A Mean-field contribution to noise

Current noise in molecular junctions: effects of the electron-phonon interaction


We study inelastic effects on the electronic current noise in molecular junctions, due to the coupling between transport electrons and vibrational degrees of freedom. Using a full counting statistics approach based on the generalized Keldysh Green’s function technique, we calculate in an unified manner both the mean current and the zero-frequency current noise. For multilevel junctions with weak electron-phonon coupling, we give analytical formulas for the lowest order inelastic corrections to the noise in terms of universal temperature- and voltage-dependent functions and junction-dependent prefactors, which can be evaluated microscopically, e.g. with ab-initio methodologies. We identify distinct terms corresponding to the mean-field contribution to noise and to the vertex corrections, and we show that the latter contribute substantially to the inelastic noise. Finally, we illustrate our results by a simple model of two electronic levels which are mutually coupled by the electron-phonon interaction and show that the inelastic noise spectroscopy is a sensitive diagnostic tool.

72.70.+m, 72.10.Di, 85.65.+h, 73.63.-b

I Introduction

Recent progress in experimental techniques, such as break junctions and scanning tunneling microscopy, allows to electrically contact single molecules, to create and manipulate atomic wires, and to investigate the electronic transport properties of these nanoscopic objects.Cuniberti et al. (2005); Agraït et al. (2003) Contrary to larger devices, atomic-scale ones usually retain their microscopic features, which are then observable in the transport spectroscopy. Apart from the purely electronic contributions, other degrees of freedom such as vibrational modes or local spins can be addressed and revealed by point-contact spectroscopyNaidyuk and Yanson (2005) (PCS) or by inelastic electron tunneling spectroscopy (IETS).Jaklevic and Lambe (1966) These techniques have been extensively used in the recent past to reveal inelastic features in the non-linear conductance due to vibrationsStipe et al. (1998); Agraït et al. (2002); Smit et al. (2002); Djukic et al. (2005); Tal et al. (2008); Frederiksen et al. (2008); Rahimi and Hegg (2009); Arroyo et al. (2010) or local spin excitations,Heinrich et al. (2004); Hirjibehedin et al. (2006); Fu et al. (2009) triggering an intense theoretical activity. So far most studies focused on the current-voltage characteristics, and present day PCS/IETS theories,Lorente and Persson (2000); Frederiksen et al. (2004); Paulsson et al. (2005); Viljas et al. (2005); de la Vega et al. (2006); Solomon et al. (2006); Sergueev et al. (2007); Frederiksen et al. (2007); Teobaldi et al. (2007); Frederiksen et al. (2007); Paulsson et al. (2008); Kristensen et al. (2009); Alducin et al. (2010); Fransson et al. (2010); Monturet et al. (2010); Patton (2010) often based on ab-initio calculations, allow to make detailed predictions for the conductance that compare favorably with experimental results.

Electronic current (shot) noiseBeenakker and Schönenberger (2003); Blanter and Büttiker (2000) is another quantity of fundamental interest in transport, representing the second cumulant of the current distribution within the full counting statistics methodology.Nazarov (1999); ? Although the measurement of higher order cumulants is experimentally challenging, shot noise in atomic contacts and molecular junctions has been already measured in the small voltage (elastic) regime,van den Brom and van Ruitenbeek (1999); Djukic and van Ruitenbeek (2006); Kiguchi et al. (2008) and there are ongoing experimental efforts to address the inelastic noise signal as well.1 These progresses made the investigation of effects due to electron-phonon (-ph) interaction on the current noise an appealing task from the theoretical point of view, with the ultimate goal of allowing for prediction for the noise in molecular junctions as reliable as those now available for the non-linear conductance.

