Abstract
We theoretically examine the effect of a single phonon mode on the structure of the frequency dependence of the ac conductance of molecular junctions, in the linear response regime. The conductance is enhanced (suppressed) by the electronphonon interaction when the chemical potential is below (above) the energy of the electronic state on the molecule.
PACS numbers: 71.38.k, 73.21.La, 73.23.b
Introduction
Molecular junctions, made of a single molecule (or a few molecules) attached to metal electrodes, seem rather well established experimentally. An interesting property that one can investigate in such systems is the interplay between the electrical and the vibrational degrees of freedom as is manifested in the IV characteristics [1,2].
To a certain extent, this system can be modeled by a quantum dot with a single effective level ε_{0}, connected to two leads. When electrons pass through the quantum dot, they are coupled to a single phonon mode of frequency ω_{0}. The dc conductance of the system has been investigated theoretically before, leading to some distinct hallmarks of the electron phonon (eph) interaction [36]. For example, the BreitWigner resonance of the dc linear conductance (as a function of the chemical potential μ, and at very low temperatures) is narrowed down by the eph interaction due to the renormalization of the tunnel coupling between the dot and the leads (the FrankCondon blockade) [4,5]. On the other hand, the eph interaction does not lead to subphonon peaks in the linear response conductance when plotted as a function of the chemical potential. In the nonlinear response regime, in particular for voltages exceeding the frequency ω_{0 }of the vibrational mode, the opening of the inelastic channels gives rise to a sharp structure in the IV characteristics. In this article, we consider the ac linear conductance to examine phononinduced structures on transport properties when the ac field is present.
Model and calculation method
We consider two reservoirs (L and R), connected via a single level quantum dot. The reservoirs have different chemical potentials, μ_{L }= μ+Re[δμ_{L}e^{iωt}] and μ_{R }= μ+Re[δμ_{R}e^{iωt}]. When electrons pass through the quantum dot, they are coupled to a single phonon mode of frequency ω_{0}. In its simplest formulation, the Hamiltonian of the electronphonon (eph) interaction can be written as , where b (c_{0}) and b^{† }() are the annihilation and the creation operators of phonons (electrons in the dot), and γ is the coupling strength of the eph interaction. The broadening of the resonant level on the molecule is given by Γ = Γ_{L }+ Γ_{R}, with , where ν is the density of states of the electrons in the leads and t_{L(R) }is the tunneling matrix element coupling the dot to the left (right) lead.
The ac conductance of the system is derived by the Kubo formula. In the linear response regime, the current is given by I = (I_{L } I_{R})/2, where
Here, is the Fourier transform of the two particle Green function,
where , with and c_{k(p) }denoting the creation and annihilation operators of an electron of momentum k(p) in the left (right) lead. The ac conductance is then given by
In this article we consider the case of the symmetric tunnel coupling, Γ_{L }= Γ_{R}. We also assume δμ_{L }=  δμ_{R }= δμ/2. The eph interaction is treated by the perturbation expansion, to order γ^{2}. The resulting conductance includes the selfenergies stemming from the Hartree and from the exchange terms of the eph interaction, while the vertex corrections of the eph interaction vanish when the tunnel coupling is symmetric. We also take into account the RPA type dressing of the phonon, resulting from its coupling with electrons in the leads [3].
Results
The total conductance is given by G = G_{0 }+ G_{int}, where G_{0 }is the ac conductance without the eph interaction, while G_{int }≡ G_{H }+ G_{ex }contains the Hartree contribution G_{H }and the exchange term G_{ex}. Figure 1 shows the conductance G as a function of ε_{0 } μ, for a fixed ac frequency ω = 0.5Γ. The solid line indicates G_{0}. The dotted line shows the full conductance G, with γ = 0.3Γ. The peak becomes somewhat narrower, and it is shifted to higher energy, which implies a lower (higher) conductance for ε_{0 } < μ (ε_{0 } > μ). However, no additional peak structure appears.
