Theory of the Franck-Condon blockade regime

Jens Koch,1 Felix von Oppen,1 and A. V. Andreev2

1Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany2Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA

�Received 13 July 2006; published 30 November 2006�

Strong coupling of electronic and vibrational degrees of freedom entails a low-bias suppression of thecurrent through single-molecule devices, termed the Franck-Condon blockade. In the limit of slow vibrationalrelaxation, transport in the Franck-Condon-blockade regime proceeds via avalanches of large numbers ofelectrons, which are interrupted by long waiting times without electron transfer. The avalanches consist ofsmaller avalanches, leading to a self-similar hierarchy which terminates once the number of transferred elec-trons per avalanche becomes of the order of unity. Experimental signatures of self-similar avalanche transportare strongly enhanced current �shot� noise, as expressed by giant Fano factors, and a power-law noise spec-trum. We develop a theory of the Franck-Condon-blockade regime with particular emphasis on the effects ofelectron cotunneling through highly excited vibrational states. As opposed to the exponential suppression ofsequential tunneling rates for low-lying vibrational states, cotunneling rates suffer only a power-law suppres-sion. This leads to a regime where cotunneling dominates the current for any gate voltage. Including cotun-neling within a rate-equation approach to transport, we find that both the Franck-Condon blockade and self-similar avalanche transport remain intact in this regime. We predict that cotunneling leads to absorption-induced vibrational sidebands in the Coulomb-blockaded regime as well as intrinsic telegraph noise near thecharge degeneracy point.

DOI: 10.1103/PhysRevB.74.205438 PACS number�s�: 73.63.�b, 81.07.Nb, 72.70.�m, 63.22.�m

I. INTRODUCTION

According to many recent experiments,1,2 transportthrough single-molecule junctions is characterized byCoulomb-blockade behavior and exhibits a Kondo effect atlow temperatures, in close analogy with experiments onquantum dots. An important distinction between transportthrough quantum dots and single molecules lies in the cou-pling to phonon degrees of freedom. While charge carrierstypically interact with a continuum of phonon modes inquantum dots, molecules are characterized by a discretespectrum of vibrational modes. Recent experiments onsingle-molecule junctions have shown that these discrete ex-citations are reflected in current-voltage �I-V� characteristicsas vibrational sidebands.3–6 This experimental effort has mo-tivated much theoretical work.7–10 It is now understood thattransport through single-molecule junctions exhibits a num-ber of qualitatively different regimes, depending on param-eters such as the strength of the electron-phonon couplingand the vibrational relaxation rate; see, e.g., Refs. 11–14.

The importance of Coulomb-blockade physics in single-molecule junctions suggests that the molecule and leads areoften weakly coupled from the point of view of electronictransport. Hence, a perturbation theory in the tunneling am-plitude between the molecule and leads should be a goodstarting point for a theoretical description. Such a perturba-tive treatment is valid if the tunneling-induced level width �is small compared to the temperature T. Then, the lowestorder in this expansion results in sequential tunneling, whichcorresponds to the transfer of one electron from a lead ontothe molecule or vice versa. Sequential tunneling is dominantas long as the electronic level is situated between the Fermienergies of the leads. By contrast, if the level is located out-side of the bias window, sequential tunneling is exponen-

tially suppressed and �next-to-leading-order� cotunnelingprocesses take over.15 This marks the onset of the Coulombblockade.16 In many experiments, a gate electrode can beused to tune the system between the sequential tunneling andthe cotunneling regime.

The coupling of electron tunneling to molecular vibra-tions adds a wealth of new physics. This coupling originatesfrom the fact that the nuclear configuration of the moleculeneeds to adjust to the addition of electrons to the molecule.Thus, electron transfer is generally accompanied by excita-tion �or deexcitation� of molecular vibrations. In the simplestcase, the electron-phonon coupling can be modeled by acharge-dependent displacement of the molecular potentialsurface.

In two recent papers,11,17 two of us have investigated theregime of strong electron-phonon coupling, where the dis-placements of the potential surfaces are large compared tothe quantum fluctuations of the nuclear configuration in thevibrational ground state. As a result, the overlap betweenlow-lying vibrational states is drastically suppressed. It isthis suppression of the Franck-Condon matrix elementswhich leads to a strong low-bias suppression of thesequential-tunneling current, which we termed theFranck-Condon �FC� blockade. It is a hallmark of the FCblockade that it cannot be lifted by means of a gate voltage,in striking contrast with other blockades such as theCoulomb blockade.

It is evident that the FC blockade is more severe for mol-ecules with fast vibrational relaxation �arising, e.g., due tointeractions with the substrate, radiation, or coupling to othervibrational modes�. The reason is that for slow vibrationalrelaxation, the transport current excites the molecular vibra-tions and transport involves excited vibrational states forwhich the suppression of the FC matrix elements is less pro-nounced. Nevertheless, the Franck-Condon blockade in the

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current-voltage characteristics exists irrespective of thestrength of vibrational relaxation.

It is thus remarkable that the transport mechanisms in thelimits of weak and strong relaxation are in fact fundamen-tally different, as revealed by current shot noise. In thesequential-tunneling approximation, strong relaxation leadsto purely sub-Poissonian current noise, as commonly ex-pected for transport in fermionic systems.18 Electrons aretransferred one by one, slightly avoiding each other due tothe Pauli principle. By contrast, for weak vibrational relax-ation transport is dominated by self-similar avalanches ofelectrons with giant Fano factors reflecting the large numberof electrons per avalanche. The occurrence of avalanches isexplained by the exponential growth of sequential-tunnelingrates with increasing vibrational quantum number. Wheneverthe molecule is in its vibrational ground state, the suppressedFC matrix elements require a long waiting time before anelectron tunnels. Most likely, the transition involves an exci-tation of the vibrations. Now, having left the vibrationalground state, the accessible transitions have faster rates, lead-ing to rapid tunneling dynamics and avalanche formation. If,eventually, the system undergoes a transition back into thevibrational ground state, the avalanche terminates and an-other long waiting time is induced. The self-similarity arises,since all previous arguments can also be applied to the firstexcited vibrational state. As an experimental fingerprint ofthe self-similarity, we have predicted a characteristicpower-law decay of the current noise as a function of fre-quency.

In this paper, we investigate the cotunneling contributionsto current and noise in the FC blockade regime. The inclu-sion of such higher-order processes is motivated by the in-sight that sequential-tunneling rates are exponentially sup-pressed in the electron-phonon coupling. By contrast,cotunneling may benefit from the participation of highly ex-cited virtual phonon states with much better overlap; see Fig.1. In this way, the relevant FC matrix elements are no longerexponentially suppressed, which may render cotunneling rel-evant despite its suppression in powers of the small quantity� /kBT. The central question which we address in this paper

is whether and, if so, in which way cotunneling affects thephenomenology of the Franck-Condon blockade as sketchedabove. We investigate this question within the rate-equationapproach, extending the formalism to include cotunnelingcorrections.

For the convenience of readers who are less interested inthe technical details, we begin our presentation in Sec. IIwith a summary of our principal results. In Sec. III wepresent the model that our analysis is based upon. The com-putational formalism and the question of regularized cotun-neling rates are addressed in Sec. IV. In particular, we dis-cuss the cotunneling rates in the FC blockade regime anddetermine their scaling with the electron-phonon coupling.Section V contains an investigation of cotunneling correc-tions to the linear conductance and nonlinear current-voltagecharacteristics. Further insight into the transport mechanismand the fate of avalanches is provided in Sec. VI, where wediscuss the current shot noise beyond the sequential-tunneling approximation. Finally, the conclusions from thisanalysis are summarized in Sec. VII. Technical details con-cerning the calculations are deferred to Appendices.

II. QUALITATIVE DISCUSSION

A. Fate of the FC blockade

The increased overlap of vibrational wave functions ex-ploited by cotunneling processes indeed makes cotunnelingcorrections significant in the FC blockade regime. In contrastto sequential-tunneling rates, we find that cotunneling ratesare not exponentially, but rather algebraically suppressed inthe electron-phonon coupling; see Eq. �14�. Consequently,for sufficiently strong electron-phonon coupling, cotunnelingdominates the linear conductance over the entire gate-voltagerange; see Fig. 4. In particular, the paradigm of a gate-controlled crossover between sequential tunneling and cotun-neling breaks down in this regime and the linear conductancedevelops a kink at the degeneracy point. Cotunneling alsosignificantly increases the low-bias current well beyond thelinear-response regime; see Fig. 6. More importantly, how-ever, we find that the remaining strong algebraic suppressionof cotunneling rates still leads to a current reduction. There-fore, a central finding is that the FC blockade as such re-mains intact.

B. Persistence of avalanches and enhanced Fano factors

Another central result is that the avalanche dynamics,found in the sequential-tunneling approximation for weakvibrational relaxation, persists in the presence of cotunnel-ing. This can be understood by noting that cotunneling isdominant whenever the molecule is close to its vibrationalground state. However, the avalanches involve excited vibra-tional states for which cotunneling is in fact a small pertur-bation. Nevertheless, the long waiting times between ava-lanches are now governed by cotunneling, leading to Fanofactors which are smaller than predicted by the sequential-tunneling approximation, although still strongly enhancedcompared to the Poissonian limit; see Fig. 9. In a similarvein, the power-law behavior of the noise power spectrum

FIG. 1. �Color online� �a� Sequential tunneling versus �b� cotun-neling for strong electron-phonon coupling. In the FC-blockade re-gime, sequential processes as depicted in �a� are strongly sup-pressed due to the exponentially small overlap of harmonicoscillator wave functions. For cotunneling processes �b�, this sup-pression is partially lifted due to contributions from highly excitedvirtual phonon states.

