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Inelastic electron transport through molecular junctions Natalya A. Zimbovskaya Department of Physics and Electronics, University of Puerto Rico-Humacao, CUH Station, Humacao, PR 00791, USA and Institute for Functional Nanomaterials, University of Puerto Rico, San Juan, PR 00931, USA (Dated: January 24, 2013) 3 1 Currently, molecular tunnel junctions are recognized as important active elements of various 0 nanodevices. This gives a strong motivation to study physical mechanisms controlling electron 2 transport through molecules. Electron motion through a molecular bridge is always somewhat affected by the environment, and the interactions with the invironment could change the energy n of the traveling electron. Under certain conditions these inelastic effects may significantly modify a J electron transport characteristics. In the present work we describe inelastic and dissipative effects intheelectron transport occurring duetothemolecularbridgevibrationsandstochastic thermally 3 activated ion motions. We intentionally use simple models and computational techniques to keep 2 a reader focused on the physics of inelastic electron transport in molecular tunnel junctions. We consider electron-vibron interactions and their manifestations in the inelastic tunneling spectra, ] l polaronic effects and dissipative electron transport. Also, we briefly discuss long-range electron l a transferreactionsin macromolecules andtheirrelation totheelectron transport throughmolecular h junctions. - s e Keywords: moleculartunneljunctions, electron-phononinteractions,inelasticelectron m tunneling spectrum, polaronic conduction, electron transfer reactions. . t a PACSnumbers: 71.18.+y,71.20-b,72.55+s m - Contents d activitiesonmolecularelectronicsreflecttheconvergence n of two trends in the fabrication of nanodevices, namely, o I. Introduction 1 the top-down device miniaturization through the litho- c graphic methods and bottom-up device manufacturing [ II. Coherent transport 2 through atom-engineering and self-assembly approach. 1 The key element and basic building block of molecular v III. Buttiker model for inelastic transport 5 electronics is a junction including two electrodes (leads) 9 6 linked by a molecule, as schematically shown in the Fig. 5 IV. Vibration-induced inelastic effects 6 1. Usually, the electrodes are microscopic large but 5 macroscopic small contacts which could be connected to 1. V. Dissipative transport 9 abatteryto providethe bias voltageacrossthe junction. 0 Suchajunctionmaybetreatedasaquantumdotcoupled 3 VI. Polaron effects: Hysteresis, switching tothechargereservoirs. Thediscretecharacterofenergy 1 and negative differential resistance 11 levelsonthedot(molecule)iscombinedwithnearlycon- : v tinuousenergyspectraonthereservoirs(leads)occurring Xi VII. Molecular junction conductance and due to their comparatively large size. long range electron transfer reactions 12 r When the voltage is applied, an electric current flows a VIII. Concluding remarks 15 through the junction. Successful transport experiments withmolecularjunctions (Ho2002,Lortscher2007,Park References 16 2000, Poot 2006, Reicher 2002, Smit 2002, Yu 2004) confirm their significance as active elements in nanodevices. These include applications as rectifiers (molecular diodes), field effect transistors (molecular triods), I. Introduction switches, memory elements and sensors. Also, these experiments emphasize the importance of thorough analy- Molecular electronics is known to be one of the most sis of the physics underlying electron transport through promisingdevelopmentsinnanoelectronics,andthe past molecular junctions. Detailed understanding of the elec- decade has seen an extraordinary progress in this field tron transport at the molecular scale is a key step to (Aviram 2002, Cuniberti 2005, Nitzan 2001). Present future device operations. Theory of electron transport 2 result in polaronic conduction (Galperin 2005,Gutierrez 2005, Kubatkin 2003, Ryndyk 2008), and they are directly related to the junction heating (Segal 2003) and (cid:5) (cid:5) (cid:9) to some specific effects such as alterations in both shape (cid:4) (cid:8) (cid:2)(cid:3) (cid:6)(cid:7) ofthe moleculeandits positionwithrespectto the leads (Komeda 2002, Mitra 2004, Stipe 1999). The effects of electron-phonon interactions may be manifested in the inelastic tunneling spectrum which presents the second derivative of the current in the junction d2I/dV ver- m (cid:2) sus the applied voltage V. The inelastic electron tunneling microscopy has proven to be a valuable method m for identification of molecular species within the con- (cid:1) ductionregion,especiallywhenemployedincombination withscanningtunnelingmicroscopyand/orspectroscopy (Galperin 2004). Inelasticity in the electron transport through molec- (cid:1) ular junctions is closely related to the dephasing effects. One may say that incoherent electron transport always includes an inelastic contribution with the possible exception of the low temperature range. FIG. 1: (Color online) Top panel: Schematic drawing of a junction including twoelectrodes and a molecule in between. The general approach to theoretically analyze electron Bottom panel: When the voltage is applied across the junc- transport through molecular junctions in the presence tion electrochemical potentials µL and µR differ, and the of dissipative/phase-breaking processes in both elec- conduction window opens up. tronic and nuclear