Modeling elastic and photoassisted transport in organic molecular wires: length dependence and current-voltage characteristics
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Modeling elastic and photoassisted transport in organic molecular wires: lengthdependence and current-voltage characteristics
J. K. Viljas,
1, 2, ∗ F. Pauly,
1, 2 and J. C. Cuevas
3, 1, 2 Institut f¨ur Theoretische Festk¨orperphysik and DFG-Center for Functional Nanostructures,Universit¨at Karlsruhe, D-76128 Karlsruhe, Germany Forschungszentrum Karlsruhe, Institut f¨ur Nanotechnologie, D-76021 Karlsruhe, Germany Departamento de F´ısica Te´orica de la Materia Condensada,Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain (Dated: October 28, 2018)Using a π -orbital tight-binding model, we study the elastic and photoassisted transport prop-erties of metal-molecule-metal junctions based on oligophenylenes of varying lengths. The effectof monochromatic light is modeled with an ac voltage over the contact. We first show how thelow-bias transmission function can be obtained analytically, using methods previously employed forsimpler chain models. In particular, the decay coefficient of the off-resonant transmission is ex-tracted by considering both a finite-length chain and infinitely extended polyphenylene. Based onthese analytical results, we discuss the length dependence of the linear-response conductance, thethermopower, and the light-induced enhancement of the conductance in the limit of weak intensityand low frequency. In general the conductance-enhancement is calculated numerically as a functionof the light frequency. Finally, we compute the current-voltage characteristics at finite dc voltages,and show that in the low-voltage regime, the effect of low-frequency light is to induce current stepswith a voltage separation determined by twice the frequency. These effects are more pronouncedfor longer molecules. We study two different profiles for the dc and ac voltages, and it is foundthat the results are robust with respect to such variations. Although we concentrate here on thespecific model of oligophenylenes, the results should be qualitatively similar for many other organicmolecules with a large enough electronic gap. PACS numbers: 73.50.Pz,85.65.+h,73.63.Rt
I. INTRODUCTION
The use of single-molecule electrical contacts for op-toelectronic purposes such as light sources, light sen-sors, and photovoltaic devices is an exciting idea. Yet,due to the difficulties that light-matter interactionsin nanoscale systems pose for theoretical and exper-imental investigations, the possibilities remain largelyunexplored. Concerning experiments, it has beenshown that light can be used to change the conforma-tion of some molecules even when they are contactedto metallic electrodes, thus enabling light-controlledswitching. Some evidence of photoassisted processes in-fluencing the conductance of laser-irradiated metallicatomic contacts has also been obtained. Theoretical in-vestigations of light-related effects in molecular contactsare more numerous, butthey are mostly based on highly simplified models,whose validity remains to be checked by more de-tailed calculations and experiments. However, forthe description of the basic phenomenology, model ap-proaches can be very fruitful, as they have been in stud-ies of elastic transport in the past. Properties of lin-ear single-orbital tight-binding (TB) chains, in partic-ular, have been studied in detail, and to a large partanalytically.
In a step towardsa more realistic description of the geometry, symmetries,and the electronic structure of particular molecules, em-pirical TB approaches such as the (extended) H¨uckel method have proved useful.
Based on a combination of density-functional calcu-lations and simple phenomenological considerations, wehave recently described the photoconductance of metal-oligophenylene-metal junctions. It was discussed howthe linear-response conductance may increase by ordersof magnitude in the presence of light. This effect canbe seen as the result of a change in the character of thetransport from off-resonant to resonant, due to the pres-ence of photoassisted processes.
Consequently, thedecay of the conductance with molecular length is sloweddown, possibly even making the conductance length-independent.
In this paper we apply a H¨uckel-type TB model ofoligophenylene-based contacts combined with Green-function methods to study the effects of monochromaticlight on the dc current in metal-oligophenylene-metalcontacts. Again we concentrate on the dependence ofthese effects on the length of the molecule. We beginwith a detailed account of the elastic transport prop-erties of the model, and show that the zero-bias trans-mission function can be obtained analytically, similarlyto simpler chain models. We demonstrate how infor-mation about the length dependence of the transmissionfunction for a finite wire can be extracted from an in-finitely extended polymer. Based on these analytical re-sults, we discuss the length dependences of the conduc-tance and the photoconductance for low-intensity andlow-frequency light. While the conductance decays ex-ponentially with length, its relative enhancement due tolight exhibits a quadratic behavior. Here we also brieflyconsider the thermopower, whose length dependence islinear. Next, we calculate numerically the zero-bias pho-toconductance as a function of the light frequency ω ,and find that the conductance-enhancement due to lightis typically very large. In particular, we show thatthe results of Ref. 5 are expected to be robust with re-spect to variations in the assumed voltage profiles. Fi-nally, we describe how the step-like current-voltage ( I - V ) characteristics are modified by light. At high ω themost obvious effect is the overall increase in the low-bias current. At low ω , additional current steps similarto those in microwave-irradiated superconducting tunneljunctions can be seen. Their separation, in our caseof symmetric junctions, is roughly 2¯ hω/e .TB models of the type we shall consider neglect vari-ous interaction effects (see Sec. V for a discussion), andthus cannot be expected to give quantitative predictions.However, the qualitative features of the results rely onlyon the tunneling-barrier character of the molecular con-tacts, which results from the fact that the Fermi energyof the metal lies in the gap between the highest-occupiedand lowest-unoccupied molecular orbitals (HOMO andLUMO) of the molecule. Thus, these features should re-main similar for junctions based on many other organicmolecules exhibiting large HOMO-LUMO gaps. Thelight-induced effects, if verified experimentally, could beused for detecting light, or as an optical gate (or “thirdterminal”) for purposes of switching.The rest of the paper is organized as follows. In Sec. IIwe describe our theoretical approach, discuss the gen-eral properties of TB wire models, and introduce theGreen-function method for the calculation of the elas-tic transmission function. Then, in Sec. III we calculatethe transmission function of oligophenylene wires analyt-ically. The decay coefficient for the off-resonant transmis-sion is extracted also from infinitely extended polypheny-lene. Following that, in Sec. IV we present our numericalresults for the conductance, the thermopower, the pho-toconductance, and the I - V characteristics. Finally, Sec.V ends with our conclusions and some discussion. De-tails on the calculation of the time-averaged current inthe presence of light are deferred to the appendixes. InApp. A a simplified interpretation of the current formulais derived, and in App. B a brief account of the generalmethod is given. Readers mainly interested in the discus-sion of the results for the physical observables can skipmost of Secs. II and III, and proceed to Sec. IV. II. THEORETICAL FRAMEWORKA. Transport formalism
Our treatment of the transport characteristics for thetwo-terminal molecular wires is based on Green’s func-tions and the Landauer-B¨uttiker formalism, or its gen- eralizations. Assuming the transport to be fully elastic,the dc electrical current through a molecular wire can bedescribed with I ( V ) = 2 eh Z dEτ ( E, V )[ f L ( E ) − f R ( E )] . (1)Here V is the dc voltage and τ ( E, V ) is the voltage-dependent transmission function, while f X ( E ) =1 / [exp(( E − µ X ) /k B T X ) + 1], µ X , and T X are the Fermifunction, the electrochemical potential, and the temper-ature of side X = L, R , respectively. The electrochem-ical potentials satisfy eV = ∆ µ = µ L − µ R , and wecan choose them symmetrically as µ L = E F + eV / µ R = E F − eV /