Since the noise is technically represented by a two-particle non-equilibrium correlation function, its direct evaluation poses a significant challenge compared to the mean current. For molecular junctions weakly coupled to leads, noise calculations based on one-level models have been put forward within the rate equation approach.Mitra et al. (2004); Koch and von Oppen (2005); ? In the opposite limit, pioneering works based on the non-equilibrium Green’s functions formalismZhu and Balatsky (2003); Galperin et al. (2006) adopted a mean-field-like approximation for the noise, thus neglecting the contributions due to the vertex corrections. A very convenient tool to overcome these limitations is the full counting statistics,Nazarov (1999); ?; Levitov and Lesovik (1993); ? since it allows to address the noise and other current cumulants, while taking consistently into account all contributions due to -ph coupling up to a given order in perturbation theory. Simultaneously with other two groups,Schmidt and Komnik (2009); Avriller and Levy Yeyati (2009) we exploited such an approach to analyze the transport properties of a simple model for molecular junctions consisting of a single resonant level symmetrically coupled to metallic leads and weakly interacting with a single phonon mode.Haupt et al. (2009) Despite its simplicity, this model can be applied to the experimentally relevant case of junctions formed by a single hydrogen/deuterium molecule between platinum leads,Smit et al. (2002) and in this case we predicted a significant inelastic contribution to the current noise.Haupt et al. (2009)

In this paper we go beyond such a simple model and we extend our theory for inelastic noiseHaupt et al. (2009) to more complex molecular junctions and to atomic wires. In fact, we consider the case of a junction formed by multiple electronic levels with asymmetric coupling to leads, and derive analytical formulas for the corrections to current and noise due to weak electron-phonon coupling. We express our result in terms of universal temperature- and voltage-dependent functions and junction-dependent prefactors. These expressions, when supplemented with ab-initio calculations to estimate microscopically the prefactors characterizing a given junction, can serve as a basis to make realistic predictions for the current noise in a relevant class of molecular and atomic-size junctions. In this respect, our work can be viewed as a direct extension of the corresponding lowest order expansion scheme developed for the non-linear conductance.Paulsson et al. (2005); Viljas et al. (2005); de la Vega et al. (2006) In addition, we identify the contributions to noise due to the vertex corrections and show that, even in limit of weak -ph coupling, they need to be taken into account in order to obtain accurate results and to comply with the fluctuation-dissipation theorem.

The paper is organized as follows. After a brief description of the model of a multi-level junction coupled to leads and weakly interacting with a number of vibronic modes (phonons) in Sec. II, we introduce the methodology of the noise calculation via extended Keldysh Green’s functions in Sec. III. In Sec. IV we consider the case of no interactions and recover the standard results for the elastic current and noise. Our original contribution is presented in Secs. V and VI, where we discuss the corrections to the current and noise due to the Hartree and the Fock diagrams, respectively. In subsection VI.3 we then illustrate our theory by a simple example of independent electronic levels which are coupled only via the -ph interaction. Finally, we conclude and give an outlook of open issues and possible extensions of the present work in Sec. VII. More technical parts of the text are deferred to 5 appendices. In addition, we make use of the Electronic Physics Auxiliary Publication Service (EPAPS) to supplement the paper with a Mathematica notebook with full expressions for the lowest order corrections to current and noise due to -ph coupling. This file is intended to be of use for interested readers in order to access directly our results without need of retyping cumbersome formulas from the main text, and it also extends the results of subsection VI.2 to the case of finite temperature.

Ii Model

The system we consider can be schematically represented as a central device region (representing the molecule or the atomic-wire) which is tunnel-coupled to non-interacting metallic leads


Neglecting for simplicity the spin degree of freedom2 the central region can be described by the following Hamiltonian


where () and () are the electron and phonon annihilation (creation) operators, respectively; is the single-particle effective Hamiltonian of the electrons moving in a static arrangement of atomic nuclei, is the Hamiltonian of free uncoupled phonons, is the -ph coupling within the harmonic approximation, and is the -ph coupling matrix for the -th phonon mode. Here, boldface notation stands for matrices in the system electronic space. The leads and tunneling Hamiltonians are given by


The states in the leads are occupied according to the Fermi distributions , with , the inverse temperature, and the chemical potential of lead-. The applied bias voltage is .

Iii Methods

iii.1 The generalized Keldysh Green’s function technique

To calculate the average current and the zero-frequency noise in the stationary regime, we employ the generalized non-equilibrium Keldysh Green’s function technique. Nazarov (1999); ? In this approach, one is interested in finding the cumulant generating function , which in the case of charge transport is defined as


where is the probability for charges to be transferred through the system during the measuring time and is a continuous parameter known as counting field. Given , the cumulants of the charge transfer distribution can be straightforwardly calculated according to the prescription


Under the assumption that the measuring time is much longer than any correlation time of the system (), the first two cumulants of are directly proportional to the average current through the system and to the zero-frequency current noise ,


which are the quantities we are primarily interested in.

