulomb staircase’’regime occurs when the coupling betweenthe bridge structure, (usually a quantumdot) and the electrodes is relatively weakcompared to the electronic self-energy onthe dot. Under these conditions, the dotcan charge multiply, lead to a ‘‘coulombstaircase.’’ In molecules, this charging willalso change the geometries; we, therefore,do not expect a straightforward Coulomb

staircase with equally spaced chargingpeaks, essentially because of vibronic cou-pling. This observation has indeed beenmade in measurements on conductive oli-gomer structures (59).

The interaction between vibrations andelectronic states is responsible for many ofthe most important phenomena in molec-ular materials, including vibrational peaksin electronic spectra, nonadiabatic pro-cesses of many types, including Jahn–Teller distortions, and the characteristicactivation processes associated with trans-port in conducting polymers (60–62).These same vibronic coupling effects areexpected to be extremely important incertain aspects of molecular electronics.In this section, we very briefly discusssuch vibronic coupling in inelastic electrontunneling spectroscopy (IETS), the mech-anistic transition from coherent tunnelingto hopping-type processes, and in some ofthe characteristic behaviors of transport inmolecular junctions, including negativedifferential resistance mechanisms andhysteretic�switching behavior.Vibronic coupling: IETS spectra and lineshapes. The IETS experiment, first devel-oped (63–67) at the Ford Laboratories inthe 1950s, consists of observing vibra-tional side bands in the I–V characteristicsof any transport medium. Because thesepeaks arise from electron�vibration cou-pling, they are expected to depend criti-cally on the nature of this coupling. Theexperiment examines the dependence ofthe transport current on the voltage, thefirst derivative (conductance), and, morespecifically, the second derivative of thecurrent. In these second-derivative IETSspectra, one sees peaks that correspond toinelastic behavior, excitation of a vibra-tional mode on the wire.

From standard textbook discussions(67), one might expect the behavior indi-cated in Fig. 4. Whenever the appliedvoltage V is large enough for an inelasticevent to occur such that the current pass-ing through a molecular wire can depositenergy corresponding to a single quantumof vibration, one should see an increase inthe rate at which current flows. This be-havior would be expected to give changesin slope of the current, changes in the val-ues of the conductance, and peaks in theIETS behavior.

It is useful to consider the moleculargeneralization of the so called Landauer–Buttiker contact time, during which thetunneling electron is actually in contactwith the molecular bridge (the time inFig. 2 between the injection and the re-lease of this electron to the opposite elec-trode). A reasonable approximation (68)can be derived on the basis of the originalButtiker argument (69), extended to dealwith a tight-binding type molecular wire,which yields the approximation


�C � N–h�EG. [3]

The contact time, �c, is inversely propor-tional to EG, the excitation gap betweenthe injection energy and the isolatedbridge frontier orbital energy, and in-creases with the number, N, of repeatgroups in the wire. This form is actually astraightforward generalization of the un-certainty principle, but it has importantphysical implications: For large gaps (forexample �-bonded systems) and shortwires, �c will be of order 10�16 s, far tooshort for significant vibronic coupling. Forsmaller gaps (�-electrons) and longerwires, the contact time can increase tobecome of the order of a vibrational pe-riod (39). One then expects adequate timefor vibronic coupling to operate. Thus, forlong bridges and �-type systems, vibroniccoupling should be more important thanfor �-type systems and short bridges, inaccordance with a number of experimen-tal observations (70, 71).

For relatively low voltages and low tem-peratures (conditions kT –h� and eV �–h�, where � is the molecular vibrationfrequency and V is the applied voltage),one can develop a perturbative descriptionof the IETS spectrum (93). Here it is as-sumed that the vibronic coupling is weakenough that no polarons (charged distor-tions on the molecular bridge) occur. Thecurrent is elastic, and the inelastic partcan be evaluated by low-order perturba-tion theory similar to the Herzberg–Telleranalysis for vibrational structure in elec-tronic spectra.

Under these conditions and from Eq. 2,the conductance can be written (A. Troisiand M.A.R., unpublished data) in thestandard Landauer-type model as

g�E� � goTr��L�E�G�E��R�E�G �E�� .

