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Molecular Wires Acting as Coherent Quantum Ratchets Jörg Lehmann,1 Sigmund Kohler,1 Peter Ha¨nggi,1 and Abraham Nitzan2 1Institut fu¨r Physik, Universität Augsburg, Universitätsstr. 1, D-86135 Augsburg, Germany 2School of Chemistry, The Sackler Faculty of Science, Tel Aviv University, 69978 Tel Aviv, Israel (Dated: February 1, 2008) 2 0 Theeffectoflaserfieldsontheelectrontransportthroughamolecularwirebeingweaklycoupled 0 totwoleadsisinvestigated. Themolecular wireactsasacoherentquantumratchetifthemolecule 2 is composed of periodically arranged, asymmetric chemical groups. This setup presents a quantum n rectifierwith afinitedc-responsein theabsenceof astatic bias. Thenonlinearcurrentisevaluated a in closed form within the Floquet basis of the isolated, driven wire. The current response reveals J multiple current reversals together with a nonlinear dependence (reflecting avoided quasi-energy 5 crossings) on both, the amplitude and the frequency of the laser field. The current saturates for 2 long wires at a nonzero value, while it may change sign upon decreasing its length. ] PACSnumbers: 85.65.+h,33.80.-b,73.63.-b,05.60.Gg l l a h More than two decades after the idea of a molecular inelastic hopping events [13, 15]. - s rectifier [1], the experimental and theoretical study of Atoms and molecules in strong oscillating fields have e such systems presently enjoys a vivid activity [2]. Re- been widely studied within a Floquet formalism[16, 17]. m cent experimental progress has enabled the reproducible Thissuggeststhefollowingprocedure: Makinguseofthe t. measurement [3, 4] of weak tunneling currents through tools that have been acquired in that area, we develop a moleculeswhicharecoupledbychemisorbedthiolgroups a formalism that combines Floquet theory for a driven m to the gold surface of the leads. A necessary ingredient moleculewith the many-particledescriptionoftransport - for future technological applications will be the possibil- through a system that is coupled to ideal leads. This d n ity to control a priori the tunneling current through the approach is devised much in the spirit of the Floquet- o molecule. Advances in that very direction have lately Markov theory [18, 19] developed for driven dissipative c been achieved by the development of a molecular field- quantum systems. Technically, we thereby work beyond [ effect transistor [5]. Typical energy scales in molecules theusualrotating-waveapproximationasfrequentlyem- 1 areintheopticalandtheinfraredregime,wherebasically ployed, such as e.g. in Ref. [20]. v all of the today’s lasers operate. Hence, lasers represent The lead-molecule model.— The total system com- 0 aninherentpossibilitytocontrolatomsormoleculesand posedofthedrivenwire,theleads,andthemolecule-lead 8 possibly currents through them. couplings is described by the Hamiltonian 4 1 A particularly intriguing phenomenon in strongly H(t)=H (t)+H +H (1) 0 wire leads wire-leads drivensystems is the so-termedratcheteffect [6, 7, 8, 9], 2 originally discovered for overdamped classical Brown- The wire itself is modelled by N atomic orbitals |ni, 0 / ianmotionincyclicasymmetricnonequilibriumsystems. n = 1,... ,N, which are in a tight-binding description t a Counterintuitivelyto the secondlawonethen observesa coupled by hopping matrix elements. The Hamiltonian m directed transport although all acting forces possess no for the electrons on the wire reads - net bias. This effect has been established as well within d the regime of dissipative, incoherent quantum Brownian Hwire(t)= Hnn′(t)c†ncn′, (2) n motion [10]. With this work we investigate the possi- nX,n′ o c bilities for quantum wires to act as