Modeling elastic and photoassisted transport in organic molecular wires : length dependence and current-voltage characteristics

Modeling elastic and photoassisted transport in organic molecular wires:

Length dependence and current-voltage characteristics

J. K. Viljas,^1,2,

*

F. Pauly,^1,2and J. C. Cuevas^3,1,2

1Institut für Theoretische Festkörperphysik and DFG-Center for Functional Nanostructures, Universität Karlsruhe, D-76128 Karlsruhe, Germany

2Forschungszentrum Karlsruhe, Institut für Nanotechnologie, D-76021 Karlsruhe, Germany

3Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid, Spain 共Received 8 January 2008; revised manuscript received 2 March 2008; published 17 April 2008兲

Using a␲-orbital tight-binding model, we study the elastic and photoassisted transport properties of metal- molecule-metal junctions based on oligophenylenes of varying lengths. The effect of monochromatic light is modeled with an ac voltage over the contact. We first show how the low-bias transmission function can be obtained analytically, using methods previously employed for simpler chain models. In particular, the decay coefficient of the off-resonant transmission is extracted by considering both a finite-length chain and infinitely extended polyphenylene. Based on these analytical results, we discuss the length dependence of the linear- response conductance, the thermopower, and the light-induced enhancement of the conductance in the limit of weak intensity and low frequency. In general, the conductance enhancement is calculated numerically as a function of 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 steps with a voltage separation determined by twice the frequency. These effects are more pronounced for longer molecules. We study two different profiles for the dc and ac voltages, and it is found that the results are robust with respect to such variations. Although we concentrate here on the specific model of oligophenylenes, the results should be qualitatively similar for many other organic molecules with a large enough electronic gap.

DOI:10.1103/PhysRevB.77.155119 PACS number共s兲: 73.50.Pz, 85.65.⫹h, 73.63.Rt

I. INTRODUCTION

The use of single-molecule electrical contacts for opto- electronic purposes such as light sources, light sensors, and photovoltaic devices is an exciting idea. Yet, due to the dif- ficulties that light-matter interactions in nanoscale systems pose for theoretical and experimental investigations, the pos- sibilities remain largely unexplored. Concerning experi- ments, it has been shown that light can be used to change the conformation of some molecules even when they are con- tacted to metallic electrodes, thus enabling light-controlled switching.¹Some evidence of photoassisted processes influ- encing the conductance of laser-irradiated metallic atomic contacts has also been obtained.² Theoretical investigations of light-related effects in molecular contacts are more numerous,^3–19but they are mostly based on highly simplified models, whose validity remains to be checked by more de- tailed calculations^20,21and experiments. However, for the de- scription of the basic phenomenology, model approaches can be very fruitful, as they have been in studies of elastic trans- port in the past. Properties of linear single-orbital tight- binding共TB兲chains, in particular, have been studied in de- tail, and to a large part analytically.^3,22–32 In a step toward a more realistic description of the geometry, symmetries, and the electronic structure of particular molecules, empirical TB approaches such as the 共extended兲 Hückel method have proven useful.^4,8,33–35

Based on a combination of density-functional calculations and simple phenomenological considerations, we have re- cently described the photoconductance of metal-oligo- phenylene-metal junctions.⁵It was discussed how the linear- response conductance may increase by orders of magnitude

in the presence of light. This effect can be seen as the result of a change in the character of the transport from off- resonant to resonant, due to the presence of photoassisted processes.^5,7,8 Consequently, the decay of the conductance with molecular length is slowed down, possibly even making the conductance length independent.^5,8

In this paper, we apply a Hückel-type TB model of oligophenylene-based contacts³⁶ combined with Green- function methods⁴ to study the effects of monochromatic light on the dc current in metal-oligophenylene-metal con- tacts. Again, we concentrate on the dependence of these ef- fects on the length of the molecule. We begin with a detailed account of the elastic transport properties of the model and show that the zero-bias transmission function can be ob- tained analytically, similarly to simpler chain models.^23,27We demonstrate how information about the length dependence of the transmission function for a finite wire can be extracted from an infinitely extended polymer. Based on these analyti- cal results, we discuss the length dependences of the conduc- tance and the photoconductance for low-intensity and low- frequency light. While the conductance decays exponentially with length, its relative enhancement due to light exhibits a quadratic behavior. Here, we also briefly consider the ther- mopower, whose length dependence is linear. Next, we cal- culate numerically the zero-bias photoconductance as a func- tion of the light frequency␻ and find that the conductance enhancement due to light is typically very large.^3,5,8In par- ticular, we show that the results of Ref.5are expected to be robust with respect to variations in the assumed voltage pro- files. Finally, we describe how the steplike current-voltage 共I-V兲 characteristics are modified by light. At high ␻^{, the} most obvious effect is the overall increase in the low-bias PHYSICAL REVIEW B77, 155119共2008兲

1098-0121/2008/77共15兲/155119共14兲 155119-1 ©2008 The American Physical Society

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current. At low␻, additional current steps similar to those in microwave-irradiated superconducting tunnel junctions^37,38 can be seen. Their separation, in our case of symmetric junc- tions, is roughly 2ប␻/e.

TB models of the type we shall consider neglect various interaction effects 共see Sec. V for a discussion兲 and thus cannot be expected to give quantitative predictions. How- ever, the qualitative features of the results rely only on the tunneling-barrier character of the molecular contacts, which results from the fact that the Fermi energy of the metal lies in the gap between the highest-occupied and lowest-unoccupied molecular orbitals 共HOMO and LUMO兲 of the molecule.