Figure 1. The ac conductance as a function of (ε_{0 } μ). The ac frequency ω = Γ. Γ_{L }= Γ_{R }and δμ_{L }= δμ_{R}. Solid line: without eph interaction. Dotted line: γ = 0.3Γ and ω_{0 }= Γ.
Next, Figure 2a shows the full ac conductance G as a function of the ac frequency ω, when ε_{0 } μ = Γ. The solid line in Figure 2a indicates G_{0}. Two broad peaks appear around ω of order ± 1.5(ε_{0 } μ). The broken lines show G in the presence of the eph interaction with ω_{0 }= 2Γ, ω_{0 }= Γ, or ω_{0 }= 0.5Γ. The eph interaction increases the conductance in the region between the original peaks, shifting these peaks to lower ω, and decreases it slightly outside this region. Figure 2b indicates the additional conductance due to the eph interaction, G_{int}, for the same parameters. Similar results arise for all positive ε_{0 } μ. Both G_{H }and G_{ex }show two sharp peaks around ω ~ ± (ε_{0 } μ) (causing the increase in G and the shift in its peaks), and both decay rather fast outside this region. In addition, G_{ex }also exhibits two negative minima, which generate small 'shoulders' in the total G. For ε_{0 }> μ, G_{int }is dominated by G_{ex}. The exchange term virtually creates a polaron level in the molecule, which enhances the conductance. The amount of increase is more dominant for lower ω_{0}. The situation reverses for ε_{0 }< μ, as seen in Figure 3. Here, G_{0 }remains as before, but the ac conductance is suppressed by the eph interaction. Now G_{int }is always negative, and is dominated by G_{H}. The Hartree term of the eph interaction shifts the energy level in the molecule to lower values, resulting in the suppression of G. The amount of decrease is larger for lower ω_{0}.
Figure 2. The ac conductance as a function of the ac frequency ω at ε_{0 } μ = Γ. (a) The total conductance when Γ_{L }= Γ_{R }and δμ_{L }= δμ_{R}. The broken lines indicate the conductance in the presence of eph interaction with γ = 0.4Γ. ω_{0 }= 2Γ, or 0.5Γ. The solid line is the 'bare' conductance G_{0}, in the absence of eph interaction. (b) The additional conductance due to the eph interaction, G_{int}(ω) = G_{H}(ω) + G_{ex}(ω), for the same parameters as in (a).
Figure 3. The conductance as a function of the ac frequency ω at ε_{0 } μ = Γ. (a) The total conductance when Γ_{L }= Γ_{R }and δμ_{L }= δμ_{R}. The broken lines indicate the conductance in the presence of eph interaction with γ = 0.3Γ. ω_{0 }= 2Γ, Γ or 0.5Γ. The solid line is the 'bare' conductance G_{0 }in the absence of eph interaction. (b) The additional conductance due to the eph interaction, G_{int}(ω) = G_{H}(ω) + G_{ex}(ω), for the same parameters as in (a).
Conclusion
We have studied the additional effect of the eph interaction on the ac conductance of a localized level, representing a molecular junction. The eph interaction enhances or suppresses the conductance depending on whether ε_{0 } > μ or ε_{0 }< μ.
Abbreviations
eph: Electronphonon.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
AU carried out the analytical and numerical calculations of the results and drafted the manuscript. OE conceived of the study. AA participated in numerical calculations. All authors discussed the results and commented and approved the manuscript.
Acknowledgements
This study was partly supported by the German Federal Ministry of Education and Research (BMBF) within the framework of the GermanIsraeli project cooperation (DIP), and by the USIsrael Binational Science Foundation (BSF).
References

Park H, Park J, Lim AKL, Anderson EH, Alivisatos AP, MacEuen PL: Nanomechanical oscillations in a singleC_{60 }transistor.
Nature (London) 2000, 407:57. Publisher Full Text

Tal O, Krieger M, Leerink B, van Ruitenbeek JM: Electron Vibration Interaction in SingleMolecule Junctions: From Contact to Tunneling Regimes.
Phys Rev Lett 2008, 100:196804. PubMed Abstract  Publisher Full Text

Mitra A, Aleiner I, Millis AJ: Phonon effects in molecular transistors: Quantal and classical treatment.
Phys Rev B 2004, 69:245302. Publisher Full Text

Koch J, von Oppen F: FranckCondon Blockade and Giant Fano Factors in Transport through Single Molecules.
Phys Rev Lett 2005, 94:206804. PubMed Abstract  Publisher Full Text

EntinWohlman O, Imry Y, Aharony A: Voltageinduced singularities in transport through molecular junctions.
Phys Rev B 2009, 80:035417. Publisher Full Text

EntinWohlman O, Imry Y, Aharony A: Transport through molecular junctions with a nonequilibrium phonon population.
Phys Rev B 2010, 81:113408. Publisher Full Text