[4]

Here g, go, �, G, and L denote, respec-tively, the conductance, the atomic unit ofconductance (12.9 k�)�1, the mixing ofthe molecular wire with the electrode onthe left (L) and right (R) sides, and theGreen’s function (72–75) for propagationof the electron through the bridge struc-ture. The couplings, �, are proportional tothe escape rate of electrons from the mol-ecule to the two electrodes. The Green’sfunction itself is taken simply as

G�1 � �E H i��0 , [5]

where H is the full-system Hamiltonian.Expanding the conductance and the

current around the geometric minimum,one can write the Green’s function as

Gij�E ,�Q�� G ij�E ,0� �2�a

G ij�Q�.

[6]

Gij� � �1

2 G ij� Q��Q�0. [7]

The geometric derivative of Eq. 7 deter-mines the importance of any particularvibrational normal coordinate, Q�, inthe IETS spectrum. In this sense, it is adimensional coupling constant for theintensities.

One generally expects that a peakwould appear in the IETS spectrum whenthe resonance condition, eV � –h��, oc-curs. The area of this peak, W�, is

W� � goT r ��L�EF�G��EF��R

�EF�G� �EF�� . [8]

Although this is an attractive formalscheme, its computational implications aredifficult: One must deal with calculatingthe current itself, the structure, the spec-tral densities �, and the full dependenceof the Green’s functions (including self-energies) on the vibronic normal mode.

Troisi (A. Triosi, personal communica-tion) considered a simplification of theproblem that leads to a tractable compu-tation and still provides both good fits tothe experiment and substantial experi-mental insight (A. Troisi and M.A.R., un-published data). He argues that one canglean a great deal of information aboutthe molecular vibronic direction and itsimportance in IETS by ignoring, in spiritof weak coupling and perturbation theory,effects arising from the spectral densities,�. He then defines a ‘‘gateway’’ orbital,which acts as a bridge between the molec-ular entity and the electrode itself. Thesegateway orbitals are labeled � and � (for

left and right), and the computationalanalysis is restricted now in considerationof a molecule, consistent with the mo-lecular bridge terminated by the � and �orbitals.

Often, one expresses the IETS spec-trum in the form of the derivativequotient:

d2I�dV2

dI�dV.

which is plotted against voltage V. Onethen expects that there will be a peak atthe resonance condition, eV � –h��. Thepeak intensity will be given by the ratio

R��EF� � �G�� E f� �2��G��EF� �2.

[9]

Physically, this perturbative treatmenttreats the inelastic component as a pertur-bation of the elastic transport, assumingthat the system is very close to equilib-rium because the vibrational frequenciesare all less 0.2 eV. In addition, the gate-way orbital approach says that the vibra-tional structure on the bridge is, to areasonable extent, separable from thespectral densities coupling the bridge tothe electrodes.

Fig. 5 shows a comparison between theexperimental IETS spectrum (71) of theoligophenyleneethynylene molecule andthe computation, carried out by usingstandard density functional theory. Thethree major bands of the IETS spectrumare very well produced by the simulation:The high-frequency signal near 2,211cm�1 is due to the triple bond symmetricstretch; the peak at 1,582 cm�1 is due tothe double bond stretch; and the C�C�Cbend, largely localized on the aromaticrings, gives the major contribution to thepeak at 1,100 cm�1. Comparable resultsare found for other molecules.

Fig. 4. Schematic illustrations of the IETS mea-surement. The cross-sections for the elastic andinelastic tunneling are (to a first, rough approxi-mation) additive, so that when the applied voltageexceeds a characteristic vibrational energy on themolecule, one expects to see changes of slope inthe current, of value in the conductance, and peaksin the IETS (second derivative of the current) struc-ture (M. A. Reed, personal communication).

Fig. 5. Calculated (upper line) (A. Troisi, personalcommunication) and measured (lower line) (71)IETS spectra of an ortho-phenylene ethynylenemolecule with a nitro substituent and gold elec-trodes. The peaks correspond to specific normalcoordinates of the molecule.