The key idea for calculating the cumulant generating function for transport through a quantum system is to modify the Hamiltonian by introducing a time-dependent counting field and to relate to the Keldysh Green’s function of the system in the presence of , i.e. to , where the evolution is due to the modified Hamiltonian. Levitov and Lesovik (1993); ?; Nazarov (1999); ? One way to accomplish this is to add a time-dependent phase to the tunneling matrix elements ,

with on the forward branch of the Keldysh contour and on the backward one, Levitov and Reznikov (2004) where is the Heaviside step-function.

Here we extend the result derived by Gogolin and Komnik Gogolin and Komnik (2006) for the Anderson model to the case in which the central region has several electronic states. Generalizing the derivation of Ref. Gogolin and Komnik, 2006 to a multilevel system, we obtain the following expression for the derivative of the cumulant generating function


where represents the Keldysh-Green’s function of the system in Keldysh space


, with the self-energy due to the modified tunneling Hamiltonian


and stands for the trace over the electronic degrees of freedom and the Keldysh space, i.e. , with being the trace in the system electronic space. The check sign indicates matrices in the Keldysh space and the superscripts correspond to the forward/backward branch of the Keldysh-contour. Note that in Eq. (10) we have used the following sign convention for the elements of the Keldysh-matrix for the self-energy


Finally, is the level broadening due to the coupling to the lead .

According to Eq. (8), the problem of evaluating current and noise (as well as any higher order cumulant of the charge transfer distribution) is reduced to the calculation of the system single-particle Green’s function . The latter can be obtained from the solution of the Dyson equation


where is the self-energy solely due to the e-ph coupling, and is the free Green’s function of the system in the presence of the leads and of the counting field but without the e-ph interaction , with


the Green’s function of the isolated dot. It is important to notice that , depending on the Green’s function of the system, is itself a function of the counting field (see Sec. III.3).

Finally, we remark in passing that for it is , i.e. in the presence of the counting field , the four Keldysh Green’s functions are all independent.

iii.2 Current and Noise

Although Eq. (8) gives access to all cumulants of the charge transfer distribution through the system, in this work we will focus only on the study of the average current and the zero frequency noise , which are the quantities most easily accessible from the experimental point of view.

The average current is directly obtained from Eq. (8) by setting


while the noise is given by


where, we have used the identity together with the Dyson equation . It turns out that the first term of Eq. (15) corresponds exactly to Eq. (30) of Ref. Souza et al., 2008, which gives the expression for the noise within a mean-field approximation (see Appendix  A). For this reason, we identify


as the mean-field contribution to noise. The remaining term constitutes the vertex correction


As we will discuss in detail in the following, the vertex correction can give a significant contribution to the total noise, comparable to the mean-field part and thus, contrary to what was done in some pioneering works,Zhu and Balatsky (2003); Galperin et al. (2006) it cannot be omitted even in the limit of weak interaction. Moreover, neglecting generally leads to violation of the fluctuation-dissipation theorem, see Appendix  D.

iii.3 Weak electron-phonon coupling

Figure 1: Diagrammatic representations of and . Here, the plain line stands for the free electronic Green’s function , and the wiggly line for the phononic one . Finally, the dot represents the -ph coupling constant .

In order to make use of Eqs. (14), (15), we need to determine the full Green’s function . Being interested in the experimentally relevant limit of weak electron-phonon coupling, we truncate the Dyson equation at the lowest (second) order in the e-ph coupling


where is the Hartree-Fock self-energy, depicted diagrammatically in Fig. 1, with


and . Above, represent the free thermalized phonon Green’s functions of the -th phonon mode

with the thermal expectation value of the mode occupation. The proper inclusion of possible heating effects on noise, involving non-equilibrium phonon occupation and its potential back-action on the electronic transport, is beyond the scope of this publication; some of the involved issues are discussed in the concluding Sec. VII.

In Eq. (19) we introduced the generalized electronic density on the two branches of the Keldysh contour () in the presence of the counting field


Note that, on the two branches of the Keldysh contour, the electronic density is given by different limits of the corresponding Green’s functions . As a consequence, even if on the forward/backward branch, for any finite value of . On the other hand, at one gets


where is the electronic density in the noninteracting case.

