Quantum dissipative effects on non-equilibrium transport through a single-molecular transistor: The Anderson-Holstein-Caldeira-Leggett model

The Anderson-Holstein model with Caldeira-Leggett coupling with environment is considered to describe the damping effect in a single molecular transistor (SMT) which comprises a molecular quantum dot (with electron-phonon interaction) mounted on a substrate (environment) and coupled to metallic electrodes. The electron-phonon interaction is first eliminated using the Lang-Firsov transformation and the spectral density function, charge current and differential conductance are then calculated using the non-equilibrium Keldysh Green function technique. The effects of damping rate, and electron-electron and electron-phonon interactions on the transport properties of SMT are studied at zero temperature.


The Model
shows the schematic description of an SMT system. A typical SMT device consists of a single-level molecule or a QD coupled to two metal leads. The molecule is assumed to have a single vibrational mode interacting with its charge by el-ph interaction. The system is embedded on an insulating substrate (yellow colour part in Fig. 1) that can be approximated as a bath of independent harmonic oscillators in the spirit of the Caldeira-Leggett model. The substrate can cause a damping effect that can be described by a linear coupling term between the local phonon field of the molecule and a set of independent harmonic oscillators of the substrate bath. For the sake of simplicity, we neglect the effects of spin on the properties of the SMT. The model Hamiltonian for the system is given by The first term H l describes the Hamiltonian for the source (l = S) and the drain ( = ) l D and is given by where H m0 is the Hamiltonian for the electronic part of the molecule and reads as with n dσ as the number operator corresponding to the electrons on the molecule with ε d as the onsite energy (that can be varied experimentally by tuning the gate voltage (V G ) and U is the local coulomb correlation strength. H vib describes the vibrational degree of freedom of the molecule of mass m 0 and frequency ω 0 and can be written as, and H vib−e represents the el-ph interaction on the molecule and is given by with λ as the el-ph coupling constant. The leads-molecule hybridization term with hybridization strength V k is given by, Finally, the damping effect is incorporated in (1) by introducing the term Eliminating the linear oscillator-bath interaction, we can write where Δ ω 2 is the shift in the square of the molecular oscillator frequency caused by the linear oscillator-bath coupling. For very large N we can replace the summation over j by an integration over ω j . Δ ω 2 can be written as where J(ω) is the spectral function for which we choose the Lorentz-Drude form: where ω c is the cutoff frequency which is much larger than the other frequencies in the system and γ is the damping rate. The shift in the molecular frequency turns out be, ω πγω ∆ = . The total Hamiltonian finally reduces to is the creation (annihilation) operator for a molecular phonon of frequency ω  0 . It may be noted that we have neglected the decoupled bath-oscillator Hamiltonian because that merely contributes a constant to the energy.

Polaron Transformation.
To investigate the effects of the polaronic interactions in the system, a Lang-Firsov where ′ V k the phonon is mediated hybridization strength, ε  d is the renormalized molecular energy level due to the polaronic effect and ε p is the polaron binding energy. Tunneling Current. In this section, we shall employ the method of Chen et al. 26 . The current expression through the interacting region coupled to two metallic leads can be expressed as 27,28 , Similarly, the occupation number of the molecule are given by, where f S,D (ε) are the Fermi distribution functions of the source and drain whose chemical potentials are related to the bias voltage (V B ) and mid-voltage 2 is the coupling strength between the molecule and the source (drain), ρ S(D) being the density of states in the source (drain) channel. Here we have considered constant density of states in the source and drain. The possible excitation energy spectrum is described by the quantity called Spectral (SP) function, which is defined as where, where the superscript '> ' ('< ') refers to greater (lesser), 'r' ('a') refers to retarded (advanced) and 0 represents the true electronic ground state of the system. Where G r(a) (ε) and G >(<) (ε) are the energy-dependent retarded (advanced) and lesser (greater) electron Green's functions of the molecule respectively. The retarded and advanced Green functions can be easily calculated using the equation of motion approach. One obtains where the retarded (advanced) self-energy S r(a) (ε) due to hybridization interaction is given by where the real part of the self-energy can be absorbed into the molecular energy level. We assume that ε Λ( ) ∼ and where the factors −φ( τ) n n 0 0 Using Keldysh formalism we can write lesser and greater Green's functions as After calculating the lesser and greater Green functions we can obtain the SP function of the molecule electron using Eq. (37).

