REVISTA MEXICANA DE FÍSICA 47 SUPLEl\U:NTO 1. 59-6~ MARZO 2001

Molecular wires as rnesoscopic systerns: sirnilarities and sorne irnportantdifferences

Vladimiro MujicaEscuela de Química, Facultad de Ciencias, Universidad Central (le Venezuela

Apartado 47/02, Caracas J020A, Venezuela

Recibido el 29 de febrero de 2000; aceptado el 31 de octubre de 2000

Wercview sorne of the cornrnon features encountered in the descriplion of Ihe current tlow through a single rnolecule ane!othcr rnesoscopicsysterns.Attention is also drawn to sorne important differences due to the nature of the molecular clcctronic structure. A oumber of currentresearch problems on Coulomb effects in the f1eldof molecul<lrwires are also discllssed.

K('ywords: Molecular wires; e1eclronictransport; Coulomb blockadc

Desde el punto de vista del transporte electrónico se revisan elementos comunes al transporte electrónico a través de una sola moléculay otros sistemas mesoscópicos. También se establece esta comparación considerando la naturaleza de la estructura electrónica molecular.Además. se discuten varios problemas de investigacionde frontera sobre efectos Coulómbicos en el campo de los alambres moleculares.

f)e.Kriptores: Alambres moleculares; transporte electrónico; bloqueo ('oulómbico

I'ACS: 6Ró5.La: 73.50.-h; 73.23.Hk

1. Introduetion

This anicle aims to review sorne of the relevant ideas ahoutIhe physics of transport in mesoscopic systems that havefounJ a very important field of application in the descriptionof electronic Iransfer through individual molecules or mole-cular wires, At the same time, we shall point to some 01'thc1110stimportant differences hetween molecular and electros-talie junctions that arise because of the specific nature of themetal-molecule interface near to an electrodc. Emphasis willalso he made in stressing Ihc connection hetween the con-cept 01'conductance in a wire and the rate of intramolecularelectron transfer and, more gencralIy, with the study ol' singleelectron events in molecules. We will also consider some 01'the eurrent researeh problems in the fielJ, partieularly thosehaving lo tia with the metal.molecule interface and Ihe inclu-sion of Coulomb effects 00 the description of the transporto

The simplest mesoscopic circuit involving a molecularwire consisls of a molecule coupled to two or more nanoe-lcclrodes, or eontacts. which in turn are connected to an ex-lernal vollage source. This basic circuit, where charge andenergy transport occurs. has been the subject of a very inten-sive research e1'fort in the last ten years in which we have heencOllsiderahly involved. It can he used as a model for molecu-lar illlaging via the Scanning Tunneling Microscope (STM),hreak junctions and molecular conductance [1-321.

A remarkahle fealure 01' the research erfort in the me-soscopic domain is tha! it has hrought up the exislence ol"a comlllon theoretical framework for transport in vcry diffc-rent syslems: quanlum deiS, lunnclingjunctions. clcctrostalicIlano-constraints, STM and AFM imaging, and break junc-tions [1-32,36]. Coneepts like eonduetanee, once only aseri-hect lo hulk matter can now be extended to a single molccule.

revealing a deep connection hetween the models used in in-tramoleeular eleetron transler (ET) [35J and those 1m trans-par! [16].

Several experimental hrcakthroughs that have taken pla-ce during the last fiftecn years are responsible for our pre-sent capabilities to directly study processes and phenomenaat Ihe molecular scale in condensed matter. The advent oflocal probes wilh atomic resolution Iike Ihe Scanning Tunne-Iing Microscope and the Atomic Force Microscopes, togetherwilh the use al' lilhographic techniques have made possible tofahricate and study struetures whose dimensions are smallerthan the mean Iree palh 01 an eleetron [32,33].

2, Length sea les in mesoseopie systems

The word mesoseopie was coined by van Kampen [36], torefer to a syslel11whose size is intermediale hetwecn micros-copie and macroscopic. Statistical physics is concerned withlhe properties of macroscopic systems in the thcnnodynarniclimil. where hoth the numhcr ofparticles f.l, and the volume01 the system V, tenJ to infinity while the partic!e densityn = N IF is kepl constan!. Microscopic systcms are studiedeither using quantum mcchanics or semi-c1assical approxi-malions to il. Usually a system approaches macroscopic he-havior onee its size is l1luch larger than al! relevant correlationlegths ~ eharacterizing quanlum phenomena [32].