2, where E F is the Fermi energy. Forstudies of dc current we always assume T L = T R = 0.Of particular experimental interest is the linear-responseconductance G dc = ∂I/∂V | V =0 , given by the Lan-dauer formula G dc = G τ ( E F ), where G = 2 e /h and τ ( E ) = τ ( E, V = 0). In most junctions based on organicoligomers, the transport can be described as off-resonanttunneling. This results in the well-known exponentialdecay of G dc with the number N of monomeric units inthe molecule. At finite voltages V , the current increasesin a stepwise manner as molecular levels begin to enterthe bias window between µ L and µ R (Ref. 24). We shallconsider both of these phenomena below.If a small temperature difference ∆ T = T L − T R atan average temperature T = ( T L + T R ) / In an open-circuit situation, where the net current I must vanish, a thermoelectric voltage ∆ µ/e is generatedto balance the thermal diffusion of charge carriers. Inthe linear-response regime the proportionality constant S = − (∆ µ/e ∆ T ) I =0 is the Seebeck coefficient. We willbriefly consider this quantity below as an example of anobservable with a linear dependence on the molecularlength N , but will not enter a more detailed discussionof thermoelectricity or heat transport.The quantity we are most interested in is the dc cur-rent in the presence of monochromatic electromagneticradiation, which we refer to as light independently of itssource or frequency ω . We model the light as an ac volt-age with harmonic time-dependence V ( t ) = V ac cos( ωt )over the contact. The current averaged over one periodof V ( t ) can be written in the form I ( V ; α, ω ) = 2 eh ∞ X k = −∞ Z dE [ τ ( k ) RL ( E, V ; α, ω ) × f L ( E ) − τ ( k ) LR ( E, V ; α, ω ) f R ( E )] . (2)Here the transmission coefficient τ ( k ) RL ( E ), for example,describes photoassisted processes taking an electron fromleft ( L ) to right ( R ), under the absorption of a total of k photons with energy ¯ hω . The parameter α = eV ac / ¯ hω describes the strength of the ac drive. It is deter-mined by the intensity of the incident light and possi-ble field-enhancement effects taking place in the metallic Σ
11 33 Σ H HH H
12 2321 32
H H H
11 22 33
FIG. 1: (Color online) A finite block chain of length N =3connected to electrodes at its two ends. This gives rise to selfenergies Σ and Σ NN on the terminating blocks. nanocontact. Again, in addition to the full I - V char-acteristics, we study in more detail the case of linearresponse with respect to the dc bias, i.e., the photocon-ductance G dc ( α, ω ) = ∂I ( V ; α, ω ) /∂V | V =0 . The argu-ments α and ω distinguish it from the conductance G dc ,although we sometimes omit α for notational simplicity.The calculation of the coefficients τ ( k ) RL/LR ( E ) is rathercomplicated in general, and we defer comments on thisprocedure to App. B. Below we shall mostly refer to anapproximate formula (see App. A) that can be expressedin terms of τ ( E ). This amounts to a treatment of theproblem on the level of the Tien-Gordon approach. The full Green-function formalism for systems involvingac driving is presented in Ref. 4.In noninteracting (non-self-consistent) models it is ingeneral not clear how the voltage drop should be di-vided between the different regions of the wire, andthe electrode-wire interfaces. A self-consistent treatmentwould be in order in particular for asymmetrically cou-pled molecules. We only concentrate on left-right sym-metric junctions, where where both the dc and ac volt-ages ( V and V ac ) are assumed to drop according to oneof two different symmetrical profiles. The symmetry ofthe junctions excludes rectification effects, such as light-induced dc photocurrents in the absence of a dc biasvoltage. However, light can still have a strong in-fluence on the transmission properties of the molecularcontact, as will be discussed below. It will be shownthat our conclusions are essentially independent of theassumed voltage profile.
B. Wire models
Below we will specialize to the case of a metal-oligophenylene-metal junction. However, to make somegeneral remarks, let us first consider a larger class ofmolecular wires that can be described as N separateunits forming a chain, where only the nearest neighborsare coupled (see Fig. 1). We only discuss the calcula-tion of the elastic transmission function τ ( E, V ) here, asthis will be the focus of our analytical considerations inSec. III. From this quantity (at V = 0), the variouslinear-response coefficients such as the conductance andthe thermopower can be extracted. Furthermore, as al-ready mentioned, it suffices for an approximate treatmentof the amplitudes τ ( k ) RL ( E ) as well. We assume a basis | χ ( α ) p i of local (atomic) orbitals,where p = 1 , . . . , N indexes the unit, while α = 1 , . . . , M p denotes the orbitals in each unit. For simplicity, thebasis is taken to be orthonormal, i.e. h χ ( α ) p | χ ( β ) q i = δ αβ δ pq . The (time-independent) Hamiltonian H ( α,β ) pq = h χ ( α ) p | ˆ H | χ ( β ) q i of the wire is then of the block-tridiagonalform H = H H H H H . . . . . . . . . H N − ,N − H N − ,N − H N − ,N H N,N − H NN , (3)where H pq with p, q = 1 , . . . , N are M p × M q matrices.(The unindicated matrix elements are all zeros.)In the non-equilibrium Green-function picture, the ef-fect of coupling the chain to the electrodes is described interms of “lead self energies”. We assume these to be lo-cated only on the terminal blocks of the chain, with com-ponents Σ and Σ NN . The inverse of the stationary-state retarded propagator for the coupled chain will thenbe of the form F = F h h h h . . . . . . . . . h N − ,N − h N − ,N − h N − ,N h N,N − F NN . (4)Here h p,p ± = − H p,p ± , h pp = E + pp − H pp , and E + = E + i + , while F = h − Σ and F NN = h NN − Σ NN .Charge-transfer effects between the molecule and themetallic electrodes shift the molecular levels with respectto the Fermi energy E F . In a TB model, these can be rep-resented by shifting the diagonal elements of H . Once atransport voltage V is applied, further shifts are induced.In our model the voltage-induced shifts will be taken fromsimple model profiles, and the relative position of E F willbe treated as a free parameter.Effective numerical ways of calculating the propagator G = F − for block-tridiagonal Hamiltonians exist. In Sec. III we shall be interested in a special case, where H p,p − = H − , H p,p +1 = H and H pp = H withthe same H = H T − and H (of dimension M p = M )for all p , describing an oligomer of identical monomericunits. In such cases also analytical progress in calculatingthe current in Eq. (1) may be possible. Once the Greenfunction G is known, the transmission function is givenby τ ( E, V ) = Tr[ Γ G N Γ NN ( G N ) † ] , (5)where Γ = − Σ and Σ ( E, V ) = Σ ( E − eV / E F lies within the HOMO-LUMO gap, re-sulting in the exponential decay τ ( E F ) ∼ e − β ( E F ) N with N , characteristic of off-resonant transport. The decay co-efficient β ( E F ) is actually independent of Σ and Σ NN .This can be seen by considering the Dyson equation G = G + G Σ G , where G and G are the Green functionof the coupled and uncoupled wires, respectively, and Σ is the matrix for the lead self-energies. Assuming that G N decays exponentially with N , then G N ≈ ( − G Σ ) − G N (6)when N → ∞ , and therefore G N decays with the sameexponent. Thus, one can in principle obtain the decayexponent from the propagator of an isolated molecule, oreven an infinitely extended polymer. In the next Sectionwe demonstrate this by extracting the decay exponentof a finite oligophenylene junction from the propagatorfor polyphenylene. We note that in doing so, we neglectthe practical difficulty of determining the correct relativeposition of E F .There are efficient numerical methods for computingthe lead self-energies for different types of electrodes andvarious bonding situations between them and the wire.Typically, the methods are based on the calculation ofsurface Green’s functions. Below we shall simply treatthe self-energies as parameters.