Because no symmetry principles wereinvolved in this calculation, propensitybehaviors (numerical selection rules) areseen in the spectrum. The totally symmet-ric representation peaks dominate. More-over, the coupling constants characterizedin Eq. 7 index the IETS behavior verywell. The appearance of peaks in theIETS spectrum, at the normal coordinatescorresponding to the fully symmetric vi-brations of the molecule, is strong evi-dence that the transport actually occursthrough the molecule.

Although the simple perturbative modeldeveloped in Eqs. 4-9 provides a goodgeneral description, there are many morespecific and challenging issues, for whichthe simple perturbative analysis may notbe adequate. The first of these involvesthe actual line shapes for the vibronic fea-tures in the spectrum. This is a compli-cated problem, requiring self-consistentsolutions of the self-energies for the vibra-tional and the electronic Green’s functions(72–75). Although these line shapes pro-vide important mechanistic insights, thisanalysis is outside our scope here. Signifi-cant effects occur when the vibronic cou-pling becomes truly strong, and the nexttwo sections discuss the mechanistic tran-sition from coherent tunneling to hoppingbehavior, and the implications of suchstrong coupling in determining important,unusual spectral features, such as negativedifferential resistance (NDR) and hyster-etic�switching transport behaviors.Mechanistic transition: Tunneling to hopping.For short molecule-transport structures,particularly those without conjugation, theLandauer–Buttiker argument of Eq. 3suggests very weak vibronic coupling andelastic transport. This sense of the Land-auer formula follows from a general non-equilibrium Green’s function approach ifonly electronic terms are considered inthe Hamiltonian. For longer bridges andsmaller gaps, the vibronic coupling shouldbecome more important.

In scanning tunneling microscope mea-surements, pioneering work by Ho’s group(76–78) has indeed shown, for a moleculeon a semiconductor surface, that an entirevibrational progression can be seen inthe IETS spectrum. This behavior couldbe analyzed by using a variant of the self-consistent Born approximation, even whenthe coupling can become strong, allowingmore than one vibrational quantum to beinvolved in the inelastic current (79).

Perhaps the most striking predicted be-havior is a change in the mechanism fromtunneling at low temperatures and shortbridges to a hopping-type (incoherent)motion for long wires and small injectionbarriers (39, 70). In molecular junctions,as in intramolecular ET reactions, oneactually expects a mixture of two mecha-nisms: In the first mechanism, electrons

simply tunnel across the barrier throughthe intervening states; as temperature in-creases, however, activated-type processesin which the electron is thermally excitedto a higher level, thus reducing the possi-ble tunneling distance, can become rele-vant (70, 80). For high fields, long wiresand small injection gaps a fully activatedprocess can occur.

As discussed in Decoherence, the quan-tum ET events occurring at low tempera-tures lead to an exponentially decreasingcurrent as a function of the interelectrodedistance in the tunneling junction (Fig. 2).Upon activation, one expects diffuse hop-ping along the bridge, so that the lengthdependence on transport would be veryshallow (generally 1�(A BR), where Aand B are constants depending on theparticular experiment, and R is the dis-tance between the electrodes) (Fig. 6).The transition from one of these mecha-nisms to another has been well character-ized in situations like intermolecular ETprocesses in �-electron species and DNA(39, 81, 82). For electron transport in ac-tual wire junctions, we know of no directobservation of the transition indicated inFig. 6; limiting regimes have clearly beenseen, from the pure tunneling behavior(49, 83–86) in small oligoalkanes andphenylene-vinylenes (all of which are tun-neling) to actual conductive polymers,which transport charge (at least at hightemperatures) essentially by activatedmechanisms (59).