degrees of freedom is based on the advanced formalisms ((Segal 2000, Skourtis 1995, Wingreen 1989,1993). These microscopic computational approaches have the advantages of being capable of pro- in molecular junctions is being developed in the last two viding the detailed dynamics information. However, decades, and main transport mechanisms are currently while considering stationary electron transport through elucidated in general terms (Datta 2005, Imry 1999). molecular junctions, one may turn to less time consum- However,progressofexperimentalcapabilitiesinthefield ing approach based on the scattering matrix formalism ofmolecularelectronicsbringsnewtheoreticalchallenges (Buttiker 1986, Li 2001), as discussed below. causing further development of the theory. Speakingoftransportmechanisms,itisusefultomake adistinctionbetweentheelasticelectrontransportwhen II. Coherent transport theelectronenergyremainsthesameasittravelsthrough the junction, and inelastic transport processes when the To better show the effects of dissipation/dephasing electronundergoesenergychangesduetoitsinteractions on the electron transport through molecular junctions with the environment. There are several kinds of pro- it seems reasonable to start from the case where these cesses bringing inelasticity in the electron transport in effects do not occur. So, we consider a molecule (pre- mesoscopicsystemsincluding molecularjunctions. Chief sented as a set of energy levels) placed in between two among these are electron-electron and electron-phonon electrodes with nearly continuous energy spectra. While scattering processes. These processes may bring signif- there is no bias voltage applied across the junction, the icant inelastic effects modifying transport properties of latter remains in equilibrium characterized by the equi- moleculardevicesandcharging,desorptionandchemical libriumFermienergy E , andthereisnocurrentflowing F reactions as well. To keep this Chapter at a reasonable throughit. Whenthebiasvoltageisapplied,itkeepsthe length we concentrate on the inelastic effects originating left and right electrodes at different electrochemical po- from electron-phonon interactions. tentials µ and µ . Then the electric current appears L R Inpracticalmolecularjunctions the electrontransport in the junction, and molecular energy levels located in isalwaysaccompaniedbynuclearmotionsintheenviron- betweentheelectrochemicalpotentials µ and µ take L R ment. Therefore the conduction process is influenced by the major part in maintaining this current. Electrons the coupling between electronic and vibrational degrees from occupied molecular states tunnel to the electrodes of freedom. Nuclear motions underlie the interplay be- inaccordancewiththevoltagepolarity,andtheelectrons tween the coherent electron tunneling through the junc- from one electrode travel to another one using unoccu- tion and inelastic thermally assisted hopping transport piedmolecularlevelsasintermediatestatesfortunneling. (Nitzan 2001). Also, electron-phonon interactions may Usually,theelectrontransportinmolecularjunctionsoc- 3 curresviahighestoccupied(HOMO)andlowestunoccu- moleculetotheelectrodes,and G istheGreen’sfunction pied (LUMO) molecular orbitalswhich workas channels matrix for the molecule whose matrix elements between forelectrontransmission. Obviously,thecurrentthrough the molecular states i and j have the form: | | the junction depends on quality of contacts between the (cid:10) (cid:11) leads and the molecule ends. However, there also ex- Gij = iE H j . (5) | − | ist the limit for the conductance in the channels. As Here, H is the molecul(cid:10)ar Hamilto(cid:11)nian including self- was theoretically shown (Landauer 1970), the maximum energy parts Σ . conductance of a channel with a single spin degenerate L,R When a molecule contacts the surface of electrodes, energy level equals: this results in charge transfer between the molecule and e2 the electrodes, and in modification of the molecule en- G = =(25.8kΩ)−1 (1) 0 π~ ergy states due to redistribution of the electrostatic po- where e istheelectronchargeand ~ isthePlanck’scon- tentialwithinthemolecule. Besides,theexternalvoltage applied across the junction brings additional changes to stant. This isa truly remarkableresultforitprovesthat the minimum resistance R =G−1 of a molecular junc- the electrostatic potential further modifying molecular 0 0 orbitals. The coupling of the molecule to the leads may tioncannotbecomezero. Inanotherwordsonenevercan also depend on the voltage distribution. So, generally, short-circuit a device operating with quantum channels. the electron transmission T inserted in the Eq. (3) and Also, the expression(1) shows that the conductance is a electrochemical potentials µ depend on the electro- quantized quantity. L,R static potential distribution in the system. To find the Conductance g in practical quantum channels associ- correctdistributionoftheelectricfieldinsidethejunction ated with molecular orbitals can take on values signif- one must simultaneously solve the Schrodinger equation icantly smaller that G , depending on the delocaliza- 0 for the molecule and the Poissonequationfor the charge tion in the molecular orbitals participating in the elec- density, following a self-consistentconvergingprocedure. tron transport. In molecular junctions it also strongly This is a nontrivial and complicated task, and signifi- depends on the molecule coupling to the leads (quality canteffortwasappliedtostudytheeffectofelectrostatic