Plugging Eqs. (19), (20) into Eq. (18), we can rewrite Eqs. (14), (15) as and , where


are the elastic current and noise, and

give the respective corrections due to electron-phonon coupling, with


and , where

is the mean-field contribution and

the vertex correction. The previous equations can be schematically expressed by the diagrams of Fig. 2.

We note that truncating the Dyson equation to the lowest order in -ph coupling, , preserves charge conservation in that order. This implies that both mean current and zero-frequency noise are constant throughout the whole circuit, in particular at both junctions between the device and the leads. Potential violations of charge conservation can only occur in the next order, i.e. , and can thus be safely neglected for any practical purposes in the considered weak coupling limit.

Finally, we observe that to the lowest order in the -ph coupling, and are simply given by a linear superposition of contributions coming from different phonon modes. As a consequence, we can restrict ourselves to the case of coupling to a single phonon mode with frequency , occupation and coupling matrix .

Figure 2: Diagrammatic representations of Eqs. (23)-(26). The plain line stands for the electronic Green’s function ; single and doubled crosses stand for and , respectively. The box represents and, finally, the crossed box stands for the derivative of the -ph self-energy with respect to the counting field .

iii.4 The extended wide band limit

The corrections to current (25) and noise (26) due to the -ph coupling involve energy integrals which can be evaluated in general only numerically. Analytical progress can still be made if one assumes the electronic structure to be slowly changing over few multiples of a typical phonon energy around the Fermi level and approximate (i) the level broadening and (ii) the non-interacting retarded/advanced Green’s function with their values at the Fermi energy Paulsson et al. (2005); Frederiksen et al. (2007); Viljas et al. (2005); de la Vega et al. (2006)

where we took as the definition of . This approximation, which we call “extended wide-band limit” (eWBL), is reasonable for systems where either the broadening due to tunneling is large (, and ), or the closest resonance energy is far away from the Fermi energy (, and ).

Within the eWBL approximation, the integration over energy of functions with compact support can be performed analytically, resulting in explicit results for the mean current and the noise as functions of the applied bias voltage and other system parameters. It should be noted however, that approximation (ii) potentially leads to problems for integrals over infinite range and, in this case, it might be necessary to lift it. Specifically, this happens in the calculation of the electron density entering the Hartree term, see Appendix  B, and in the evaluation of the real parts of the retarded/advanced Fock self-energy via Kramers-Kronig relations, see Appendix  C.

Iv Elastic current and noise

For sake of completeness, before discussing the corrections to and due to the -ph coupling, we consider briefly the results for the elastic current and noise.

In the eWBL approximation, the elastic current is simply proportional to the voltage


with , while the noise is given by


where we have introduced the function .

The eigenvalues of the matrix give the “PIN-code” of transmission eigen-channels of the molecule connected to leads (without -ph interaction), and Eqs. (27), (28) are indeed equivalent Meir and Wingreen (1992) to the standard results for current and noise in a non-interacting system derived within the scattering theory. Büttiker (1992); Blanter and Büttiker (2000) However, is not equal to the matrix product of the transmission amplitudes of the scattering theory (it cannot be as is in general non-hermitian, for example). The two matrices are related though by a similarity (non-unitary) transformation, which among others ensures Meir and Wingreen (1992) With this caveat in mind, for sake of simplicity we will nevertheless call the transmission matrix in the rest of this paper. The construction of the scattering eigenstates within the NGF formalism is described in detail in Ref. Paulsson and Brandbyge, 2007.

We now turn our attention to the corrections to the current and noise induced by the -ph interaction. In order to make the discussion as clear as possible, we will consider the contributions coming from the Hartree and the Fock diagrams separately.