III. PHENYL-RING-BASED WIRES
In this Section we discuss a special case of the typeof wire model introduced above, describing an oligomerof phenyl rings coupled to each other via the para ( p )position. The bias voltage V is assumed to be zero.In the special case that we will consider, the inver-sion of Eq. (4) can then be done analytically with thesubdeterminant method familiar from elementary linearalgebra. Below, we first use this method for cal-culating the propagator of the finite-wire junction andderive the decay exponent β ( E ) of the transmission func-tion at off-resonant energies. After that we rederivethe decay exponent by considering an infinitely extendedpolymer of phenyl rings. A. Oligo- p -phenylene junction Our model for the oligophenylene-based molecularjunction is depicted in Fig. 2. Within a simple π -electronpicture, the electronic structure of the oligophenylenemolecule can be described with a nearest-neighbor TBmodel with two different hopping elements − γ and − η (Ref. 52). Here − γ is for hopping within a phenyl ring,between the p orbitals oriented perpendicular to thering plane, while − η describes hopping between adjacentrings. Due to the symmetry of the orbitals, the mag-nitude of η depends on the angle ϕ between the ringsproportionally to cos ϕ (Ref. 53). We shall assume that η = γ cos ϕ , and thus | η | ≤ γ . In this way the naturalenergy scale of the model is set by γ alone. ( α ) ( α ) ( α )
11 1 2 3 33
Σ ε ε ε Σ −η −η −γ−γ −γ−γ −γ−γ
FIG. 2: (Color online) A finite chain of length N =3 connectedto electrodes at its two ends. This gives rise to self energies Σ and Σ NN on the end sites. The nearest-neighbor hop-pings inside the ring ( − γ ) and between the rings ( − η ) aredifferent. The lower part indicates also the numbering of the M = 6 carbon atoms within a ring. The ring-tilt angle ϕ can be controlled to some ex-tent using side groups. For example, two side groupsbonded to adjacent phenyl rings can repel each other ster-ically, thus increasing the corresponding tilt angle. Infact, even the pure oligophenylenes in the uncharged statehave ϕ = 30 ◦ − ◦ due to the repulsion of the hydro-gen atoms. However, side groups can introduce also“charging” or “doping” effects, which shift the molecularlevels. For definiteness, we number the M = 6 carbon atomsof a phenyl ring according to the lower part of Fig. 2. Thecorresponding orbitals appear in the basis in this order.Thus the blocks in Eq. (3) are H q,q = ǫ (1) q − γ − γ − γ ǫ (2) q − γ − γ ǫ (3) q − γ − γ ǫ (4) q − γ − γ ǫ (5) q − γ − γ − γ ǫ (6) q (7)for q = 1 , . . . , N and H q,q − = − η , (8)with H q +1 ,q = [ H q,q +1 ] T . Here the onsite energies ǫ ( α ) q may be shifted non-uniformly to describe effects of pos-sible side-groups. For simplicity, we shall consider allphenyl rings to have a similar chemical environment, andthus all onsite energies are taken to be equal.As a first step we note that, assuming ǫ ( α ) q = ǫ q forall α , the eigenvalues for the Hamiltonian H qq of theisolated unit are ǫ q − γ , ǫ q + γ , ǫ q − γ , ǫ q + γ , ǫ q − γ , ǫ q +2 γ ,while the corresponding orthonormalized eigenvectors are1 √ , − , , − , , T , √ , , − , − , , T , √
12 ( − , − , − , , , T , √
12 (2 , − , − , − , − , T , √ , , , , , T , √ − , , , − , − , T . (9)The first two of the eigenstates have zero weight on thering-connecting carbon atoms 1 and 6. Therefore, theseeigenstates do not hybridize with the levels of the ad-jacent rings and consequently cannot take part in thetransport. This will be seen explicitly in the derivationof the propagator. We note that these results can alsobe used to determine a realistic value for the hopping γ from the HOMO-LUMO splitting of benzene. Below we shall only consider the analytically solvablecase, where all onsite energies are set to the same value.We choose this value as our zero of energy: ǫ ( α ) q = 0 for all q = 1 , . . . , N and α = 1 , . . . , M . Later on we shall relaxthis assumption in order to describe externally applied dcand ac voltage profiles. In the absence of such voltages,the inverse propagator [Eq. (4)] consists of the blocks h p,p = h , h p,p − = h − , and h p,p +1 = h , where h = E + γ γ γ E + γ γ E + γ γ E + γ γ E + γ γ γ E + , h − = η (10)and h = [ h − ] T . The leads are assumed to coupleonly to the terminal carbon atoms, thus making the self-energies 6 × Σ = Σ L · · ·
00 0 · · · , Σ NN = · · · · · · R . (11)We also define the symbol “tilde” (˜), which means thereplacement of the first column of a matrix by η followedby zeros. For example˜ h = η γ γ E + γ E + γ γ E + γ γ E + γ γ γ E + . (12) For the evaluation of Eq. (5), we only need the com-ponent G ,MN = [ G N ] M . Using the subdeterminantsof F = G − , we have G ,MN = ( − MN +1 det[ F ( M N | F ] . (13)Here O ( i, . . . , k | j, . . . , l ) is the submatrix of O obtainedby removing the rows i, . . . , k and columns j, . . . , l . Weshall also denote by L and R the “leftmost” and “right-most” row or column of a matrix. Thus, for exampledet[ F ( M N | F ( R | L )]Let us first concentrate on the denominator of Eq.(13). It is easy to see that det[ F ] can be written interms of determinants related to the inverse Green func-tion F = G − of the uncoupled wire as follows det[ F ] = det[ F ] − Σ L det[ F ( L | L )] − Σ R det[ F ( R | R )]+ Σ L Σ R det[ F ( L, R | L, R )] . (14)Furthermore, due to the symmetry of the molecule,det[ F ( R | R )] = det[ F ( L | L )]. Thus we are left with calcu-lating three types of determinants. It can be shown that,for 1 < n < N , all of them satisfy a recursion relation ofthe form (cid:18) D ( n ) ˜ D ( n ) (cid:19) = ( E − γ ) Y (cid:18) D ( n − ˜ D ( n − (cid:19) = ( E − γ ) (cid:18) a − cc b (cid:19) (cid:18) D ( n − ˜ D ( n − (cid:19) . (15)For example, in the calculation of det[ F ], we have D ( n ) =det[ F ( n ) ] and ˜ D ( n ) = det[ ˜ F ( n ) ], where the additionalsuperscript ( n ) on the matrices denotes the number ofthe M × M diagonal blocks. The elements of the matrix Y are given by a = ( E − γ )( E − γ ) b = − η ( E − γ ) c = ηE + ( E − γ ) . (16)Only the initial condition ( n = 1) and the last step of therecursion ( n = N ) will differ for the three determinants.The recursion relations can be solved by calculating Y n explicitly, which can be done by diagonalizing Y . Theeigenvalues of Y are λ , = ( a + b ∓ p ( a − b ) − c ) / v , = a − b ∓ p ( a − b ) − c c , ! T . (17)Then, if V = ( v , v ) and Λ = diag( λ , λ ), we have Y n = V Λ n V − . The result is Y n = y ( n )11 y ( n )12 y ( n )21 y ( n )22 ! , (18)where the components are given by y ( n )11 = ( λ n − λ n )( b − a ) + ( λ n + λ n ) p ( a − b ) − c p ( a − b ) − c y ( n )22 = ( λ n − λ n )( a − b ) + ( λ n + λ n ) p ( a − b ) − c p ( a − b ) − c y ( n )12 = − y ( n )21 = c ( λ n − λ n ) p ( a − b ) − c . (19)Using these, we can now write explicit expressions for thethree required determinants. For det[ F ], the recursioncan be started at n = 1 with the initial conditions D (0) =1 and ˜ D (0) = 0 and carried out up to n = N . The resultis det[ F ( N ) ] = ( E − γ ) N y ( N )11 . (20)The other two determinants require special initial andfinal steps, and the results aredet[ F ( N ) ( L | L )] =( E − γ ) N y ( N )21 /η det[ F ( N ) ( L, R | L, R )] =( E − γ ) N [ y ( N − c − y ( N − b ] /η . (21)Next, we consider the determinant in the numeratorof Eq. (13), det[ F ( N ) ( R | L )] = det[ F ( N ) ( R | L )]. It caneasily be shown that it satisfies the recursion relationdet[ F ( N ) ( R | L )] = 2 ηγ ( E − γ ) det[ F ( N − ( R | L )](22)and sodet[ F ( N ) ( R | L )] = 2 N ( ηγ ) N ( E − γ ) N /η. (23)Now, the Green function of Eq. (13) can be written as G ,MN = − (2 ηγ ) N /ηy ( N )11 + Σ LR y ( N )21 /η + Σ L Σ R (cid:16) y ( N − c − y ( N − b (cid:17) /η , (24)where we used the shorthand Σ LR = Σ L + Σ R .It is notable that the common ( E − γ ) N factors can-celed out from the final propagator. These factors ap-parently correspond to the two eigenvectors of h [Eq.(9)] having zero weight on the ring-connecting atoms 1and 6. The cancellation is a manifestation of the physicalfact that such localized states cannot contribute to thetransport through the molecule. In the infinite polymerto be discussed below, these states appear as completelyflat bands in the band structure.To conclude this part, we point out that for E insidethe HOMO-LUMO gap [more precisely, when ( a − b ) − c >
0] the eigenvalues λ , are real-valued and the de-cay exponent of the transmission τ ( E ) for large N is con-trolled by the one with a larger absolute value. Since (a)(b) N −η −γ−γ −γ−γ−γ−γ FIG. 3: (Color online) Phenyl-ring chains: (a) a periodicchain with N units and (b) an infinite chain. Case (b) isobtained from (a) in the limit N → ∞ . inside the gap E ≈
0, we find that λ > λ >
0. Then,using Eq. (5) and omitting N -independent prefactors, thedecay of the transmission for large N follows the law τ ( E ) ∼ (cid:20) λ ( E )2 ηγ (cid:21) − N = e − N ln[ λ ( E ) / (2 ηγ )] . (25)Thus the decay exponent is given by β ( E ) = 2 ln[ λ ( E ) / (2 ηγ )] . (26)We note that for resonant energies, oscillatory depen-dence of τ ( E ) on N can be expected, instead, and forlimiting cases also power-law decay is possible. Next,we shall reproduce the result for the decay exponent byconsidering an infinitely extended polymer.