of contacts), as was remarked before. The total resistance r = g−1 includes contributions from contact and potential distribution on the electron transport through molecules (Damle 2001, Di Ventra 2000, Galperin 2006, molecularresistances,andcouldbewrittenas(Wingreen Lang 2000, Mujica 2000, Xue 2001,2003,2004). Here we 1993): put these detailed considerations aside, and we use the 1 1 T simplified expression for the electrochemical potentials: r = 1+ − . (2) G T 0 (cid:18) (cid:19) µ =E +η eV; µ =E (1 η)eV (6) L F R F Here, T is the electron transmission coefficient which | | − − | | generally takes on values less than unity. where the parameter η indicates how the bias voltage The general expression for the electric current flowing is distributed between the electrodes. Also, we assume through the molecular junction could be obtained if one thatinsidethe moleculethe externalelectrostaticfieldis calculates the total probability for an electron to travel screened due to the charge redistribution, and the elec- betweentwoelectrodesatacertaintunnelenergy E and tron transmission is not sensitive to the changes in the then integrates the latter over the whole energy range voltage V. Although very simple, this model allows to (Datta 1995). This results in the well known Landauer analyzethemaincharacteristicsoftheelectrontransport expression: throughmolecularjunctions. Withinthismodelonemay write down the following expression for the self-energy e I = π~ T(E) fL(E)−fR(E) dE. (3) parts ((D’Amato 1990): Z (cid:2) (cid:3) τ∗ τ Here, fL,R(E) are Fermi distribution functions for the (Σ ) = ik,β kj,β . (7) β ij electrodes with chemical potentials µ , respectively. E ǫ +is L,R k − k,β The values of µ differ from the equilibrium energy X L,R EF , and they are determined with the voltage distri- Here, β ∈ L,R, τik,β is the coupling strength between bution inside the system. Assuming that the coherent “i”-thmolecularstateand “k”-thstateontheleft/right tunneling predominates in the electron transport, the lead, ǫk,β are the energy levels on the electrodes, and s electron transmission function is given by (Datta 1997, is a positive infinitesimal parameter. Assuming that the Samanta 1996): molecule is reduced to a single orbital with the energy E (a single-site bridge), the Green’s function accepts T(E)=2Tr ∆ G∆ G+ (4) 0 L R the form: { } where the matrices ∆L,R represent imaginary parts of 1 G(E)= . (8) self-energy terms ΣL,R describing the coupling of the E E0 ΣL ΣR − − − 4 FIG. 2: (Color online) Coherent electron transmission (left FIG. 3: (Color online) Coherent electron transmission (left panel) and current (right panel) versus bias voltage applied panel) and current (right panel) through a junction with a across a molecular junction where the molecule is simulated fiveelectronicstatesbridge. Thecurvesareplottedfor ∆L= by a single electronic state. The curves are plotted assuming ∆R =0.1eV, b=0.3eV, E0=−0.5eV, T =30K. ∆L =∆R =0.1eV, E0=−0.5eV, T =30K. near E =E . Asthecouplingstrengthens,thetransmis- 0 Accordingly,inthiscaseonemaysimplifytheexpression sion minimum values increase. Now, the current voltage (4) for the electron transmission: curvesexhibitasequenceofsteps. Thelongeristhechain the more energy levels it possesses, and the more steps 4∆ ∆ L R T(E)= . (9) appear in the I V curves. (E E0)2+(∆L+∆R)2 − − The second example concerns the electron transport To elucidate the main features of electron transport through a carbon chain placed between copper elec- through molecular junctions we consider a few exam- trodes. In this case, as well as for practical molecules a ples. In the first example we mimic a molecule as a one- preliminary step in transport calculations is to compute dimensional chain consisting of N identical hydrogen- the relevant molecular energy levels and wave functions. like atoms with nearest neighbors interaction. We as- Usually, these computations are carried out employing sume that there is one state per isolated atom with the quantumchemistrysoftwarepackages(e.g. GAUSSIAN) energy E , andthatthecouplingbetweenthe neighbor- or density-functional based software. Also, a proper 0 ing sites in the chain is characterized by the parameter treatment of the molecular coupling to the electrodes is b. Such a model was theoretically analyzed by D’Amato necessaryfor itbringschangesinto the molecularenergy and Pastawskiand in some other works (see e.g. Mujica states. For this purpose one may use the concept of an 1994). BasingontheEqs. (4),(5),itcouldbeshownthat “extended molecule” proposed by Xue, Datta and Rat- forasingle-sitechain (N =1) the electrontransmission ner (Xue 2001). The point of this concept is that only reveals a well distinguished peak at E = E ,, shown in a few atoms on the surface of the metallic electrode are 0 theFig. 2Theheightofthispeakisdeterminedwiththe significantly disturbed when the molecule is attached to coupling of the bridge site to the electrodes. The peak the latter. These atoms are located in the immediate inthe electrontransmissionarisesbecause the molecular vicinity of the molecule end. Therefore one may form orbital E =E worksasthe channel/bridgeforelectron a system consisting of the molecule itself and the atoms 0 transportbetweentheleads. Similarpeaksappearinthe from the electrode surfaces perturbed by the molecular conductance g = dI/dV. Assuming the symmetric volt- presence. This system is called the extended molecule age distribution (η =1/2) the peak in the conductance and treated as such while computing the molecular or- islocatedat V = 2E . Asforthecurrentvoltagechar- bitals. In the considered example the extended molecule 0 ± acteristics,theydisplaystep-likeshapeswiththestepsat included four copper atoms at the each side of the car- V = 2E . Whenthe chainincludesseveralsitesweob- bon chain. The results for electron transport are shown 0 ± tainasetofstates(orbitals)forourbridgeinsteadofthe intheFig. 