a coherent quantum where the fermionic operators c , c† annihilate, respec- n n : ratchet, i.e.we consider the coherentquantumtransport v tively create, an electron in the atomic orbital |ni. The throughmolecularwireswithasaw-toothlikelevelstruc- i influence of the laser field is given by a periodic time- X ture of the orbital energies when subjected to the influ- dependence of the on-site energies yielding a single par- r ence of a strong laser field. a ticleHamiltonianofthestructureHnn′(t)=Hnn′(t+T), Recent theoretical descriptions of molecular conduc- whereT =2π/Ωis determinedbythe angularfrequency tivity are based on a scattering approach [11, 12], or as- Ω of the laser field. sumethattheunderlyingtransportmechanismisanelec- The orbitals at the left and the right end of the trontransferreactionfromthedonortotheacceptorsite molecule, that we shall refer to as donor and acceptor and that the conductivity can be derived from the cor- states,|Di=|1iand|Ai=|Ni,respectively,arecoupled responding reaction rate [13]. It has been demonstrated to ideal leads (cf. Fig. 1) by the tunneling Hamiltonian thatbothapproachesyieldidenticalresultsinalargepa- rameter regime [14]. Within the high-temperature limit, Hwire-leads = (VqLc†qLcD+VqRc†qRcA)+h.c., (3) X the electron transport on the wire can be described by q 2 and mutatis mutandis for the net current through the |N−1i right contact. Equation (5) expresses the current by the |3i ES expectation values h[c ,c˜ (t − τ,t)] i and hc†c i. We |2i n n + n n emphasize that these quantities depend on the dynam- EB ics of the isolated wire and are thus influenced by the driving. µL |1i |Ni µR Floquet decomposition.— Let us now focus on the (donor) (acceptor) driven molecule decoupled from the leads. Since its Hamiltonian is periodic in time, Hnn′(t)=Hnn′(t+T), T =2π/Ω,we cansolvethe time-dependentSchrödinger equation within a Floquet approach, i.e. we make use FIG. 1: Level structure of the wire ratchet with N = 8 of the fact that there exists a complete set of solu- atomic sites, i.e. Ng = 2 asymmetric molecular groups. The tions of the form |Ψ (t)i = e−iǫαt/h¯|Φ (t)i with the bridgelevelsare EB abovethedonorandacceptorlevelsand α α are shifted by ±ES/2. quasi-energies ǫα. The so-called Floquet modes |Φα(t)i obey the time-periodicity of the driving field and can thus be decomposed in a Fourier series: |Φ (t)i = α where c (c ) annihilates an electron in state Lq (Rq) ∞ e−ikΩt|Φ i. These obey the quasienergyequa- qL qR k=−∞ α,k ontheleft(right)lead. Inthefollowingwewillassumea tPion [16, 21, 22] wide-bandlimit Γ =(2π/¯h) |V |2δ(ǫ−ǫ ), L/R q qL/R qL/R d i.e. an energy-independent coupPling strength. |niHnn′(t)hn′|−i¯h |Φα(t)i=ǫα|Φα(t)i. (6) The leads themselves aremodelledby grand-canonical (cid:16)nX,n′ dt(cid:17) ensembles of electrons with the Hamiltonian H = leads (ǫ c† c +ǫ c† c )andelectro-chemicalpoten- To make use of the knowledge about the driven q qL qL qL qR qR qR molecule that we obtain from Floquet theory, we define tPials µ . Then the only non-trivial expectation values L/R the Floquet representationof the fermionic creation and ofleadoperatorsarehc† c i=f(ǫ −µ ),whereǫ is qL qL qL L qL annihilation operators by the time-dependent transfor- thesingleparticleenergyofthestateqLandcorrespond- mation ingly for the right lead. Here, f(ǫ) = (1+eǫ/kBT)−1 is the Fermi function. c (t)= hΦ (t)|nic , (7) α α n Perturbationtheory.