B. Poly- p -phenylene For comparison with the “correct” evaluation of thepropagator and the decay coefficient for a finite chain,let us consider the propagator for an infinitely extendedpolymer. To describe the polymer, we start from a finitechain with periodic boundary conditions. Neglecting cur-vature effects, the latter actually represents a ring-shapedoligomer, as depicted in Fig. 3(a).Let us first consider the eigenstates of the periodicchain. The Hamiltonian H ( α,β ) pq = h χ ( α ) p | ˆ H | χ ( β ) q i is ofthe general form H = H H H − H − H H . . . . . . . . . H − H H H H − H , (27)where H , ± are the M × M matrices ( M = 6) of Eqs.(7) and (8), with ǫ ( α ) q = 0. (Again, only nonzero elementsare indicated.) The normalized eigenvectors ψ ( n ) p ( k ) sat-isfying X q H pq ψ ( n ) q ( k ) = E ( n ) ( k ) ψ ( n ) p ( k ) (28)are of the Bloch form ψ ( n ) q ( k ) = e ikqd φ ( n ) ( k ) / √ N , where φ ( n ) ( k ) are the normalized eigenvectors of H ( k ) = e ikd H + H + e − ikd H − (29)with the eigenvalue E ( n ) ( k ), and n = 1 , . . . , M . Due tothe finiteness of the wire, the k values are restricted to k µ = 2 πµ/N d , where µ is an integer and d is the latticeconstant (the length of a single phenyl-ring unit).The spectral decomposition of the (retarded) propaga-tor g ( E ) = ( E + − H ) − of the chain is of the form g ( α,β ) pq ( E ) = X µ,n h χ ( α ) p | ψ ( n ) ( k µ ) ih ψ ( n ) ( k µ ) | χ ( β ) q i E + − E ( n ) ( k µ ) , (30)with the Bloch states | ψ ( n ) ( k µ ) i = 1 √ N ⌊ N/ ⌋ X p = −⌈ N/ ⌉ +1 e ik µ pd M X α =1 φ ( n ) α ( k µ ) | χ ( α ) p i . (31)In the limit of large N [Fig. 3(b)], we can use N − P µ → ( d/ π ) R π/d − π/d dk to turn the summation into an integralover the first Brillouin zone. In this case, there are M = 6bands with energies E (1 , ( k ) = ± γE (3 , ( k ) = ± √ p η + 5 γ − B ( k ) E (5 , ( k ) = ± √ p η + 5 γ + 2 B ( k ) , (32)where B ( k ) = 12 p ( η + 3 γ ) + 16 ηγ cos( kd ) . (33)Clearly we have the symmetries E (1) ( k ) = − E (2) ( k ), E (3) ( k ) = − E (4) ( k ), and E (5) ( k ) = − E (6) ( k ). For n =1 , φ (1 , ( k ) are as in Eq. (9), i.e., independentof k and completely localized on atoms α = 2 , , , p = q , they do not contribute to the propaga-tor in Eq. (30). For n = 3 , , ,
6, the vectors are verycomplicated, but they are not needed in the following.To compare with the result of Sec. III A, we should nowcalculate, for example, the component g (1 , pq . However,expecting the decay exponent to be independent of α and β , we consider the simpler case Tr[ g pq ] = P α g ( α,α ) pq . Dueto the orthonormality P α φ ( m ) α φ ( n ) ∗ α = δ mn , the depen-dence on the vector components then drops out. Thus,for p = q X α g ( α,α ) pq = 4 EA d π Z π/d − π/d dk e ikd ( p − q ) A − B ( k ) , (34) where we defined A = E −
12 ( η + 5 γ ) , (35)such that E − [ ǫ (3 , ( k )] = A ± B ( k ). Defining now z = e ikd , the integral can be turned into a contour integralaround the contour | z | = 1 X α g ( α,α ) pq = − EA πiηγ I | z | =1 dz z p − q ( z − z + )( z − z − ) , (36)where the poles z ± are determined from the equation z − [4 A − ( η + 3 γ ) ](8 ηγ ) − z + 1 = 0. They aregiven by z ± = 4 A − ( η + 3 γ ) ηγ ± s(cid:20) A − ( η + 3 γ ) ηγ (cid:21) − z + = 1 /z − , and we choose the signs so that z − is inside the contour | z | = 1. In addition to this,assuming that p < q , there is a pole of order q − p at z = 0. The integral can then be evaluated using residuetechniques, with the result X α g ( α,α ) pq = 2 EAηγ z p − q + z + − z − . (38)This leads to an exponential decay of the propagator withgrowing q − p >
0, when E is off-resonant (in whichcase z ± are real-valued). Using this result, we can givean estimate for the decay of the transmission function[Eq. (5)] through a finite chain of length N by replacing G ,MN with Tr[ g N ] /M . This yields τ ( E ) ∼ [ z + ( E )] − N = e − N ln[ z + ( E )] , (39)and thus the exponent β ( E ) = 2 ln[ z + ( E )] . (40)It can be checked that this result is, in fact, equal to theresult [Eq. (26)] obtained for the finite chain.It is thus seen explicitly that the decay coefficient ofthe off-resonant transmission does not in any way dependon the coupling of the molecule to the leads. It should bekept in mind, however, that the relative position of E F within the HOMO-LUMO gap depends on the electrode-lead coupling and the charge transfer effects. This infor-mation is still needed for predicting the decay exponent β ( E F ) of the conductance.The analytical results presented in this and the previ-ous section can be used for understanding the behavior ofthe transmission function upon changes in the parame-ters. For example, it should be noted that when η is madesmaller, the band gap around E ≈ β ( E ) grows. Inthis way, the conductance of a molecular junction canbe controlled, for example, by introducing side groups tocontrol the tilt angles ϕ between the phenyl rings. d / 6 d / 3 d / 3 d / 3 (a) d L (b) ( z ) P BA z FIG. 4: (Color online) (a) The coordinates of the carbonatoms in the direction z along the molecular wire. The leftelectrode is at z = 0 and the length of a phenyl-ring unit is d .(b) Relative variation of the onsite energies for two differentvoltage profiles, A and B. The profile function P ( z ) describeshow the harmonic voltage V ( t ) = V + V ac cos( ωt ) is assumedto drop over the junction, the voltage at z being given by V ( z, t ) = V ( t ) P ( z ). IV. PHYSICAL OBSERVABLES ANDNUMERICAL RESULTS
In this Section we present numerical results based onour model. Throughout, we employ the “wide-band” ap-proximation for the lead self-energies, such that Σ L ( E ) = − i Γ L / R ( E ) = − i Γ R /
2, with energy-independentconstants Γ
L,R . Furthermore we only consider the sym-metric case Γ L = Γ R = Γ. First we briefly describehow we generalize the theory, as presented above, to takeinto account static and time-dependent voltage profiles.Then we concentrate on near-equilibrium (or “linear-response”) properties, using as examples the conduc-tance, the thermopower, and the conductance enhance-ment due to light with low intensity and frequency. Inthis case, knowledge of the zero-bias transmission func-tion calculated above is sufficient, and we can discuss thelength dependence of the transport properties in a simpleway. After that we consider the dc current in the pres-ence of an ac driving field of more general amplitude andfrequency, first concentrating on the case of infinitesimaldc bias, and finally on the I - V characteristics. A. Voltage profiles