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remains valid, but is reduced in the frequency range overwhich it applies; see Fig. 10.

C. Absorption-induced vibrational sidebands

For weak vibrational relaxation, cotunneling leads to astriking new feature: the appearance of additional vibrationalsidebands inside the Coulomb-blockaded region; see Fig. 7.Inside the Coulomb-blockaded regime, molecular vibrationscan be excited by inelastic cotunneling. Once molecular vi-brations are excited, sequential-tunneling processes becomepossible when accompanied by absorption of a phonon. Thisleads to new vibrational sidebands which were in fact ob-served in experiments on suspended carbon nanotubes.19 In-terestingly, such absorption-induced sidebands are expectedto be a generic feature which is not limited to strongelectron-phonon coupling.

D. Inherent telegraph noise

For strong vibrational relaxation, avalanche dynamics isruled out and current noise remains sub-Poissonian in thesequential-tunneling approximation. Surprisingly, we findthat cotunneling causes a strong enhancement of currentnoise at low bias due to inherent telegraph noise; see Figs. 2and 8. Traces of this telegraph behavior are also found forweak vibrational relaxation, rendering this effect robust withrespect to the relaxation rate of the system. Its origin is theunusual situation that cotunneling rates are larger thansequential-tunneling rates, Wcot�Wseq. Thus, sequential tun-neling leads to a slow switching between the two chargestates n=0 and 1, which modulates the fast cotunneling dy-namics. The cotunneling rates in these two states are differ-ent in general, which leads to two distinct current states. Thisis reflected by large Fano factors F��Wcot /Wseq�1, where�Wcot denotes the difference of cotunneling rates in the twocharge states.

III. MODEL

The Anderson-Holstein Hamiltonian constitutes the mini-mal model which captures the Coulomb blockade and theKondo effect, as well as vibrational sidebands.9–12,20–23 Sev-eral assumptions and simplifications enter into this model: �i�Transport is assumed to be caused by tunneling through onespin-degenerate orbital of the molecule. �ii� One mode ofmolecular vibrations is taken into account within the har-monic approximation. These ingredients result in the Hamil-tonian H=Hmol+Hleads+HT with

Hmol = �dnd + Und↑nd↓ + ��b†b + ����b† + b�nd �1�

describing the electronic and vibrational degrees of freedomof the molecule,

Hleads = �a=L,R

�p,

��ap − a�cap† cap �2�

the noninteracting leads, and

HT = �a=L,R

�p,

�tapcap† d + H.c.� �3�

the tunneling between the leads and molecule. �iii� In addi-tion, electronic relaxation is taken to provide the shortesttime scale in the problem, so that the metallic electrodes aredescribed by Fermi seas in thermal equilibrium at all times.

The validity and various extensions of this model in thecontext of transport through single molecules have been dis-cussed, e.g., in Refs. 9, 10, 12, and 24–26. However, theinclusion of generalizations would only obscure the centralpoints of the present paper, and we therefore restrict our-selves to the model constituted by Eqs. �1�–�3�. For simplic-ity, we consider the case of symmetric voltage splitting; i.e.,the bias voltage V enters the left and right Fermi energies asL,R= ±eV /2. �Note that all energies are measured with re-spect to the zero-bias Fermi energy.� All results are derivedfor the wideband limit with constant densities of states �L,Rand momentum-independent tunneling matrix elements ta.

There are four different energy scales relevant to theanalysis of transport regimes of the Anderson-Holsteinmodel: the phonon energy ��, the tunneling-induced levelwidth �=2��a�a � ta�2, and the energy scales set by theKondo temperature TK and the ambient temperature T. Here,we are interested in the “high-temperature” limit of themodel; i.e., we study the regime where kBTK, �� ��, kBT.In this case, the tunneling Hamiltonian HT may be treatedperturbatively, and it has been shown that the equation ofmotion for the reduced density matrix of the molecule re-duces to simple rate equations.10

The electron-phonon coupling, parametrized by the di-mensionless coupling parameter �, can be treated exactly bymeans of the Lang-Firsov canonical transformation.27 Thistransformation amounts to renormalizations of the single-particle energy �d→�d�=�d−�2�� and of the charging en-ergy U→U�=U−2�2�� �polaron shift�, and it introduces atranslation operator into the tunneling matrix elements, ta→exp�−��b†−b��ta. This shift �� in the normal coordinateof the harmonic oscillator precisely corresponds to the factthat the equilibrium distances between the nuclei of the mol-

FIG. 2. Coarse-grained current I�t� as obtained by Monte Carlosimulation for equilibrated phonons at strong electron-phonon cou-pling ��=5�, with bias and gate voltage fixed to eV=0.8�� and�d=0.5��, respectively. The time dependence of the current clearlyreveals the current telegraph noise caused by a slow switching be-tween the two molecular charge states via sequential tunneling�kBT=0.05��, �a=0.02��; coarse graining over 300 events�.

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ecule change whenever an additional electron is present inthe molecule. The transformed Hamiltonian reads H�=Hmol�+Hleads� +HT�, with Hleads� =Hleads and

Hmol� = �d�nd + U�nd↑nd↓ + ��b†b , �4�

HT� = �a=L,R

�p,

�tape−��b†−b�cap† d + H.c.� . �5�

In the following, we will proceed with the transformedHamiltonian and drop all primes.

The system is expected to display qualitatively differentbehavior depending on the vibrational relaxation rate as wellas the strength of the electron-phonon coupling. Throughoutthis paper, we will focus on the case of strong electron-phonon coupling, ��1,28 and non-negative effective charg-ing energies, U−2�2�� 0. The latter inequality is alwayssatisfied for sufficiently strong Coulomb repulsion betweenelectrons inside the molecular orbital. Unless explicitlystated otherwise, we will concentrate on the limit U→�,applicable whenever U is large compared to all other rel-evant energy scales.

IV. FORMALISM

In this section, we discuss the formalism which consti-tutes the basis for our analysis. The main input for the trans-port calculations consists of the transition rates induced byelectron tunneling. In the following subsection, we discusshow to obtain these rates, especially in the case of cotunnel-ing where the naive application of Fermi’s golden rule leadsto a divergence. The subsequent subsection explains the rate-equation approach for calculating the steady-state current aswell as the noise spectrum.

A. Transition rates

Regarding the tunneling Hamiltonian HT as a perturba-tion, one can proceed to calculate rates for transitions �i�→ �f� by an expansion of the T-matrix and applying Fermi’sgolden rule,

�if =2�

��f �T�i��2��Ei − Ef�

=2�

��f �HT + HTG0HT + ¯ �i��2��Ei − Ef� . �6�

Here, G0= �Ei−Hmol−Hleads+ i��−1 is the free retardedGreen’s function. Generally, the initial and final states �i� and�f� still involve degrees of freedom of the leads. These areeliminated by integrating over the energies of particles andholes generated in the Fermi seas during the process, takinginto account thermal occupations by f or 1− f factors, wheref is the Fermi function. As a result, one arrives at rates Wwhich are labeled by molecular degrees of freedom only. �Ofcourse, the rates still include a label for the participatingjunction�s�.�

To lowest order in HT, one obtains sequential-tunnelingprocesses, which transfer one electron from a lead onto themolecule or vice versa. For example, the rate for a process

changing the charge number of the molecule from n=0 ton=1 by tunneling across junction a, and simultaneouslychanging the number of excited phonons from q to q�, isgiven by

Wqq�;a01 = s�0,1�

�a

��Mqq��

2fa��d + �q� − q� ��� . �7�

Here, fa���= f��−a� denotes the Fermi function for lead a.The symbol Mqq� denotes the FC matrix elements for a pho-non transition q→q�, given by the overlap of two harmonicoscillator wave functions �, spatially displaced by the dis-tance �x=2��osc. Here, �osc=x2�q=0= �� /M��1/2 definesthe oscillator length. Introducing q=min�q1 ,q2� and Q=max�q1 ,q2�, the matrix elements read

Mq1q2=

−�

�

dx�q1

* �x + �x��q2�x�

= �sgn�q2 − q1��q1−q2�Q−qe−�2/2 q!

Q!Lq

Q−q��2� , �8�

where Lmn �x� denotes the generalized Laguerre polynomial.

Our choice of jointly treating the spin-up and spin-downstates as the singly occupied state n=1 requires the inclusionof a spin factor s. The definitions s�0,1��2 and s�1,0��1reflect the fact that rates for processes n=0→1 are twice aslarge as rates for n=1→0 due to the spin degeneracy of thestate n=1. The rate for the analogous process n=0→1 canbe obtained from Eq. �7� by swapping the spin factor andsubstituting fa by �1− fa�, i.e.

Wqq�;a10 = s�1,0�

�a

��Mqq��

2�1 − fa��d − �q� − q� ���� . �9�

In next-to-leading order in HT, cotunneling processes aregenerated. These transfer one electron from lead a to lead b,while the electronic occupation of the molecule changes onlyvirtually in the intermediate state. Based on Eq. �6�, the cor-responding rates are obtained as

Wqq�;ab00 =

s�0,0�2��

�a�b d���q�

Mq�q�Mqq�*

� − �d + �q − q�� ���2

fa���

��1 − fb�� + �q − q�� ���� , �10�

Wqq�;ab11 =

s�1,1�2��

�a�b d���q�

Mq�q�Mqq�*

�d − � + �q� − q�� ���2

fa���

��1 − fb�� + �q − q�� ���� . �11�

Here, the spin factors are s�0,0�=s�1,1��2, which followsfrom an analysis of the relevant contributions as illustrated inFig. 3. For cotunneling through the empty molecule, n=0→0, there is an incoherent addition of the processes trans-ferring a spin-up or a spin-down electron, respectively. Forn=1→1 the cotunneling transition may either leave the mol-ecule’s spin state invariant or cause a spin flip, again result-ing in two incoherent processes. In both cases n=0 and 1, the

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rates of the two contributions are identical and hence can beabsorbed into the spin factor.