Thus, these features should remain similar for junctions based on many other organic molecules exhibiting large HOMO-LUMO gaps. The light-induced effects, if verified experimentally, could be used for detecting light, or as an optical gate共or “third terminal”兲for purposes of switching.

The rest of the paper is organized as follows. In Sec. II, we describe our theoretical approach, discuss the general properties of TB wire models, and introduce the Green- function method for the calculation of the elastic transmis- sion function. Then, in Sec. III, we calculate the transmission function of oligophenylene wires analytically. The decay co- efficient for the off-resonant transmission is extracted also from infinitely extended polyphenylene. Following that, in Sec. IV, we present our numerical results for the conduc- tance, the thermopower, the photoconductance, and the I-V characteristics. Finally, Sec. V ends with our conclusions and some discussion. Details on the calculation of the time- averaged current in the presence of light are deferred to the Appendixes. In Appendix A, a simplified interpretation of the current formula is derived, and in Appendix B, a brief account of the general method is given. Readers mainly in- terested in the discussion of the results for the physical ob- servables can skip most of Secs. II and III and proceed to Sec. IV.

II. THEORETICAL FRAMEWORK A. Transport formalism

Our treatment of the transport characteristics for the two- terminal molecular wires is based on Green’s functions and the Landauer-Büttiker formalism, or its generalizations. As- suming the transport to be fully elastic, the dc electrical cur- rent through a molecular wire can be described with

I共V兲=2e

h

冕

^{dE␶共E,V兲关fL共E兲−fR共E兲兴. 共1兲}

Here, V is the dc voltage and ␶共E,V兲 is the voltage- dependent transmission function, while fX共E兲= 1/关exp共共E

−␮X兲/k_BT_X兲+ 1兴,␮X, andT_Xare the Fermi function, the elec- trochemical potential, and the temperature of sideX=L,R, respectively.³⁹ The electrochemical potentials satisfy eV

=⌬␮⁼␮L−␮R, and we can choose them symmetrically as

␮L=E_F+eV/2 and␮R=E_F−eV/2, whereE_Fis the Fermi en- ergy. For studies of dc current, we always assume T_L=T_R

= 0. Of particular experimental interest is the linear-response conductanceGdc=兩⳵I/⳵V兩V=0, given by the Landauer formula

Gdc=G₀␶共EF兲, where G₀= 2e²/h and ␶共E兲=␶共E,V= 0兲. In most junctions based on organic oligomers, the transport can be described as off-resonant tunneling. This results in the well-known exponential decay ofGdcwith the numberNof monomeric units in the molecule.⁴⁰At finite voltagesV, the current increases in a stepwise manner as molecular levels begin to enter the bias window between␮Land␮R共Ref.24兲. We shall consider both of these phenomena below.

If a small temperature difference⌬T=TL−TR at an aver- age temperatureT=共TL+TR兲/2 is applied, heat currents and thermoelectric effects can arise.^36,41,42In an open-circuit situ- ation, where the net currentI must vanish, a thermoelectric voltage⌬␮/eis generated to balance the thermal diffusion of charge carriers. In the linear-response regime, the proportion- ality constant S= −共⌬␮/e⌬T兲I=0 is the Seebeck coefficient.

We will briefly consider this quantity below as an example of an observable with a linear dependence on the molecular length N but will not enter a more detailed discussion of thermoelectricity or heat transport.

The quantity we are most interested in is the dc current in the presence of monochromatic electromagnetic radiation, which we refer to as light independently of its source or frequency␻. We model the light as an ac voltage with har- monic time dependence V共t兲=Vaccos共␻t兲 over the contact.

The current averaged over one period ofV共t兲can be written in the form^3,4,43

I共V;␣,␻兲=2e h

兺

k=−⬁

⬁

冕

^{dE关␶RL共k兲共E,V;␣,␻兲fL共E兲}

−␶LR共k兲共E,V;␣^,␻兲fR共E兲兴. 共2兲 Here, the transmission coefficient ␶RL共k兲共E兲, for example, de- scribes photoassisted processes taking an electron from left 共L兲to right共R兲, under the absorption of a total ofkphotons with energy ប␻. The parameter ␣^=eVac/ប␻ describes the strength of the ac drive.⁴⁴It is determined by the intensity of the incident light and possible field-enhancement effects tak- ing place in the metallic nanocontact.⁴⁵Again, in addition to the fullI-V characteristics, we study in more detail the case of linear response with respect to the dc bias, i.e., the pho- toconductance G_dc共␣^,␻兲=兩⳵^I共V;␣^,␻兲/⳵^V兩V=0. The argu- ments␣ and␻ distinguish it from the conductanceGdc, al- though we sometimes omit ␣ for notational simplicity. The calculation of the coefficients␶RL共k兲/LR共E兲is rather complicated in general,⁴ and we defer comments on this procedure to Appendix B. Below, we shall mostly refer to an approximate formula共see Appendix A兲that can be expressed in terms of

␶共E兲. This amounts to a treatment of the problem on the level of the Tien-Gordon approach.^3,37,46 The full Green-function formalism for systems involving ac driving is presented in Ref.4.