When considering finite dc or ac biases within a non-selfconsistent TB model that cannot account for screen-ing effects, one of the obvious problems is how to choosethe voltage profile. Throughout the discussion, we shallrefer to two possible choices, as depicted in Fig. 4. Theyare in some sense limiting cases, and the physically mostreasonable choice should lie somewhere in between. Pro-file A assumes the external electric fields to be completelyscreened inside the molecule, such that the onsite energiesare not modified, while B corresponds to the complete ab-sence of such screening. In both cases, we can write thetime-dependent onsite energies as ǫ ( α ) p ( t ) = eV ( t ) P ( z ( α ) p ), where z ( α ) p are the distances of the carbon atoms fromthe left metal surface, and V ( t ) = V + V ac cos( ωt ). Incase A, P ( z ) = 0 inside the junction, while in case B P ( z ) = ( L − z ) / (2 L ), where L = N d + d/ I - V characteristics can be calculated based on the knowledgeof the zero-bias transmission function in the absence oflight, τ ( E ). As discussed in App. A, the current is givenby I ( V ; α, ω ) = 2 eh ∞ X l = −∞ h J l (cid:16) α (cid:17)i Z dEτ ( E + l ¯ hω ) × [ f L ( E ) − f R ( E )] . (41)The low-temperature zero-bias conductance then takesthe particularly simple form G dc ( α, ω ) = G ∞ X l = −∞ h J l (cid:16) α (cid:17)i τ ( E F + l ¯ hω ) . (42)Here l indexes the number of absorbed or emitted pho-tons, J l ( x ) is a Bessel function of the first kind (oforder l ), and α = eV ac / ¯ hω is the dimensionless pa-rameter describing the strength of the ac drive. Notethat G dc ( α, ω = 0) = G dc ( α = 0 , ω ) = G τ ( E F ) = G dc . Equation (41) may equally well be written in theform I ( V ; α, ω ) = ∞ X l = −∞ h J l (cid:16) α (cid:17)i I ( V + 2 l ¯ hω/e ) , (43)where I ( V ) is the I - V characteristic in the absence oflight [Eq. (1)]. Below, the results from these formulas arecompared to the numerical results for profile B.In Fig. 5 we plot the zero-bias transmission functionsfor wires with N between 1 and 7. Notice that the fourenergy bands numbered 3-6 in Eq. (32) are all visible,being separated by the HOMO-LUMO gap at E/γ ≈ E/γ ≈ ± .
7. Here we use theparameters Γ /γ = 5 . ϕ = 40 ◦ (i.e. η/γ ≈ . E F /γ = − .
4. These values are closeto those used in Ref. 36, where they were extracted from afit to results for gold-oligophenylene-gold contacts basedon density-functional theory (DFT). We shall continueto use them everywhere below. A DFT calculation forthe HOMO-LUMO splitting of benzene, together withthe results preceding Eq. (9), yields the hopping γ ≈ d = 0 .
44 nm, and the largest ac electric fields V ac /L considered will be on the order of 10 V/m. The photonenergies ¯ hω will mainly be kept below the energy of theHOMO-LUMO gap of the oligophenylene. -2 -1 0 1 2E / γ -6 -4 -2 τ ( E ) N=1 N=7E F FIG. 5: (Color online) Transmission functions for theoligophenylene wires with lengths N = 1 , , ,
7. The param-eters are Γ /γ = 0 . ϕ = 40 ◦ , E F /γ = − .
4, as discussed inthe text. -8 -6 -4 -2 G d c / G N S / ( k B π T / e γ ) N ( ∆ G d c / G d c ) / ( α h _ ω / γ ) (a)(b) (c) FIG. 6: (Color online) Dependence of observables on the num-ber of units N : (a) Conductance, (b) Seebeck coefficient, and(c) the light-induced relative conductance-enhancement. Thecircles correspond to values extracted from the τ ( E ) function[Fig. 5], using Eqs. (44), (45), and (46). The red lines cor-respond to the simple order-of magnitude estimates of Eqs.(47), (48), and (49), with the analytically calculated β ( E ). In(c) the crosses ( × for profile A and + for profile B) show nu-merical results with the finite values α = 0 . hω/γ = 0 . B. Near-equilibrium properties
Let us start by illustrating the usefulness of the ana-lytical results of Sec. III with a few examples. We con-centrate on low temperatures and small deviations fromequilibrium. In addition to the linear-response conduc-tance G dc = G τ ( E F ) , (44)we shall consider the thermopower, or Seebeck coefficient.At low enough temperature T , this is given in terms of the zero-bias transmission function τ ( E ) as S = − π k B T e τ ′ ( E F ) τ ( E F ) , (45)where prime denotes a derivative. Thus it measures thelogarithmic first derivative of the transmission functionat E = E F . The sign of this quantity carries informa-tion about the location of the Fermi energy within theHOMO-LUMO gap of molecular junction. The thirdquantity we shall consider is the photoconductance. Inthe limit α ≪ hω/γ ≪ τ ( E ) andthe Bessel functions in Eq. (42) (see App. A) to lead-ing order in these small quantities, yielding G dc ( ω ) = G τ ( E F )+ G ( α ¯ hω ) τ ′′ ( E F ) /
16. Defining then the light-induced conductance correction ∆ G dc ( ω ) = G dc ( ω ) − G dc ( ω = 0), where G dc ( ω = 0) = G dc = G τ ( E F ), therelative correction becomes∆ G dc ( α, ω ) G dc = ( α ¯ hω ) τ ′′ ( E F ) τ ( E F ) . (46)We thus see that this quantity gives experimental ac-cess to the second derivative of the transmission functionat E = E F . Note that in this approximation, whichcan be seen as an adiabatic or “classical” limit, theconductance correction depends only on the driving fieldthrough the ac amplitude V ac = α ¯ hω/e .As discussed above, it is reasonable to assume that forlarge enough N , the transmission function τ ( E ) satisfiesthe exponential decay law τ ( E ) ∼ C ( E ) e − β ( E ) N (47)at the off-resonant energies E ≈ E F . Let us furthermoreassume that C ( E ) is only weakly E -dependent. Then itis clear that the Seebeck coefficient will have the followingsimple linear dependence on N (Refs. 28,36): S ∝ τ ′ ( E F ) /τ ( E F ) ∼ − β ′ ( E F ) N. (48)In contrast, the light-induced conductance correction sat-isfies a quadratic law∆ G dc ( ω ) /G dc ∝ τ ′′ ( E F ) /τ ( E F ) ∼ − β ′′ ( E F ) N + [ β ′ ( E F )] N . (49)Deviations from these laws can follow from the energy-dependence of C ( E ).In Fig. 6 we demonstrate these length dependenceswithin our model for the oligophenylene junctions. Thecircles connected by lines show the results based on thetransmission functions of Fig. 5, using Eqs. (44), (45),and (46). The separate solid lines are the estimates ofEqs. (47), (48), and (49), based on the analytic result for β ( E ). The result for ∆ G dc ( ω ) /G dc is furthermore com-pared with some example results for finite α and ω , using α = 0 . hω/γ = 0 .