4. Again,weobserveacomb-likeshapeofthe single state E =E , and their number equals the num- electron transmission corresponding to the set of trans- 0 berofsitesinthechain. Allthesestatesarethechannels portchannelsprovidedbythemolecularorbitals,andthe for the electron transport. Correspondingly, the trans- stepwise I V curveoriginatingfromthe latter. Trans- − mission reveals a set of peaks as presented in the Fig. 3. port calculations similar to those described above were The peaks are located within the energy range with the repeatedly carried out in the last two decades for vari- width 4b around E = E . The coupling of the chain ouspracticalmolecules(seee.g. Xue2001,2003,Galperin 0 endstotheelectrodesaffectsthetransmission,especially 2006, Zimbovskaya 2002,2003,2008). 5 (cid:1) (cid:1) (cid:1) (cid:4) (cid:4)(cid:1) (cid:2) (cid:4) (cid:2) (cid:3) (cid:8) (cid:3) (cid:2)(cid:3) (cid:7) (cid:1)(cid:1) (cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:5)(cid:6) (cid:4) (cid:2) (cid:2) (cid:3) (cid:3) (cid:1)(cid:1) (cid:1) (cid:1)(cid:1) (cid:1) (cid:5)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6) (cid:5) (cid:7)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6) (cid:7) (cid:5)(cid:2)(cid:9)(cid:2)(cid:10)(cid:11)(cid:12)(cid:6)(cid:10) FIG. 5: (Color online) Schematic drawing illustrating inelas- FIG. 4: (Color online) Coherent electron transmission (left ticelectrontransportthroughamolecularjunctionwithinthe panel)and current(right panel)throughacarbon chain cou- Buttikermodel. pled to the copperleads at T =30K. i,k label the channels for the transport: 1 i, k 4. III. Buttiker model for inelastic transport ≤ ≤ J′ = T J . (10) i ik k An important advantage of the phenomenological k X model for the incoherent/inelastic quantum transport Thecoefficients T intheselinearcombinationsarema- proposed by Buttiker (Buttiker 1986) is that this model ik trix elements of the transmission matrix which are re- couldbeeasilyadaptedtoanalyzevariousinelasticeffects latedtotheelementsofthescatteringmatrix S, namely: in the electron transport through molecules (and some T = S 2. The matrix S expresses outgoing wave othermesoscopicsystems)avoidingcomplicatedandtime ik | ik| amplitudes b′,b′,a′,a′ in terms of the incident ones consuming advanced methods such as based on the non- 1 2 3 4 b ,b ,a ,a . To provide the charge conservation in the equilibrium Green’s functions formalism (NEGF). 1 2 3 4 system, the net currentin the channels 3 and 4 linking Here, we present the Buttiker model for a simple the systemwith the dephasingreservoirmustbe zero,so junction including two electrodes linked by a single-site we may write: molecular bridge. The bridge is attached to a phase- randomizing electron reservoir, as shown in the Fig. 5. J +J J′ J′ =0. (11) Electrons tunnel from the electrodes to the bridge and 3 4− 3− 4 vice versa via the channels 1 and 2. While on the The transmission for the quantum transport could be bridge,anelectroncouldbescatteredintothechannels 3 definedastheratiooftheparticlefluxoutgoingfromthe and 4 withacertainprobability ǫ. Suchelectronarrives systemandthatoneincomingtothelatter. SolvingEqs. at the reservoir where it undergoes inelastic scattering (10), (11) we obtain: accompanied by phase-breaking and then the reservoir reemits it back to the channels 3 and 4 with the same T(E)= J2′ =T + K1·K2 (12) 21 probability. So, within the Buttiker model the electron J1 2R transport through the junction is treated as the combi- where: nation of tunnelings through the barriers separating the molecule from the electrodes and interaction with the K =T +T ; K =T +T ; 1 31 41 2 23 24 phase-breaking electron reservoir coupled to the bridge R=T +T +T +T . (13) site. The key parameter of the model is the probability 33 44 43 34 ǫ which is closely related to the coupling strength be- Forthejunctionincludingthesingle-sitebridgethescat- tweenthe bridgesite andthe reservoir. When ǫ=0 the tering matrix S has the form (Buttiker 1986): reservoir is detached from the bridge, and the electron transportis completely coherentand elastic. Within the r +α2r αt t βt αβt r 1 2 1 2 1 1 2 opposite limit (ǫ = 1) electrons are certainly scattered 1 αt t r +α2r αβr t βt S =  1 2 2 1 1 2 2 . into the reservoir which results in the overall phase ran- Z βt1 αβr1t2 β2r1 αr1r2 α − domization and inelastic transport.  