In noninteracting 共non-self-consistent兲 models, it is, in general, not clear how the voltage drop should be divided between the different regions of the wire and the electrode- wire interfaces. A self-consistent treatment would be in or- der, in particular, for asymmetrically coupled molecules. We only concentrate on left-right symmetric junctions, where both the dc and ac voltages共VandV_ac兲are assumed to drop according to one of two different symmetrical profiles. The

symmetry of the junctions excludes rectification effects, such as light-induced dc photocurrents in the absence of a dc bias voltage.^3,9,45However, light can still have a strong influence on the transmission properties of the molecular contact, as will be discussed below. It will be shown that our conclu- sions are essentially independent of the assumed voltage pro- file.

B. Wire models

Below, we will specialize to the case of a metal- oligophenylene-metal junction. However, to make some gen- eral remarks, let us first consider a larger class of molecular wires that can be described as N separate units forming a chain, where only the nearest neighbors are coupled共see Fig.

1兲. We only discuss the calculation of the elastic transmission function␶共E,V兲here, as this will be the focus of our analyti- cal considerations in Sec. III. From this quantity 共atV= 0兲, the various linear-response coefficients such as the conduc- tance and the thermopower can be extracted. Furthermore, as already mentioned, it suffices for an approximate treatment of the amplitudes␶RL共k兲共E兲 as well.

We assume a basis兩␹p共␣兲典of local共atomic兲orbitals, where p= 1 , . . . ,Nindexes the unit, while␣= 1 , . . . ,Mpdenotes the orbitals in each unit.⁴⁷For simplicity, the basis is taken to be orthonormal, i.e., 具␹p共␣兲兩␹q共␤兲典=␦␣␤␦pq. The 共time-indepen- dent兲HamiltonianH_pq^{共␣,␤兲}=具␹p共␣兲兩Hˆ兩␹q共␤兲典of the wire is then of the block-tridiagonal form

H=

冢

^{HH1121 HH1222 HN−1,N−2H23 HHN−1,N−1N,N−1 HHN−1,NNN}

冣

^{, 共3兲}

where Hpq with p,q= 1 , . . . ,N are Mp⫻Mq matrices. 共The unindicated matrix elements are all zeros.兲

In the nonequilibrium Green-function picture, the effect of coupling the chain to the electrodes is described in terms of “lead self-energies.”⁴⁸We assume these to be located only on the terminal blocks of the chain, with components ⌺11

and⌺NN. The inverse of the stationary-state retarded propa- gator for the coupled chain will then be of the form

F=

冢

^{Fh2111 hh1222 hN−1,N−2h23 hhN−1,N−1N,N−1 hFN−1,NNN}

冣

^{. 共4兲}

Here, h_p,p⫾1= −H_p,p⫾1, hpp=E₊1pp−Hpp, and E₊=E+i0⁺, whileF₁₁=h₁₁−⌺₁₁andF_NN=h_NN−⌺NN. Charge-transfer ef-

fects between the molecule and the metallic electrodes shift the molecular levels with respect to the Fermi energyE_F. In a TB model, these can be represented by shifting the diago- nal elements of H. Once a transport voltage V is applied, further shifts are induced. In our model, the voltage-induced shifts will be taken from simple model profiles, and the rela- tive position ofE_F will be treated as a free parameter.

Effective numerical ways of calculating the propagator G=F⁻¹for block-tridiagonal Hamiltonians exist.^49,50In Sec.

III, we shall be interested in a special case, where H_p,p−1

=H₋₁,H_p,p+1=H₁, andH_pp=H₀with the sameH₁=H₋₁^T and H₀共of dimension Mp=M兲for allp, describing an oligomer of identical monomeric units. In such cases also, analytical progress in calculating the current in Eq. 共1兲 may be pos- sible. Once the Green’s functionG is known, the transmis- sion function is given by⁴⁸

␶共E,V兲= Tr关⌫11G_1N⌫NN共G1N兲^†兴, 共5兲 where⌫11= −2 Im⌺11and⌺11共E,V兲=⌺11共E−eV/2兲, for ex- ample.

Typically,EF lies within the HOMO-LUMO gap, result- ing in the exponential decay␶共EF兲⬃e^{−␤共EF兲N}withN, charac- teristic of off-resonant transport. The decay coefficient␤共E_F兲 is actually independent of⌺11and⌺NN. This can be seen by considering the Dyson equationG=G+G⌺G, whereGand G are the Green’s function of the coupled and uncoupled wires, respectively, and ⌺ is the matrix for the lead self- energies. Assuming that G1N decays exponentially with N, then

G_1N⬇ 共1−G11⌺11兲⁻¹G1N 共6兲 whenN→⬁, and thereforeG_1Ndecays with the same expo- nent. Thus, one can, in principle, obtain the decay exponent from the propagator of an isolated molecule, or even an in- finitely extended polymer. In the next section, we demon- strate this by extracting the decay exponent of a finite oli- gophenylene junction from the propagator for polyphenylene. We note that in doing so, we neglect the practical difficulty of determining the correct relative posi- tion ofEF.

There are efficient numerical methods for computing the lead self-energies for different types of electrodes and vari- ous bonding situations between them and the wire. Typically, the methods are based on the calculation of surface Green’s functions.⁵¹Below, we shall simply treat the self-energies as parameters.