It is crucial to note that, in general, the cotunneling ratescannot be directly evaluated from Eqs. �10� and �11�. Theseexpressions diverge due to second-order poles from the en-ergy denominators. This divergence, related to the infinitelifetime of the virtual intermediate state within a purely per-turbative approach, has been pointed out before.29,30 A regu-larization scheme has been developed in order to extract thecorrect cotunneling rates,30,31 which we summarize here, de-ferring the details to Appendix A.

The basis for the regularization of cotunneling rates isgiven by two observations. �i� Second-order perturbationtheory clearly misses the fact that the intermediate state ob-tains a finite width �� due to the tunneling. This widthshould enter the energy denominator as an imaginary part,shifting the pole away from the real axis. �ii� For a specificcotunneling transition �i�→ �f� at finite temperature, the finalstate �f� can also be reached from the initial state �i� by twosequential processes.

Specifically, the regularization proceeds as follows. First,a level width ��� is introduced in the energy denominators.With the poles shifted away from the real axis, the integralcan indeed be carried out. The resulting expression is thenexpanded in powers of the level width �. The leading-orderterm is proportional to 1/�. When combining this with theprefactor �a�b of the rates Eqs. �10� and �11�, the overallorder of this term is found to be identical to a sequential-tunneling term. Following point �ii�, this term must be disre-garded to avoid double-counting of sequential processes. Thenext-to-leading order in the � expansion is a �0 term. Thisterm gives the regularized expression for the cotunnelingrate. The full regularization procedure is carried out in detailin Appendix A, and the resulting rates are given in Eqs. �A1�and �A3�.

It is interesting to note that these “cotunneling rates” maybecome negative. This result can in fact be expected by con-sidering the simple case of a single-level resonant tunnelingmodel without charging energy. For this model, the exactresult for the current in the T→0 limit is

I = sgn�eV�2e

h

�L�R

�L + �R�

a=L,Ra arctan�a�eV� − 2�d

�� ,

�12�

where we identify the index L �R� with a= +1 �a=−1�. Ex-panding this in powers of �, we find that the second-orderterm corresponding to cotunneling is given by

Icot = −e

h�L�R

�eV��eV/2�2 − �d

2 . �13�

This expression indeed becomes negative, whenever thelevel is located inside the bias window; i.e., cotunneling re-duces the sequential tunneling. Our regularization procedurecombined with the rate-equation approach exactly repro-duces these expressions, as shown in detail in Appendix B. Itis worth noting that, alternatively, the formalism developedby König, Schoeller, and Schön may be used to circumventthe necessity of regularization; see, e.g., Ref. 32. We find thatadditional corrections due to level shifts and broadening cap-tured by their approach are irrelevant in the case ��kBTconsidered here. In this limit, both approaches are expectedto give identical results.

In the FC-blockade regime, the dominant contributions tothe cotunneling rates �Eqs. �10�, �11�, �A1�, and �A3�� arisefrom highly excited virtual phonon states. According to theFranck-Condon principle, the wave function overlap reachesa maximum for transitions that occur �approximately� verti-cally in the diagram of Fig. 1�b�. For transitions originatingin the vibrational ground state q=0, the “optimal” overlap isobtained for virtual states q� of the order of q���2. Deepinside the FC blockade, the resulting poles ���d±�2���see Eqs. �10� and �11�� are typically located well beyond thebias window. As a consequence of that, negative cotunnelingrates are essentially irrelevant in the FC-blockade regime.33

For an understanding of cotunneling rates it is importantto note that there is not a single virtual state q� resulting inmaximal overlap, but rather an extended range �q�� ofexcited states centered around q���2; see Fig. 1�b�. By ap-plying Sterling’s formula to the expression �8� for the FCmatrix element M0��2�, we obtain �M0��2��4�1/2��2, so thatthe contribution from the intermediate state q�= ��2��� isgiven by

�a�b

��

kBT+eV

�2��d+�2���2 . Carrying out the coherent sum over

the effective range �q�� and subsequent squaring leads toan additional factor of �2, resulting in the approximation

W00;LR00 �

�L�R

��

kBT + eV

��d + �2 ���2 , �14�

valid for 0�eV��2��. A more detailed derivation of thisexpression is given in Appendix C. By contrast to thesequential-tunneling rates, Eq. �14� establishes that cotunnel-ing rates are not exponentially but rather algebraically sup-pressed. In particular, for �d=0, we have W00;LR

00 ��−4.

B. Current and shot noise

As shown, e.g., in Refs. 10 and 34, the steady-state cur-rent and the current noise can be computed via the rate-

FIG. 3. �Color online� Relevant contributions to the cotunnelingrates in the U→� limit. For both charge states n=0 and 1 there aretwo incoherent contributions with either a spin-up or a spin-downelectron being transferred across the molecule.

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equation formalism. In the first step, the steady-state occupa-tion probabilities Pq

n are obtained from the rate equations

0 =dPq

n

dt= �

n�,q�

�Pq�n�Wq�q

n�n − PqnWqq�

nn�� −1

��Pqn − Pq

eq�q�

Pq�n � .

�15�

The last term is added on a phenomenological basis anddescribes relaxation of the vibrations towards the equilibriumdistribution Pq

eq=e−q��/kBT�1−e−��/kBT� on a time scale �. Inthe following, we will mainly investigate the two limitingcases of equilibrated phonons ��→0, distribution fixed toequilibrium� and unequilibrated phonons ��→�, nonequilib-rium distribution entirely determined through tunneling dy-namics�.

It is convenient to reexpress Eq. �15� in terms of a matrixequation of the form 0=WP, where W is a coefficient matrixcontaining all rates and the vector P consists of all steady-state probabilities. �See Appendix D for further details.� Fora systematic expansion in orders of the tunneling, we writeW=W�1�+W�2�+¯ as well as P=P�0�+P�1�+¯. Substitutingthis into the rate equation, we obtain up to second order theequations

W�1�P�0� = 0, �16�

W�1�P�1� = − W�2�P�0�. �17�

The normalization condition for P leads to tr P�0�=�iPi�0�=1

and tr P�1�=0. Here and in the following, the “trace” of avector always denotes the sum of its components. Similarly,we expand the steady-state current

I = I�1� + I�2� + ¯

= tr WI�1�P�0� + tr�WI

�1�P�1� + WI�2�P�0�� + ¯ . �18�

Here, WI denotes the reduced coefficient matrix containingonly current-carrying processes; see Appendix D. Thelowest-order current I�1� is the sequential-tunneling current.The second-order contribution I�2� contains the cotunnelingcurrent and corrections to the sequential current due tochanges of the probability distribution by cotunneling.

A method for evaluating the current shot noise

S��� = 2 −�

�

d�ei���I���I�0�� − I�2� �19�

within the rate-equation formalism has been developed byKorotkov in Ref. 34. This method can be generalized to in-clude higher-order processes beyond sequential tunneling, aswe show in detail in Appendix D. It is an important questionwhether the inclusion of negative “cotunneling rates” intothis noise formalism is justified. In Appendix B, we explic-itly verify for the resonant-tunneling model that this is in-deed the case. The exact solution for the resonant-tunnelingmodel allows for an evaluation of the zero-frequency noisevia the relation

S =2e2

h dE��

a

T�E�fa�E��1 − fa�E��

+ T�E��1 − T�E���fL�E� − fR�E��2�; �20�

see Refs. 18 and 35. We find that the leading and next-to-leading order expansion in � /kBT of the resulting noise isexactly reproduced by the generalized Korotkov formalismincluding negative cotunneling corrections.

V. CURRENT-VOLTAGE CHARACTERISTICS

A. Linear conductance

For discrete electronic levels weakly coupled to two elec-trodes via tunneling junctions, the linear conductance isknown to develop peaks whenever an electronic level isaligned with the Fermi energies of the leads. These peaks arewell described by sequential-tunneling processes within therate-equation formalism. Away from the peaks, thesequential-tunneling conductance is exponentially sup-pressed in the parameter �d /kBT and cotunneling contribu-tions become important. In the following, we show thatstrong electron-phonon coupling destroys this gate-controlled crossover between sequential tunneling and cotun-neling at low temperatures kBT� ��, rendering cotunnelingdominant over wide parameter ranges.

1. Sequential-tunneling conductance

We start by considering the lowest-order contributionsto the conductance in the limit of low temperatures��kBT� ��. In this case, only the vibrational ground stateis occupied and any real phonon excitations are negligible.This only leaves the rates W00

01 and W0010 for consideration.

Employing the rate-equation approach to this simple situa-tion, the linear conductance for infinite U is obtained as

Gseq = −2e2

�

�L�R

�L + �R

f���d�1 + f��d�

e−�2. �21�

In the absence of vibrations, this result can be found, e.g., inRef. 36. The crucial point here is the fact that the presence ofvibrations leads to an exponential suppression of the linearconductance in �2, even at the peak value. As discussed inRef. 36, it is interesting to note that in distinction to the U=0 case, the conductance peak position is not at �d=0 butvaries with temperature as �d=kBT ln 2 /2. This shift origi-nates from the fact that the sequential-tunneling rates n=0→1 are twice as large as the rates n=1→0 due to the spindegeneracy.