—Whilethedynamicsoftheleads X n and the wire, including the external driving, will be c = hn|Φ (t)ic (t). (8) n α α treated exactly, we take the wire-lead Hamiltonian as a X α perturbation into account. From the Liouville-von Neu- mann equation i¯h̺˙(t) = [H(t),̺(t)] for the total den- The backtransformation(8)follows fromthe mutualor- sityoperator̺(t)oneobtainsbystandardtechniquesthe thogonality and the completeness of the Floquet states master equation at equal times [16]. It is now straightforward to prove that c˜ (t−τ,t)=c (t)exp(iǫ τ/¯h). This yields a spec- α α α i traldecompositionofEq.(5), whichmakes itpossible to ̺˙(t)=− [H (t)+H ,̺(t)] (4) wire leads ¯h evaluate the time andenergy integrationstherein. Aver- ∞ aging I (t) over the driving period [23] leads to our first 1 L − ¯h2 Z dτ[Hwire−leads,[Hwire−leads(t−τ,t),̺(t)]]. main result, namely the average current 0 e eΓ I¯= L hΦ |DihD|Φ if(ǫ +k¯hΩ−µ ) The tilde denotes operators in the interaction picture ¯h Xh α,k α,k α L with respect to the molecule and the lead Hamilto- αk (9) nian without the molecule-lead coupling, X(t,t′) = − hΦα,k′+k|DihD|Φβ,k′iRαβ,k , U†(t,t′)XU (t,t′), U is the propagator without the Xβk′ i 0 0 0 e coupling. The net (incoming minus outgoing) electrical whereR (t)=hc†(t)c (t)i= e−ikΩtR . Herewe current through the left contact is given by the time- αβ α β k αβ,k haveusedthefactthattheR P(t)areexpectationvalues αβ derivative of the electron number in the left lead multi- of a dissipative, periodically driven system. Therefore, plied by the electron charge −e. From Eq. (4) follows in they share in the long-time limit the time-periodicity of the wide-band limit the expression the driving field and can be represented by a Fourier ∞ series. Γ I (t)=etr[̺˙(t)N ]=e L Re dτ dǫei(ǫ+µL)τ/h¯ The remaining task in computing the stationary cur- L L π¯h Z Z rent is to find the Fourier coefficients R at asymp- αβ,k 0 totic times. For that purpose, we derive from the mas- ×f(ǫ) [c ,c˜†(t−τ,t)] −eΓ c†c (5) 1 1 + L 1 1 ter equation (4) an equation of motion for the Rαβ(t). (cid:10) (cid:11) (cid:10) (cid:11) 3 Since all coefficients of the master equation as well as its asymptotic solution are T-periodic, we can split it intoitsFouriercomponentswhichhavetosatisfyEq.(4) separately. Finally, we obtain for the R the inhomo- αβ,k geneous set of equations Γ] i e (ǫ −ǫ +k¯hΩ)R (10) 3 ¯h α β αβ,k − 0 Γ 1 L [ = hΦβ,k′′+k′|DihD|Φβ′,k′′+kiRαβ′,k′ ¯I 2 Xk′ nβX′k′′ + hΦα′,k′′+k′|DihD|Φα,k′′+kiRα′β,k′ αX′k′′ −f(ǫα+k′Ω−µL)hΦβ,k′−k|DihD|Φα,k′i −f(ǫβ +k′Ω−µL)hΦβ,k′|DihD|Φα,k′+ki A[∆] o + Γ ,µ ,|DihD| → Γ ,µ ,|AihA| . L L R R FIG. 2: Time-averaged current (in arbitrary units) through (cid:0) (cid:1) (cid:0) (cid:1) a molecular wire that consists of Ng bridge units as a func- Forthetypicalparametervaluesusedbelow,alargenum- tion of the driving strength A. The bridge parameters are ber of side-bands contributes significantly to the Fourier EB =10∆, ES =∆, the driving frequency is Ω=3∆/¯h, the decomposition of the Floquet modes |Φα(t)i. Numerical coupling to the leads is chosen as ΓL = ΓR = 0.1∆/¯h, and convergenceforthe solutionofthe masterequation(10), the temperature is kBT = 0.25∆. The arrows indicate the however,isalreadyobtainedbyusingafewside-bandsfor drivingamplitudes used in Fig. 3. the decomposition of R (t). This keeps the numerical αβ effortrelativelysmallandjustifies the useofthe Floquet representation (8). Yet we are able to treat the prob- neighboring sites, a driving amplitude A = ∆ is equiva- lem beyond the usual rotating-wave-approximation [20], lent to an electrical field strength of 2×106V/cm. which in certain parameter regimes turns out to be cru- Figure2showsthestationarytime-averagedcurrentI¯. cial [24]. In the limit of a very weak laser field, we find I¯∝E A2 S Ratchetwire.