III. PHENYL-RING-BASED WIRES

In this section, we discuss a special case of the type of wire model introduced above, describing an oligomer of phe- nyl rings coupled to each other via the para 共p兲 position.³⁶ The bias voltageVis assumed to be zero. In the special case that we will consider, the inversion of Eq. 共4兲 can then be done analytically with the subdeterminant method familiar from elementary linear algebra.23,24,27,32 Below, we first use this method for calculating the propagator of the finite-wire junction and derive the decay exponent␤共E兲of the transmis- sion function at off-resonant energies. After that, we rederive

1 2 3

Σ₁₁ H H Σ₃₃

H H

12 23

21 32

H₁₁ H₂₂ H₃₃

FIG. 1.共Color online兲A finite block chain of lengthN= 3 con- nected to electrodes at its two ends. This gives rise to self-energies

⌺11and⌺NNon the terminating blocks.

the decay exponent by considering an infinitely extended polymer of phenyl rings.

A. Oligo-p-phenylene junction

Our model for the oligophenylene-based molecular junc- tion is depicted in Fig.2. Within a simple␲-electron picture, the electronic structure of the oligophenylene molecule can be described with a nearest-neighbor TB model with two different hopping elements −␥^{and −}␩共Ref.52兲. Here, −␥^is for hopping within a phenyl ring, between the p orbitals oriented perpendicular to the ring plane, while −␩^describes hopping between adjacent rings. Due to the symmetry of the orbitals, the magnitude of␩depends on the angle␸^between the rings proportionally to cos␸共Ref. 53兲. We shall assume that ␩⁼␥^cos␸^{, and thus} 兩␩兩艋␥. In this way, the natural energy scale of the model is set by␥alone.

The ring-tilt angle ␸ can be controlled to some extent using side groups. For example, two side groups bonded to adjacent phenyl rings can repel each other sterically, thus increasing the corresponding tilt angle.^53,54In fact, even the pure oligophenylenes in the uncharged state have ␸

= 30° – 40° due to the repulsion of the hydrogen atoms.^36,53 However, side groups can introduce also “charging” or “dop- ing” effects, which shift the molecular levels.⁵⁵

For definiteness, we number theM= 6 carbon atoms of a phenyl ring according to the lower part of Fig.2. The corre- sponding orbitals appear in the basis in this order. Thus, the blocks in Eq.共3兲are

H_q,q=

冢

^{⑀−−000q共1␥ ⑀␥兲 −−000q共2兲␥␥ −⑀−000q共3␥␥兲 −−⑀000q共4兲␥␥ −⑀−000q共5␥␥ ⑀兲 −−000q共6兲␥␥}

冣

^{, 共7兲}

forq= 1 , . . . ,N, and

H_q,q−1=

冢

0 0 0 0 0 −␩

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

冣

^{, 共8兲}

withH_q−1,q=关Hq,q−1兴^T. Here, the on-site energies⑀q共␣兲may be shifted nonuniformly to describe effects of possible side groups.³⁶For simplicity, we shall consider all phenyl rings to have a similar chemical environment, and thus all on-site energies are taken to be equal.

As a first step we note that, assuming⑀q共␣兲=⑀qfor all␣^{, the} eigenvalues for the HamiltonianHqqof the isolated unit are

⑀q−␥^,⑀q+␥^,⑀q−␥^,⑀q+␥^,⑀q− 2␥^{, and}⑀q+ 2␥, while the cor- responding orthonormalized eigenvectors are

冑

14共0,− 1,1,− 1,1,0兲^T, 1

冑

4共0,1,− 1,− 1,1,0兲^T,

冑

112共− 2,− 1,− 1,1,1,2兲^T, 1

冑

12共2,− 1,− 1,− 1,− 1,2兲^T,

冑

16共1,1,1,1,1,1兲^T, 1

冑

6共− 1,1,1,− 1,− 1,1兲^T. 共9兲 The first two of the eigenstates have zero weight on the ring- connecting carbon atoms 1 and 6. Therefore, these eigen- states do not hybridize with the levels of the adjacent rings and consequently cannot take part in the transport. This will be seen explicitly in the derivation of the propagator. We note that these results can also be used to determine a real- istic value for the hopping␥from the HOMO-LUMO split- ting of benzene.³⁶

Below, we shall only consider the analytically solvable case, where all on-site energies are set to the same value. We choose this value as our zero of energy: ⑀q共␣兲= 0 for all q

= 1 , . . . ,Nand␣= 1 , . . . ,M. Later on, we shall relax this as- sumption in order to describe externally applied dc and ac voltage profiles. In the absence of such voltages, the inverse propagator 关Eq. 共4兲兴 consists of the blocks h_p,p=h₀, h_p,p−1

=h₋₁, andh_p,p+1=h₁, where

h₀=

冢

^{E␥␥000+ E␥ ␥␥000+ E00␥0+ E0␥0␥ ␥0+ E00␥0+ E000␥␥+}

冣

^,

(α) (α) (α)

1 2 3

11 1 2 3

Σ ε ε ε 33

Σ

1 3

2 4

5 6

−η −η

−γ

−γ −γ

−γ −γ

−γ

FIG. 2.共Color online兲A finite chain of lengthN= 3 connected to electrodes at its two ends. This gives rise to self-energies⌺11and

⌺NNon the end sites. The nearest-neighbor hoppings inside the ring 共−␥兲and between the rings共−␩兲are different. The lower part indi- cates also the numbering of theM= 6 carbon atoms within a ring.

h₋₁=

冢

^{0 0 0 0 0}0 0 0 0 0 0^␩ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

冣

^{, 共10兲}

andh₁=关h−1兴^T. The leads are assumed to couple only to the terminal carbon atoms, thus making the self-energy 6⫻6 matrices of the form

⌺11=

冢

^{⌺] ] 00L 000 ¯¯0 0000}

冣

^{, ⌺NN=}

冢

^{0000 ¯¯ ] ]0 000 ⌺00R}

冣

^.

