N. Renaud, M. A. Ratner and C. Joachim- A Time-Dependent Approach to Electronic Transmission in Model Molecular Junctions

8/3/2019 N. Renaud, M. A. Ratner and C. Joachim- A Time-Dependent Approach to Electronic Transmission in Model Molecula

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Published: February 16, 2011

r 2011 American Chemical Society 5582 dx.doi.org/10.1021/jp111384d|J. Phys. Chem. B 2011, 115, 55825592

ARTICLE

pubs.acs.org/JPCB

A Time-Dependent Approach to Electronic Transmission in ModelMolecular Junctions

N. Renaud,*, M. A. Ratner,*, and C. Joachim*,

Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United StatesNanoscience Group & MANA Sattelite CEMES/CNRS, 29 rue J. Marvig, BP 4347, 31055 Toulouse Cedex, France

ABSTRACT: We present a simple method to compute the transmission coefficient of aquantum system embedded between two conducting electrodes. Starting from thesolution of the time-dependent Schrodinger equation, we demonstrate the relationship

between the temporal evolution of the state vector, |(t) , initially localized on oneelectrode and the electronic transmission coefficient, T(E). We particularly emphasize therole of theoscillation frequency andthe decay rate of |(t) inthelineshapeofT(E). Thismethodis applied to thewell-known problems of thesingle impurity, two-site systems andthe benzene ring, where it agrees with well-accepted time-independent methods and givesnew physical insight to the resonance and interference patterns widely observed inmolecular junctions.

I. INTRODUCTION

Molecular electronics

1

has become an intensefi

eld of researchfollowing the suggestion of the first molecular rectifier in the mid1970s.2-4 To propose an alternative solution to silicon-basedcircuits, several propositions have been made to implement aBoolean function using the electronic conduction of a singlemolecule,either utilizing the molecule like a classical circuit5-8 orusing its internal degrees of freedom.9-14

Many methods have thus been proposed to compute theelastic conductance of a single molecule when connected toseveral leads.15,16 Following the Lippman-Schwinger ap-proach,17 most of them are based on the solution of the time-independent Schrodinger equation to compute the transmissioncoefficient, noted T , between electronic eigenstates of the twoelectrodes. The low-bias current flowing from one electrode

to the other is then obtained using the Landauer-Buttikerformula.18 Several methods can then be used to compute thiselectronic transmission through a molecular junction. Theelectron scattering quantum chemistry (ESQC) method, devel-oped by Sautet et al.,19-21 is based on the transfer matrixformalism where the molecular states are mapped at the endonto each electrode and where the continua formed by theperiodic electrodes are exactly treated using spatial propagators.This powerful method allows one, for example, to predict andanalyze images obtained via scanning tunnelling microscopy.22-24

Later, Mujica et al. proposed a Green-function-based methodapplied to the Lowdin partitioning of the system where the self-energies of theelectrodes shift thediagonal matrix element of the

molecular Hamiltonian.25,26 Many other Green-function-basedmethods have been proposed28-35 and compared with ESQC36

that can itself be re-expressed in terms of surface Greenfunctions.37 Very different approaches can be implemented tocompute the transmission matrix such as the recent graph-theoretical-based method proposed by Pickup et al.38 that usesa source/sink potential to model the electrodes.39 However,starting from a time-independent point of view, a clear physicalinterpretation in terms of electron trajectories is hard to obtainfrom these methods. This interpretation, helpful to understandthe mechanism underlying the interference patterns observed inmolecular junctions,40-42 can be made afterward studying theresidue of the Green function,43 converting the transmissioncoefficient to a rate expression for electron transfer,44-46 ordecomposing the transmission matrix into local contributions.47

A formulation of the transmission coefficient from the time-dependent Schrodinger equation could a priori give a completedescription of the electron trajectory during the tunnellingprocess and would provide an intuitive interpretation of theresonance and interference patterns observed in a molecular

junction. Bar-Joseph et al. proposed such an approach but onlyfor a single quantum state trapped between the two electrodes

Special Issue: Shaul Mukamel Festschrift

Received: November 30, 2010Revised: January 12, 2011


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5583 dx.doi.org/10.1021/jp111384d |J. Phys. Chem. B 2011, 115, 55825592

The Journal of Physical Chemistry B ARTICLE

and without comparing their results to previous theory norstudying the behavior of the wave vector during the tunnellingprocess.48 Ness et al. also proposed a time-dependent point of

view but expressed their result in terms of Green functions,preventing a different interpretation from those obtained withtime-independent methods.49 Recently, Sanchez et al.50 andSubtonik et al.51 proposed independently similar solutions basedon the steady states of the Liouville equation where the continuaformed by the electrodes act as relaxation terms on a smallnumber of bath states interacting with the molecule. Other time-dependent methods have been proposed,52-57but to the best ofour knowledge, none of them has been used to understand the

nature of resonance and interference patterns and their relation-ship with the evolution of the time-dependent state vector of thesystem during the tunnelling process.