αβt1r2 βt2 αr1r2 α β2r2   −  WithintheButtikermodeltheparticlefluxesoutgoing  (14) from the junctions J′ could be presented as linear com- Here, Z = 1 α2r r , α = √1 ǫ, β = √ǫ and r i − 1 2 − 1,2 binations of the incoming fluxes J where the indexes and t are the amplitude reflection and transmission k 1,2 6 IV. Vibration-induced inelastic effects Interaction of electrons with molecular vibrations is knowntobeanimportantsourceofinelasticcontribution to the electron transport through molecules. Theoret- ical studies of vibrationally inelastic electron transport through molecules and other similar nanosystems (e.g. carbon nanotubes) were carried out over the past few yearsbyalargenumberofauthors(Cornaglia2004,2005, Donarini 2006, Egger 2008, Galperin 2007,Gutierrez 2006,Kushmerick2004,Mii2003,Rynduk2007,Siddiqui 2007, Tikhodeev 2004, Troisi 2006, Zazunov 2006, 2007, FIG.6: (Color online) Electron transmission (left panel) and Zimmerman 2008). Also, manifestations of the electron- current (right panel) computed within theButtikermodel at vibroninteractionswereexperimentallyobserved(Agrait various values of the dephasing parameter ǫ, namely: ǫ=0 2003, Djukic 2005, Lorente 2001, Qiu 2004, Repp 2005, (dotted lines), ǫ = 0.5 (dashed lines), and ǫ = 1 (solid Segal 2001, Smit 2004, Tsutsui 2006, Wang 2004, Wu lines). The curves are plotted assuming that the molecule is 2004, Zhitenev 2002). To analyze vibration induced ef- simulated by a single orbital with E0 = −0.5eV at ∆L = fectsontheelectrontransportthroughmolecularbridges, ∆R =0.1eV, T =30K. one must assume that molecular orbitals are coupled to the phonons describing vibrations. While on the bridge, electronsmayparticipateintheeventsgeneratedbytheir coefficients for the two barriers. Later the expressionfor interactions with vibrational phonons. These events inthis matrix suitable for the case of multi-site bridges in- volvevirtualphononemissionandabsorption. Forrather cludingseveralinelasticscattererswasderived(Li2001). strong electron-phonon interaction this leads to the ap- Assuming for certainty the charge flow from the left pearance of metastable electron levels which could par- to the right we may write down the following expression ticipate in the electron transport through the junctions, (Zimbovskaya 2005): bringing an inelastic component to the current. As a result, vibration induced features occur in the differential g(E)(1+α2)[g(E)(1+α2)+1 α2 T(E)= − (15) molecular conductance dI/dV and in the inelastic tun- 2 g(E)(1 α2)+1+α2 (cid:3) neling spectrum d2I/dV2. This was observedin the ex- − periments(seee.g. Qiu2004,Zhitenev2002). Sometimes where (cid:2) (cid:3) evencurrentvoltagecurvesthemselvesexhibitextrastep ∆L∆R originatingfromthe electron-vibroninteractions(Djukic g(E)=2 . (16) s(E E0)2+(∆L+∆R)2 2005). − Particular manifestations of electron-vibrionic effects Now, the electron transmission strongly depends on the in the transport characteristics are determined by the dephasing strength ǫ. As shown in the Fig. 6 coherent relation of three relevant energies. These are the cou- transmission (ǫ = 0) exhibits a sharp peak at E = E 0 pling strengths of the molecule to the electrodes ∆ , which gives a step-like shape to the volt-ampere curve, L,R the electron-phonon coupling strength λ and the ther- as was discussedin the previous section. In the presence malenergy kT (k is the Boltzmanconstant). When the of dephasing the peak gets eroded. When the ǫ value molecule is weakly coupled to the electrodes (∆ approaches 1 the I V curve becomes linear, corrobo- L,R − λ,~Ω) and the temperature is low (kT ∆ ), th≪e rating the ohmic law for the inelastic transport. L,R ≪ electron transfer through the junction may give rise to a Within the Buttiker’smodel ǫ is introducedas a phe- strong vibrational excitation, and one may expect pro- nomenological parameter whose relation to the micro- nounced vibrational resonance structure in the electron scopic characteristics of the dissipative processes affect- transmission to appear. Correspondingly, extra steps ing electron transport through molecular junctions re- shouldoccurinthe I V curves. Propertheoreticalcon- mains uncertain. To further advance this model one − sideration of the electron transport in this regime could shouldfindouthowtoexpress ǫ intermsoftherelevant becarriedoutwithintheapproachproposedbyWingreen microscopic characteristics for various transport mecha- et al (Wingreen 1989). Here, we employ a very simple nisms. Thisshouldopenthewayto makealinkbetween semi-quantitative approximationwhich, nevertheless,al- the phenomenological Buttiker model and NEGF. Such lowstoqualitativelydescribethisstructureincludingthe attempt was carried out in recent works (Zimbovskaya effectofhigherphononharmonicsonthetransportchar- 2005,2008)where the effect of stochastic nuclear motion acteristics. on the electron transport through molecules was analyzed. We consider a junction including a single-site bridge 7 FIG. 7: (Color online) The electron transmission versus energy (left panel) and the conductance versus voltage (right FIG. 8: (Color online) The inelastic electron tunneling spec- panel) for a junction with the molecular bridge simulated trum plotted against the bias voltage at ~Ω = 0.22eV, λ = by a single electronic state weakly coupled to the leads: 0.3eV. Left panel: ∆L = ∆R = 0.01eV, E0 = 0.5eV. ∆L = ∆R = 0.01eV, E0 = 0.5eV. Solid lines are plotted Right panel: ∆L = ∆R = 0.5eV, E0 = 0.5eV (solid line), assuming thatthebridgeis coupledtoasingle phononmode E0=−0.5eV (dashed line). (~Ω = 0.22eV, λ = 0.3eV). Dashed lines correspond to the coherent electron transport. which is coupled to a single vibrational mode with the ∆ ~Ω. This agrees with the results of the earlier frequency Ω. An electron on the bridge may virtually L,R ≪ theoreticalworks(Wingreen1989)aswellaswiththeex- absorb several phonons which results in the creation of periments (Qiu 2004, Zhitenev 2002). Phonon-induced a set of metastable states with the energies E = E˜ + n 0 peaks in the transmission give rise to the steps in the n~Ω(n=0,1,2,...). Here,theenergy E˜ isshiftedwith 0 I V curves andrather sharpfeatures (peaks and dips) respect to E0 due to the electron-phonon interaction. in−the