Although these transitions are perhapsthe most striking indication of vibroniccoupling and its effects in changing ETmechanism in molecular wires and theyhave been extensively discussed theoreti-cally, direct experimental confirmation isstill incomplete. In actual applications, thecoherent and incoherent regimes could bequite important: Simply because the trans-port is incoherent does not necessarily

mean that substantial energy is dissipatedas heat.Polarons, NDR, and hysteric�switching behav-ior in molecular junctions. Vibronic effectsin molecular transport junctions havebeen stressed very recently, both theoreti-cally and experimentally. As indicatedabove, localization of the electrons canlead to incoherent tunneling processes.Datta and collaborators (87) have veryrecently extended earlier work concerningthe effects of image forces on molecularwire transport.

Suppose that the two interfaces are in-equivalent; for example, suppose that themolecule were more strongly coupled tothe left A than to the right B electrode inFig. 2. Then, localized charge on the mol-ecule can get closer to the left electrode,building a greater image-type stabilizationthere and yielding an asymmetry in theoverall self-consistent electrostatic poten-tial. Formally, one can rewrite the Green’sfunction for transport through the molec-ular wire as

G�1�E� � �E F �L �R� .

[10]

Here, the last two terms are the elec-tronic self-energies arising from interac-tion between the molecule and the leftand right electrodes, whereas F is theFock operator, representing the self-consistent one-electron Hamiltonianthat appears for any given applied fieldin any given geometry.

One can construct a self-consistent so-lution to this problem, based on a com-plete neglect of differential overlap-typesemiempirical argument (88). A self-con-sistent electrostatic potential can be foundthat depends on the change in electronicdensity on the bridge. This expression en-ters the Fock operator and can be in turnrepresented as

V�� VLaplace VPoisson V image�� .

[11]

The Poisson part of the equation is ap-proximated by using the complete neglectof differential overlap potential, whichcontains the charging and the screeningeffects (� is the charge density). Laplaceand image potentials are calculated byusing a finite element method, withboundary conditions set by the localchemical potentials in the presence of theapplied electrostatic potential so that anyasymmetry in the potential profile can beincluded in an explicit fashion in terms ofthe self-consistent overall potential.

Application of polaron theory for de-vice possibilities in molecular transportjunctions, in particular for NDR andswitching behavior, has been developed

Fig. 6. Schematic expected behavior for intramo-lecular ET reactions and for conductance as a func-tion of the length of the molecular bridge. Thetotal rate is the sum of injection hopping (incoher-ent motion), which is weakly distance-dependent,and coherent tunneling, which depends exponen-tially on distance.


by Galperin and coworkers (89). Theirsimple model consists of a wire repre-sented by a single site, containing a singleelectronic state and a single harmonicoscillator. This wire is placed between twoelectrodes, and the vibration on the mole-cule is allowed to interact with phonons inthe environment (which could be elec-trodes or a solvent). The Hamiltonian canthen be expressed as

H � �0c0 c0 �

L,R

�kak ak

�k

� Vkc0 ck h .c .�

Mc0 c0�a a�

��

��b� b� U��b�

b��

� �a a�� a a�0. [12]

Here the terms �0, �0, �k, �� are theelectronic energy level on the wire,the vibrational frequency on the wire, theelectronic state energies in the electrodes,and the vibrational frequency of the pho-nons, respectively. The coupling constantsVk, U�, and M are respectively the injec-tion tunneling matrix element betweenthe molecular wire and the bridge, thephonon�vibration interaction, and the vi-bronic coupling on the molecule. Finally,

c0, a, ck and b� are the destruction opera-tors for the bridge electronic state, bridgevibrational state, bulk electronic state, andenvironmental phonon, respectively. ThisHamiltonian is characteristic of linearcoupling to form a polaron plus an injec-tion model at the interface.

In the spirit of the Born–Oppenheimerapproximation, it is possible to separatethe electronic and vibrational parts of theHamiltonian. The electronic term thenreads

Hel � �0 c0

c0

�k

��kck ck Vkc0

ck h .c .� ,

[13]

Here, the renormalized site energy �0 de-pends on the occupation of the site, n0.The renormalized energy is

�0�n0� � �0 2�reorgn0, [14]

where the reorganization energy isgiven by

�reorg � M2�0��02 ��2�2� ,

[15]

with � being the phonon lifetime.In the wide-band limit and at steady

state, one can rewrite the occupation onthe level in terms of a contour integralover the lesser Green’s function in theform

n0 � � d�

2�

�LfL�E� �RfR�E�

�E �02�n0��

2 ��2�2 .