05 (see below). Although Eq. (46)was derived above by assuming the profile A, the resultappears to be rather well satisfied for profile B as well.0 -2 -1 G d c ( ω ) / G h _ ω / γ -3 -2 -1 G d c ( ω ) / G h _ ω / γ N=1 N=2N=3 N=4 α =2.0 α =0.5 (a) (b)(c) (d) FIG. 7: (Color online) Zero-bias conductance for differentdriving frequencies ω and driving strengths α = eV ac / ¯ hω .Panels (a)-(d) are for N = 1 , . . . ,
4. The solid lines correspondto profile A, and the dashed lines to profile B. The lower pairof curves is for α = 0 .
5, and the upper pair for α = 2 . -6 -5 -4 -3 -2 -1 G d c ( ω = . γ / h _ ) / G AB FIG. 8: (Color online) Dependence of the conductance on N . Circles represent the conductance in the absence of light,while the squares are for light with ¯ hω/γ = 0 . α = 1 . C. Zero-bias conductance at finite drivefrequencies and amplitudes
Next we consider the zero-bias photoconductance G dc ( ω ) for light whose frequencies and intensities arenot restricted to the adiabatic limit. We have discussedthis case previously, based on DFT results for gold-oligophenylene-gold contacts. There, however, the anal-ysis was based solely on the simple formula of Eq. (42).Here we show that those results are not expected tochange in an essential way within a more refined theory,since the results of our TB model are not very different for the two voltage profiles A and B. This is seen in Fig.7, where we show G dc ( ω ) for N = 1 , . . . , ω for two values of α , and for both profiles. The re-sults for profile A again follow from Eq. (42), but theresults for B require a more demanding numerical calcu-lation (see App. B). In both cases the effect of light is toincrease the conductance considerably. The physical rea-son is that the photoassisted processes, where electronsemit or absorb radiation quanta, brings the electrons toenergies outside of the HOMO-LUMO gap, where thetransmission probability is higher. This happens when¯ hω exceeds the energy difference between the Fermi en-ergy and the closest molecular orbital, in this case theHOMO. The main difference between the two profiles isthat in case B, the sharp resonances at some frequenciesare smeared out, and thus the light-induced conductanceenhancement tends to be smaller. The increase can stillbe an order of magnitude or more.The dependence of this effect on the length of themolecule is still illustrated in Fig. 8, where the conduc-tances in the absence of light and in the presence of lightwith ¯ hω/γ = 0 . α = 1 . N . While the conductance in the absence of light hasa strong exponential decay, in the presence of light thisdecay is much slower. For profile A the conductance ac-tually oscillates periodically, while in the case of profileB the oscillations are superimposed on a background ofslow exponential decay. In the DFT-based results theoscillations were not present, or at least not visible forthe cases N = 1 , . . . , π -orbital contributions, as well asuses the wide-band approximation.The results of Fig. 8 can also be stated in terms of therelative conductance-enhancement ∆ G dc ( ω ) /G dc . Forlarge α and ω , the increase of this quantity with N isexponential for both profiles A and B. This should becontrasted with the quadratic behavior for small α and ω [Eq. (49)]. Thus, the fact that the results indicatedby the crosses in Fig. 6 exceed the result of Eq. (46) isunderstandable. D. Current-voltage characteristics
Finally we discuss the effects of light at finite voltages V . Let us first consider the properties of the I - V charac-teristics in the absence of light. Examples are shown inFig. 9(a) for the case N = 5. They consist of consecutivesteps, which appear every time a new molecular levelcomes into the bias window between µ L and µ R . Thesesteps are seen as the peaks 1 and 2 in the differential con-ductance dI/dV shown in Fig. 9(b). The first one occursroughly at the voltage V = 2( E F − E HOMO ) /e , where E HOMO is the energy of the HOMO. The factor 2 arisesfrom the symmetric division of the voltages with respectto the molecular energy levels. In the case of profile B,the currents tend to be smaller than for profile A, but1 e I / ( G γ ) γ ( d I / d V ) / G γ -3 -2 -1 ( d I / d V ) / G (b)(a) N=5 lightlight (c) light 2 h _ ω / γ A B1 2
FIG. 9: (Color online) (a) I - V characteristics ( N = 5) withand without light for profiles A (solid lines) and B (dashedlines). Results in the presence of light with α = 1 . hω/γ = 0 .
075 are indicated with an arrow. (b) The corre-sponding differential conductances. (c) Same as (b), but con-centrating on the low-bias regime and on a logarithmic scale.The vertical dashed lines indicate the approximate positionsof the main peak and the light-induced side peaks. They areall separated by 2¯ hω/e in voltage. -1 -0.5 0 0.5 1 1.5E / γ -5 -4 -3 -2 -1 τ ( E , V ) γ E F FIG. 10: (Color online) The voltage-dependent transmissionfunction at three voltages for the wire with N = 5 and profileB. For profile A the result is independent of voltage and equalto τ ( E, V = 0). the current steps occur at roughly the same voltages. Itshould also be noticed that for profile B, a small negativedifferential conductance is present following some of thesteps. The origin of this is the localization of the molec-ular eigenstates due to the dc voltage ramp, which sup-presses the transmission resonances. This can be seen eV / γ -2 -1 ( d I / d V ) / G eV / γ eV / γ -3 -2 -1 ( d I / d V ) / G eV / γ (a) (b)(c) (d)N=1 N=2N=3 N=4light FIG. 11: (Color online) Same as Fig. 9(c) but for wires with N = 1 , . . . , in the voltage-dependent transmission functions τ ( E, V )in Fig. 10.In the presence of light, the step structure of the I - V curves is modified. For profile A, the results follow simplyfrom Eq. (41) or (43), but for profile B a fully numeri-cal treatment is again needed. In Fig. 9 the results for α = 1 . hω/γ = 0 .