In its original meaning, mesoscopic systems were studieJto understand the macroscopic Iimit and how it is achievedby building up largcr amI larger c1usters to go from the mole-cule to the hulk. In current rescarch, many novel phenomenaexist that are intrinsic lOmesoseopic systems. An example isprovided hy small conducting systems ofthe kind considered

611 VLA[)IMIRO MUJICA

If an obslacle, ror instance a moleeule, is inserted ioto(he channel, lhe transmission probahility T becomes smallerIhan one and the conductance is reduced aceordingly lo

Landauer formalislll in the simple version presenled hereaSSUlllCS Lhat ooth Ihe lempcrature ami the voltage are verysmall and there are no incoherenl processes involved in Ihetransport. These constraints can he released in generalizedversions 01' ¡he theory [.12, 43J.

The resislancc associated 10 (3) can he written as [311:

(4)

(3 ).,e-

'J = -T.. rrh

l' = rrh (1 + 1 - T) = rrh (1+ !!.)c2 T el, T'

where R is the reflection coefllcient. Equalion (4) makes itexplicit Ihat in the simple case con sude red here the tOlal re-sistance associated to a channel consists of a contact and amolecular resistance.

A very convenienl way lo extcnd Landauer formalism tomolecular circuils is the use of scattering theory [6,7]. Usingthis formalism we have derived an expressiotl ror lhe conduc-tance lhat can he used in bOlh Ihe linear ami non linear voltageregimes alllllhat include the elfeel ol' the rcservoirs in a morecomplete fashiotl. In this model the conduclance for a circuitconsisting of 1wo eleclrolles and a molecular wire aLlached 10Ihe elcctrodcs al sites 1 and N Ihrough a chemisortive bondis givell hy [11):

3. Lalldaucr Iransmission model 01'conduclan-ce and ils exlensions lo molecular wires

herc which silow cohcecot quantum transport; that ¡s, an clcc-¡ron can propagalc across lhe whole syslem withall! inclasticscatlcring lhcrchy prcscrving rhase mcmory l321.

Roughly spcaking, ror a systcm of size L, thefe are fourimportanl Icngths. in ¡ncrcasing magnitudc, lhat charactcrizcthe sySICIll and define Ihe typc of lransport: lhe Fcrmi wa-vclcnglh '\F' lhe elcctronic mean free path le. lhe coherenccIcnglh ~ and Ihe localizalion Icnglh LcP0 Por L hctwccn Arami L{,'J lhe systclll is considcrcd to hc mcsoscopic and lhetransporllllcchanism goes [mm ballistic to diffusivc and lhenlocalizcd. dcpcnding on whcthcr AF ::; L ::; le; le :S L :::;€01' ~ :s L :S L1>' respectively. In aH these rcgimes. transponmust he dcscrihcd quantum mechanically. For L > L¡p' lhesyslem is considered 10 he macroscopic and Bollzmann equa-¡ion is a good approximation Lo lhe transport equation [32].

From sil.c considcrations only, it is clear that transportthrough a molccule will involve the three transporl mecha-nisms. But a Illolccule has an clectronic structurc 01' its ownthaL must he taken ¡nto account for the dcscription 01' Ihe cu-rren!. This Illeans that considering eleclron lransport throughan individual mnleculc hrings in a much richer structure hc-cause 01' tile discrcle naturc 01' Lhe energy spectrum and theract IhaL upon charging a molccu1c virtually hecomes a newchcmical spccies [1G,20-28].

This i~Ihe conl!uctancc 01' an ideal hallistic channcl withlransmission cqual to unity. The potenlial drop, associatedwith Ihe resistancc ,. :::::y-I, occurs al the cOllnections lothe rcscrvoir. Therefore. (2) specifies a limiling value nI' IhecOlllacl resis1ancc lhat is kllown as Sharvin limit [411.

wlu:re .i is the net currenl flow anJ V is lhe external vo11a-ge. Por an ideal conducting channel, wilh no irregularities orscaltering Illcchanisllls along ils lenglh, anJ wilh Ihe addilio-nal asslllllplion that lhe IUhe is narrow enough so that onlythe lowest 01' the lransverse eigenstates in lhe channel has itsenergy helow lhe Pertni level 01' Ihe conlacls, Ihe resultingconductance is given hy

I.andaucr 132, 37-40] prohably advanccd lhe mosl inllucntialwmk in the study 01' chargc Iransport in mesoscopie syslelll.The hasic idea in Landauers approach is to associatc eon-ducLance with lransmission through the intereleeLrode region.This approach is especially suitahle l'or mesoscopic lransportalld rcquircs the use 01' the full quanlum mechanical maehi-!lcr)' sincc no\\' the carriers can have a coherent history wilhinthe sample.