inelastic tunneling spectrum (IETS). The latter The difference in these energies E is called a polaronic p are shown in the Fig. 8 (left panel) and they resemble shift and could be estimated as E =λ2/~Ω (Gutierrez p those obtained using proper NEGF based calculations 2006, Ryndyk 2007, Wingreen 1989). At weak coupling (Galperin 2004). of the bridge state to the electrodes the lifetime of these metastable states is long enough for them to serve as When the molecule is strongly coupled to the elec- channels for the electron transmission. Therefore, one trodes (∆ λ) and the temperature is still low may roughly approximate the transmission as a sum of (kT λL,,~RΩ≫), electron-vibrionic interaction effects contributions from all these channels. The terms in the ≪ are less pronounced. Both current and conductance sum have the form similar to the well-known expression are weakly affected by the electron-phonon coupling for the coherent transmission (see Eq. (9)). However, (Galperin 2007, Tikhodeev 2004). However, the IETS every term includes the factor P(n) which corresponds features remain distinguishable. These features appear to the probability of the metastable state to appear. So, at the threshold V = ~Ω/e which corresponds to the we obtain: | | opening of a channel for inelastic transport. To ana- P(n) lyze IETS in a simplest way one may use the result for T(E)=2∆ ∆ . L R (E E˜ n~Ω)2+(∆ +∆ )2 electron transmission derived within the Buttiker model Xn − 0− L R(17) where the dephasing parameter ǫ is expressed in terms Here (Cizek 2004): of the relevant energies, namely: 1 λ2 n λ2 P(n)= exp . (18) n! 2~2Ω2 −2~2Ω2 (cid:18) (cid:19) (cid:18) (cid:19) Γ ph ǫ= (19) The phonon induced peaks in the transmission are dis- 2(∆ +∆ )+Γ L R ph playedintheFig. 7alongwiththetransmissionpeakfor the coherent transport through a single-site bridge. As expected,thecouplingofelectronicdegreesoffreedomto thevibrationalmotionsplitsthesinglepeakinthecoher- where Γ =2Im(Σ ), end Σ istheself-energyterm ph ph ph ent transmission into the set of smaller peaks associated originatingfromtheelectron-phononinteraction. Basing withvibrationallevels. Thepeakscouldberesolvedwhen on the nonequilibrium Green’s function formalism, the 8 expression for Γ was derived in the form (Mii 2003): glanceone may expect the net currentthroughthe junc- ph tion to increase at the threshold. Indeed, the inelastic Γ (E)= 2πλ2 dωρ(ω) N(ω)[ρ (E ~ω)+ρ (E+~ω)]contribution to the current increases from zero to a cer- ph el el − tain nonzero value at this threshold for the channel for Z +(1+N(ω))([1(cid:8) n(E ~ω)]ρ (E ~ω) inelastic transport opens up. However, more thorough el − − − +n(E+~ω)ρ (E+~ω)) studies show that both elastic and nonelastic contribu- el tions to the net current undergo changes at the inelas- +N(ω) [1 n(E+~ω)]ρ (E+~ω) − el tic tunneling threshold, andthe elastic currentcouldde- +n(E (cid:0)~ω)ρel(E ~ω) . (20) crease there, as was first shown by Perssonand Baratoff − − (Persson 1987). Moreover, this decrease in the elastic Here, ρ (E) and ρ (ω) are th(cid:1)e(cid:9) phonon and elec- el ph currentmayoverweighthecontributioncomingfromthe tron densities of states, respectively; N(ω) is the Bose- inelastic channel. Depending on the relative value of the Einstein distribution function at the temperature T, elasticandinelasticcontributionstothenetcurrentnear and: threshold, the IETS revealsa peak or a dip at the corre- 1 spondingvoltage. Experimentscorroboratethevarietyin n(E)= f (E)+f (E) . (21) 2 L R theIETStakenformolecularjunctions(Hahn2000,Dju- (cid:2) (cid:3) kic 2005, Wang 2004, Zhitenev 2002). The shape of the While considering a junction with a single-site bridge signal is very sensitive to the characteristics of the junc- statecoupledtothesinglevibrationalmode,wecanwrite tionsuchasthe positionofthe electronicstate,electron- the following expressions for ρ and ρ : ph el phonon and molecule-to leads coupling strengths, and 1 γ the vibrionic frequency. For instance, at a very strong ρph(ω)= π~(ω Ω)2+γ2, (22) coupling of the molecular bridge to the leads, it could so − happen that the backscattered by the negatively biased electrode electrons whose energies belong to the conduc- ρ (E)= 1 ∆L+∆R (23) tion window between µL and and µR, are locally de- el π(E E0)2+(∆L+∆R)2 pletednearthe junction. Inthis situationthe openingof − the inelastic channel may cause the increased reflection where the polaron shift is neglected, and the parameter (otherwise forbidden by the Fermi exclusion) leading to γ characterizes the broadening of the maximum in ρ ph the decrease in the conduction. Consequently, the de- at ω = Ω due to the interaction of the vibrionic mode scribed scenario should result in the dip in IETS signal. with the environment. At low temperatures we may sig- Also, as concluded in the recent work of Ryndyk and nificantly simplify the expression for Γ . Within the ph Cuniberti (Ryndyk 2007) the above discussed sideband conduction window µ <E <µ we get: R L phonon-induced features in the electron spectral density (µL−E)/~ couldgiverisetothecorrespondingfeaturesinthediffer- Γ (E) πλ2 dωρ (ω)ρ (E+~ω) ential conductance dI/dV and IETS assuming that the ph ph el ≈ (Z0 molecule coupling to the leads is not too strong. Con- (E−µR)/~ tributions from these sideband features may be respon- + dωρph(ω)ρel(E ~ω) . (24) sible for the shape of the IETS signal at the threshold Z0 − ) of the inelastic tunneling channel. These contributions could produce an extra inelastic signal, as well. The