We propose here a simple model to compute the conductanceof a molecular junction starting from the time-dependent Schro-dinger equation where the continua formed by the electrodes aredescribed by a Fano model.58-62 Our solution is compared totime-independent methods for validationand to time-dependentquantities to give physical insight into the conductance lineshape. The article is organized as follows: In section II, wepresent the method when applied to thesingle impurity problem

where analytical expressions for all the time-dependent popula-tions can be derived. The concept of resonance is discussed here.The generalization of the method to larger systems is then

presented in section III using a Bloch criterion63-65

to obtainaccurate approximative solutions for the evolution of the state

vector. In section IV,we study different configurationsof two-sitesystems and give physical insight into the different interferencepatterns observed. Finally, in the last section, our method isapplied to thenetwork of a benzene ring where theinterferenceand resonance patterns are easily explained using the conceptsestablished in the previous sections.

II. ANALYTICAL CASE: THE SINGLE IMPURITYPROBLEM

We begin with the system represented in Figure 1 where twoidentical linear chains composed of N states each interact

through a single quantum state, |.66,67

We suppose that onlythe last site of each electrode interacts with | though a icoupling. This system has been widely studied, so previousresults will help to confirm our own. Consider the solution ofthe time-dependent Schrodinger equation (taking p = 1):

jt e - iHtja 1where H is the Hamiltonian of the entire system describing thetwo electrodes and the impurity. The initial state, |a, is chosenas an eigenstate with an energyE of the left electrode. The choiceof this particular |a is discussed below. Due to the interaction

between the left electrode and the impurity, |a is not aneigenstate ofH. Therefore, the time-dependent state vector,

|(t) , can propagate and eventually reach a state at leastlargely localized in the right electrode. Since we are interestedin elastic tunnelling, the target state, labeled |b, is chosenas an eigenstate of the right electrode with the same energy, E,as |a.

The coupling strengths a , between | and |a , and b,between | and|b, are dominant features in the characteristics

of |(t). With the simple electrodes used here, these couplingsare26,67

i viffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 -E - e0

2h

2s2

with = (2/(N 1))1/2, where Nis the number of states in theelectrode. e0 and h are, respectively, the energy of each site andcoupling strength between two neighboring sites in the electrode(see Figure 1)and are set in the following to e0 =0eVand h = -2eV. Therefore, even if the impurity interacts strongly with theelectrodes,a and b remain very weak due to the delocalizationof |a and |b over the large number, N , of sites in eachelectrode.

To describe the two quasi-continua formed by the electrodesin a finite way, we will use the Fano model58-60 and the effectiveHamiltonian it leads to.61,62 In this effective Hamiltonian, theenergy of any state interacting with one electrode is shifted by animaginary quantity. This non-Hermitian modification of theHamiltonian accounts for the wave function leaking in thequasi-continuum through the impurity. Since | interacts withthe two continua, its energy is shifted by = (a b), with i =-2i

2F(E), where F(E) = |1/2h|(1 - ((E-e0)/2h)

2)1/2 is theelectrodes density of states.67 The effective FanoHamiltonian ofthe system shown in Figure 1 is finally given by

This effective Hamiltonian, sketched in Figure 2, is equivalentto the one derived by Mujica et al., if we only consider theimaginary part of the continuums self-energy.25,26 Using thiseffective Hamiltonian, it is possible to solve eq 1 exactly and togive analytical expressions for the projected wave vector on thedifferent states: Ca(t) = a|(t), Cb(t) = b|(t) , andC(t) = |(t)

Cat

a

2

a2 b2 b2=a2 cos2 e - ie X =2te

- X-=2t

sin2 e - ie - X =2teX- =2te- iEt 4

Cbt - aba2 b2 1 - cos2 e - ie X =2te - X-=2t

- sin2 e - ie - X =2teX - =2te - iEt 5

Ct 1= ffiffi2p cos sin e - ie X =2te - X - =2t- e- ie - X =2teX-=2te- iEt 6

Figure1. Tight binding model of a single-stateimpurity, |, embeddedbetween two conducting electrodes. The site energy of each site in theelectrode is e0 , and the site-site coupling between two neighboringstates is h. The last state of each electrode interacts with the impurity

through the coupling strength a and b.