共11兲 We also define the symbol “tilde”共˜兲, which means the re- placement of the first column of a matrix by␩followed by zeros. For example,

h˜

0=

冢

^{␩ ␥ ␥00000 E0␥00+ E␥000+ E0␥00␥ ␥+ E␥000+ E000␥␥+}

冣

^{. 共12兲}

For the evaluation of Eq.共5兲, we only need the component G_1,MN=关G1N兴1M. Using the subdeterminants of F=G⁻¹, we have

G_1,MN=共− 1兲^MN+1det关F共MN兩1兲兴

det关F兴 . 共13兲

Here,O共i, . . . ,k兩j, . . . ,l兲 is the submatrix of O obtained by removing the rows i, . . . ,k, and columns j, . . . ,l. We shall also denote byLandRthe “leftmost” and “rightmost” rows or column of a matrix, respectively. Thus, for example, det关F共MN兩1兲兴= det关F共R兩L兲兴.

Let us first concentrate on the denominator of Eq.共13兲. It is easy to see that det关F兴can be written in terms of determi- nants related to the inverse Green’s functionF=G⁻¹ of the uncoupled wire as follows:²³

det关F兴= det关F兴−⌺Ldet关F共L兩L兲兴−⌺Rdet关F共R兩R兲兴 +⌺L⌺Rdet关F共L,R兩L,R兲兴. 共14兲 Furthermore, due to the symmetry of the molecule, det关F共R兩R兲兴= det关F共L兩L兲兴. Thus, we are left with calculat- ing three types of determinants. It can be shown that, for 1

⬍n⬍N, all of them satisfy a recursion relation of the form

冉

^{DD˜共共nn兲兲}

冊

^{=共E+2−␥2兲Y}

冉

^{DD˜共共n−1n−1兲兲}

冊

=共E₊²−␥²兲

冉

^ac ⁻b^c

冊 冉

^{DD˜共共n−1n−1兲兲}

冊

^{. 共15兲}

For example, in the calculation of det关F兴, we have D^共n兲

= det关F^共n兲兴 andD˜共n兲= det关F˜共n兲兴, where the additional super- script 共n兲 on the matrices denotes the number of the M

⫻Mdiagonal blocks. The elements of the matrixYare given by

a=共E₊²−␥²兲共E₊²− 4␥²兲, b= −␩²共E₊²−␥²兲,

c=␩^E+共E₊²− 3␥²兲. 共16兲 Only the initial condition共n= 1兲 and the last step of the re- cursion 共n=N兲 will differ for the three determinants. The recursion relations can be solved by calculatingYⁿexplicitly, which can be done by diagonalizingY. The eigenvalues ofY are␭1,2=共a+b⫿

冑

共a−b兲²− 4c²兲/2, while the共unnormalized兲 eigenvectors are

v_1,2=

冉

^a−b⫿

^冑

^{共a2c−b兲2− 4c2,1}

冊

^{T. 共17兲}

Then, if V=共v1,v₂兲 and ⌳= diag共␭1,␭2兲, we have Yⁿ

=V⌳ⁿV⁻¹. The result is

Yⁿ=

冉

^yy11共21共ⁿn^兲兲 ^yy^12共₂₂^共nn兲兲

冊

^{, 共18兲}

where the components are given by y₁₁^共n兲=共␭1

n−␭2

n兲共b−a兲+共␭1 n+␭2

n兲

冑

共a−b兲²− 4c² 2

冑

共a−b兲²− 4c² , y₂₂^共n兲=共␭₁ⁿ−␭₂ⁿ兲共a−b兲+共␭₁ⁿ+␭₂ⁿ兲

冑

共a−b兲²− 4c²

2

冑

共a−b兲²− 4c² , y₁₂^共n兲= −y₂₁^共n兲= c共␭₁ⁿ−␭₂ⁿ兲

冑

共a−b兲²− 4c². 共19兲 Using these, we can now write explicit expressions for the three required determinants. For det关F兴, the recursion can be started at n= 1 with the initial conditions D^共0兲= 1 and D˜共0兲

= 0 and carried out up ton=N. The result is

det关F^共N兲兴=共E₊²−␥²兲^Ny₁₁^共N兲. 共20兲 The other two determinants require special initial and final steps, and the results are

det关F^共N兲共L兩L兲兴=共E₊²−␥²兲^Ny₂₁^共N兲/␩^,

det关F^共N兲共L,R兩L,R兲兴=共E₊²−␥²兲^N关y₂₁^共N−1兲c−y₂₂^共N−1兲b兴/␩^2. 共21兲

Next, we consider the determinant in the numerator of Eq.

共13兲, det关F^共N兲共R兩L兲兴= det关F^共N兲共R兩L兲兴. It can easily be shown that it satisfies the recursion relation

det关F^共N兲共R兩L兲兴= 2␩␥³共E₊²−␥²兲det关F^共N−1兲共R兩L兲兴 共22兲 and so

det关F^共N兲共R兩L兲兴= 2^N共␩␥³兲^N共E₊²−␥²兲^N/␩^. 共23兲 Now, the Green’s function of Eq.共13兲can be written as

G_1,MN= −共2␩␥³兲^N/␩

y₁₁^共N兲+⌺LRy₂₁^共N兲/␩⁺⌺L⌺R共y₂₁^共N−1兲c−y₂₂^共N−1兲b兲/␩^2, 共24兲 where we used the shorthand⌺LR=⌺L+⌺R.