lat- Omitting from consideration the coupling of the phonon terappearsasanadditionalpeakordipinthedifferential mode to the environment (γ 0) we may easily carry → conductance. out integrationover ω in the Eq. (24), and we arrive at the result: The question of current decrease/increase at the phonon excitation threshold which corresponds to the Γ (E) πλ2 ρ (E+~Ω)θ(µ ~Ω E) peak/dip in the IETS is not completely answered so ph el L ≈ − − +ρ(cid:8)el(E ~ω)θ(E µR ~Ω) (25) far, and the appropriate theory is being developed (Bal- − − − seiro 2006,Egger 2008). Nevertheless, it is presently un- where θ(x) is the step function. (cid:9) derstood that three relevant energies, namely, molecule- Substituting the approximation for Γph given by Eq. electrode couplings ∆L,R, electron-phonon coupling (24) or Eq. (25) into the expression (19) for the dephas- strength λ and the phonon energy ~Ω play very im- ing parameter ǫ and employing the earlier result (15) portant part in determining the shape of the IETS sig- fortheelectrontransmission,wemaycalculatetransport nal. VaryingtheseparametersonemayconverttheIETS characteristics. Theadoptedsimplifiedapproach,aswell signal from a peak to a dip and vice versa. asNEGFbasedcalculations,showsthatthe IETSsignal Atfinite temperaturesmolecularvibrationsalwaysoc- appears at the threshold determined by the frequency of cur in the presence of stochastic nuclear motions. These the vibrational mode (see Fig. 8, right panel). At first motions could be described as a phonon thermal bath. 9 Coupling of vibrational modes to this bath further af- coupled to the phonon bath, and this coupling is spec- fects the electron transport causing energy dissipation. ified by the spectral function. The particular form of The dissipative processes must be taken into account to ρ (ω) may be found basing on the molecular dynamics ph properly analyze the effects of electron-phonon interac- simulations. However,toqualitativelystudytheeffectof tions in the electron transport. Also, the displacements dissipationonthe electrontransportonecanemploy the ofionsinvolvedinthemolecularvibrationsareaccompa- expression (Mahan 2000): nied with the changes in the electrostatic field inside the molecule. This could give rise to polaronic effects in the ω ω ρ (ω)=λ exp (26) ph electron transport. We discuss these issues in the next ωc (cid:18)−ωc(cid:19) sections. where the parameter λ characterizes electron-phonon couplingstrength,and ω is the cutofffrequencyforthe c bath related to its thermal relaxation time τ =ω−1. V. Dissipative transport c c To illustrate possible effects of dissipation on the electron transport we return back to our simple model Electron transport through molecular junctions is al- where the molecular bridge is represented by a single ways accompanied by stochastic nuclear/ion motions in state. Now, we assume that this state is coupled to the closeenvironment. Interactionsoftravellingelectron the phonon bath. This model is hardly appropriate with these environmentalfluctuations causes energy dis- to properly analyze dissipative effects in the electron sipation. The importance of dissipative effects depends transport through practical molecules for the molecule on severalfactors. Among these factors the temperature length is very important for dissipative effects to be and the size and complexity of the molecular bridge are pronounced. Nevertheless, it still could serve to basi- predominating. The temperature determines the inten- cally outline main features of the dissipative electron sity of the nuclear motions, and the size of the molecule transport through molecular junctions. Also, the pro- determines so-called contact time, that is the time for posedmodelcouldbeusefultoanalyzeelectrontransport electron to travel through the junction and, in conse- in doped polyacetylene/polyaniline-polyethylene oxide quence, to contact the environment. It was shown by nanofibers (Zimbovskaya 2008). These conducting poly- Buttiker and Landauer (Buttiker 1985) that the contact mers could be treated as some kind of granular metals time is proportional to the number of sites (subunits) in where metallic-like regions (grains) are embedded in the the molecule providing intermediate states for the elec- poorly conducting medium of disorderly arranged poly- tron tunneling. For small molecules at low temperatures mer chains (MacDiarmid 2001). While in metallic state, the contact time is shorter than characteristic times for the intergrain electron transport in these nanofibers is fluctuationsintheenvironment,so,theeffectofthelatter mostly provided by electron tunneling through interme- ontheelectrontransportisnotverysignificant. Onemay diate states on the polymer chains between the grains expectsmallbroadeningofthe moleculeenergystates to (Prigodin 2002). In this case the contact time could be occur which brings a moderate erosion of the peaks in longenoughfor the effects ofdissipationto be wellman- the electron transmission and steps in the I V char- − ifested which justifies the adoption of the above model. acteristics. Again, one may carryout transport calculations using On the contrary, in large sized molecules such as pro- Eq. (15) for the electron transmission, and expressing teins and DNA, electron transport is accompanied by ǫ in terms of of the relevant energies. Substituting Eq. strongenergydissipation. Thesignificanceofthesystem- (26)intheexpression(20)wemaycalculate Γ (E). The environment interactions in macromolecules was recog- ph