Conductance . .'/, is dellned hy the simple relation

(6)

(5)

(7)\<!

\~(.)= '\' _1,_,-J,. Lc-_,"O'

l' - ...."

with H lhe molecular Hamiltonian. 1/.1 and 11N the intcrfa-cing orhilals on the len and righL ends 01' the wire, EF IheFcnni encrgy,l} the ellergy variable, ~('/) representing Ihefull self-cnergy tcrm that conlains the effect of (he electro-des and 'ó'J(N) lhe imaginary part orthe self energy (speclraldellsity) associated with the end sitcs and their coupling lo Ihereser\'oirs. For elccLmdes descrihed hy Ilon-inleracting stalesthe sclf el1crgy for sitc l is dellned by

whcrc lhe Sllm mns ovc!' Lhe intinite slates in Ihe reservoir andFII, is the cOllpling hetwccn state 1 in the wire and lhe p-thstale in the reservo ir. A similar expression holds rol' si te N .

Equatioll (5) has heen llscd hy us anO other re-searchcrs to compute Ihe conductance 01' real molecu~les 11:3-1;).18, l!).,l-l~WL to sludy nonlinear cffects and (oinclllde Ihe local variatioll of lhe eXlernal t1eld Ihrough Ihe useof a self-consislcllt pmcedurc 10 solve Poisson ami Schrbdin-gel' equations silllultaneously. Grcen's fUllctiolls lechniques01' the type associaled wilh (5) constitutc a powerful 1001 to

(1)

(2)

( _ J.J - V'

.,e-r¡ =-.. ¡rJ¡

Ni'I'. Atex. F,~\.. 47 SI (2001) 59-63

SIMULATION OF MOl.ECULAR mANSITIONS USING CLASSICAL TRAJECTORIES 61

-l. Cllullllllh drccts amI currcn! rcsarchprohlcllls

\vilh fl()(l\) hcing the density nI"states in the Illctallic clcctro-des. Thesc two cqllatiolls exhihil the close COllllcclion hctwe-en lhe underlying physical Illechanisllls involved in the Iwophen()111ena.

where Tf)A is an elfeclivc clectronic coupling hctwcen IhedOllor (O) amI acccptor (A) sites and (JFC is a Frank-Condonwcig.hteJ dcnsily 01'slates. Equatioll (8) should he comparedwil!l lhal ror the conductancc in lhe ¡¡mil (Ir low voltagc amitcmperalure [6,7)

4.1. Where is lhe Fermi energy?

4.2. ~lany.l){)d~' eff{'cts cm the currl'nt and Coulomhhhll'kade

The energy difference helwcen the Fermi energy of a me-tal and lhe occupied levels of the lype 01'organic moleculesthal are cOllll11onlyused as wires is typically of several eYwhen the molecule is not in conlact wilh the metal surface.Under the conditions of current mcasurcment expcriments, itis clear that Ihe molecular levels are going lo he inllllcncedhy Ihc presence 01' the metal ami thal one should speak ofIhe comhined mctal-molecule states. However, speclroscopicevidence and lhe cxpcriments themselves. suggest that theIlloleculc preserves. to a large extenl, its identity ano that lheIllodilkalion 01' lhe molecular levcls occurs mostly lhroughchargc transfcr fmm nr lo the melal. This leads lo a situalionin which the Fermi level ¡¡es somcwhat belween lhe originalHaMO allll LUMa molecular levels. whieh. to a first appro-ximation Ihat neglects internal displacemenls, scem 10 havecxpericnccd a quasi-rigid energy shift [mm their original po-sition 118, -!D].

Given lhe resonanl nalure of the Iransport phenornenon.slTlallchangcs in the position of the Fcrmi (injcclion) encrgyIranslates inlo huge changes in the conductancc. Any quanli-lative compulation of lhe currenl would then requirc a muchIllore accurale calculation of Ihe posilion ofthc Fermi energy.This. is, in tmn. a difficlIlt and suhlle quantul11 chemical pro-hlcm mainly hccausc olle nceds and accurale rcpresenlalion01'"melal" ano "ll1olcculc" eleclrons.

Gcnera1ly speaking. Ihe Fenni energy is going to be si.milar 10 Ihat of lhe hare melal ami one could expecl that acluster calculation involving Ihe \Vire and a numher of mc-lal aloll1s would render a valuc that converges rapidly as afUllclion 01'thc cluster size. Such cxpectalion is indeed horneout by Ihe cakulations reported 011Ref. IR using a Hmtree-Fock model for a clusler of gold and 0', o' xylyl dithiol andhelll.cnc-I ,4-dithiol.lt is found that the energy ofthe HOMOlevel varies fmm -G.I.ID7 eY ror a cluster with 3 goJd atomslo -(j ..18DO cV ror 1:3 gold ¡¡IOmS, which compares rcasona-bly \Vel1\vith Ihc Fcrllli cllcrgy 01'gold.