V Corrections due to the Hartree diagram

v.1 Current

We start by considering the contributions to the current coming from the Hartree diagram. After integrating Eq. (25) in the eWBL approximation we obtain




with the noninteracting electron density [ Eq. (22)] and . The correction is therefore a smooth function of the voltage with no features at the phonon emission threshold. For this reason has been often discarded in previous works on the effects of -ph interaction on the current. Frederiksen et al. (2004); Paulsson et al. (2005); Frederiksen et al. (2007); Paulsson et al. (2008); Viljas et al. (2005); de la Vega et al. (2006); Galperin et al. (2006); Fransson and Galperin (2010)

It should be noticed, however, that is generally non-linear in , since can be a (smooth) function of the applied bias voltage. Such a voltage dependence is nevertheless rather weak in the eWBL (see Appendix  B), and in such a case it is possible to interpret Eq. (29) as a quasi-elastic correction to an effective transmission matrix , i.e. because of the -ph coupling, the current is not proportional to the bare transmission coefficient but rather to .

v.2 Noise

Figure 3: Diagrammatic representations of and . Here, plain and wiggly lines stand for the electronic and phononic Green’s functions, respectively. The single cross stands for and the doubled one for . Finally, the dot represents the -ph coupling constant .

The mean-field contributions and the vertex correction to noise due to the Hartree diagram can be schematically represented by the diagrams in Fig. 3, which are the result of inserting the Hartree self-energy from Fig. 1 into appropriate diagrams in Fig. 2. In the usual eWBL, takes the simple form


Analogously to the current , this contribution has a simple interpretation in terms of the renormalization of the transmission matrix introduced above . This can be seen easily, as Eq. (31) corresponds exactly to the contribution of order to the elastic shot-noise of a system with transmission matrix :

where is given in Eq. (28).

Making use of the cyclic invariance of the trace, the vertex correction can be rewritten as


where we have used the fact that , with . Performing the integrals over energy in the usual eWBL approximation3 one obtains

with . Contrary to , Eq. (32) has no simple interpretation in terms of an effective transmission coefficient and it represents a distinctive contribution to noise coming from the Hartree term. From the physical point of view, it stems from the coupling of occupations of the electronic levels with the current fluctuations.Gogolin and Komnik (2006); Hershfield (1992)

We note, however, that in the case of a system with a single electronic level symmetrically coupled to leads. Therefore in this particular case the correction to noise induced by the Hartree term is given by alone.

Vi Corrections due to the Fock diagram

vi.1 Current

We now turn our attention to the corrections to current induced by the Fock diagram. Integrating Eq. (25) in the usual eWBL approximation, we obtain




with the spectral density. All the involved quantities depend only on the properties of the system at the Fermi level and can be determined by ab-initio calculations. Paulsson et al. (2005); Frederiksen et al. (2007); Viljas et al. (2005); de la Vega et al. (2006)

The voltage dependence of is carried by the functions




where is the Hilbert transform. Eq. (33) is in agreement with the result of Viljas et al.Viljas et al. (2005) while a term is missing in Refs. Paulsson et al., 2005; Frederiksen et al., 2007. Such a discrepancy is further discussed in Appendix  C.

The functions and give contributions to which are even/odd in bias, respectively (see Fig. 4). The term proportional to vanishes in the case of symmetric coupling to the leads, and it is typically much smaller than the contribution proportional to , even for asymmetric junctions. Frederiksen et al. (2007); Paulsson et al. (2008) Moreover, experimentally measured conductance curves are usually very weakly asymmetric under reversal of and at present it is unclear if the asymmetry is caused by phonons or by other effects.

At low temperature (, ), the main contribution to is therefore given by the first two terms of Eq. (33) alone. The first of these terms, linear in , is a quasi-elastic correction that, similarly to , contributes to an effective transmission matrix . The second one has instead a threshold behavior at the phonon emission energy, see Fig. 4, and it is responsible for the jump in the conductance observed in IETS and PCS experiments.

The sign of the conductance step at the phonon emission threshold (positive or negative) depends on the coefficient , and it has been discussed in detail in Refs. Paulsson et al., 2005; Viljas et al., 2005; de la Vega et al., 2006. As a rule of thumb, in the case of a molecular junction with low (high) bare transmission , inelastic -ph scattering results in an increase (decrease) of the conductance above the phonon emission threshold.

In the case of a system with a single electronic level symmetrically coupled to the leads via , reduces to , where is the transmission coefficient. In this case, the crossover from an increase to a decrease in the conductance is predicted to occur at Paulsson et al. (2005); de la Vega et al. (2006) This behavior has been explored and confirmed experimentally in Ref. Tal et al., 2008.