2. Cotunneling conductance

We now consider the next-to-leading-order contributionsto the conductance. Remaining in the limit of low tempera-tures ��kBT� ��, the only two additional processes areelastic cotunneling through the empty and singly occupiedmolecule with corresponding rates W00;aa�

00 and W00;aa�11 , re-

spectively. The resulting conductance due to elastic cotunnel-ing is given by

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Gcot = e�

�V� �n=0,1

P0n�W00;LR

nn − W00;RLnn ��

V=0

�2e2

h

�L�R

1 + f��d�� 1 − f��d���d + �2 ���2 +

2f��d���d − �2 ���2� .

�22�

Here, the apparent poles at �d= ±�2�� are exponentiallysuppressed due to the Fermi function factors. In fact, they arean irrelevant artifact of our simple approximation in Eq. �14�.The correct treatment via the regularization procedure elimi-nates this flaw.

By contrast to the �e−�2suppression of the sequential-

tunneling conductance, the cotunneling conductance is onlyalgebraically suppressed as ��−4. This is a strong indicationthat, in the regime of strong electron-phonon coupling, co-tunneling will dominate the linear conductance, even at theconductance peak. Representative numerical results confirm-ing this expectation are shown in Fig. 4�a�. For kBT� ��,the cotunneling conductance Gcot is dominant. In this inter-esting case, the conductance develops a kink feature at �d=0 broadened over a gate-voltage range given by kBT, whichcan be understood from Eq. �22�. Away from the peak maxi-mum, Gcot roughly decays as ���d+const�−2; see Fig. 4�b�.This is consistent with Eq. �22�. For larger temperatureskBT�0.2��, real excitation of phonons becomes important.Such excitations, which we have neglected in our approxi-mations �21� and �22�, open up additional channels for se-quential tunneling. The increase of sequential contributionsis evident from the high-temperature curve in Fig. 4 �in red/gray color�, which shows the recurrence of a sequential-

tunneling peak on top of the cotunneling background.

B. Nonlinear current-voltage characteristics

1. Sequential current

For strong electron-phonon coupling, we have establishedin Ref. 11 that the nonlinear I-V develops a low-bias gap,essentially eliminating the conventional crossing at the de-generacy point; cf. Fig. 7�b�. For equilibrated phonons, it isnot difficult to estimate the sequential-tunneling current inthe FC blockade regime. The main input is that �i� for lowtemperatures kBT� �� and strong relaxation, the initial vi-brational state is the phonon ground state q=0 for each tun-neling process and �ii� the squared Franck-Condon matrixelements �M0q�2 strongly increase with q up to values q��2.

Due to �ii�, the current is dominated by tunneling eventswith maximal change in the phonon number, �q= ��eV � /2���, as long as �eV � /2����2. Hence, �i� allowsus to approximate the current via the rates W0,�q; a

01 as

Iseq � sgn�V�e

�

2�a�a�

2�a + �a�e−�2 �2�q

�q!, �23�

with �a ,a��= �L ,R� or �R ,L� depending on the bias sign.Here, we have assumed �d=0 for simplicity, which can inprinciple always be arranged by adjusting the gate voltage.After averaging over steps, one obtains for voltages largerthan temperature

I�V� �e

�

2�a�a�

2�a + �a�

e−�2��eV�/��−1

���eV�/2 �� + 1/2�sgn�V� , �24�

where the denominator involves the gamma function ��x�.The comparison between the asymptotic form Eq. �24�

and the actual rate-equation solution is depicted in Fig. 5.Deviations of the asymptotic form from the actual solutionare observed when the inequality 2 �eV � ��2�� graduallybreaks down. Then, the true current assumes higher values ascompared to Eq. �24�, since assumption �ii� is no longervalid; i.e., not only one but several phonon channels yieldsignificant contributions to the current.

FIG. 4. �Color online� Linear conductance as a function of gatevoltage in the FC-blockade regime ��=4, �a=0.02���. �a� A com-parison of the sequential-tunneling conductance �dashed lines� andthe conductance including cotunneling corrections �solid lines�clearly shows the dominance of cotunneling contributions at lowtemperatures. At �d�0, cotunneling results in a conductance kinkfor kBT� ��. Sequential contributions become relevant for tem-peratures kBT�0.2�� that allow thermal-induced occupation ofexcited phonon states. �b� The plot of G−1/2 reveals the inversequadratic decrease of the cotunneling conductance as a function ofgate voltage.

FIG. 5. �Color online� Comparison between the sequential-tunneling current �black curve� and the asymptotic formula �24��red/gray curve� for equilibrated phonons in the Franck-Condon-blockade regime ��d=0, kBT=0.05��, �L=�R�.

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For unequilibrated phonons, where the FC blockade leadsto self-similar avalanches of electrons,11,17 an estimate forthe sequential current is less trivial. The avalanche dynamicsmakes it necessary to include several phonon channels. Inparticular, following the arguments of Ref. 17, the meanelectron number per avalanche as well as the mean numberof subavalanches within each avalanche will depend on biasand gate voltage, as well as on the coupling strength �.

2. Nonlinear I-V including cotunneling corrections

We now investigate the cotunneling corrections to thecurrent-voltage characteristics in the FC-blockade regime.Representative results for both equilibrated and unequili-brated vibrations are depicted in Figs. 6 and 7.

We first focus on the case of equilibrated phonons. TheI-V plots for �d=0, Figs. 6�a� and 6�c�, illustrate the impor-tance of cotunneling contributions well beyond the linearresponse regime. Cotunneling corrections significantly in-crease the current for bias voltages �eV � ��2��, the reasonbeing again the increased overlap for oscillator states in co-tunneling processes with highly excited intermediate states.It is a crucial point, however, that cotunneling does not leadto a breakdown of the FC blockade. The blockade remainsintact, since cotunneling rates are still algebraically sup-pressed at strong electron-phonon coupling. This suppressionremains strong due to the rather large exponents in the scal-

ing with �; see Eq. �14�. An additional change caused by thedominance of cotunneling rates is a drastic reduction of stepfeatures in the current; see especially Fig. 6�c�. When plot-ting the differential conductance dI /dV on a logarithmicscale �see Fig. 7�, one observes that small steps persist on topof the �much larger� cotunneling background and form theusual system of vibrational sidebands.

An important low-bias feature beyond the sequential-tunneling approximation consists of two steps in dI /dV foreV= ± ��. These steps mark the onset of inelastic cotunnel-ing. The emergence of such steps is not restricted to the FCblockade and has been discussed, e.g., in Ref. 8. Additionalsteps at bias voltages equal to larger integer multiples of ��can be observed when going to even lower temperatures thanchosen in Figs. 6 and 7.

For unequilibrated phonons, we find that cotunneling cor-rections again dominate the FC blockade region of thecurrent-voltage characteristics; see Fig. 6�b� and 6�d�. How-ever, relative to the equilibrated case, the corrections are lessdrastic. In particular, vibrational steps in I-V remain wellvisible and the bias range where cotunneling dominates isnarrower than in the equilibrated case. This can be explainedby the persistence of avalanche dynamics generated by se-quential tunneling. Within the FC blockade, important con-tributions to the current stem from rather featureless cotun-neling events. However, phases of pure cotunneling are stillinterrupted by avalanches. During those, the system attainsphonon excitations high enough to make sequential tunnelingdominate over cotunneling. Hence, sequential tunneling re-tains a significant role in the unequilibrated case.

Figure 7 demonstrates an additional striking consequenceof cotunneling: For unequilibrated phonons, additional vibra-tional sidebands appear in the Coulomb-blockade region.The mechanism which generates these additional sidebandsis the following. The Coulomb-blockade regime, wheretransport is conventionally dominated by cotunneling, allowsfor excitation of vibrations through inelastic cotunneling assoon as the bias voltage exceeds the phonon energy. Forweak vibrational relaxation, such excitations decay slowlyand therefore remain relevant for subsequent tunnelingevents. The crucial point is that the usual exponential sup-pression for sequential tunneling in the Coulomb blockadeapplies to processes which leave the vibrational stateunchanged,37 but not to processes gaining energy throughphonon deexcitation. Thus, phonon excitations �induced byinelastic cotunneling� facilitate the recurrence of sequentialtunneling accompanied by phonon deexcitation in theCoulomb-blockaded region. Furthermore, the onset of inelas-tic cotunneling at �eV � = �� and the resulting absence of vi-brational excitations for lower bias voltages explain whyabsorption-induced sidebands must terminate at �eV � = ��and cannot enter the low-bias region.

This interplay between inelastic cotunneling and sequen-tial tunneling results in the observed absorption-induced vi-brational sidebands,38 resembling experimental findings intransport through suspended carbon nanotubes.19 We empha-size that the absorption-induced vibrational sidebands are notspecific to the case of strong electron-phonon coupling andpoint out that their presence may be exploited to estimate thevibrational relaxation rate in molecular junctions.19

FIG. 6. �Color online� Nonlinear current-voltage characteristicsin the FC-blockade regime. Each graph shows the steady-state cur-rent I�1�+ I�2� �solid curves� and the sequential current I�1� �dashedcurves� for comparison. The electron-phonon coupling and vibra-tional relaxation strength are chosen as �a� �=4, equilibrated, �b��=4, unequilibrated, �c� �=5, equilibrated, and �d� �=5, unequili-brated. The results unambiguously indicate the dominance of cotun-neling contributions in the low-bias FC-suppressed regime �param-eters: �d=0, kBT=0.05��, �a=0.02���.