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The angle comes from the definitionof theSU(3) matrix whichdiagonalizes the Hamiltonian given by eq 3 and is given by

sin - 1

E -

e X2

- i - X -

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE - e X2 - i - X-2

2

2vuut

0BBBBBBB@

1CCCCCCCA

7

with = (a2 b2)1/2 and

X( 12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2 -2422 4e22

q( e2 - 2 42

1=28

The transmission coefficient, T, that the initial state localized inthe left electrode reaches the right electrode is defined by thetemporal mean value ofCb(t) given by eq 5. A major simplifica-tion can be made here. Due to the weak value ofa and b, thedefinition of leads to = /2. An accurate approximateexpression for Cb(t) can then be obtained neglecting the weakcos2 term, leading to

Cbt= a1 - e - itet 9with a = ab/(a

2 b2), = 1/2(|e| - X), and = 1/2-(X-)e 0. The very weak values ofa andb actasa filter on

Cb(t) keeping only the dominant Fourier components. Usingeq 9, the integral ofCb(t) does not converge. To solve thisproblem, thefinite lifetime of|(t) due to relaxation of thewave

vector within the electrodes is often introduced.17,66 Supposingthat this lifetime follows an exponential law with rate parameter, the temporal mean value ofCb(t) is given by

Cbt Z

0e-tCbt dt a

i

i - i 10

The same transformation has been used by Ness et al.49 tocompute the mean value of |Cb(t)|

2 , where they justify theintroduction of the exponential term as a convergence require-ment.Theelectronic transmission coefficientfrom|a to |b isthen given byT= |Cb(t)|

2 and reads

T a2 3

2 2

2 2 - 2 11

It is possible to extract the value of from physical arguments,imposing the resonance condition T(E = e) = 1. To simplify thisderivation, we impose E = 0 as the reference energy and supposethesystemin the tunnellingregime, i.e., |h| . |i|. In this regime,the transmission coefficientis approximatedbyT= (a/)2(2

2). This approximation is justified by the rapid decay of and off resonance. With these approximations and after somelengthy algebra, eq 11 becomes

Te 12

a2b

2

e2 2 12

Imposing T(e = 0) = 1 leads to

ab

13

Since eq 13 is independent of the parameters defining theimpurity, this expression will also be used for larger systems inthe following.

The probabilityT(e) given by eq 12 is plotted in Figure 3b fori = -1/4 eV and h = -2 eV. On the same figure are plotted the

Figure 2. Representation of the Hamiltonian given in eq 3. The initialstate, |a, and the target state, |b, are eigenstates of the left and rightelectrode, respectively. These two eigenstates are associated with thesame energyE. They both interact with | through weak couplings aand b. The interaction between the impurity and the two continua ismade by the dissipative Fano terms denoted a and b.

Figure 3. (a) Population of |b in thesingle impurity case forE =0eVandthreevaluesofe: e =0eV(black), e = 0.5 eV (blue), and e =1(red)eV.Theother parameters are as follows: a = b = 0.5 eV, h = -2 eV, and = 0.2. During the evolution, half of the population leaks in the continua and the otherhalf oscillates between |a and |b through |. The transmission coefficient is proportional to the area belowe

-t |Cb(t)|2. As shown in the inset,the population of the central state decreases when e moves away from the resonance. (b) Transmission given by eq 12 (plain line), the ESQC method(diamond points), and the Green-function-based method25,26 (dashed blue line) computed for h = -2 eV and i = -0.25 eV.


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transmission coefficient for the same system calculated with theESQC method19and the Green-function-based method.25 Thesethree curves present the same Lorentzian shape centered on E =e, and very good agreement among the three is seen. Further-more, the method presented here gives access to importantphysical quantities such as the population of the different statesinvolved in the electron transfer through the junction. Valuable

physical information can be obtained comparing eq 11 to thetarget state population given by

jCbtj2 a21 e2t - 2et cost 14The parameters and , which define the T(e) line shape, areclearly identified as the oscillation frequency and the exponentialenvelope parameterof the target state population. The oscillationamplitude, a, only modifies the mean value ofT(e) but does notaffect its profile. This definition of the transmission coefficientagrees with Bardeens work68 that proposed a relation betweenthe T(E) and the square of the oscillation frequency between thetwo reservoirs.

The target state population, |Cb(t)|2 given by eq 5, is plotted

in Figure 3a fore

= 0, 0.5, and 1 eV with

= 0.2 anda

=b

=0.5 eV. Due to the dissipative term in the Hamiltonian, half of thepopulation leaks in the continua during the temporal evolution.

When the steady state is reached, the other half is equallydistributed on |a and |b and the population of | is null.This explains why |Cb()|

2 converges to 1/4 in this particularconfiguration. This peculiar behavior does not affect the calcula-tion ofT(e), since the exponential term e-treaches zero rapidly.

As already proposed in the literature,54 only the first oscillationsofCb(t) are involved in the computation ofT(e): the functione-t acts as a band-pass filter on the target state population.