075 are shown as the curves indi-cated with arrows. In Fig. 9(a) it is seen that the currentfor voltages below the steps is increased, and decreasedabove them. This removes the negative differential con-ductance present in the case of profile B. These changesare associated with the appearance of additional currentsteps. Here we concentrate only on the additional stepsin the low-bias regime at voltages
V < ∼ V , as the relativechanges are largest there. Fig. 9(c) shows the differentialconductance on a logarithmic scale in this voltage region.It can be seen that there are multiple extra peaks belowthe main peak, all of which are separated by voltages2¯ hω/e from each other. These peaks are “images” of themain peak at V = V , and are easily understood based onEq. (43). For profile B all the peaks are moved to slightlysmaller voltages and their spacing is reduced, since finitevoltages tend to also suppress the transmission gap (seeagain Fig. 10). Notice that, in contrast to high dc bi-ases [Fig. 9(a,b)], in the low-bias regime [Fig. 9(c)] theresults depend only weakly on the choice of the voltageprofile. Thus the predictions of the model appear to berobust. To observe the side steps, the radiation frequencyshould be large enough such that the steps are not “lost”under the broadening of the main steps. On the otherhand, it should be small enough to have at least one steppresent. Thus, if the voltage broadening of the mainstep at V = V is approximately ∆ /e , then we require∆ < ∼ ¯ hω < E F − E HOMO .Figure 11 additionally shows the low-bias differentialconductances for N = 1 , . . . ,
4, with other parameterschosen as in Fig. 9(c). It is seen that the effects of lightquickly become weaker, as the length of the moleculedecreases. In the case N = 4, small side peaks are still2observed. Larger effects could be obtained by increasingthe parameter α .Similar-looking additional steps are visible in the I - V characteristics of an extended-H¨uckel model for xylyl-dithiol in Ref. 8. Despite the differences in magnitudes ofparameters, and slight asymmetries in the geometries, itis likely that some of those steps have essentially the sameorigin as explained above. However, the most striking re-sult in that reference was the overall order-of-magnitudeincrease in the current. V. CONCLUSIONS AND DISCUSSION
In this paper we have studied a π -orbital tight-binding model to describe elastic and photoassistedtransport through metal-molecule-metal contacts basedon oligophenylenes. In contrast with simpler linear chainmodels that have previously been studied in great detail,our model describes a specific molecule, and its parame-ters can be directly associated with quantities obtainablefrom DFT simulations, for example. Models of this typecan be of value in analyzing the results of more detailed ab-initio or DFT calculations, and in making at leastqualitative predictions in situations where such calcula-tions would be prohibitively costly.We first showed that at zero voltage bias the model canbe studied analytically in a similar fashion as the simplerlinear chain models. In particular, we derived an expres-sion for the decay exponent of the off-resonant trans-mission function. We then discussed the length depen-dence of the dc conductance, the thermopower, and therelative light-induced conductance enhancement in thecase of light with a low intensity ( α ) and low frequency( ω ). The conductance enhancement was found to scalequadratically with length. For large α and ω , the rel-ative enhancement increases exponentially with length.Finally it was shown, by numerical calculations, that thecurrent-voltage characteristics are modified in the pres-ence of light by the appearance of side steps with a volt-age spacing 2¯ hω/e . We demonstrated that the predic-tions of the model are robust with respect to variations inthe assumed voltage profiles. This provides further sup-port for our previous results on the photoconductance. In our work, only symmetrical junctions with sym-metrical voltage profiles were studied. Asymmetries canmodify our results through the introduction of rectifica-tion effects, and can change the positions of the light-induced current steps. The experimental observation ofadditional steps with a spacing related to the frequencyof the light would nevertheless provide more compellingevidence for the presence of photoassisted transport thana conductance enhancement alone. The latter can alsohave other causes. We note that the light-induced current steps are similarto the steps observed in current-voltage characteristicsof microwave-irradiated superconducting tunnel junc-tions, where they result from photoassisted quasiparticle tunneling.
In that case, the main difference is thatthe energy gap necessary for the effect is located in themacroscopic electrodes, while the transmission throughthe tunnel barrier depends only weakly on energy andvoltage. As a result, the current steps have a voltagespacing of precisely ¯ hω/e . These effects are exploited inthe detection of microwaves in radioastronomy. Sim-ilarly, one may imagine properly engineered molecularcontacts as detectors of light in the infrared or visiblefrequency range.In terms of our model, to increase the chances of ob-serving the light-induced current steps, the aim shouldbe to minimize the broadening ∆ /e of the first maincurrent step at voltage V , and to maximize α . Also, awire with a large enough V should be used. The broad-ening ∆ is related to the sharpness of the transmissionresonances, and thus to the length of the molecule andits coupling to the electrodes, described by Γ. A decreaseof Γ, however, increases the importance of Coulomb cor-relations. Their effect on photoassisted transport hasrecently been discussed within simple models. In-crease of α through the light intensity, in turn, increasesthe heating of the electrodes and the excitation of lo-cal molecular vibrations. These may affect the geome-try through thermal expansion and structural deforma-tions, but will also give rise to an incoherent componentto the current. At high enough photon energies, alsothe direct excitation of electrons on the molecule may be-come important. The relaxation of such excitations dueto various mechanisms (creation of electron-hole pairs inthe electrodes, spontaneous light emission) should thusalso be considered. Also conformational changes of themolecule are possible. Finally, a proper treatment ofscreening effects on the molecule and in the electrodes,the excitation of plasmons, and their role in the fieldenhancement are other issues that should be studied inmore detail.Of course, for the investigation of most of these is-sues, noninteracting models of the type presented aboveare not sufficient. Strong time-dependent electric fieldsmay have effects that can only be captured by self-consistent theories taking properly into account the elec-tron correlations due to Coulomb interactions. Theseinteractions may influence the electronic structure in away that would, at least, require the parameters of ourmodel to be readjusted in the presence of the light.Even the geometry of the junction can become unsta-ble, and so it should in principle be optimized with thelight-induced effects included. Time-dependent density-functional theory is showing some promise for the treat-ment of such problems. In addition to DFT, alsomore advanced computational schemes are being devel-oped to handle correlation effects.
A systematic in-vestigation of the optical response of metal-molecule-metal contacts, and thus the testing of the predictionsof the simple models, remains an important goalfor future research.3
Acknowledgments
This work was financially supported by the HelmholtzGemeinschaft (Contract No. VH-NG-029), by the DFGwithin the Center for Functional Nanostructures, and bythe EU network BIMORE (Grant No. MRTN-CT-2006-035859). F. Pauly acknowledges the funding of a YoungInvestigator Group at KIT.
APPENDIX A: SIMPLIFIED FORMULA FORTHE TIME-AVERAGED CURRENT
Consider the expression Eq. (2) for the time-averaged(or dc) current. The coefficient τ ( k ) RL ( E ), for example, isthe sum of the transmission probabilities of all transportchannels taking the electron from energy E on the leftto energy E + k ¯ hω on the right. That is, for k > k <
0) it describes electron transmission under the ab-sorption (emission) of k photons. Assuming the wide-band approximation and the voltage profile A, Eq. (2)can be written in the more transparent forms of Eqs. (41)and (43). This can be demonstrated rigorously using theequations of App. B, but it is instructive to consider thefollowing simpler derivation. The idea is the same as inthe “independent channel approximation” of Ref. 7.For now, we allow the ac voltage drops at the L and R lead-molecule interfaces to be asymmetrical. Thus wedefine the quantities α L and α R , satisfying α = α L − α R . Since for profile A there is no voltage drop on themolecule, electronic transitions only occur at the lead-molecule interfaces. Thus the transmission coefficients τ ( k ) RL ( E ) are given by τ ( k ) RL ( E ) = ∞ X l = −∞ [ J l − k ( α R )] τ ( E + l ¯ hω ) [ J l ( α L )] , (A1)where [ J l ( α L )] is the probability for absorbing (emit-ting) l photons on the left interface and [ J l − k ( α R )] theprobability for emitting (absorbing) l − k photons on theright interface. The propagation between the interfacesoccurs elastically at the intermediate energy E + l ¯ hω ,according to the transmission function τ ( E ). A similarexpression holds for τ ( k ) LR ( E ). Using these and the sumformula P ∞ k = −∞ [ J k ( x )] = 1, Eq. (2) leads to I ( V ; α, ω ) = 2 eh ∞ X l = −∞ Z dEτ ( E + l ¯ hω ) × n [ J l ( α L )] f L ( E ) − [ J l ( α R )] f R ( E ) o . (A2)Equation (41) follows by setting α L = α/ α R = − α/
2, and the equivalent form of Eq. (43) follows bychanging summation indices and integration variables.Similarly, other suggestive forms may be derived.