energy dissipation effects are more distinctly prononced nized long ago in studies of long range electron transfer at moderately high temperatures, so we assume kT reactions. Inthesereactionselectronstravelbetweendis- ~ω . Then the main contribution to the integral over≫ω tantsitesonthemoleculecalledadonorandanacceptor. c in the Eq. (20) originates from the low frequency region It was established that when an electron initially local- ω ω ; and we obtain the following approximation: ized on the donor site moves to the acceptor site with ≪ c a lower energy, the energy difference must be dissipated 8kTλΓ Γ (E)= (27) to the environment to provide the irreversibility of the ph (E E )2+Γ2 0 transfer (Garg 1985). − Ausualwaytotheoreticallyanalyzedissipativeeffects where Γ = ∆ +∆ + 1Γ (E). Solving the obtained L R 2 ph in the electron transport through molecules is to intro- equationfor Γ andusingtheEq. (19)weobtain(Zim- ph duce a phonon bath representing the random motions bovskaya 2008): in the environment. In general, there is no one/several dominatingmodesinthebath. Instead,thebathischar- 1 ρ2 1+ 1+ρ2 ǫ= (28) acterizedbyacontinuousspectralfunctionofallrelevant 24 E−E0(cid:0) 2+p1 1+ (cid:1) 1+ρ2 3 phononmodes ρph(ω). Theelectronsaresupposedtobe ∆L+∆R 2 (cid:16) (cid:17) (cid:0) p (cid:1) 10 (cid:1)(cid:2)(cid:3)(cid:4)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:2) (cid:10)(cid:11)(cid:8) (cid:14)(cid:2)(cid:8) (cid:9) (cid:12)(cid:13) FIG.9: (Coloronline)Schematicdrawingofamolecularjunc- tionwherethemolecularbridgeiscoupledtothephononbath FIG. 10: (Color online) Left panel: The electron conduc- via side chains. tance versus voltage for a junction with a single electronic state bridge directly coupled to the phonon thermal bath. The curves are plotted assuming E0 = 0.4eV, ∆L = ∆R = 0.1eV, T = 30K, λ = 0.3eV (solid (line), λ = 0.05eV where ρ2 =32kTλ/(∆ +∆ )2. (dashed line). Right panel: Electron transmission through L R Voltage dependencies of the conductance computed the junction in the case when the bridge state interacts with thephononbathviathesidechaincoupledtothebridgestate basing on Eqs. (3), (15), (28) are presented in the Fig. with the coupling parameter w. The curves are plotted as- 10(leftpanel). Onemayseethatatlowvaluesofthebias suming ∆L = ∆R = ∆, w/∆ = 20, E0 = 0, λ = 0.3eV voltagethe electronscouplingto the phononbathbrings (solid line), λ=0.05eV (dashed line). an enhancement in the conduction. The effect becomes reversedasthevoltagegrowsaboveacertainvalue. This happens because the phonon-induced broadening of the molecular level (the bridge) assists the electron trans- of the bath (Gutierrez 2005). As a result, a small finite density of phonon-induced states appears inside the gap port at small bias voltage. When the voltage rises this effect is surpassed by the scattering effect of phonons supporting electron transport at low bias voltage. So which resists the electron transport. The significance of again, the environment induces incoherent phonon assisted transport through molecular bridges. For illus- theelectron-phononinteractionsisdeterminedbythera- tio of the coupling constant λ and the self-energy terms tration we show here the results of calculations carried describing the bridge coupling to the electrodes. The out for a toy model with a single-site bridge with a side chain attached to the latter. The side chain is supposed phonon bath makes a significant effect on the transport characteristics when λ>∆ . to be coupled to the phonon bath. The results for the L,R electron transmission are displayed in the Fig. 10 (right A dissipative electron transport through large DNA panel). Weseethattheoriginalbridgestateat E =0 is moleculeswasstudiedboththeoreticallyandexperimen- completely damped but two new phonon-induced states tally(Gutierrez2005,Xu2004). Theoreticalstudieswere emerge nearby which could support electron transport. basedonamodelwherethemoleculewassimulatedbya Animportantcharacteristicsofthedissipativeelectron tight-bindingchainofsiteslinkingtheelectrodes,andat- transport through molecular junctions is the power loss tached side chains. Electrons are allowed to travel along in the junction, that is the energy flux from electronic the bridge chain and to hop to the near by side chains. into phononic system. Assuming the current flow from These chains are coupled to the phonon bath providing the left to the right, this quantity may be estimated as the energy dissipation (see Fig. 9). Although proposed the sum of the energy fluxes Q at the left and right for the specific kind of poly(dG)-poly(dC) molecules, L,R terminals (leads): this model seems to be quite generic and useful for a larger class of macromolecules. P =Q +Q . (29) L R Severalcouplingregimestothebathmaybeanalyzed. The most preferred regime for dissipative effects to ap- One may expressthe energy fluxes in terms ofrenormal- pearisthestrong-couplinglimitdefinedbythecondition izedcurrentsattheelectrodes I˜L,R(E) whicharedefined λ/ω >1. Within this regime the characteristictime for as follows (Datta 2005): c the electron bath interactions is much shorter than typ- e I = I˜ (E)dE. (30) ical electron time scales. Consequently, the bath makes L,R π~ L,R a significant impact on the molecule electronic struc- Z Then Q may be presented in the form: ture. New bath- induced states appear in the molecular L,R spectrum inside the HOMO-LUMO gap. However,these 1 Q = EI˜ (E)dE. (31) states are strongly damped due to the dissipative action L,R π~ L,R Z