In the time-dependent approach depicted above, a resonancein T(E) appears as a maximum oscillation frequency of the targetstate population, which is obtained here for e = 0. At this energy,

this oscillation frequency becomes (e = 0) = (a2

b2

)1/2

,whereas it is proportional to 2 off resonance. The envelopeparameter,, varies also with e. Its absolute value is maximal fore = 0 where it becomes 22/, whereas it is proportional to 4 offresonance. This explains the evolution of the target statepopulation obtained at resonance where the oscillatory term iscompletely damped by the envelope. At resonance, |Cb(t)|

2 ismaximum and the transmission coefficient presents a peak.

The coefficient C(t) = |(t) , given by eq 6, is also avaluablequantity. As canbe observedin theinset of Figure3a, themaximal amplitude of |C(t)|

2 is larger at resonance than offresonance. This amplitude also increases when the amplitude ofthe imaginary Fano terms decreases. However, this amplituderemains close to zero which is characteristic of superexchange

oscillations between the initial and the target state.54,15,69In a time-dependent calculation, the choice of the initial state

is a key feature. An initial state localized on one single atomicorbital of the left electrode49 or a state uniformly distributed overall these atomic orbitals55 is often used. In such situations, sincethe initial state is not an eigenstate of the unperturbed electrode,|(t) propagates in the left electrode subspace due to strongsite-site coupling rather than oscillating smoothly from oneelectrode to the other. Then, the oscillation amplitude of anystate localized in the right electrode remains extremely low, andthe definition of a transfer coefficient becomes tricky. Using aneigenstate of the unperturbed left electrode solves this problemand allows us to obtain Rabi-like oscillations of the wave function

between the two electrodes. The principal drawback of thisspecific initial state is the very slow oscillation rate obtained

between |a and |b which is proportional to and 2,

respectively, at and off resonance. For example, with a = b =10-1 eV, i.e., when |a and |b are delocalized over roughly100 orbitals, the time needed for the initial state to be entirelytransferred to the target state is around 22 fs at resonance and

increases dramatically off

resonance.

III. GENERALIZATION TO LARGER SYSTEMS

The same time-dependent analysis can be made for moleculeswith several sites. Following the same model used above, theHamiltonian used to compute T(E) is

where is a coupling matrix containing the weak couplingstrength i between |a, |b and the molecular states |j.H0 is theunperturbed Hamiltonian of themolecule, and isthedissipative Fano matrix. In the general case, this matrix cancontain off-diagonal elements when several molecular statesinteract with the same electrode.62

For these systems, the time-dependent Schrodinger equationcannot generally be solved analytically. Thanks to the small

values of the coupling strength between the scattering states andthe central molecule in the tunnel junction, a very accuratesolution can be easily obtained through a Bloch criterion.13,63,65

Suppose that the initial and target states are resolved on theeigenbasis of the Hamiltonian by |a = Pnan|n and |b =P

nbn|n , where {..., |n , ...} are the eigenvectors of theHamiltonian. The function Cb(t) = b|e -iHtt|a can then

be written as

Cbt X

n

anbn exp

- int 16

where n is the nth eigenvalue of the Hamiltonian. Due to thetopology of thesystem, |a and|b aremainly localizedon twoeigenstates, labeled | and |- , and associated with theeigenvalues(. Numerically, thetwo eigenstates | and|-can be found easily, since they maximize the Bloch criterion:

n, m ajPnjbbjPmja 17

with Pk = |

k

k|. The general expression for Cb(t) cantherefore be simplified to

Cbt= aeffexp - i t - exp - i - t 18where aeff=

Pn|anbn| is the amplitude of the effective oscillation.

The mean value of eq 18 is calculated following the transforma-tion introduced in eq 10, and the transfer coefficient is given bythe square modulus of this mean value. With the same approx-imations used for the single impurity case, T(E) of a multistatesystem is given by

TE aeff2 3

2 22

19


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with given by eq 13. In eq 19, and are given by = Re-(- - ) and = |Im(- - )|. These parameters are,respectively, the oscillation frequency and the decay rate of|Cb(t)|

2 = aeff2e-2tsin2(t). This method provides a black

box allowing us to compute the transmission coefficient of anysystem tracking the dominant subspace formed by the eigenvectors|(. On the basis of an effective model, this method does notgive access to the projection of a wave vector on the differentstates. These quantities, very useful to understand the patterns inthe T(E) line shape, will be approximated using the Dysonexpansion of the time-dependent Schrodinger equation.

