Effect of phonons on the ac conductance of molecular junctions
© Ueda et al; licensee Springer. 2011
Received: 16 August 2010
Accepted: 9 March 2011
Published: 9 March 2011
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 electron-phonon 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
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 I-V 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 (e-ph) interaction [3–6]. For example, the Breit-Wigner resonance of the dc linear conductance (as a function of the chemical potential μ, and at very low temperatures) is narrowed down by the e-ph interaction due to the renormalization of the tunnel coupling between the dot and the leads (the Frank-Condon blockade) [4, 5]. On the other hand, the e-ph 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 I-V characteristics. In this article, we consider the ac linear conductance to examine phonon-induced 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 electron-phonon (e-ph) 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 e-ph interaction. The broadening of the resonant level on the molecule is given by Γ = ΓL + ΓR, 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.
In this article we consider the case of the symmetric tunnel coupling, ΓL = ΓR. We also assume δμ L = - δμ R = δμ/2. The e-ph interaction is treated by the perturbation expansion, to order γ 2. The resulting conductance includes the self-energies stemming from the Hartree and from the exchange terms of the e-ph interaction, while the vertex corrections of the e-ph 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 .
We have studied the additional effect of the e-ph interaction on the ac conductance of a localized level, representing a molecular junction. The e-ph interaction enhances or suppresses the conductance depending on whether ε 0 > μ or ε 0 < μ.
This study was partly supported by the German Federal Ministry of Education and Research (BMBF) within the framework of the German-Israeli project cooperation (DIP), and by the US-Israel Binational Science Foundation (BSF).
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