—Weconsideramolecularwirethatcon- (not shown). This behavior is expected from symmetry sists of a donor and an acceptor site and N asymmetric g considerations: Theasymptoticcurrentmustbeindepen- molecular groups (cf. Fig. 1). Each of the N = 3N +2 g dentofanyinitialphaseofthedrivingfieldandtherefore orbitals is coupled to its nearest neighbors by a hopping is an even function of the field amplitude A. This indi- matrix elements ∆. The laser field renders each level cates that the ratcheteffect canonly be obtainedfrom a oscillating in time with a position dependent amplitude. treatmentbeyondlinearresponse. Forstronglaserfields, Then the time-dependent wire Hamiltonian reads we find that I¯is almost independent of the wire length. If the driving is intermediately strong, I¯ depends in a Hnn′(t)=[En−Acos(Ωt)xn]δnn′ −∆(δn,n′+1+δn+1,n′), short wire sensitively on the driving amplitude A and (11) the number of asymmetric molecular groups N : even g the sign of the current may change with N , i.e. we find wherex =(N+1−2n)/2isthescaledpositionofsiten, g n a current reversal as a function of the wire length. For the energy A equals electron charge multiplied by laser fieldstrengthanddistancebetweentwoneighboringsites, long wires that comprise five or more wire units, the average current becomes again length-independent, as can and∆isthehoppingmatrixelement. Theenergiesofthe be seen from Fig. 3. This identifies the observed current donorandthe acceptororbitalsareassumedtobe atthe levelofthe chemicalpotentialsofthe attachedleads and reversal as a finite size effect. The fact that I¯converges to a finite value if the number of wire units is enlarged, since no voltage is applied, E = E = µ = µ . The 1 N L R bridge levels E lie at E and E ±E /2, respectively, demonstratesthatthedissipationcausedbythecoupling n B B S to the leads is sufficient to establish the ratchet effect in as sketched in Fig. 1. We remark that for the sake of the limit of long wires. In this sense, no on-wire dissipa- simplicity intra-atomic dipole excitations are neglected within our model Hamiltonian (11). tion is required. In our numerical studies, we use the hopping matrix Figure4depictstheaveragecurrentvs.thedrivingfre- element ∆ as the energy unit; in a realistic molecule, ∆ quency Ω, exhibiting resonance peaks as a striking fea- is of the order 0.1eV. Thus, our chosen wire–lead hop- ture. Comparisonwiththequasi-energyspectrumreveals ping rate Γ = 0.1∆/¯h yields eΓ = 2.56×10−5Ampère thateachpeakcorrespondstoanon-linearresonancebe- andΩ=3∆/¯hcorrespondstoalaserfrequencyinthein- tween the donor/acceptor and a bridge orbital. While frared. Notethatforatypicaldistanceof2˚Abetweentwo the broaderpeaksath¯Ω≈E =10∆matchthe 1:1res- B 4 efficient for larger wire systems. With this formalism at hand, we have establishedthe possibility of using molec- A=7∆ ular wires as coherent quantum ratchets. As a finite size effect,thewiremayexhibitacharacteristiccurrentrever- sal as a function of (decreasing) length; upon increasing Γ] the wire length the average current rapidly converges to e 3 A=3∆ a finite value. The strong dependence on the driving pa- − 0 rameters in turn admits a tailored quantum control of 1 ¯I[ current with respect to both its sign and its magnitude. A=5∆ Inparticular,thisdrivenmolecularwirewiththedistinc- tive multiple current reversal feature encompasses new prospectstopumpandshuttleelectronsonthenanoscale inana priori manner. Ourphysicalestimatesshowthat a realizationof a molecularwire ratchetindeed is within N g experimental reach. This work has been supported by Sonderforschungs- FIG.3: Time-averagedcurrent(inarbitraryunits)asafunc- bereich486ofthe Deutsche Forschungsgemeinschaftand tionofthenumberofbridgeunitsNg forthelaseramplitudes by the Volkswagen-Stiftung under grant No. I/77 217. indicatedinFig.2. AllotherparametersareasinFig.2. 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