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Nonequilibrium resonant spectroscopy of molecular vibrons Dmitry A. Ryndyk and Gianaurelio Cuniberti Institut fu¨r Theoretische Physik, Universität Regensburg, D-93040 Germany (Dated: February 6, 2008) Quantum transport through single molecules is essentially affected by molecular vibrations. We 7 0 investigate the behavior of the prototype single-level model with intermediate electron-vibron cou- 0 pling and arbitrary coupling to the leads. We have developed a theory which allows to explore 2 this regime via the nonequilibrium Green function formalism. We show that the nonequilibrium resonant spectroscopy is able to determine the energies of molecular orbitals and the spectrum of n molecularvibrations. Ourresultsarerelevanttoscanningtunnelingspectroscopyexperiments,and a demonstrate the importance of the systematic and self-consistent investigation of the effects of the J vibronicdynamics onto thetransport through single molecules. 4 ] l Molecular electronics [1, 2, 3] is the most promising We analyze here the electron-vibronmodel developing l a development of nano-electronics, raising new theoretical the preliminary results obtained in our previous publi- h challengesandcallingfornewmethods. Theunderstand- cations [14, 15]. We consider a molecule coupled to free - ing of quantumelectrontransportatthe molecularscale conduction electrons in the leads by a usual tunneling s e is a key step to future devices. Recently, the interac- Hamiltonian. Furthermore, the electrons are coupled to m tion of electrons with molecular vibrations attracted an vibrational modes. The full Hamiltonian is the sum of . enormous attention unveiled by electron transport ex- the molecularHamiltonian Hˆ , the Hamiltonians of the t M a periments through single molecules [4, 5, 6, 7, 8, 9]. In leads Hˆ , the tunneling Hamiltonian Hˆ describing R(L) T m scanningtunnelingspectroscopy(STS)experimentsclear the molecule-to-lead coupling, the vibron Hamiltonian - signatures of the electron-vibron interaction have been Hˆ includingelectron-vibroninteractionandcouplingof d V observed [10, 11, 12, 13]. In these experiments both the vibrations to the environment (describing dissipation of n o electron-vibron coupling and the electron hybridization vibrons) c are well beyond the perturbation limit, so that a the- [ ory beyond the linear response or master equation ap- Hˆ =HˆM +HˆV +HˆL+HˆR+HˆT. (1) 1 proaches is necessary. In this Letter, we present such a A molecule is described by a set of localized states |αi v theory and discuss the vibronic features in the current- 6 voltage curves of molecular junctions as a spectroscopic with energies ǫα and inter-orbital overlap integrals tαβ 8 by the following model Hamiltonian: tool. 0 01 Resonant electron transport through a molecular or- HˆM(0) = (ǫα+eϕα(t))d†αdα+ tαβd†αdβ, (2) 7 bitalwithelectron-vibroninteractionatfinite voltagesis Xα αX6=β 0 essentiallydifferentfromthewellknowninelasticelectron / tunneling spectroscopy (IETS). Indeed, in the IETS sit- where d†α,dα are creation and annihilation operators in t a uation one has typically the multi-channel tunneling be- thestates|αi, andϕα(t)isthe(self-consistent)electrical m tweentwo bulk metals with continuousdensity of states, potential. The index α is used to mark single-electron - andanincrease ofthe conductance is observedwhenthe states(atomic orbitals)including the spindegreeoffree- d biasvoltageexceedsthethresholddeterminedbythefre- dom. The parameters of a tight-binding model could be n o quency of vibrational mode. As we shall see below, in determined by ab initio methods [3]. This is a compro- c resonant tunneling through molecular levels coupled to : vibrons,thesituationisverydifferent: inadditiontothe v i usual IETS signal, the inelastic signal can be observed X as an extra peak in the differential conductance, or just r as a decrease of the conductance. The resulting signal a is determined by the energy of the electronic state, the L G e G R vibronicfrequency,theelectron-vibroncoupling,andthe L 0 R molecule-to-leadcoupling. Thepositionofthe molecular energylevelscouldbe changedby the externalgatevolt- age, but in the STS experiments the application of the gate voltage is not possible, instead the tip-to-molecule w distance can be changed, also changing the molecule-to- 0 tipcoupling[11]. Thusitisveryimportanttoinvestigate quantum transport at arbitrary coupling to the leads. The simplest model, which describes properly these pe- culiarities, is a single electron with energy ǫ interacting Figure 1: (Color online) Schematic picture of the considered 0 with a single vibronwith frequency ω (Fig.1), andcou- fluctuatingsingle-level model. 