[16]

Here fL and fR are the Fermi functions onthe left and right electrodes, respectively,with the spectral density

� � �R �L. [17]

Eq. 17 is the self-consistency condition:The population, n0, on the bridge deter-mines the polaron shifted energy �0,which in turn redetermines n0. This condi-tion becomes quite straightforward in thelimit of T � 0 at equilibrium, �R � �L.Then the integral can be performed, andthe result is

n0 �1�

arctan�x� 12

n0 ��x

4�reorg

�0 �

2� reorg. [18]

Solving these equations self-consistentlyyields either one or three roots. Whenthree roots are seen, a typical hysteresissituation occurs, arising from multistability

(highly reminiscent of the mean-fieldtreatment of magnetic systems in general).Hysteresis loops might be attainable, asshown in Fig. 7.

When �0 lies in the window between �Land �R, one would normally expect reso-nant tunneling to occur. If, however,polaron formation lowers the energy suffi-ciently by reorganization effects that �drops below the chemical potential of thelow-voltage electrode, then the currentshould abruptly turn off, as is seen in theNDR experiments. Fig. 8 shows such aprediction arising from the mean fieldmodel, compared with the experimentallyobserved structure. Both the temperaturedependence and the voltage dependenceof the calculated NDR spectrum agreewith that reported in the experimentalliterature.

In this overview, we have focused onthree topics (largely unexplored) of mo-lecular electronics that go beyond simpletransport junctions, like the question ofimproving the efficiency of a tunnel junc-tion as a source of electron prepared in aquantum state superposition, the notionof the superexchange transport junction asa guide for those electrons pointing to-ward quantum information, and the roleof vibronic coupling and inelasticity in theintramolecular events determining thephysical characteristics of this guiding.The first and the last of these topics willcertainly comprise major foci of the fieldin the near future.

Fig. 7. Computed (Lower) (89) and measured(Upper) (90) hysteretic behavior in the conduc-tance spectrum of a conjugated molecular struc-ture. Note that Right examines only the positivevoltage sweep regime.

Fig. 8. Measured (Upper) (91) and calculated(Lower) (89) negative differential resistance fea-ture in the conductance measurement of thesketched molecule. The model explains the resultbased on charging of the molecular junction.


Note Added in Proof. While this manuscriptwas in production, an article was publishedthat offers a specific view of some aspectsof actual molecular-based computingschemes (92).

C.J. thanks the NanoScience Group at theCentre National de la Recherche Scienti-fique (Toulouse, France) and his partners in

the European Bottom-Up Nanomachinesproject, the Consortium for HamiltonianIntramolecular Computing, the Atomic andMolecular Manipulation as a new Tool forScience and Technology network, and thePico-Inside consortium for many illuminatingdiscussions. M.A.R. thanks his many collabo-rators, particularly A. Nitzan, V. Mujica, A.Troisi, M. Galperin, Z. Bihary, and Y. Xue,and the Department of Defense Multidisci-

plinary University Research Initiatives�De-fense University Research Initiative on Nano-technology Program, the Defense AdvancedResearch Projects Agency Applications ofMolecular Electronics Program, the NationalScience Foundation�Network for Computa-tional Nanotechnology program, the Chemis-try Divisions of the National Science Founda-tion, and the Office of Naval Research forsupport.