It is notable that the common共E₊²−␥²兲^Nfactors canceled out from the final propagator. These factors apparently cor- respond to the two eigenvectors ofh₀ 关Eq.共9兲兴having zero weight on the ring-connecting atoms 1 and 6. The cancella- tion is a manifestation of the physical fact that such localized states cannot contribute to the transport through the mol- ecule. In the infinite polymer to be discussed below, these states appear as completely flatbands in the band structure.

To conclude this part, we point out that forE inside the HOMO-LUMO gap关more precisely, when共a−b兲²− 4c²⬎0兴, the eigenvalues␭1,2are real valued and the decay exponent of the transmission␶共E兲for largeNis controlled by the one with a larger absolute value. Since inside the gapE⬇0, we find that ␭2⬎␭1⬎0. Then, using Eq. 共5兲 and omitting N-independent prefactors, the decay of the transmission for largeNfollows the law

␶共E兲 ⬃

冋

^␭22␩␥^共E兲3

册

^{−2N=e−2Nln关␭2共E兲/共2␩␥3兲兴. 共25兲}

Thus, the decay exponent is given by

␤共E兲= 2 ln关␭2共E兲/共2␩␥³兲兴. 共26兲 We note that for resonant energies, oscillatory dependence of

␶共E兲 onN can be expected, instead, and for limiting cases also power-law decay is possible.³²Next, we shall reproduce the result for the decay exponent by considering an infinitely extended polymer.

B. Poly-p-phenylene

For comparison with the “correct” evaluation of the propagator and the decay coefficient for a finite chain, let us consider the propagator for an infinitely extended polymer.

To describe the polymer, we start from a finite chain with periodic boundary conditions. Neglecting curvature effects, the latter actually represents a ring-shaped oligomer, as de- picted in Fig.3共a兲.

Let us first consider the eigenstates of the periodic chain.

The HamiltonianH_pq^{共␣,␤兲}=具␹p共␣兲兩Hˆ兩␹q共␤兲典is of the general form

H=

冢

^{HHH−101 HH10 HH−11 HH−10 HHH−110}

冣

^{, 共27兲}

whereH_0,⫾1are theM⫻M matrices共M= 6兲of Eqs.共7兲and 共8兲, with ⑀q共␣兲= 0. 共Again, only nonzero elements are indi- cated.兲The normalized eigenvectors␺p共n兲共k兲satisfying

兺

q

Hpq␺q共n兲共k兲=E^共n兲共k兲␺p共n兲共k兲 共28兲 are of the Bloch form ␺q共n兲共k兲=e^ikqd␾^共n兲共k兲/

冑

N, where

␾^共n兲共k兲are the normalized eigenvectors of

H共k兲=e^ikdH₁+H₀+e^−ikdH₋₁ 共29兲 with the eigenvalueE^共n兲共k兲, andn= 1 , . . . ,M. Due to the fi- niteness of the wire, the k values are restricted to k_␮

= 2␲␮/Nd, where␮is an integer anddis the lattice constant 共the length of a single phenyl-ring unit兲.

The spectral decomposition of the 共retarded兲 propagator g共E兲=共E+1−H兲⁻¹ of the chain is of the form

g_pq^{共␣,␤兲}共E兲=

兺

_␮,n ^{具␹共␣兲p 兩␺共n}_E^兲共k₊₋^␮兲典具_E^共n␺兲_共k^共n兲_␮^共k_兲^{␮兲兩␹q共␤兲典, 共30兲}

with the Bloch states 兩␺^共n兲共k_␮兲典= 1

冑

N_p=−

兺

^N_N^/_/2+1^{2 eik␮pd}_␣=1

兺

^{M ␾␣共n兲共k␮兲兩␹p共␣兲典. 共31兲}

In the limit of large N 关Fig. 3共b兲兴, we can use N⁻¹兺_␮

→共d/2␲兲兰_−␲/␲/dd

dkto turn the summation into an integral over the first Brillouin zone. In this case, there are M= 6 bands with energies

E^共1,2兲共k兲= ⫾␥^,

E^共3,4兲共k兲= ⫾ 1

冑

2

冑

_␩²+ 5␥²− 2B共k兲,

(a)

(b) ^{1 2}

3 4

N

−η −γ

−γ

−γ −γ

−γ −γ

FIG. 3. 共Color online兲Phenyl-ring chains:共a兲 a periodic chain withNunits and共b兲an infinite chain. Case共b兲is obtained from共a兲 in the limitN→⬁.

E^共5,6兲共k兲= ⫾ 1

冑

2

冑

_␩²+ 5␥²+ 2B共k兲, 共32兲 where

B共k兲=1

2

冑

共␩^{2+ 3}␥²兲²+ 16␩␥³cos共kd兲. 共33兲 Clearly, we have the symmetries E^共1兲共k兲= −E^共2兲共k兲, E^共3兲共k兲

= −E^共4兲共k兲, and E^共5兲共k兲= −E^共6兲共k兲. For n= 1 , 2, the bands are completely flat, and the corresponding eigenvectors␾^共1,2兲共k兲 are as in Eq.共9兲, i.e., independent ofkand completely local- ized on atoms␣= 2 , 3 , 4 , 5. Thus, forp⫽q, they do not con- tribute to the propagator in Eq. 共30兲. For n= 3 , 4 , 5 , 6, the vectors are very complicated, but they are not needed in the following.