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FIG. 7. �Color online� �a� Differential conductance dI /dV as a function of gate and bias voltage for �=4 and �a=0.02��, plotted witha logarithmic color scale. The low-temperature plots �left-hand side� show a step in dI /dV due to the onset of cotunneling �horizontal featureat eV= ± ���. For unequilibrated phonons, inelastic cotunneling gives rise to phonon absorption-induced vibrational sidebands inside theCoulomb-blockade region �whose borders are marked by dashed lines�. For comparison, the plots in �b� show the corresponding results

within the sequential-tunneling approximation which misses the absorption-induced sidebands �G�= e2

�

�L�R

�L+�R�.

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Interestingly, phonon excitation due to inelastic cotunnel-ing is the only viable mechanism for absorption-induced vi-brational sidebands up to second order in the tunneling. It iscrucial to note that thermally induced phonon excitationswill in general not lead to visible additional sidebands, butrather to a smooth smearing of the Coulomb diamond; seeFig. 7�b�. This is explained by the fact that vibrational side-bands are clearly visible only for kBT� ��, and in this casethermally activated phonon excitations are negligible.

VI. CURRENT SHOT NOISE

The current shot noise serves as an important tool foracquiring information about the transport dynamics of thesystem beyond the steady-state current. In particular, we areinterested in the Fano factor F and noise spectrum S��� forunequilibrated vibrations to determine the fate of the self-similar avalanches in the presence of cotunneling correc-tions. We first turn to the regime of strong vibrational relax-ation and show that, even in this case, cotunneling leads tointeresting new effects.

A. Zero-frequency noise for equilibrated vibrations:Telegraph noise

As shown in Ref. 11, sequential tunneling results in con-ventional sub-Poissonian Fano factors for equilibratedphonons; see the dashed curves in Fig. 8. Electrons are trans-ferred one by one, interacting with each other only via thecharging energy U, which leads to the suppression of F be-low 1. As depicted in Fig. 8 by the black curves, the inclu-sion of cotunneling processes radically changes this picture.The overall Fano factor remains sub-Poissonian for �d=0.Remarkably, it switches to super-Poissonian behavior fornonzero �d in a certain bias range. The magnitude of Fstrongly depends on the coupling strength �, as well as gateand bias voltage, and reaches values as large as 100 for�=5.

More insight into this surprising enhancement of noise,which even occurs in the low-bias regime where no phononscan be excited, is obtained by Monte Carlo simulations. Forthe parameters of the large maximum of the Fano factor inFig. 8 ��=5, �d=0.5���, Fig. 2 shows the coarse-grainedcurrent as a function of time. These data clearly show theslow switching between a low- and a high-current state, ontop of much more rapid dynamics within each current state.The underlying reason for this telegraph noise is the roleinversion of sequential-tunneling and cotunneling rates in theFC blockade; i.e., rates for cotunneling are larger than thosefor sequential tunneling.

In the following, we give a detailed explanation of thetelegraph behavior. Denoting the molecular state by �n ,q�, astraightforward inspection shows that for �d=0.5��and eV=0.8��, the only relevant transitions are �i��0,0�→ �1,0� and �1,0�→ �0,0� due to sequential tunnelingand �ii� �0,0�→ �0,0� and �1,1�→ �1,1� due to cotunneling.As explained in Sec. IV, the cotunneling transitions are muchfaster than the sequential transitions in the FC-blockade re-gime, such that W00

00,W0011�W00

01,W0010. The key point in the

emergence of the telegraph behavior is the fact that the ratesfor cotunneling through the empty and through the singlyoccupied dot are generally different, W00

00�W0011. This can be

understood from Eqs. �10� and �11�, noting that �d enters theenergy denominator with a positive sign for n=1, but with anegative sign for n=0. For �d=0, the two cotunneling ratesare identical, and indeed no super-Poissonian noise is de-tected in this case. However, for nonzero �d, the two ratesbecome different. As a consequence, the rapid cotunnelingdynamics in the charge states n=0,1 leads to different meancurrents eW00

00 and eW0011, respectively. The rare sequential

transitions effect the slow switching between these two cur-rent states.

The Fano factor resulting from the current telegraph noiseis directly related to the excess charge transmitted in thehigh-current state during the typical residence time in thecorresponding charge state, i.e. F��Wcot /Wseq. Here,�Wcot= �W00

00−W0011� denotes the difference of the relevant co-

FIG. 8. �Color online� Excess-noise Fano fac-tor for equilibrated phonons with �=4 �top row�and �=5 �bottom row�. For nonzero �d we ob-serve large super-Poissonian Fano factors due tocurrent telegraph noise �kBT=0.05��, �a

=0.02���.

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tunneling rates. Due to the role inversion of cotunneling andsequential tunneling, we have Wcot�Wseq so that the result-ing Fano factor may become large. A more quantitative cal-culation of the Fano factor can be achieved by employingthe generalization of Korotkov’s method34 described in Ap-pendix D. Along these lines, we can derive approximate ex-pressions for the zero-frequency shot noise and its Fanofactor,

S�f = 0� � 4e2W0001W00

10�W0000 − W00

11�2

�W0001 + W00

10�3 , �25�

F � 6W00

01W0010�W00

00 − W0011�2

�W0000 + 2W00

11��W0001 + W00

10�3 ; �26�

see Appendix E for details. This approximation is validwhenever W00

00,W0011�W00

01,W0010 and W00

00�W0011. Counting the

orders of cotunneling versus sequential-tunneling rates in thenumerator and denominator of the Fano factor, we find thatthe Fano factor is indeed of the order of �Wcot /Wseq.

B. Zero-frequency noise for unequilibrated vibrations:Persistence of avalanches

For unequilibrated vibrations, the sequential-tunneling dy-namics in the FC-blockade regime is described by self-similar avalanches of electrons.11,17 The large Fano factorsdue to sequential tunneling shown in Fig. 9 �dashed curves�reflect the large number of electrons per avalanche. Inthe �d=0 case, the Fano factor is exactly given by F=2Ni�,i.e. twice the mean number of electrons per level-0 ava-lanche. More generally, the Fano factor can be obtained byevaluating the fluctuations of the waiting times ti betweenlevel-0 avalanches and the fluctuations of the electron num-ber Ni in each avalanche,17

F = Ni�ti

2� − ti�2

ti�2 +Ni

2� − Ni�2

Ni�. �27�

Before turning to the cotunneling-induced corrections, westudy the effect of nonzero �d, which was not discussed inRefs. 11 and 17 in detail. Interestingly, a nonzero �d can leadto even larger Fano factors as compared to the �d=0 case;see Fig. 9. Especially large Fano factors are observed for lowbiases above the threshold eV� �� whenever the electroniclevel �d is roughly aligned with the Fermi energy of the rightor left lead.

At first view, the resulting Fano factor enhancement issurprising since the mean number of electrons per avalancheis decreased for such a level configuration. For an under-standing of this decrease, consider the case of level align-ment with the left Fermi energy. Then, phonon excitationscan only occur through tunneling in the right junction.Events in the junction L can only decrease the phonon state�or leave it unchanged�. Thus, every second tunneling eventwill inhibit phonon excitations. With avalanches benefitingfrom the possibility of reaching highly excited phonon states,the mean number of electrons per avalanche is reduced ascompared to the case where the level is centered between thetwo Fermi levels. This decrease in Ni� is confirmed byMonte Carlo simulations.

The somewhat counterintuitive result that the Fano factornevertheless becomes larger can be traced back to the sig-nificantly enhanced fluctuations in waiting times ti, whichdirectly contribute to the Fano factor; see Eq. �27�. In thefollowing, we explain this enhancement of waiting-time fluc-tuations. Any generation-0 avalanche may terminate either inthe charge state n=0 or n=1, and the corresponding prob-abilities are denoted p0 and p1. After the avalanche termina-tion, the system undergoes a waiting period ti. The crucialpoint is that for level alignment with one Fermi energy, thiswaiting time strongly depends on the charge state n. Startingin the state n=0, transitions in the left junction do not allowphonon excitations. By contrast, for the initial state n=1 andtransitions across the right junction, the excitation ofphonons is favored �due to the increase of FC matrix ele-ments towards highly excited vibrational states�. Accord-ingly, the relevant two rates after avalanche termination are

FIG. 9. �Color online� Excess-noise Fano fac-tor for unequilibrated phonons with �=4 �toprow� and �=5 �bottom row� with kBT=0.05��and �a=0.02��. Due to the coexistence of co-tunneling and avalanches, the Fano factor is re-duced as compared to the sequential-tunnelingapproximation. The super-Poissonian peak ob-served at low bias for �=5 and �d�0 signals theoccurrence of current telegraph noise.

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dramatically different, W0001�W01

10, and the waiting times inthe two cases will differ accordingly,

ti�n=0�� = �W00

01�−1 � ti�n=1�� = �W01

10�−1. �28�

Consequently, the probability distribution of waiting times isnot given by a single exponential anymore, but is well ap-proximated by

p�ti� � p0W0001exp�− W00

01ti� + p1W0110exp�− W01

10ti� . �29�

While the �d=0 case �with a simple exponential distribution�results in �ti

2�− ti�2� / ti�2=1, we now find

ti2� − ti�2

ti�2 �2 − p0

p0, �30�

valid for p0�W0001/W01

10. By Monte Carlo simulations weconfirm that p0 satisfies 1� p0�W00

01/W0110 for realistic pa-

rameters. Therefore, Eq. �30� explains the enhancement ofthe Fano factor due to the increased waiting-time fluctua-tions.