Two-Site Systems: Interferences. A two-site system, withtwo states |1 and |2 of energies e1 and e2 , can interact indifferent ways with the electrodes and therefore can lead todifferent T(E) line shapes. The longitudinal and transversesituations are represented in Figure 4. Their respective elec-tronic transmissions have been calculated following the meth-od presented in the previous section. The Fano matrices in thelongitudinal (L) and transverse (T) configuration are,respectively,

with i = -2i2F(E). Then, for each value of E , the dominant

eigenvalues of the Hamiltonian are computed using the Blochcriteria and then used in eq 19 to compute T(E). The result ofthis calculation is represented in Figure 4 and compared with theESQC calculation. Both calculations have been performed usingthe same electrodes as in Figure 1 with h = -2 eV and fordifferent values of a and b. The dimer system used here ischaracterized bye1 = e2 = 0 and R = -1 eV.Very good agreement

between the two methods is seen.The conduction of the serial configuration is the typical

multiple-resonance line shape of a simple two-atom wire. As

for the single impurity case, when the energyE is resonant withan eigenenergy, (, of the dimer, the oscillation frequency of|Cb(t)|

2 becomes (E = () = (a2 b2)1/2 and the

transmission coefficient is equal to unity. In between these twovalues, the oscillation frequency is proportional to 2 and thetransmission decreases dramatically.

Frequency-Related Interference. The transverse configura-tion presents a deep interference in between the two resonances.

While this interference is well-known,40,44,70 the time-dependentmethod presented here allows new insight into the physicalprocess leading to this effect. Since this interference appears forany value of the coupling terms, a and b, we study the simplestsituation where these couplings are small enough for theimaginary part of the Hamiltonian to be neglected. In thissituation and assuming a = b = , the Hamiltonian of thetransverse configuration reads

with( = 1/2[(e1 e2) ( ((e1 - e2)2 4R2)1/2], - = cos ,and = sin , where = 1/2 tan

-1[2R/(e1 - e2)]. The dotsin eq 21 are null matrix elements not represented to emphasizethe structure of the system. Using the Dyson expansion of thetime-dependent Schrodinger equation, one can obtain the ex-pression of the projected wave function on the dimer eigenstates,C

((t) = (|(t):

C ( t=(

(1 - cos ( t - i sin ( teiEt 22

with ( = E - (. The probability amplitude Cb(t) = b|(t) , can be obtained using the second order of the

Figure 4. Two-site system in (a) longitudinal and (b) transverse configuration. The electronic transmission of these two systems calculated with ourmethod (plain curves) and the ESQC method (dashed curves). The calculation has been made for R = -1 eV and e1 = e2 = 0 for both cases, even ifrepresented differently in part b. Theelectrodes arethe same as in Figure1 with h = -2 eV.The interactions between theelectrodes andthe sites area =b = 0.25 (blue lines), 0.5 eV (black lines), and 1 eV (red lines). The vertical dashed lines represent the position of the molecular eigenstates.


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Dyson expansion:

Cbt= - iXn (

n

Zt0

e - inCn d

X

n (

n2

n

1 - eint

n it

" #eiEt 23

The termP

n(n/n)2(1 - eint) reflects the presence of high

frequency components with a very small amplitude in thedevelopment ofCb(t). The term

Pn(n

2/n)tis the first-orderdevelopment of the function sin(

Pn(n

2/n)t), which is a lowfrequency component with amplitude equal to 1. This sinusoidalfunction smooths Cb(t) with Rabi-like oscillations. The oscilla-tion frequency of this term is therefore the dominant oscillationfrequency used to compute T(E). This frequency is given by

E Xn (n

2

n Xna

jH

jnn

jH

jb

E - n 24which is the Green function of the dimer projected on the initialand target states used in most of the time-independent methodsto compute the electronic conduction.25,36,37 In the transverseconfiguration, this term is given by

E 2 E - cos2 - - sin2 Q

n (E - n

0B@

1CA 25

Therefore, forE = cos2- sin2, the dominant oscillation

frequency vanishes, leading to an interference pattern in T(E). In

the simple case e1 = e2 = 0, represented in Figure 4, thisinterference is located at E = 0.Equation 23 is not really convenient to understand what

interferes with what to obtain the interference pattern. A moreconvenient formulation for Cb(t) can be obtained neglecting the

weak high-frequency terms in eq 23, leading to Cb(t) =i sin(t)eiEt. Developing sin(t)Cb(t) becomes

Cbt= ImY

n (exp i

han

n3 hnb

t

" #eiEt 26

where han/n = a|H|n/(E - n) is the oscillation ampli-tude ofCn(t) and hnb = n|H|b is the coupling strength

between |b and |n. Cb(t) appears as the product ofoscillating terms, each coming from one eigenstate of the dimersystem. The oscillation frequency of the nthtermisgivenbytheoscillation amplitude ofC

n

(t) multiplied by the interactionstrength between |b and |n. For these interferences tooccur, it is therefore not required that all the Cn(t) oscillate inphase, out of phase, or with any phase relation, to obtain theinterference, as long as their amplitudes, weighted by thestrength with which |b includes each oscillation, cancel eachother.