E\'en lhnugh ah ;'1;1;0 Harlrec-Pock calculations repre-senl a substantial improvelllent over semi.empirical melhodslike l!lose reported on references 14 and 15, they are unablelo represenl adeqllately the aninily levels (virtual orbitals) ofIhe cluster "nti yield a 100 large value for the HaMO-LUMa.011 Ihe othe!"hand, any linilc cluster calculation fails to repre-sent appropriately Ihe self-encrgy contribulion 01' the melal.\Ve are currenll)' working in de\'eloping a density functionalapproach to Ihe cluster calculation combined with a repre.senlalion of lhe metal as a scmi-inllnitc surface with perindichOllndary (Olltiilions in thc two directions perpendicular 10lhe ahsorption axis ¡!jOI.

This is a \'Cry important aspect on which much work is nce-dedo The currenl fnnnalism is hased on a single-eleclron vi('w

(9 )

(X)

describe an open system wbere Ihe IOlal HamilIonian has be-en partitioncd so that lhe rcscrvoirs modify lhe molecular Ha-millonian through Ihe sclf-cncrgy lerm Ihal represen! Ihe ill-fluence of Ihe contacts. This upproach takcs its mosl generallonn in the Keldysh fonnalism [31,47,481.

Thc use of lile scatlcring dcscription pcnnils a natural in.Icrprctation 01' molecular orhitals. or dclocalizcd molcculmstnlcs as lhe analoguc of lhe transversal channcls in electros-latie junctions. Thc conductancc associatcd wilh a molecularchannel can be mucll lowcr than lhe correspondíng one fora hallistic channcl hccausc il dcpcnds 011lhe spccific nalurc01' lile chcmical bono hctwccn lhe wire and Ihe melal surfa-cc. This corrcsponds lo Ihe non resollan! rcgimc 17. 28}. Thcresonant regimc. on Ihc other hanJ. is cntirely analogous lolhe hallistic oehavior anJ the wire-eleclrode conductance he.comes equal to the Sharvin conductance lllulliplied hy (henllmher 01' degenerale molecular channcls. These ideas nrefundamental to Ihe underslanding of the I-V curves ill\'ol.ving molecular wires 17.28, .tGl,

Another rewarding aspect of Ihe use of the scattering I"or-malisll1 is thal il permils estahlishing a comlllon frameworkfor lhe prohlems 01'intramolecular ET rcactions and l11olecll.lar condllctance at Imv hias vollage. The main dilTerence islhat Ihe eleclronic conlinua in the conductance prohlcm arercplaced hy a vihronic continuum in the ET prohlern. This ap.proach pcrrnils recovering thc convcntional Marcus expres-sion for Ihe non adiahatic. intrarnolecular ET rate 11 G, ;151

Some 01' Ihe mns.l interesling aspects of currenl rcscarch inmolecular \Vires are related to (he propl..'r indusion of Cou-lomb elfec[s in lhe descriplion 01"Ihe curren!. They have lo dowilh lhe very nalllre of the lllelal-lllo1t:cllle inlcrl~lcc ano also'!"'ilhIhe raet that Ihe Illolecule ilself has ;:In cleclfOnic struc-tUTe01'ils own thal can hardly he Illotlclcd as a free-elcctrongas.

Rel'.Mo. rú. 47 SI (200 I ) 59-b~

62 VLADlMIRO MUJleA

or clcctronlransport. which essentially assurncs (haL the clcc-lron tllllllcling Ihrough (he moleculc cxpcricnccs a constant.frozcn pOlcntial. The validity of this approximation hinges0111he fael that clectron residence times are assurncd lo heshOrlcr than a typical nuclear v¡bration. which sccms lo be arcasonablc lirsl approximation. Howevcr, corrc)ation cffectsliJ..c ¡hose involvcd in the intriguing possibility oí" molecularCoulomh hlockade would prccludc Ihe use orthe one-particlepielurc HS. 51].

Thc dcscriptioll 01' Coulomb hlockade in a lllo1cculc\voultl. al the vcry Icast. rcquirc the consitlcration of lunnc-Iing through slructurcs with variable numbcr of clcctrons. 011

Ihe nthcr hand. chargc transfcr in a mctal-molecule interfacemost likely invoives only a frac(ion of (he clectronic charge,so a eaklllalion wi(h an in(eger nurnher of electrons wouldnol nceessarily he appropriate.