To compare with the result of Sec. III A, we should now calculate, for example, the component g_pq^共1,6兲. However, ex- pecting the decay exponent to be independent of␣^and␤^{, we} consider the simpler case Tr关gpq兴=兺_␣g_pq^{共␣,␣兲}. Due to the ortho- normality 兺_␣␾_␣^共m兲␾_␣^共n兲*=␦mn, the dependence on the vector components then drops out. Thus, forp⫽q,

兺

_␣ ^{gpq共␣,␣兲= 4EA}₂^d_␲

冕

_−␲/d

␲/d

dk e^{ikd共p−q兲}

A²−B²共k兲, 共34兲 where we defined

A=E₊²− 1

2共␩^{2+ 5}␥²兲, 共35兲 such thatE₊²−关⑀^共3,5兲共k兲兴²=A⫾B共k兲. Defining nowz=e^ikd, the integral can be turned into a contour integral around the con- tour兩z兩= 1,

兺

_␣ ^{gpq共␣,␣兲= −}_2␲^2EA_i␩␥³

冖

_兩z兩=1

dz z^p−q

共z−z₊兲共z−z₋兲, 共36兲 where the poles z_⫾ are determined from the equation z²

−关4A²−共␩^{2+ 3}␥²兲²兴共8␩␥³兲⁻¹z+ 1 = 0. They are given by z_⫾=4A²−共␩^{2+ 3}␥²兲²

16␩␥^{3 ⫾}

冑 冋

^{4A2−16共␩}␩␥^{2+ 33 ␥2兲2}

册

^{2− 1}

共37兲 such that z₊= 1/z₋, and we choose the signs so that z₋ is inside the contour 兩z兩= 1. In addition to this, assuming that p⬍q, there is a pole of orderq−patz= 0. The integral can then be evaluated using residue techniques, with the result

兺

_␣ ^{g共␣,␣兲pq =2EA}_␩␥³_z^z+p−q

+−z₋. 共38兲

This leads to an exponential decay of the propagator with growingq−p⬎0 whenE is off-resonant共in which case z_⫾ are real valued兲. Using this result, we can give an estimate for the decay of the transmission function关Eq.共5兲兴through a finite chain of lengthNby replacingG_1,MNwith Tr关g_1N兴/M.

This yields

␶共E兲 ⬃ 关z+共E兲兴^−2N=e^{−2Nln关z+共E兲兴}, 共39兲 and thus the exponent

␤共E兲= 2 ln关z+共E兲兴. 共40兲 It can be checked that this result is, in fact, equal to the result 关Eq.共26兲兴obtained for the finite chain.

It is thus seen explicitly that the decay coefficient of the off-resonant transmission does not in any way depend on the coupling of the molecule to the leads. It should be kept in mind, however, that the relative position of EF within the HOMO-LUMO gap depends on the electrode-lead coupling and the charge-transfer effects. This information is still needed for predicting the decay exponent␤共EF兲of the con- ductance.

The analytical results presented in this and the previous section can be used for understanding the behavior of the transmission function upon changes in the parameters. For example, it should be noted that when␩is made smaller, the band gap aroundE⬇0 becomes larger, and at the same time the decay exponent␤共E兲grows. In this way, the conductance of a molecular junction can be controlled, for example, by introducing side groups to control the tilt angles␸ ^between the phenyl rings.^36,53

IV. PHYSICAL OBSERVABLES AND NUMERICAL RESULTS

In this section, we present numerical results based on our model. Throughout, we employ the “wide-band” approxima- tion for the lead self-energies, such that⌺L共E兲= −i⌫L/2 and

⌺R共E兲= −i⌫R/2, with energy-independent constants ⌫L,R. Furthermore we only consider the symmetric case ⌫L=⌫R

=⌫. First, we briefly describe how we generalize the theory, as presented above, to take into account static and time- dependent voltage profiles. Then, we concentrate on near- equilibrium 共or “linear-response”兲 properties, using as ex- amples the conductance, the thermopower, and the conductance enhancement due to light with low intensity and frequency. In this case, knowledge of the zero-bias transmis- sion function calculated above is sufficient, and we can dis- cuss the length dependence of the transport properties in a simple way. After that, we consider the dc current in the presence of an ac driving field of more general amplitude and frequency, first concentrating on the case of infinitesimal dc bias and finally on theI-Vcharacteristics.

A. Voltage profiles

When considering finite dc or ac biases within a non-self- consistent TB model that cannot account for screening ef- fects, one of the obvious problems is how to choose the voltage profile. Throughout the discussion, we shall refer to two possible choices, as depicted in Fig.4. They are in some sense limiting cases, and the physically most reasonable choice should lie somewhere in between. Profile A assumes the external electric fields to be completely screened inside the molecule, such that the on-site energies are not modified, while B corresponds to the complete absence of such screen- ing. In both cases, we can write the time-dependent on-site energies as⑀p共␣兲共t兲=eV共t兲P共z_p^共␣兲兲, wherez_p^共␣兲 are the distances of the carbon atoms from the left metal surface, and V共t兲

=V+Vaccos共␻t兲. In case A, P共z兲= 0 inside the junction,

while in case BP共z兲=共L− 2z兲/共2L兲, whereL=Nd+d/3 is the distance between the two metal surfaces.