Amplitude-Related Interference. Other configurations can be considered to connect two sites to the electrodes. Onesolution is represented in Figure 5 where |2 interacts with

both |a and |b whereas |1 interacts only with |a. In thiscase, the Fano matrix is given by

with i = -2i2F(E). The off-diagonal matrix element comes

from the fact that both |1 and |2 interact with the leftelectrode and are therefore indirectly coupled through thiscontinuum.62 The electronic conduction of this dangling config-uration has been calculated using both the method presentedabove and the ESQC method for e1 = -e2 = 1 eV with the sameelectrodes as in Figure.1. Very good agreement appears onceagain between the two methods, and both show an interferencelocated atE = e1. To understand theoriginof this interference, letus write the Hamiltonian of this configuration and its block-

diagonalization in the {|a,|1} subspace, with a = b = :

with E( = 1/2[(E e1) ( ((E - e1)2 42)1/2], - = cos, = - sin , and = 1/2 tan

-1(2/E - e1). Once again,the Dyson expansion of Cb(t) is very useful to understand

Figure 5. (a)Two-level systemwith oneconductive state (|2) andone dangling state (|1). (b) The electronictransmission of this two-levelsystem,calculated with our method (plain) and the ESQC method (dashed). The calculation has been made for e1 = -e2 = 1 eV. The electrodes are the same asin Figure 1 with h = -2 eV and i = 0.25 eV (blue lines), 0.5 eV (black lines), and 1 eV (red lines). The vertical dashed lines represent the energeticposition of the two states.


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the interference pattern observed at T(E = e1). This expansionleads to

Cbt= R 1 - e- ie2 -Et

e2 - E- R

1 - e - iE -E - tE -E -

"

- 1 - e - iE -E t

E - E e - iEt 29

with R =-2/(E-- e1)and =

2/(E- e1).Togetasimplepicture, let us express |Cb(t)|

2when |1 is at and offresonancewith E. When E = e1 , the population of the target state can besimplified as Cbtj2=

2

4sin2t 30

whereas off resonance this population is given byCbtj2= 1

2

E - e1

E - - e2 2

1 - cos22

E - e1t

! !31

It is interesting to note that, even if |1 does not connect the twoelectrodes, it induces a resonance of |Cb(t)|

2, since its oscillationfrequency is maximum at E = e1, where it equals , and drops to2 off resonance. The interference observed at E = e1 isconsequently not a frequency-related interference like the oneobserved in the transverse configuration. On the contrary, theamplitude of |Cb(t)|

2 is minimum at resonance where it scaleslike 2 and increases to unity off-resonance. This drastic drop ofthe oscillation amplitude is responsible for the interferencepattern observed at E = e1. The origin of this drop in amplitudeis easy to understand writing the Dyson expansion ofC1(t).Dueto the delocalization of |a over | and |-, this expansioncontains a zero order term and reads

C1t= 2i cos sin e- iEe2=2t sin0t - sin C2 - t- cos C2 t 32

with 0 = 1/2((E - e1)2 42)1/2 and C

(

(2)(t) given by thesecond order of the Dyson expansion terms:

C2(

t X

n (

(jan(En - e2

1 - eiE ( Ent

E ( En -1 - eie2 - E ( t

e2 - E (

!33

In the vicinity of the resonance, when E = e1, |a and |1 arequasi-degenerate and the zero-order term in eq 32 is far moreimportant than the second-order terms. The population of |1then reads

jC1tj2 =42 sin20t

E - e12 4234

It is then clear that when E = e1 almost all the initial population istransferred from |a to |1. This leads to the very smalloscillation amplitude of |Cb(t)|

2 and consequently to the inter-ference observed in Figure 5. Off resonance, since E - e1 . ,theamplitudeofC1(t) remains very weak and |(t) can slowlyreach the target state.

The two different types of interferences are schematized inFigure 6, the frequency-related interference where the oscillationfrequency of |Cb(t)|

2 drops to 0 and the amplitude-relatedinterference where the oscillation amplitude of |Cb(t)|

2 dropsdramatically. Theresonanceand interference patterns studied forsimple pedagogical systems in this section will be very useful inthe next section to understand the T(E) line shape of more

complex systems.

IV. MOLECULAR JUNCTION: THE BENZENE RING

The resonance and interference patterns studied in the pre-vious sections are widely observed in molecular junctions. Let usexamine the well-known case of the network of a benzene ring

between twoconducting leads in a tight binding model.38,40-44,70

In this Huckel-like approximation, the direct coupling betweenthe two electrodes is neglected and therefore does not blur theintrinsic conduction of the molecule. We use the same electrodesas in Figure 1, with h = -2 eV; the two electrodes will beconnected either in the ortho- or meta-substituted position.These calculations are made for different values of the couplingstrength a = b, which allows us to compare our result with the

ESQC calculation in the tunnelling regime (i = -0.25 eV), thepseudoballistic regime (i = -2 eV), and the supercoupledregime (i = -4 eV).