The inclusion of cotunneling contributions to the currentshot noise results in the black curves depicted in Fig. 9.Again, at low biases which do not allow real phonon excita-tions, we observe traces of telegraph noise. �See the �=5plots for nonzero �d in Fig. 9.� For higher biases which fa-cilitate avalanche dynamics, we find that cotunneling gener-ally reduces the Fano factor as compared to pure sequentialtunneling. The reason for this reduction is the coexistence ofelectron avalanches �leading to large Fano factors� with co-tunneling, which results in Fano factors of the order of unityaway from the telegraph noise regime. We emphasize thatdespite the cotunneling-induced noise reduction, the overallFano factor remains large compared to unity, signaling thepersistence of electron bunching in avalanches.

C. Noise spectrum for unequilibrated vibrations: Self-similarity of avalanches

Probing the self-similarity of avalanches requires one togo beyond the zero-frequency noise. In Refs. 11 and 17 twoof us have shown that the characteristic signature of the self-similarity within the sequential-tunneling approximation isan approximate power-law behavior of the noise as a func-tion of frequency f , S� f−1/2. In Fig. 10 we present the re-sults for the noise power spectrum including cotunneling cor-rections.

The central result is that the power-law scaling remainsvalid. Only the frequency range over which it applies is re-duced by cotunneling. In particular, we observe a larger low-frequency cutoff when taking cotunneling into account. Thisresult is consistent with the picture of coexistence of cotun-neling and sequential avalanches: In the sequential-tunnelingpicture, the onset of the power law at low frequencies isessentially given by the strongly suppressed rate for escapingfrom the vibrational ground state, i.e. f �W01

01/10. Going be-yond sequential tunneling, we have shown that transitionsclose to the vibrational ground state are dominated by cotun-neling which does not contribute to the avalanche dynamics.

Accordingly, the onset of self-similarity and hence thepower-law behavior is shifted towards higher phonon levelswith corresponding higher rates. This is reflected by a largerlow-frequency cutoff for the power law in the noise spec-trum.

VII. CONCLUSIONS

In conclusion, we have developed a complete theory ofthe Franck-Condon blockade including cotunneling correc-tions. This regime, caused by a strong coupling to thebosonic degrees of freedom of molecular vibrations, empha-sizes the stark contrast between transport through molecularjunctions and more conventional nanostructures such asquantum dots. In particular, the intriguing finding thatelectron-phonon coupling may result in a significant bunch-ing of electrons into a hierarchy of self-similar avalanchesdominating the transport constitutes a novel collective effectunknown from electron transfer through quantum dots.

In distinction to the Coulomb-blockade scenario in quan-tum dots, where the crossover between sequential tunnelingat the conductance peaks and cotunneling away from thepeaks may be probed by tuning the gate voltage, we find thatthe FC-blockade regime features the dominance of cotunnel-ing contributions in the entire low-bias region. The underly-ing reason for this result is the enhanced overlap betweenvibrational states, which is exploited by highly excited vibra-tional states in cotunneling processes. We have shown thatthe FC blockade and avalanche-type transport persist despitethe cotunneling corrections. Furthermore, we have estab-lished that cotunneling leads to a number of remarkable ef-fects missed by the sequential-tunneling approximation. Insummary, our central results are: �i� Apart from horizontaldI /dV steps due to inelastic cotunneling, the most importantconsequence of cotunneling is the appearance of additionalabsorption-induced vibrational sidebands inside the

FIG. 10. �Color online� Noise power spectrum for unequili-brated phonons with �=4 �left� and �=5 �right� with kBT=0.05��, �a=0.02��, �d=0, and eV=3��. The approximatepower-law behavior reflecting the self-similarity of avalanches per-sists in the presence of cotunneling. The dominance of cotunnelingduring sequential waiting times provides a long-time �low-frequency� cutoff for the power law.

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Coulomb-blockade regime, relevant for unequilibrated vibra-tions. This may also serve as an experimental measure of thevibrational relaxation rate. �ii� The role inversion betweencotunneling and sequential tunneling may cause current tele-graph noise whenever the electronic level is not centeredbetween the left and right Fermi energy. This is reflected instrongly enhanced Fano factors, independent of the vibra-tional relaxation strength.

For a more detailed summary of results we refer thereader to the summary provided in Sec. I. In closing, weemphasize the appeal of realizing shot-noise measurementsin experiments with single-molecule devices. These couldprovide essential information about the transport mode,which cannot be accessed by current-voltage characteristics.Experimental efforts in this direction are under way in sev-eral research groups.

ACKNOWLEDGMENTS

We would like to thank M. E. Raikh, M. R. Wegewijs, andK. A. Matveev for valuable discussions. We gratefully ac-knowledge hospitality by the University of Washington�J.K.� and the Weizmann Institute of Science �J.K. andF.v.O.�, made possible by the EU–Transnational Access pro-gram �RITA-CT-2003-506095�. This work was supported inpart by the DFG through Sfb 658 and Spp 1243, by theStudienstiftung des dt. Volkes, by the German-Israeli ProjectCooperation �DIP�, and by the David and Lucille PackardFoundation.

APPENDIX A: REGULARIZED COTUNNELING RATES

The regularized cotunneling rates can be expressed interms of digamma and trigamma functions,31

Wq→q�;ab0→0 =

2�a�b

2�� ��r

�Mq�rMqr�2J�a,b − �q − q�� � �,�d − �q − r� � ��

+ �r�s

Mq�rMqr* Mq�s

* MqsI�a,b − �q − q�� � �,�d − �q − r� � �,�d − �q − s� � ��� , �A1�

Wq→q�;ab1→1 =

2�a�b

2�� ��r

�Mq�rMqr�2J�a,b − �q − q�� � �,�d + �q� − r� � ��

+ �r�s

Mq�rMqr* Mq�s

* MqsI�a,b − �q − q�� � �,�d + �q� − r� � �,�d + �q� − s� � ��� . �A2�

Here, the regularized expressions I and J can formally be obtained by evaluating the �→0 limit after subtracting 1/� terms,

I�E1,E2,�1,�2� = lim�→0

Re dE f�E − E1��1 − f�E − E2��1

E − �1 − i�

1

E − �2 + i�

=nB�E2 − E1�

�1 − �2Re���1/2 + i��E2 − �1�/2�� − ��1/2 − i��E2 − �2�/2��

− ��1/2 + i��E1 − �1�/2�� + ��1/2 − i��E1 − �2�/2��� , �A3�

J�E1,E2,�� = lim�→0

� dE f�E − E1��1 − f�E − E2��1

�E − ��2 + �2 − O�1/���=

�

2�nB�E2 − E1�Im����1/2 + i��E2 − ��/2�� − ���1/2 + i��E1 − ��/2��� . �A4�

Our numerical implementation of the polygamma function �has been based on a routine written by Kölbig, available inthe CERN Program Library.39,40

APPENDIX B: EXACTLY SOLVABLE MODEL

We consider the simple case of a single spin-degenerate,noninteracting level coupled to two leads. Due to the absence

of interaction in this model, spin only results in trivial factorsof 2 and may be neglected. We have picked this trivialmodel, since it is exactly solvable, e.g. using a one-particlescattering approach.

1. Current

The current is given by

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I =e

h dE

�L�E��R�E��L�E� + �R�E�

Ad�E��fL�E� − fR�E�� , �B1�

where Ad denotes the spectral function of the dot,

Ad�E� = − 2 Im GdR�E� =

��E��E − �d�2 + ���E�/2�2 , �B2�

and the energy broadening of the level is ��E�=�L�E�+�R�E� with �a�E�=2��a�E� � ta�2. �Note that any possibleenergy shift due to the tunnel coupling has been absorbedinto �d.� In the wideband limit and for symmetric voltagesplitting, the solution reads

I =e

h

�L�R

�L + �R dE

�

�E − �d�2 + ��/2�2 �fL�E� − fR�E��

=e

�

�L�R

�L + �R

1

�Im�

a

a��1

2+

��

4�+ i

�

2���d + aeV/2�� .

�B3�

Here, � denotes the digamma function. The exact result canbe expanded in orders of �, which gives

I =e

�

�L�R

�L + �R�fL��d� − fR��d��

+e

h

�

2��L�RIm�

a

a���1

2+ i

�

2���d + aeV/2�� + O��3� .

�B4�

For the weak-coupling case we can alternatively apply therate-equation approach. In this case, the stationary current iscalculated via I= IL=−IR with

IL = IL�seq� + IL

�cot�

= e�P0WL+ − P1WL

−� + e�P0�WLR00 − WRL

00 � + P1�WLR11 − WRL

11 �� .

�B5�

Here, Wa+ �Wa

−� gives the rate for a sequential-tunneling pro-cess in junction a where one electron enters �leaves� the dot.Similarly, WLR

nn �WRLnn � denotes the rate for a cotunneling pro-

cess with initial and final charge state n where one electron iscoherently transferred from the left to the right lead �or viceversa�. The stationary probabilities for the empty and occu-pied dot P0 and P1 are obtained from the rate equations

0 =dP0

dt= P1W− − P0W+, P0 + P1 = 1, �B6�

where W±�WL±+WR

±. It is important to note that in this case,there are no cotunneling contributions to the rate equations:Elastic cotunneling does not change the charge state of thedot and hence does not affect the probability distribution.