0 pled to two noninteracting equilibrium electrodes. 2 Refs. [17, 18, 19, 20]. There the exact solution in the single-particleapproximationwasderived,ignoringcom- pletely the Fermi sea in the leads. At strong electron- vibron couplings and weak couplings to the leads the satellites of the main resonant peak are formed in the ) spectral function (Fig.2). The question which remains e( A is whether these side-bands can be observed in the differential conductance. New theoretical treatments were 0 2 e4 6 8 presented in Refs.[21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. In parallel, the theory of inelastic resonant tunneling inscanning tunneling spectroscopywasdeveloped [33, 34, 35]. Theessentialprogressincalculationoftransportprop- 0 2 4 6 8 10 e erties in the strong electron-vibron interaction limit has beenmadewiththehelpofthemasterequationapproach [23, 26, 31]. This method, however, is valid only in the Figure 2: (Color online) Spectral function at different limit of very weak molecule-to-lead couplings and ne- electron-vibron couplings: λ/ω0 = 0.4 (black), λ/ω0 = 1.2 glects all spectral effects, which are the most important (blue, dashed), and λ/ω0 = 2 (red); at ǫ0/ω0 = 5, ΓL/ω0 = at finite coupling to the leads. ΓR/ω0 =0.1. Intheinsertthespectralfunctionatλ/ω0 =1.2 is shown at finite voltage, when the level is partially filled. Our approach is based on the nonequilibrium Green Energies are in unitsof ~ω0. function technique[36, 37, 38], which is now a standard method in mesoscopic physics and molecularelectronics. We follow the formulationpioneeredby Meir, Wingreen, and Jauho[39, 40, 41]. This method was also further mise,whichallowsustoconsidercomplexmoleculeswith developed in Refs.[24, 27, 28, 29, 32, 35]. The case of a relatively simple model. intermediate and strong electron-vibron couplings at fi- The isolated single-level electron-vibron model is de- nite tunneling rates is the most interesting, but also the scribed by the Hamiltonian (~=1) most difficult. Only the approaches by Flensberg [24], and Galperin, Ratner, and Nitzan [32] exist, both start- Hˆ =(ǫ +eϕ )d†d+ω a†a+λ a†+a d†d, (3) M+V 0 0 0 ing from the exact solution for the isolated system and (cid:0) (cid:1) then switching on tunneling as a perturbation. Our ap- where the first and the second terms describe the free proachisexactfornon-interactingsystemswitharbitrary electron state and the free vibron, and the third term is tunneling coupling, and is approximate in the strong electron-vibron minimal coupling interaction. electron-vibron coupling limit without tunneling. Still, Theelectricalpotentialofthemoleculeϕ playsanim- 0 its advantage is clear in the truly nonequilibrium formu- portant role in transport at finite voltages. It describes lation. theshiftofthemolecularlevelbythebiasvoltage,which Thecurrentintheleft(i=L)orright(i=R)contact isdividedbetweentheleftlead(tip),therightlead(sub- tothemoleculeisdescribedbythewell-knownexpression strate), and the molecule as ϕ =ϕ +η(ϕ −ϕ ) [16]. 0 R L R The Hamiltonians of the right (R) and left (L) leads ie dǫ J = Tr Γ (ǫ−eϕ ) G<(ǫ)+ read i=L,R ~ Z 2π i i (cid:8) (cid:0) (6) Hî=L(R) = (ǫikσ +eϕi)c†ikσcikσ, (4) fi0(ǫ−eϕi) GR(ǫ)−GA(ǫ) , Xkσ (cid:2) (cid:3)(cid:1)(cid:9) wheref0(ǫ)istheequilibriumFermidistributionfunction i ϕi(t) are the electrical potentials of the leads. Finally, intheleadiwiththechemicalpotentialµi,andthelevel- the tunneling Hamiltonian width function Γ (ǫ)≡Γ (ǫ) is i=L(R) iαβ HˆT = Vikσ,αc†ikσdα+h.c. (5) Γiαβ(ǫ)=2π Vikσ,βVi∗kσ,αδ(ǫ−ǫikσ). (7) X X(cid:16) (cid:17) X i=L,Rkσ,α kσ Thelesser(retarded,advanced)Greenfunctionmatrix describes the hopping between the leads and the molecule. A direct hopping between two leads is ne- of a nonequilibrium molecule G<(R,A) ≡G<(R,A) can be αβ glected. found from the Dyson-Keldysh equations in the integral Though the model described above has a long his- form tory, the many questions are not answered up to now. GR(ǫ)=GR(ǫ)+GR(ǫ)ΣR(ǫ)GR(ǫ), (8) While the isolated electron-vibron model can be solved 0 0 exactly by the so-called polaron or Lang-Firsov trans- G<(ǫ)=GR(ǫ)Σ<(ǫ)GA(ǫ), (9) formation, the coupling to the leads produces a true many-body problem. The inelastic resonant tunneling or from the corresponding equations in the differential of electrons coupled to phonons was first considered in form (see Refs.14, 15 and references therein). 