1. Petty, M. C., Bryce, M. R. & Bloor, D. (1995)Introduction to Molecular Electronics (OxfordUniv. Press, Oxford).

2. Aviram, A., ed. (1992) Molecular Electronics: Sci-ence and Technology (Am. Inst. Phys., CollegePark, MD)

3. Aviram, A. & Ratner, M. A., eds. (1998) Ann. N.Y.Acad. Sci. 852, 1–370.

4. Jortner, J. & Ratner, M. A., eds. (1997) MolecularElectronics (Blackwell Scientific, Oxford).

5. Reed, M. A. & Lee, T., eds. (2003) MolecularNanoelectronics (Am. Sci., Stevenson Ranch, CA).

6. Reimers, J., Picconatto, C., Ellenbogen, J. &Shashidhar, R., eds. (2003) Ann. N.Y. Acad. Sci.1006, 1–330.

7. Cuniberti, G., Fagas, G. & Richter, K., eds. (2005)Introducing Molecular Electronics (Springer, Berlin).

8. Seideman, T. & Guo, H. (2003) J. Theor. Comp.Chem. 2, 439–458.

9. Salomon, A., Cahen, D., Lindsay, S., Tomfohr, J.,Engelkes, V. B. & Frisbie, C. D. (2003) Adv. Mater.15, 1881–1890.

10. Cuberes, M. T., Schlittler, R. R. & Gimzewski,J. K. (1996) Appl. Phys. Lett. 69, 3016–3018.

11. Chuang, I. L., Vandersypen, L. M. K., Zhou, X. L.,Leung, D. W. & Lloyd, S. (1998) Nature 393,143–146.

12. Joachim, C. & Gimzewski, J. K. (1997) Chem.Phys. Lett. 265, 353–357.

13. Aviram, A. & Ratner, M. (1974) Chem. Phys. Lett.29, 277–283.

14. Carter, F. (1983) J. Vac. Sci. Technol. B 1, 959–968.15. Joachim, C., Gimzewski, J. K. & Aviram, A. (2000)

Nature 408, 541–548.16. Yi, J. & Cuniberti, G. (2003) Ann. N.Y. Acad. Sci.

1006, 306–311.17. Joachim, C. (2002) Nanotechnology 13, R1–R7.18. Datta, S. (2005) Quantum Transport: Atom to

Transistor (Cambridge Univ. Press, New York).19. Nitzan, A. & Ratner, M. (2003) Science 300,

1384–1389.20. Troisi, A. & Ratner, M. A. (2003) in Molecular

Nanoelectronics, eds. Reed, M. A. & Lee, T. (Am.Sci., Stevenson Ranch, CA).

21. Heath, J. R. & Ratner, M. A. (2003) Phys. Today56 (3), 43–49.

22. Ebling, M., Ochs, R., Koentopp, M., Fischer, M.,von Hanisch, C., Weigend, F., Evers, F., Weber,H. B. & Mayor, M. (2005) Proc. Natl. Acad, Sci.USA 102, 8815–8820.

23. Ellenbogen, J. C. & Love, C. (2000) Proc. IEEE 88,386–426.

24. Walter, D., Neuhauser, D. & Baer, R. (2006)Chem. Phys. 299, 139–145.

25. Fiurasek, J., Cerf, N. J., Duchemin, I. & Joachim,C. (2004) Physica E 24, 161–172.

26. Moresco, F. & Gourdon, A. (2005) Proc. Natl.Acad. Sci. USA 102, 8809–8814.

27. Guisinger, N. P., Yoder, N. L. & Hersam, M. C.(2005) Proc. Natl. Acad. Sci. USA 102, 8838–8843.

28. Nazin, G. V., Wo, S. W. & Ho, W. (2005) Proc.Natl. Acad. Sci. USA 102, 8832–8837.

29. Repp, J., Meyer, G., Stojkovic, S., Gourdon, A. &Joachim, C. (2005) Phys. Rev. Lett. 94, 02680–02683.

30. Guisinger, V. P., Greene, M. E., Basu, R., Baluch,A. S. & Hersam, M. C. (2004) Nano Lett. 4, 55–59.

31. Langlais, V., Schlittler, R. R., Tang, H., Gourdon,A., Joachim, C. & Gimzewski, J. K. (1999) Phys.Rev. Lett. 83, 2809–2812.

32. Bonifazi, D., Spillmann, H., Kiebele, A., de Wild,M., Seiler, P., Cheng, F. Y., Guntherodt, H. J.,Jung, T. & Diederich, F. (2004) Angew. Chem. Int.Ed. 43, 4759–4763.