For x ≪ l > J ± l ( x ) ≈ ( ± x/ l /l ! − ( ± x/ l +2 / ( l + 1)!. This can be used inthe limit α ≪
1, ¯ hω/γ ≪ APPENDIX B: GREEN’S-FUNCTION METHODFOR THE TIME-AVERAGED CURRENT
Here we outline the Green-function method usedfor obtaining the results for voltage profile B. Consideragain the dc current of Eq. (2). In the case of a har-monic driving field, it is reasonable to assume the exis-tence of time-reversal invariance, in which case we havethe symmetry τ ( k ) LR ( E ) = τ ( − k ) RL ( E + k ¯ hω ) . (B1)The current expression of Eq. (8) in Ref. 4 was derivedunder this assumption, and that result can be broughtinto the form of Eq. (2). Using the notation of thatreference, the coefficients can be written τ ( k ) RL ( E ) = Tr ω [ ˆ G ( E )ˆΓ ( k ) R ( E ) ˆ G † ( E )ˆΓ (0) L ( E )] τ ( k ) LR ( E ) = Tr ω [ ˆ G ( E )ˆΓ ( k ) L ( E ) ˆ G † ( E )ˆΓ (0) R ( E )] , (B2)where the hats denote the extended “harmonic”matrices and Tr ω a trace over them. In particular, ˆ G is the matrix for the retarded propagatorˆ G ( E ) = [( ˆ E − H ˆ1) − ˆ W − ˆΣ L ( E ) − ˆΣ R ( E )] − , (B3)where H is the Hamiltonian of the wire in the ab-sence of voltage profiles. The matrix ˆ E is defined by[ ˆ E ] m,n = ( E + m ¯ hω ) δ m,n , where m and n are the har-monic indices. Using the wide-band approximation forthe electrodes, the matrices ˆΣ X and ˆΓ ( l ) X are given by[ ˆΣ X ] m,n ( E ) = δ m,n Σ X [ˆΓ ( l ) X ] m,n ( E ) = J m − l ( α X ) J n − l ( α X ) Γ X , (B4)with X = L, R and α L,R = ± α/
2. Here Σ X is the self-energy matrix of lead X (extended to the size of H ), and Γ X = − Σ X . The matrix ˆ W includes the effect of theprofiles for the voltage V ( t ) = V + V ac cos( ωt ). If W ( t ) = W dc + W ac cos( ωt ) is a diagonal matrix consisting of theonsite energies ǫ ( α ) p ( t ), then[ ˆ W ] m,n = W dc δ m,n + 12 W ac ( δ m − ,n + δ m +1 ,n ) . (B5)In this formalism, the time-reversal invariance amountsto ˆ G and ˆΓ ( k ) L,R being symmetric, i.e. ˆ A T = ˆ A .Equation (B1) can then be proved by using therelations [ ˆ G ] m + k,n + k ( E ) = [ ˆ G ] m,n ( E + k ¯ hω ) and[ˆΓ ( l ) X ] m + k,n + k ( E ) = [ˆΓ ( l − k ) X ] m,n ( E + k ¯ hω ). We note thatˆΓ ( l ) X is defined with a different sign of l than in Ref. 4.4 ∗ Electronic address: [email protected] S. J. van der Molen, H. van der Vegte, T. Kudernac,I. Amin, B. L. Feringa, and B. J. van Wees, Nanotech-nology , 310 (2006). D. C. Guhr, D. Rettinger, J. Boneberg, A. Erbe, P. Lei-derer, and E. Scheer, Phys. Rev. Lett. , 086801 (2007). S. Kohler, J. Lehmann, and P. H¨anggi, Phys. Rep. ,379 (2005). J. K. Viljas and J. C. Cuevas, Phys. Rev. B , 075406(2007). J. K. Viljas, F. Pauly, and J. C. Cuevas, Phys. Rev. B ,033403 (2007). J. Buker and G. Kirczenow, Phys. Rev. B , 245306(2002). A. Tikhonov, R. D. Coalson, and Y. Dahnovsky, J. Chem.Phys. , 10909 (2003). A. Tikhonov, R. D. Coalson, and Y. Dahnovsky, J. Chem.Phys. , 567 (2002). M. Galperin and A. Nitzan, Phys. Rev. Lett , 206802(2005). I. Urdaneta, A. Keller, O. Atabek, and V. Mujica, Int. J.Quant. Chem. , 460 (2003). I. Urdaneta, A. Keller, O. Atabek, and V. Mujica, J. Phys.B: At. Mol. Opt. Phys. , 3779 (2005). E. R. Bittner, S. Karabunarliev, and A. Ye, J. Chem. Phys. , 034707 (2005). S. Welack, M. Schreiber, and U. Kleinekath¨ofer, J. Chem.Phys. , 044712 (2006). C. Liu, J. Speyer, I. V. Ovchinnikov, and D. Neuhauser, J.Chem. Phys. , 024705 (2007). G.-Q. Li, M. Schreiber, and U. Kleinekath¨ofer, Europhys.Lett. , 27006 (2007). J. Lehmann, S. Kohler, P. H¨anggi, and A. Nitzan, Phys.Rev. Lett. , 228305 (2002). M. Galperin and A. Nitzan, J. Chem. Phys. , 234709(2006). P. A. Orellana and M. Pacheco, Phys. Rev. B , 115427(2007). U. Harbola, J. B. Maddox, and S. Mukamel, Phys. Rev. B , 075211 (2006). S. Kurth, G. Stefanucci, C.-O. Almbladh, A. Rubio, andE. K. U. Gross, Phys. Rev. B , 035308 (2005). M. Galperin and S. Tretiak, arXiv:0712.1166. H. M. McConnell, J. Chem. Phys. , 508 (1961). V. Mujica, M. Kemp, and M. A. Ratner, J. Chem. Phys. , 6849 (1994). V. Mujica, M. Kemp, A. Roitberg, and M. Ratner, J.Chem. Phys. , 7296 (1996). A. Nitzan, Annu. Rev. Phys. Chem. , 681 (2001). D. Segal, A. Nitzan, and P. H¨anggi, J. Chem. Phys. ,6840 (2003). Y. Asai and H. Fukuyama, Phys. Rev. B , 085431 (2005). D. Segal, Phys. Rev. B , 165426 (2005). A. Painelli, Phys. Rev. B , 155305 (2006). S. K. Maiti, Chem. Phys. , 254 (2007). J. K. Tomfohr and O. F. Sankey, Phys. Rev. B , 245105(2002). V. Mujica, M. Kemp, and M. A. Ratner, J. Chem. Phys. , 6856 (1994). M. P. Samanta, W. Tian, S. Datta, J. I. Henderson, andC. P. Kubiak, Phys. Rev. B , R7626 (1996). H. Dalgleish and G. Kirczenow, Phys. Rev. B , 245431(2006). C. A. Stafford, D. M. Cardamone, and S. Mazumdar, Nan-otechnology , 424014 (2007). F. Pauly, J. K. Viljas, and J. C. Cuevas, arXiv:0709.3588. P. K. Tien and J. P. Gordon, Phys. Rev. , 647 (1963). A. H. Dayem and R. J. Martin, Phys. Rev. Lett. , 246(1962). In general τ ( E, V ) will also depend on T = ( T L + T R ) / T = T L − T R . However, we will either consider ∆ T = 0,or assume a linear-response regime with respect to ∆ T and∆ µ such that the dependence on ∆ T does not play a role.Furthermore, we concetrate on the limit of low tempera-tures and will thus neglect the dependence on T as well. H. B. Akkerman and B. de Boer, J. Phys.: Condens. Mat-ter , 013001 (2008). M. Paulsson and S. Datta, Phys. Rev. B , 241403(R)(2003). P. Reddy, S.-Y. Jang, R. A. Segalman, and A. Majumdar,Science , 1568 (2007). A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B , 5528 (1994). M. Wagner and W. Zwerger, Phys. Rev. B , R10217(1997). S. Grafstr¨om, J. Appl. Phys. , 1717 (2002). G. Platero and R. Aguado, Phys. Rep. , 1 (2004). The electron spin is not considered explicitly in the basis.It only appears as the factor 2 in G and the expressionsfor electrical current. S. Datta,
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