Figure 4: Upper panel: Plots of the dependencies of the functions and on the applied bias voltage (Eqs. (35) and (36), respectively). Lower panel: Same as above, but for the derivatives and . In both panels .

vi.2 Noise

We finally address the corrections to noise due to the Fock diagram , which are schematically represented by the diagrams in Fig. 5.

After lengthy but straightforward calculations, integration over energy in the usual eWBL approximation leads to analytic results for as functions of the applied bias voltage. The final expressions are, however, rather cumbersome and, for simplicity, we consider here only the limit of zero temperature . The complete expressions for at finite temperature are given in the supplementary material, 4 while the limit is discussed in Appendix  D in relation to the fluctuation-dissipation theorem.

In the limit of zero temperature, we obtain




where are given in Eqs. (34a), (34b),






is the zero temperature limit of Eq. (36). The corrections to noise can then be divided into a symmetric term, which is even in bias, and an antisymmetric one, which contains the Hilbert transform and yields an odd contribution. We notice that, while is a continuous function, its derivative shows logarithmic divergencies at .Entin-Wohlman et al. (2009); Egger and Gogolin (2008) These zero-temperature divergencies are, however, an artifact of treating the phonons as non-interacting modes, and they are regularized either by finite temperature or if any broadening of the phonon spectrum is taken into account.Egger and Gogolin (2008); Entin-Wohlman et al. (2009) This issue, however, goes beyond the scope of this work.

Figure 5: Diagrammatic representations of and . As in Fig. 3, plain and wiggly lines stand for the electronic and phononic Green’s functions, respectively. The single cross stands for and the doubled one for . Finally, the dot represents the -ph coupling constant .

At zero temperature, the symmetric contribution to is a piece-wise linear function of . At low voltages, , it is given by the first term of Eq. (VI.2) alone. Following the same reasoning as for Eq. (31), this linear contribution can be directly interpreted in terms of the renormalization of the transmission , consistently with the sub-threshold correction to the current. Above the phonon emission threshold, , inelastic processes come into play and their contribution to the noise is given both by the second term of Eq. (VI.2) and by the vertex correction Eq. (38). It is important to notice that these two contributions are in general of the same order (see below the Sec. VI.3 for a demonstrative example), so that the latter cannot be discarded.

Experimentally, is often measured directly by a lock-in technique. Such a quantity shows at the phonon emission threshold a sharp and distinguishable jump on top of a featureless background due to the elastic and quasi-elastic contributions. Therefore, we define here the inelastic noise signal as the difference of the plateau values of the noise derivative just above and below the jump


with accounting for the finite jump width at finite temperatures. At low enough temperatures, terms proportional to give a very small contribution to the inelastic noise signal due to the symmetric shape of around (for details see Appendix  E) and we can then approximate


i.e. at low temperatures carries the structural information about the junction given by the terms with the threshold behavior at the phonon emission energy.

vi.3 Independent electronic levels

We now consider a toy model for molecular junctions, in which we assume the electronic levels to be mutually coupled only via the -ph interaction. In this case, the relevant matrices in the system electronic space are given by

and , where , and is the number of electronic levels involved in the transport. Under the further assumption that each channel is symmetrically coupled to the leads (), the prefactors vanish identically and Eqs. (VI.2), (38) can be rewritten in a particularly suggestive form in terms of the transmission probabilities of the individual levels


where we have introduced the dimensionless coupling constants . For , Eqs. (44) reduce directly to the result of Refs. Haupt et al., 2009; Schmidt and Komnik, 2009; Avriller and Levy Yeyati, 2009. The voltage dependence of and is presented in Fig. 6 for the case of a systems with only two levels. We notice that (see also Eq.44b), meaning that the vertex corrections correspond to processes that lead to a suppression of the noise through the system. Moreover, Fig. 6 evidences that the contributions to the noise due to the vertex corrections can be of the same order of magnitude as the mean-field ones, and that they generally need to be taken into account in order to make accurate predictions for the phonon-assisted current noise.

In terms of the transmission coefficients of the different channels, the inelastic noise signal is given by


with . Depending on the values of and , can be either positive or negative and it is in general very sensitive to the parameters of the junction as illustrated in Fig. 7, again for the case of a system with only two levels. Here we plot as a function of the transmission coefficients for different values of the -ph coupling matrix elements. As general features we notice that is always positive when