33. Datta, S., Tian, W., Hong, S., Reifenberger, R.,Henderson, J. I. & Kubiak, C. P. (1997) Phys. Rev.Lett. 79, 2530–2533.

34. Moresco, F., Gross, L., Alemani, M., Rieder,K. H., Tang, H., Gourdon, A. & Joachim, C.(2003) Phys. Rev. Lett. 91, 036601.

35. Liu, S., Maoz, R. & Sagiv, J. (2004) Nano Lett. 4,845–851.

36. Rosei, F., Schunack, M., Jiang, P., Gourdon, A.,Joachim, C. & Besenbacher, F. (2002) Science 296,328–331.

37. McConnell, H. M. (1961) J. Chem. Phys. 35, 508–515.38. Halpern, J. & Orgel, L. E. (1960) Discuss. Faraday

Soc. 29, 32–41.39. Nitzan, A. (2001) Ann. Rev. Phys. Chem. 52, 681–750.40. Joachim, C. (1987) J. Phys. A 20, L1149–L1155.41. Launay, M. P. (2001) Chem. Soc. Rev. 30, 386–397.42. Galperin, M., Segal, D. & Nitzan, A. (2000)

J. Chem. Phys. 111, 1569–1579.43. Nakamura, Y., Chen, C. D. & Tsai, J. S. (1997)

Phys. Rev. Lett., 79, 2328–2331.44. Devoret, M. H., Esteve, D. & Urbina, C. (1992)

Nature 360, 547–553.45. Buttiker, M., Imry, Y., Landauer, R. & Pinhas, S.

(1985) Phys. Rev. B 31, 6207–6215.46. Joyez, P., Esteve, D. & Devoret, M. H. (1998)

Phys. Rev. Lett. 80, 1956–1959.47. Doyen, G. (1993) J. Phys. C 5, 3305–3312.48. Lee, H. & Levitov, L. S. (1996) Phys. Rev. B 53,

7383–7391.49. Wold, D. J. & Frisbie, C. D. (2001) J. Am. Chem.

Soc. 123, 5549–5556.50. Beenakker, C. W. J. & van Houten, H. (1991) in

Solid State Physics: Semiconductor Heterostructuresand Nanostructures, eds. Ehrenreich, H. & Turn-bull, D. (Academic, New York), pp. 1–228.

51. Kaun, C.-C., Guo, H., Grutter, P. & Lennox, R. B.(2004) Phys. Rev. B 70, 195309.

52. Joachim, C. (1987) J. Chem. Phys. 116, 339-349.53. Magoga, M. & Joachim, C. (1998) Phys. Rev. B 57,

1820–1823.54. Pendry, J. B. (2000) Phys. Rev. Lett. 85, 3966–3969.55. Lahamidi, A. & Joachim, C. (2003) Chem. Phys.

Lett. 381, 335–339.56. Joachim, C. & Gimzewski, J. K. (1995) Euro. Phys.

Lett. 30, 409–414.57. Xue, Y. & Ratner, M. A. (2005) Int. J. Quantum

Chem. 102, 911–924.58. Yablonovitch, E. (1989) Science 246, 347–351.59. Kubatkin, S., Danilov, A., Hjort, M., Cornil, J.,

Bredas, J.-L., Stuhr-Hansen, N., Hedegard, P. &Bjornholm, T. (2003) Nature 425, 698–701.

60. Kuznetsov, A. M. (1995) Charge Transfer in Phys-ics, Chemistry and Biology (Gordon & Breach,New York).

61. Kuznetsov, A. M. & Ulstrup, J. (1998) ElectronTransfer in Chemistry and Biology: An Introductionto the Theory (Wiley, New York).