The profile B is more complicated, because the voltage ramp breaks the homogeneity of the wire. In this case, the current must be calculated with the method outlined in Ap- pendix B. In the case of profile A, however, theI-V charac- teristics can be calculated based on the knowledge of the zero-bias transmission function in the absence of light,␶共E兲.

As discussed in Appendix A, the current is given by^3,46,56 I共V;␣^,␻兲=2e

h

兺

l=−⬁

⬁

冋

^Jl

^冉

^␣2

^冊 册

²

^冕

^{dE␶共E+lប␻兲关fL共E兲−fR共E兲兴.}

共41兲 The low-temperature zero-bias conductance then takes the particularly simple form^4,5

Gdc共␣^,␻兲=G₀

兺

l=−⬁

⬁

冋

^Jl

^冉

^␣2

^冊 册

^{2␶共EF+lប␻兲. 共42兲}

Here,lindexes the number of absorbed or emitted photons, Jl共x兲 is a Bessel function of the first kind 共of order l兲, and

␣^=eVac/ប␻ is the dimensionless parameter describing the strength of the ac drive. Note that G_dc共␣^,␻^{= 0}兲=G_dc共␣

= 0 ,␻兲=G₀␶共EF兲=Gdc. Equation 共41兲 may equally well be written in the form^37,57

I共V;␣^,␻兲=_l=−⬁

兺

^⬁

冋

^Jl

^冉

^␣2

^冊 册

^{2I0共V+ 2lប␻/e兲, 共43兲}

whereI₀共V兲 is theI-V characteristic in the absence of light 关Eq. 共1兲兴. Below, the results from these formulas are com- pared to the numerical results for profile B.

In Fig.5, we plot the zero-bias transmission functions for wires withN between 1 and 7. Notice that the four energy bands numbered 3–6 in Eq.共32兲are all visible, being sepa- rated by the HOMO-LUMO gap at E/␥⬇0 and the addi- tional gaps at E/␥⬇⫾1.7. Here, we use the parameters

⌫/␥^{= 5.0,}␸^{= 40°}共i.e.,␩/␥⬇0.77兲, and set the Fermi energy toEF/␥= −0.4. These values are close to those used in Ref.

36, where they were extracted from a fit to results for gold- oligophenylene-gold contacts based on density-functional

theory 共DFT兲. We shall continue to use them everywhere below. A DFT calculation for the HOMO-LUMO splitting of benzene, together with the results preceding Eq.共9兲, yields the hopping ␥⬇3 eV. The length of a phenyl-ring unit is approximately d= 0.44 nm, and the largest ac electric fields Vac/Lconsidered will be on the order of 10⁹V/m. The pho- ton energiesប␻will mainly be kept below the energy of the HOMO-LUMO gap of the oligophenylene.

B. Near-equilibrium properties

Let us start by illustrating the usefulness of the analytical results of Sec. III with a few examples. We concentrate on low temperatures and small deviations from equilibrium. In addition to the linear-response conductance

Gdc=G₀␶共EF兲, 共44兲 we shall consider the thermopower, or Seebeck coefficient.

At low enough temperatureT, this is given in terms of the zero-bias transmission function␶共E兲as28,41,58,59

S= −␲^2kB 2T 3e

␶⬘^共EF兲

␶共EF兲 , 共45兲 where prime denotes a derivative. Thus, it measures the loga- rithmic first derivative of the transmission function at E

=E_F. The sign of this quantity carries information about the location of the Fermi energy within the HOMO-LUMO gap of molecular junction.⁴¹The third quantity we shall consider is the photoconductance. In the limit␣Ⰶ1 andប␻/␥Ⰶ1, we can expand ␶共E兲 and the Bessel functions in Eq. 共42兲 共see Appendix A兲to leading order in these small quantities, yield- ing Gdc共␻兲=G₀␶共EF兲+G₀共␣ប␻兲²␶⬙^共EF兲/16. Defining then the light-induced conductance correction ⌬Gdc共␻兲=Gdc共␻兲

−Gdc共␻= 0兲, where Gdc共␻= 0兲=Gdc=G₀␶共EF兲, the relative correction becomes

⌬G_dc共␣^,␻兲

G_dc =共␣ប␻兲² 16

␶⬙共EF兲

␶共E_F兲 . 共46兲 We thus see that this quantity gives experimental access to thesecond derivativeof the transmission function atE=EF. Note that in this approximation, which can be seen as an d/ 6

d/ 3d/ 3d/ 3

(a) ^d

L

(b) B P(z)

A

z

FIG. 4.共Color online兲 共a兲The coordinates of the carbon atoms in the directionzalong the molecular wire. The left electrode is at z= 0 and the length of a phenyl-ring unit isd.共b兲Relative variation of the on-site energies for two different voltage profiles, A and B.

The profile function P共z兲 describes how the harmonic voltage V共t兲=V+V_accos共␻t兲is assumed to drop over the junction, the volt- age atzbeing given byV共z,t兲=V共t兲P共z兲.

-2 -1 0 1 2

E /γ 10^-6

10^-4 10^-2 10⁰

τ(E)

N=1

N=7 E_F

FIG. 5. 共Color online兲Transmission functions for the oligophe- nylene wires with lengths N= 1 , 3 , 5 , 7. The parameters are ⌫/␥

= 0.5,␸= 40°, andE_F/␥= −0.4, as discussed in the text.