In this connection. the use al' density functional theoryfor the description 01' hoth the electronic s(ruc(ure and (he cu-rrent could be very appealing. Density funelional lheory is,in principIe. a very general way (o descrihe ho(h lhe eleclro-nic strllcture 01"solids and molecllles and also offers a directand simple way (o includc an ex(ernal eleclrostic potenlial 01"Ihe kind lIscd in a molecular circui( [52-541. Unforluna(ely,a cOllsistenl way to compute the current in the spirit 01' a hy-drodynamic analogy (o electronic (ransport is stilllacking alt-hough sOllle work in this direction has heen done [551. Also(JI' importance are some references on time-dependen( densi(yI"unclional (heory (52-56].

\Ve have also pursued a less amhitious approach that ke-cps Ihe e1ectron prohlem at Ihe Hartree-Fock level hut allowstúr ti fractional occupation numher that takes into accouot (het.:l1argc Iransferred from or to the melal. The occupation isself-collsislcntly dClermined using the self-encrgics and a ki-IlClic cqualion ror the steady-state current 1571.

4,3. Elct'trnstatic spatial profile

Much rcla(ed to Ihe considerations aboye. is the issue 01' whatis (he appropriate description of the elec(rosta(ic potentialacross lhe molecular junction. Thc solution lo Poisson equa-(ion in the ahsence of the molecule is a potcn(ial rampo Inthe presence of the molecule (wo important cffccts arise thatmodify lhe ramp picture: First is (he image poten ti al feh byt.:harges c10se to the me(al surfacc. Second is the ¡nfluencethal tile applied potential has on the molecular charge distri-

1. ~1.C. Pcuy. f\1.R. Brycc, and D. Bloor./lIlrodllctioll lo Molecu-lar Ekclmllic.\'. (Oxford University Press, Ncw York. 19(5)

2. iilol('(,II/ar l:'/aITOllics. ediled by J. Jortner amI M.A. Ralncr.(Blackwell. Lundon, 1997).

:1 Molecular Ekctronic.{: Science (lmi Tt'Cllllology. cdílcd hy A.Av;ram and M.A. Ratner, (Ann. NY. Acad. Sci .. Ncw York.199X) Vol. X52.

bution, which, aCl'ording lo Maxwell cquations, modifics ina self-consistent way Ihe pOlential.

This is a suhject in which so me progress has heen madein the lasl years [ 19.491. hut lhat requires much more effortto recancile lhe IwO. arparently differenl. dcscriptions 01' (hepotential. Its imporlance slems from the fac( that. togctherwith the accurate detenninalion of the Fermi energy. il cons-titutes the most important factor in determining the nature ofthe Intensity- Vohagc curves.

The main result that emerges from thcsc studics is tha(depending on (he pattcrn 01"chargc localization on the mo-leculc tlle eleclroslalic potcnlial changes drastically. It goesfrom (he one corrcsponding (o a point charge. which rcsel1l-hles a simple 1/,. imagc po(cntial, lO a function similar lOa double layer poten(ial of (he kind ohserved in the vicinity01' an elcc(rolic. Thc spatial profile can be charac(erizcd hya voltage division factor 1.19) that charac(erizes caeh mClal-moleeule interface thal in turn depends strongly on the localaspects 01"(he chcmical hondo Por identical contacls. the po-tential drops symlllctrically Oll ho(h interfaces, whereas fordifferenl contat.:ts, say olle fOl"lning a strong chemisorptivebond and the otiler i.I weak one, most of the poten(ial dropoccurs in Ihe late!" interface where lhe resistance is loca(ed.

5. Conc!usions all(l linal remarks

The stlldy of transpon in molecular wires is part 01"the fas-cinating field of single molecule response. As new experi-men(al tcchniqlles ¡¡re dcvelopcd (ha( allO\\/ prohing indivi-dual Illolecllles. ncw l]lIestiolls arise that demand a close in-teraclioll betwccll thcory ami experimcnt.

Not Illentiollcd in lhis short review are issllcs regarding(he illlporlance 01"electron-phonon interaction in molecularconductancc [581 alHl Ihe !lew and exciting developmen(sconcerning transport in magnetic domains ami spin-sensitivechiralmolecllles [591.

Final1y, the real technological challenge lies in lIsing (heproperties of molecular wires as controllable elements ror de-vice designo Much cfl"ort is curren(ly under way in that direc-(ion.

Acknowledgments

Collahoralioll hetwecn Universidad Central de Venezuela andNorthwestcrn Universi(y is supported by NSF-Conicit.

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