Sequential-tunneling contributions. In the first step, wecalculate the stationary probabilities and the sequential-tunneling current. The transition rates obtained from Fermi’sgolden rule are

Wa+ =

1

��afa��d�, Wa

− =1

��a�1 − fa��d�� . �B7�

From substituting this into Eq. �B6� one obtains

P0 =W−

W− + W+ =�a

�a�1 − fa��d��

�L + �R, �B8�

P1 =W+

W− + W+ =�a

�afa��d�

�L + �R, �B9�

and upon substituting into Eq. �B5�, the sequential-tunnelingcurrent is found to be

I�seq� =e

�

�L�R

�L + �R�fL��d� − fR��d�� . �B10�

Indeed, this expression is identical with the leading-ordercurrent contribution from the expansion of the exact solu-tion; see Eq. �B4�.

Cotunneling contributions. In the second step, we calcu-late the cotunneling current by evaluating the cotunnelingrates

Wabnn =

�a�b

2�� d�� 1

� − �d�2

fa����1 − fb���� . �B11�

After regularization, we find

Wabnn =

�

4�2

�a�b

�nB�b − a�Im����1

2+ i

�

2��b − �d��

− ���1

2+ i

�

2��a − �d��� . �B12�

Substituting this together with Eqs. �B8� and �B9� into theexpression for the cotunneling current from Eq. �B5� andusing nB�x�+nB�−x�=−1 leads to

I�cot� =e

h�L�R

�

2�Im�

a

a���1

2+ i

�

2���d + aeV/2�� .

�B13�

By comparison with Eq. �B4� one verifies that this is identi-cal with the next-to-leading-order term of the expansion ofthe exact solution.

2. Zero-frequency noise

Our starting point for the exact noise calculation is thetwo-terminal formula for the zero-frequency noise,

S =2e2

h dE��

a

T�E�fa�E��1 − fa�E��

+ T�E��1 − T�E���fL�E� − fR�E��2�; �B14�

see Refs. 18 and 35. The transmission coefficient T�E� isknown to be proportional to the spectral function Ad�E�. The

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exact relation between them can be extracted easily by com-paring Eq. �B1� to the Landauer formula

I� =e

h dE T�E��fL�E� − fR�E�� . �B15�

This leads to T�E�=Ad�E��L�R /�.

The integrations can be carried out by invoking the Fou-rier transform for the Lorentzian. The energy integration thenturns into a Fourier transform of the Fermi factors, whichmay be carried out by contour integration. The remainingtime integration leads to polygamma functions. The exactresult for the zero-frequency noise is given by

S =2e2

h � 2�L�R��L2 + �R

2��3 coth��eV/2�Im�

a

a��1

2+ i

�

2���d + aeV/2� +

��

4��

+2��L�R�2

�2

�

2�coth��eV/2�Im�

a

a���1

2+ i

�

2���d + aeV/2� +

��

4��

+2��L�R�2

��3 Re�a

���1

2+ i

�

2���d + aeV/2� +

��

4�� −

��L�R�2

��2

�

2�Re�

a

���1

2+ i

�

2���d + aeV/2� +

��

4��� . �B16�

Expanding this expression in �, we find in leading and next-to-leading order

S�1� =2e2

h�2��L�R��L

2 + �R2�

�3 �fR��d� + fL��d� − 2fL��d�fR��d�� +4���L�R�2

�3 �fL��d��1 − fL��d�� + fL��d��1 − fL��d���� ,

�B17�

S�2� =2e2�L�R

h

�

2�coth��eV/2�Im�

a

a���1

2+ i

�

2���d + aeV/2�� . �B18�

Again, we can compare this expansion with the results fromthe rate-equation approach. Employing the formalism de-scribed in Appendix D, we obtain the occupation probabili-ties

P0 =�a

�a�1 − fa�

�L + �R, P1 =

�a�afa

�L + �R, �B19�

the steady-state current

I�/e = � �L�R

�L + �R�fL − fR� + wLR − wRL� , �B20�

and the vectors

uLL = � P0��LfL/ � + wLR + wRL�P1��L�1 − fL�/ � + wLR + wRL�

� , �B21�

uRR = � P0��RfR/ � + wLR + wRL�P1��R�1 − fR�/ � + wLR + wRL�

� , �B22�

uRL = uLR = �P0�wLR + wRL�P1�wLR + wRL�

� , �B23�

yL = �− P1�L�1 − fL�/ � + P0�wLR − wRL�P0�LfL/ � + P1�wLR − wRL�

� , �B24�

yR = �P1�R�1 − fR�/ � + P0�wLR − wRL�P0�RfR/ � + P1�wLR − wRL�

� , �B25�

wL = � �LfL/ � + wLR − wRL

− �L�1 − fL�/ � + wLR − wRL� , �B26�

wR = � − �RfR/ � + wLR − wRL

�R�1 − fR�/ � + wLR − wRL� . �B27�

A lengthy but straightforward calculation shows that the rate-equation results for the zero-frequency noise in leading andnext-to-leading order are identical with Eqs. �B17� and�B18�.

APPENDIX C: APPROXIMATE EVALUATIONOF COTUNNELING RATES

We derive an approximate expression for the elastic co-tunneling rate W00;ab

00 , valid for strong electron-phonon cou-pling ��1. Our starting point is the �unregularized� rate asobtained by Fermi’s golden rule; see Eq. �10�. This expres-sion can be greatly simplified by noting that only a few termsin the q� sum give significant contributions, which is due tothe behavior of Franck-Condon matrix elements for strongelectron-phonon coupling. Specifically, the relevant matrixelements read

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�M0q�2 =�2qe−�2

q!

�1

2�exp�− �2 + 2q ln � + q − q ln q −

1

2ln q� ,

�C1�

where we have used Sterling’s approximation for the facto-rial in the last step. By inspection for ��1, we find that theargument of the exponential function is strictly negative andexhibits a single maximum at q��2. We obtain a reasonableapproximation for the strong-coupling FC matrix elementsby expanding the argument to second order in q around themaximum q=�2, which leads to the Gaussian

�M0q�2 �1

2��exp�−

�q − �2�2

2�2 � . �C2�

Consequently, as a function of q the �squared� FC matrixelement �M0q�2 exhibits a single peak of width �� centeredat q=�2. Accordingly, in Eq. �10� we only need to accountfor contributions from these summands. To leading order in1/�, we can then neglect the variation of 1 /q� within theGaussian peak; i.e., setting q�=�2 in the energy denominatorand noting that �� � � ��d−�2��� for bias and gate voltagessmall compared to the polaron shift, we obtain the approxi-mation

W00;ab00 �

�a�b

��

kBT + eV

��d + �2 � ��2��q�

�M0q��2�2

��a�b

��

kBT + eV

��d + �2 � ��2

�� −�

�

dq1

2��exp�−

�q − �2�2

2�2 ��2

��a�b

��

kBT + eV

��d + �2 � ��2 . �C3�

APPENDIX D: FORMALISM FOR CALCULATINGCURRENT AND SHOT NOISE

We review Korotkov’s technique for noise calculations,Ref. 34, and slightly generalize his formalism in order toincorporate higher-order processes such as cotunneling.Throughout this section, we will keep wording and formal-ism as general as possible to stress the wide applicability ofthis approach.

We consider a system consisting of some central structure,such as a quantum dot or molecule, and source and drainelectrodes coupled to the central structure. We assume thatthe state space of the center is discrete and finite, and wedenote it by

S � �1,2, . . . ,N� . �D1�

Then, the system’s dynamics originates from transitions be-tween pairs of these states, �i�→ �f�, which are associated

with specific transition rates Wif 0. Mathematically speak-ing, the dynamics can be classified as a time-dependent Mar-kov process, and it is fully characterized by the master equa-tion

�

�tP�f ,t�i� = �

k�S�P�k,t�i�Wkf − P�f ,t�i�Wfk� , �D2�

with initial condition P�f , t=0 � i�=�if. Here, P�f , t � i� denotesthe conditional probability for the system to occupy the state�f� at time t, given that the initial state at time t=0 was �i�.Note that, due to our choice of labeling processes and corre-sponding rates with the center degrees of freedom only, ratesWii with identical initial and final states exist and need to betaken into account. �A relevant example is given by elasticcotunneling processes.�

It is convenient to translate Eq. �D2� into a compact ma-trix notation. This can be achieved by the definitions

pi�t� � „P�1,t�i�,P�2,t�i�, . . . ,P�N,t�i�…T, �D3�

and

W =�− �k�1

W1k W21 W31 ¯

W12 − �k�2W2k W32 ¯

W13 W23 − �k�3W3k ¯

� �

� ,

�D4�

where all k sums run over the state space S. Now, the masterequation can be summarized as

�

�tpi�t� = Wpi�t� , �D5�

with initial condition pi�t=0�= ei, having the formal solution

pi�t� = eWtei. �D6�

Under rather general conditions, typically satisfied in physi-cal applications, one can show that for long times t, the prob-ability distribution converges to a unique stationary distribu-tion P. Instead of performing the limit

P = limt→�

pi�t� �independent of i� , �D7�

one can solve the stationary master equation

WP = 0, �D8�

with the condition that the sum over all components of thevector P be 1 �normalization�, which we write as tr P=1.

1. Current

Electron transfer between the electrodes and the centralsystem gives rise to a current, whose time average I� �thestationary current� must be identical in the left and rightjunctions, i.e. I�= IL�= IR�, due to charge conservation.�Here and in the following, ·� always denotes time averag-

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ing.� At this point, we may fix the sign of the current onceand for all, defining the current to be positive for an electrontransfer from the left to the right.