In each configuration, each electrode interacts with a given pzorbital of the network of the molecule. The left electrodeinteracts always with |1 and the right electrode with |X withX= 2 in the ortho-configuration, represented in Figure 7a, andX=3inthemeta-configuration.Inourmodel,thisleadstoaweakcoupling between |a and |1 and between |b and |X. TheFano matrix introduces imaginary diagonal elements that shiftthe energies of |1 and |X byi = -2i

2F(E). The transmis-

sion coefficient of these two systems has been calculated follow-ing the method presented in this article and compared with the

Figure 6. Graphical representation of the two interference patterns observed on the dimer system. (a) Frequency-related interference: the oscillationfrequency ofCb(t) vanishes when

Pnann = 0 with an being the oscillation amplitude ofCn(t). (b) Amplitude-related interference: when E = e1, the

initial population is entirely transferred from |a to the dangling resonant eigenstate, |1. Theamplitude ofCb(t) drops then to2, whereas it equals 1

when E 6 e1.


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Figure7. (a) Schematic representationof the system usedto compute thetransfercoefficientofabenzeneringwhenthetwoelectrodesareattachedinameta substituted configuration. (b) Population of the target state for different values ofE: 2 eV (black), 0.5 eV (blue), and 0 eV (red) with = 0.25, i =1/2 eV, and h = -2 eV. In the resonant case, the high oscillation frequency leads to a high conductance, whereas the low oscillation frequency obtainedoff resonance leads to a low value of the transfer coefficient.

Figure 8. Electronic conduction of a benzene ring in between two conducting electrodes. These calculations are performed by the time-dependent method presented here (solid line) and by the ESQC method (dashed line) for comparison. The electrodes are connected either inortho (left column) or meta (right column) position. Three regimesare investigated:tunnelling with i = -0.25 eV (top row), pseudoballisticwithi = -2 eV (middle row), and supercoupled with i = -4 eV (bottom row). The vertical dashed lines represent the energetic position of thebenzenes eigenstates.


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ESQC calculation. To ease the interpretation, the energy of thepz orbitals has been set as the reference energy, i.e., 0 eV, and thecoupling between two neighboring pz orbitals normalized to -1eV. With these parameters, the eigenvalues of the Hamiltonianare simplye6 = -e1 = 2 eV and e5 = e4 = -e2 = -e3 = 1 eV. Asobserved in Figure 8, good agreement is found between the twomethods for the different values ofa and b.

Already observedfor thesingle impurity andthe dimer system,the resonances appear clearly and still correspond, as illustratedin Figure 7b, to a maximum of the oscillation frequency reached

when E equals the eigenvalues of the benzene network. Asbefore, resonance corresponds to a maximum of the oscillationfrequency between |a and |b.

Interference patterns also appear in both configurations.40

Several arguments have been made to explain these interferencessuch as a phase opposition between topological electronicpathways.44 They have been studied in detail using a Greenfunction formalism43 and a local current description.47 Theconcepts establised in the previous sections are very useful tounderstand the nature of these interferences. Let us examine themeta-configuration neglecting theimaginarypart of theHamiltonian

to facilitate the calulations. Block-diagonalizing the Hamiltonianof the system on the molecular level subspace, the Hamiltonianof the meta-configuration becomes

where {..., |n, ...} are the eigenstates of the benzene ring and1 = -2eV,2 =3 = -1eV,4 =5 = 1 eV, and6 = 2 eV are itseigenenergies. Two eigenstates, |3 and |4 associated,respectively, with the energies -1 and 1 eV, are only connectedto one of the two electrodes. As for the dimer system, thesedangling eigenstates induce amplitude-related interferences

when E = (1 eV, as seen in Figure 8. The initial population isthen completely transferred from |a to |3 ifE = -1 eV or to|4 ifE = 1 eV. At these energies, the oscillation amplitude ofCb(t) remains very weak and an amplitud-related interferenceappears in the T(E) line shape.

The degenerancies at E = (1 eV lead to a competitionbetween a resonant and an interfering pathway at each of theseenergies. In the tunnelling regime, the interactions between theelectrodes and the system are not strong enough to splitsignificantly the two degenerate states. During the temporalevolution, the initial population then splits between the twopathways. A resonance involving half of the initial population isthen observed on the target state. This explains whyT(E = ( 1eV) is lower than 1 in the tunelling regime. Increasing theinteractions between the electrodes and the system splits thedegenerancies at E = (1 eV, suppressing the competition

between the resonant and interfering pathways. The ampli-tude-related interference is not screened by a resonance and

clearly appears on the T(E) profile. The same situation isobserved in the ortho-configuration but not in the para-config-uration, where |3 and |4 are disconnected from bothelectrodes.