62. Barbara P. F., Meyer, T. J. & Ratner, M. A. (1996)J. Phys. Chem. 100, 13148–13168.

63. Jaklevic, R. C. & Lambe, J. (1966) Phys. Rev. Lett.17, 1139–1140.

64. Adkins, L. J. & Phillips, W. A. (1985) J. Phys. CSolid State Phys. 18, 1313–1346.

65. Wolf, E. L. (1985) Principles of Electron TunnelingSpectroscopy (Oxford Univ. Press, New York).

66. Hipps, K. W. & Mazur, U. J. (1993) J. Phys. Chem.97, 7803–7814.

67. Hansma, P., ed. (1982) Tunneling Spectroscopy(Plenum, New York).

68. Nitzan, A., Jortner, J., Wilkie, J., Burin, A. L. & Ratner,M. A. (2000) J. Phys. Chem. B 104, 5661–5665.

69. Buttiker, M. & Landauer, R. (1985) Phys. Scr. 32, 429.70. Datta, S. (2004) Nanotechnology 15, S433–S451.71. Kushmerick, J. G., Lazorcik, J., Patterson, C. H.,

Shashidhar, R., Seferos, D. S. & Bazan, G. C.(2004) Nano Lett. 4, 639–642.

72. Galperin, M., Ratner, M. A. & Nitzan, A. (2004)J. Chem. Phys. 121, 11965–11979.

73. Mii, T., Tikhodeev, S. G. & Ueba, H. (2003) Phys.Rev. B 68, 205406.

74. Lundin, U. & McKenzie, R. H. (2002) Phys. Rev.B 66, 075303.

75. Meir, Y. & Wingreen, N. S. (1992) Phys. Rev. Lett.68, 2512–2515.

76. Stipe, B. C., Rezaei, M. A. & Ho, W. (1998)Science 280, 1732–1735.

77. Lee, H. J. & Ho, W. (1999) Science 286, 1719–1722.78. Gaudioso, J., Lauhon, L. J. & Ho, W. (2000) Phys.

Rev. Lett. 85, 1918–1921.79. Galperin, M., Ratner, M. A. & Nitzan, A. (2004)

Nano Lett. 4, 1605–1611.80. Burin, A. L. & Ratner, M. A. (2003) J. Polym. Sci.

Part B: Polym. Phys. 41, 2601–2621.81. Jortner, J. & Bixon, M. (1999) in Advances in

Chemical Physics, eds. Prigogine, I. & Rice, S.(Wiley, New York), p. 106.

82. Rice, S. A., Berry, R. S. & Nitzan, A., eds. (2004)Israel J. Chem. 44, 1–408.

83. Reichert, J., Weber, H. B., Mayor, M. & Lohney-sen, H. V. (2003) Appl. Phys. Lett. 82, 4137.

84. Wang, W., Lee, T., Kretzschmar, I. & Reed, M. A.(2004) Nano Lett. 4, 643–646.

85. Smit, R. H. M., Noat, Y., Untiedt, C., Lang, N. D.,Hemert, M. C. V. & van Ruitenbeek, J. M. (2002)Nature 419, 906–909.

86. Reichert, J., Ochs, R., Beckmann, D., Weber,H. B., Mayor, M. & Lohneysen, H. V. (2002) Phys.Rev. Lett. 88, 176804.

87. Zahid, F., Ghosh, A. W., Paulsson, M., Polizzi, E.& Datta, S. (2004) arXiv: cond-mat�0403401.

88. Ratner, M. A. & Schatz, G. C. (2002) Introductionto Quantum Mechanics in Chemistry (Prentice–Hall, Englewood Cliffs, NJ).

89. Galperin, M., Ratner, M. A. & Nitzan, A. (2005)Nano Lett. 5, 125–130.

90. Blum, A. S., Kushmerick, J. G., Long, D. P.,Patterson, C. H., Yang, J. C., Henderson, J. C.,Yao, Y., Tour, J. M., Shashidhar, R. & Ratna,B. R. (2005) Nat. Mater. 4, 167–172.

91. Chen, J., Reed, M. A., Rawlett, A. M. & Tour,J. M. (1999) Science 286, 1550–1552.

92. Remacle, F., Heath, J. R. & Levine, R. D. (2005)Proc. Natl. Acad. Sci. USA 102, 5653–5658.

93. Troisi, A., Nitzan, A. & Ratner, M. A. (2003) J.Chem. Phys. 118, 5782–5788.


Documents

Molecular Electronics