Some caution is now in place regarding the nature of thetransitions �i�→ �f�. In general, there may be several differentprocesses which cause a transition between the central sys-tem’s states �i� and �f�, and they differ from each other in theresulting leads’ states. In order to elucidate this situation, wegive one concrete example. Consider a system whose centralsystem is simply a single electronic level. Then, cotunnelingwill result in a transition n→n, i.e. the electronic occupationof the center remains invariant. However, there are severalincoherent contributions to this transition: The cotunnelingtransition can result in a transfer of one electron from the leftto the right or vice versa. In principle, the tunneling back andforth between only one lead and the center is possible aswell.

This example reveals that, in general, each transition �i�→ �f� may be made up of several elementary processes con-tributing to this transition. Accordingly, the rates Wif are asum of the rates of the corresponding elementary processes�,

Wif = ��

Wif���. �D9�

�Of course, the notion of elementary processes only comesabout due to our choice of expressing all results in terms ofthe states of the central system alone.�

Generally, not all elementary processes contribute to thecurrent, and each process which does can transfer a number nof electrons either in the positive or negative direction. Tak-ing this into account, we define the quantity sif ,�

a as follows:If the elementary process � belonging to the transition �i�→ �f� gives a current contribution in junction a �a=L ,R�,then sif ,�

a gives the number of electrons transferred acrossjunction a within this process, and its sign reflects whetherthe transfer occurs in the positive or negative direction. Forall processes without current contribution, sif ,�

a vanishes.Now the stationary current in junction a can be written as

Ia�/e = �i,f�S

��

sif ,�a PiWif

��� = tr WIP , �D10�

where �WI�ij =e��sji,�a Wji

���.For the analysis of the current shot noise, we are also

interested in the explicit time dependence of the current.Viewing the system dynamics as a Markov process consist-ing of quasi-instantaneous jumps between different states, wecan define the times tn

�if ,�� �n=1,2 , . . . , � � at which the pro-cess � belonging to the transition �i�→ �f� takes place. Then,the time-dependent current in junction a can formally bewritten as a sum of � functions according to

Ia�t� = e �i,f�S

��,n

sif ,�a ��t − tn

�if ,��� . �D11�

Taking the time average of this expression and comparingwith Eq. �D10�, we obtain the relation

�n

��t − tn�if ,���� = PiWif

���. �D12�

2. Noise

Our central goal consists of the derivation of a compactand general expression for the current noise defined by

Sab��� = 2 −�

�

d�ei���Ia���Ib�0�� − I�2� . �D13�

The starting point for our considerations is the current-current correlator Ia�t�Ib�t���. When substituting Eq. �D11�into the correlator, we obtain a multiple sum with each sum-mand being proportional to the average of a product of two �functions:

Ia�t�Ib�t��� = e2 �i,f�S

��,n

�i�,f��S

�,m

sif ,�a si�f�,

b

���t − tn�if ,�����t� − tm

�i�f�,��� . �D14�

As the set of all times tn�if ,�� forms the time lattice of the

Markov process, the times are considered to be pairwise

distinct. Thus, we have tn�if ,��� tm

�i�f�,� for �n , if ,��� �m , i�f� ,�. Accordingly, the correlator can be separatedinto autocorrelation and cross-correlation contributionsIa�t�Ib�t���a= Ia�t�Ib�t���a+ Ia�t�Ib�t���x, where

Ia�t�Ib�t���a = e2 �i,f�S

��,n

sif ,�a sif ,�

b

���t − tn�if ,�����t� − tn

�if ,���� , �D15�

Ia�t�Ib�t���x = “nondiagonal terms”. �D16�

We first turn to the evaluation of the autocorrelations. Usingthe fact that ��t− tn

�if ,�����t�− tn�if ,����=��t− t����t− tn

�if ,���� andapplying relation �D12�, we obtain

Ia�t�Ib�t���a = e2��t − t�� �i,f�S

��

sif ,�a sif ,�

b PiWif���.

�D17�

The cross-correlation terms now involve only distinct pro-cesses at different times. These contributions can be dealtwith in the following way. Let us assume that t t�. By usingI=dQ /dt, we note that

Ia�t�Ib�t���x =dQa�t�dQb�t���

dtdt�, �D18�

and we only get a contribution to the time average if thereare current-contributing processes �causing charge variationsdQ� in junction b during �t� , t�+dt�� and in junction a during�t , t+dt�, which leads us to

Ia�t�Ib�t���x = e2 �i,j,k,f�S

�,�

sij,�a skf ,

b PiWij���p�k,t − t��j�Wkf

��.

�D19�

We are now in the position to give an intermediate result forthe bracketed expression in Eq. �D13�. We find

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�Ia����Ib�0�� = e2���� �i,f�S

��

sif ,�a sif ,�

b PiWif���

+ e2 �i,j,k,f�S

�,�

sij,�a skf ,

b PiWij���

��p�k,��j� − Pk�Wkf��. �D20�

This expression is valid for � 0. For ��0 we exploit thesymmetry

�Ia����Ib�0�� = �Ia�0��Ib�− ��� . �D21�

Now, the last missing piece for an evaluation of the noiseconsists of the determination of the Fourier transform of theterm

Gkj��� = �����p�k,��j� − Pk� . �D22�

Noting that p�k ,� � j�= �eW��kj and defining the matrix V byVkj = Pk, we can rewrite Eq. �D22� in matrix form as G���=�����eWt−V�, and its Fourier transform is found to be

G��� = 0

�

dt ei���eW� − V� = − �W + i�1�−1 +1

i�V , �D23�

which can directly be evaluated at all finite frequencies. �The�→0 will be discussed below.� Thus, the noise can be ex-pressed as

Sab��� = 2e2 �i,f�S

��

sif ,�a sif ,�

b PiWif���

+ �2e2 �i,j,k,f�S

�,�

sij,�a skf ,

b PiWij���Gkj���Wkf

��

+ �same term with a ↔ b�*�= 2e2�tr uab + wbG���ya + waG���*yb� . �D24�

Here we have defined the vectors uab, ya, and wb by

�uab�i = �f�S

��

sif ,�a sif ,�

b PiWif���, �D25�

�ya� j = �i�S

��

sij,�a PiWij

���, �D26�

�wb�k = �f�S

�

skf ,b Wkf

��. �D27�

In order to evaluate the zero-frequency limit �→0, wemake use of the following properties of the relevant matri-ces. First, we observe that GP=0. Second, one easily con-firms that Vx=0 for any traceless vector x. Third, we notethat any well-behaved rate-equation matrix W �which has aunique steady-state solution� is invertible in the subspace oftraceless vectors. Thus, in Eq. �D24� we may modify thevector ya by adding some multiple of the stationary distribu-tion P, leaving the result invariant.

We choose

�ya� j � �ya� j − Ia�Pj/e , �D28�

which makes ya a traceless vector. Consequently, the zero-frequency noise can be evaluated via

Sab�� = 0� = 2e2�tr uab − wbW−1ya − waW

−1yb� . �D29�

APPENDIX E: ZERO-FREQUENCY NOISE AND FANOFACTOR FOR THE TELEGRAPH REGIME

According to the simplest situation where telegraph noiseoccurs, we take into account the transitions �1� �0,0�→ �1,0� and �1,0�→ �0,0� due to sequential tunneling and�2� �0,0�→ �0,0� and �1,1�→ �1,1� due to cotunneling. Atfinite bias the transport is essentially unidirectional, so thatwe restrict ourselves to LR cotunneling, as well as sequentialtunneling �0,0�→ �1,0� via the left junction and �1,0�→ �0,0� via the right junction. In order to avoid irrelevantindices, we will here denote the rates as W+, W− for sequen-tial tunneling and W0, W1 for cotunneling. The rate-equationmatrix then reads

W = �− W+ W−

W− − W+ � , �E1�

and the resulting steady-state occupation probabilities for theempty and singly occupied molecule are

P0 =W−

W+ + W− , P1 =W+

W+ + W− . �E2�

Note that due to the spin degeneracy of the singly occupiedstate, the sequential-tunneling rates obey W+=2W−, leadingto P0=1/3 and P1=2/3. Neglecting sequential-tunnelingrates as compared to cotunneling rates where appropriate, wefind that the vectors uaa�, ya, and wa are given by

uaa� = �P0W0

P1W1 � = ya, �E3�

wa = �W0

W1 � . �E4�

The steady-state current is

I� = P0W0 + P1W1 =1

3W0 +

2

3W1, �E5�

so that ya is obtained as

yL = yR =W+W−�W0 − W1�

�W+ + W−�2 � 1

− 1� . �E6�

Finally, we have

W−1ya = −W+W−�W0 − W1�

�W+ + W−�3 � 1

− 1� . �E7�

For the noise, we therefore find SLL=SRR=SLR=SRL, and thefinal results are

S�� = 0� � 4e2W+W−�W0 − W1�2

�W+ + W−�3 , �E8�

F � 6W+W−�W0 − W1�2

�W0 + 2W1��W+ + W−�3 . �E9�

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37 More generally, the suppression applies to all sequential pro-cesses which leave the vibrational state unchanged or increasethe excitation level.

38 The behavior of these absorption-induced vibrational sidebands,which are not specific to the case of strong electron-phononcoupling, deserves further investigations and will be publishedelsewhere �M. C. Lüffe, J. Koch, and F. von Oppen �unpub-lished��.

39 K. S. Kölbig, Comput. Phys. Commun. 4, 221 �1972�.40 K. S. Kölbig, http://cern.ch/cernlib

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