Other interference patterns appear in Figure 8. The Dysonexpansion of the time-dependent Schrodinger equation allowsone to express the functions Cn = n|(t) and provide a

simple explanation of these interference patterns. These func-tions read

Cnt=han

n1 - einte- iEt 36

with haj = a|H|j. As before, j = E -j, where j is the jtheigenvalue of the benzene ring. The expression of the target stateamplitude probability is then given by

Cbt= -X6n1

hanhnb

n

1 - eint

n it

" #e- iEt 37

with hnb = n|H|b. As for the dimer system, the termPn(hanhnb/n)t is the first order in the Taylor expansion ofsin(

Pn(hanhnb/n)t); its oscillation frequency is given by(E)=P

n(hanhnb/n). The expression of(E) in the ortho and metaconfigurations is

OE 161 2 - 4 - 6 -

2E2 - 2nE - n 38

ME 161 - 2 - 4 6

2E

nE - n 39

with i = 2/(E -i). The interferences located at E = ((2)

1/2

and E = 0 for the ortho- and meta-configuration, respectively, areconsequently frequency-related interference as observed in the

transverse confi

guration of the dimer system. This assertion isverified numerically in Figure 7b, where |Cb(t)|2 for the meta-

configuration remains null when E = 0. The ortho-configurationis particularly interesting, since at E = ((2)1/2 the oscillationfrequencies of theCn(t) arenoteven commensurable with eachother and yet an interference is obtained on Cb(t). Thisemphasizes the non-necessity for phase matching between theoscillation on the molecular states to obtain an interferencepattern.

Thanks to the preliminary study of the single impurity and thedimer system, the interpretation of the T(E) profile for the

benzene ring becomes extremely easy, since it is a simple super-position of resonances and frequency- or amplitude-relatedinterferences. The method presented in this article gives a rather

simple physical explanation of the process involved in a molec-ular junction and can easily be applied to larger molecules. Verydifferent conclusions relative to the origin of interference pat-terns in a benzene ring have recently been drawn.43 On the basisof the Green function element beween the two carbons atomsconnected to the elecrodes, rather than between states inside thetwo electodes, this analysis leads to different classification for theinterference patterns. The interferences located at E = (1 eVand ((2)1/2 eV form the first class of interference, labeledmultipath zeroes. The remaining interference, located at E = 0eV in the meta configuration, is referred to as a resonance zero.The former are related to interfering spatial pathways in themolecule, whereas the latter requires a pole in the self-energy of


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one of the pathways. Despite this strong difference of interpreta-tion, the spectra obtained by both methods agree perfectly.

V. CONCLUSION

Good agreement has been demonstrated between the time-dependent method used in this article and well accepted time-

independent calculations of the electronic transmission coeffi

ci-ent. The time-dependent method allows us to grasp the intrinsicphenomena taking place inside the junction. The resonances areinterpreted as high frequency oscillations between eigenstates ofthe electrodes leading to a maximal transfer probability. Inter-ference can have several origins: it canbe frequency-related inter-ferences where the oscillation frequency of |Cb(t)|

2 cancel out oramplitude-related interference where the oscillation amplitude of|Cb(t)|

2 drops drastically. To obtain a frequency-related inter-ference, it is not necessary for the population of the moleculareigenstates to oscillate in phase, out of phase, or with any phaserelation. Only the amplitude of these oscillations and the inter-action of each molecular eigenstate with the |b are relevant.The amplitude-related interferences require the presence of a

dangling eigenstate only connected to one electrode. In such acase, the initial population is entirely transferred from |a to thiseigenstate and the oscillation amplitude of |Cb(t)|

2 remains verylow. We have also shown that in most of the cases the populationof the molecular states remains almost null as expected in thesuper exchange picture of the tunnelling process. However, atresonance and for extremely weak interactions between theelectrodes and the central system, these states are significantlypopulated and electron-electron interactions should be consid-ered. This approach can now be generalized taking into accountthese electron interactions or inelastic effects71,72 occurring inthe junction.

AUTHOR INFORMATION

Corresponding Author*E-mail: [email protected] (N.R.); [email protected] (M.A.R.); [email protected] (C.J.).

ACKNOWLEDGMENT

This article is in honor of Shaul Mukamel, scientist, friend,scholar,and teacher. N.R. and C.J. would like to thank the E. U.Commission and the Japans ministry of education, culture,sports, science and technology (MEXT) for their financialsupport through, respectively, the ICT PICO-INSIDE integratedproject andthe WPI MANA program. N.R. was supported as partof the Non-Equilibrium Energy Research Center (NERC), an

EnergyFrontier Research Center fundedby theU.S. Departmentof Energy, Office of Science, Office of Basic Energy Sciencesunder Award Number DE-SC0000989.

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Documents

N. Renaud, M. A. Ratner and C. Joachim- A Time-Dependent Approach to Electronic Transmission in Model Molecular Junctions