3 1.0 0.12 0.8 ] ] 0 0 G G0.08 [ 0.6 [ V V d 0.4 d / /0.04 I I d d 0.2 0.0 0.00 -8 -4 0 4 8 -4 0 4 8 Voltage Voltage Figure 3: Differential conductance of a symmetric junction Figure4: Differentialconductanceofanasymmetricjunction (η = 0.5, ΓR = ΓL) at different molecule-to-lead coupling, (η = 0, ΓR = 20ΓL) at different molecule-to-lead coupling, fromΓL/ω0 =0.1(lowercurve)toΓL/ω0 =10(uppercurve), from ΓR/ω0 =0.2 (lowercurve)toΓR/ω0 =4(uppercurve), λ/ω0 =1, ǫ0/ω0 =2. Voltage is in theunits of ~ω0. λ/ω0 =2, ǫ0/ω0 =5. The voltage is in theunits of ~ω0 Here ΣR,< = ΣR,<(T) + ΣR,<(T) + ΣR,<(V) is the tion, a suppression of the conductance at V ≃ ±~ω L R 0 total self-energy of the molecule composed of the tun- takes place as a result of inelastic scattering of the co- neling (coupling to the left and right leads) self-energies herently transformed from the left lead to the right lead ΣR,<(T) ≡ΣR,<(T) = V∗ GR,<V ,andthe electrons. In the asymmetric junction, the usual IETS j=L,R jαβ kσ jkσ,α jkσ jkσ,β vibronic self-energy ΣPR,<(Vn) ≡ ΣαRβ,<(V). Heore we as- ivnoclrtaeagseinVg ≃of−th~eωco,ntdhuisctfeaantcuereisisobwseearvkedanadtcaannebgeatoivbe- 0 sume,thatvibronsareinequilibriumandarenotexcited served only in the incoherent tail of the resonant con- by the current, so that the self-consistent Born approxi- ductance. We conclude, that the vibronic contribution mation is a good starting point. The corresponding self- to the conductance can be distinguished clearly in both energies can be found in Refs.[14, 15, 26, 27, 28, 29, 35, coherent and tunneling limits. 38, 41]. NowletusdiscusstheSTSexperiments[10,11,12,13]. Forthesingle-levelmodelallequationsaresignificantly Here we concentrate mainly on the dependence on the simplified. Combining J and J the expression for L R tip-to-molecule distance [11]. When the tip (left lead in the current canbe written for energy independent Γ L(R) our notations) is far from the molecule, the junction is (wide-band limit) as strongly asymmetric: Γ ≪ Γ and η → 0, and the L R e Γ Γ conductanceissimilartothatshowninFig.4. Whenthe J = 2π~Γ L+RΓ Z dǫA(ǫ) f0(ǫ−eϕL)−f0(ǫ−eϕR) . tipisclosetothemolecule,thejunctionisapproximately R L (cid:2) (cid:3) (10) symmetric: ΓL ≈ ΓR and η ≈ 0.5, and the conductance Itlooks assimple as the Landauer-Bu¨ttikerformula,but curve is of the type shown in Fig.3. We calculated the it is not trivial, because the spectral density A(ǫ) = transformation of the conductance from the asymmetric −2ImGR(ǫ) now depends on the distribution function of tosymmetriccase(Fig.5). Itisonenewfeatureappeared electrons inside the fluctuating molecule and hence the in asymmetric case due to the fact that we started from applied voltage, ϕ =−ϕ =V/2 [14]. A typical exam- a finite parameter η = 0.2 (in the Fig. 4 η = 0), namely L R ple of the spectral function at zero voltage is shown in a single peak at negative voltages, which is shifted to Fig. 2. At finite voltage it is calculated self-consistently. smallervoltageinthesymmetricjunction. Theformand Before considering the experiments, let us discuss a behavior of this peak is in agreement with experimental general picture of the vibronic transport in symmet- results [11]. ric and asymmetric molecular bridges coupled to the Inconclusion,atweakmolecule-to-lead(tip,substrate) electrodes, provided in experiments by the MCBJ and coupling the usual vibronic side-band peaks in the dif- STM technique, respectively. The differential conduc- ferential conductance are observed; at stronger coupling tance, calculated at different molecule-to-lead coupling, to the leads (broadening) these peaks are transformed is shown in Fig. 3 (symmetric) and Fig. 4 (asymmetric). into step-like features. A vibronic-induced decreasing At weak coupling, the vibronic side-band peaks are ob- of the conductance with voltage is observed in high- served,reproducingthe correspondingpeaksinthe spec- conductancejunctions. TheusualIETSfeature(increas- tral function (Fig. 2). 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