Results and Discussion
In the present calculation, all energies are measured in units of phonon energy ω 0 which is set equal to 1. Furthermore, the coupling of the molecule with the source and that with the drain are considered symmetric. In our calculation we have taken ε = = , Γ = . V 0 02 d G and eV m = 0.1. Our main aim is to study the damping effect of the substrate on the properties of the SMT system. In Fig. 2, we show the variation of the SP function A(ε) of a SMT with ε for different values of the damping rate γ and a given value of the el-ph coupling constant λ (λ = 0.6). The inset shows the A(ε) vs. ε behavior for the case: λ = γ = 0, which is a simple Lorentzian with a single resonant peak at ε d = 0. The el-ph interaction induces polaronic effects that renormalize the SMT parameters and shift the ε d = 0 peak of the SP function towards red and also make them sharper. Most importantly, the SP function also develops side peaks at ε ω ∓ n d 0 in the presence of the el-ph interaction. These, so called, phonon side bands in the SP function at zero temperature represent the phononic excitation energy levels created by the electrons tunneling on to the molecule by absorbing or emitting phonons. Due to the damping effect of the substrate the phonon frequency gets renormalized to ω  0 . As the damping rate increases, the heights of the phonon side bands decrease and broaden. This suggests that as the damping rate increases, the occupation probability of the phonon side bands decreases.
In Fig. 3 we present the results for the normalized tunneling current J(eV B , eV m ) of the SMT system as a function of bias voltage for different values of γ in the presence of the el-ph interaction. The normalized tunneling current for the case: λ = γ = 0 is shown in the inset for comparison. As the damping rate increases the current also increases. To see the effect of el-ph interaction on current, we plot, in Fig. 4, the current as a function of λ for different values of γ at constant bias voltage V B . As λ increases, the current decreases smoothly but rapidly and becomes zero at a critical value of λ, say λ c . This is easy to understand physically, though mathematically, Eq. (25) explicitly shows that the current decreases exponentially with λ. At the same time as λ increases, the separation between the molecule energy level and the chemical potential of source increases. As γ increases, λ c is also found to increase.
In Fig. 5, we present the results for the differential conductance G(eV B , eV m )(= dJ/dV B ) as a function of the bias voltage. In Fig. 5(a) we show the behaviour of G in the absence of the el-ph interaction and the damping effect. The symmetry in the conductivity peaks is clearly visible. In Fig. 5(b) we show the behavior of G in the presence of el-ph interaction (λ = 0.6) for different values of γ. The el-ph interaction and damping have visible independent effects on G. The effect of the el-ph interaction is two-fold. First, it sharpens the conductivity peaks and secondly and more importantly it gives rise to new satellite peaks that originate because of phonon-assisted tunneling transport. To  show the effect of el-ph interaction on the differential conductance explicitly, we plot in Fig. 6, G/G 0 as a function of λ for different values of γ. For a given γ, the differential conductance has a peak at a certain value of λ and as γ increases the peak reduces in height and also shifts towards right. Both shifting and reduction in height of the conductance peaks are the expected behavior.
In Fig. 7(a,b) we plot J and G respectively as a function of the el-el interaction strength U. As U increases, both J and G decrease, while they increase with increasing γ. These variations can be explained in the following way. Due to the Coulomb blockade effect the onsite coulomb correlation opposes the double occupancy on the molecule as a result of which the current decreases with increasing U, while as γ increases, the effective phonon frequency decreases leading to an increase in the current and the conductance.
Finally, we make a three-dimensional plot for J as a function of both γ and λ in Fig. 8(a) and for G as a function of γ and λ in Fig. 8(b).

Conclusion
In this work we have considered a SMT system in which a molecule or a quantum dot is placed on a substrate coupled to two metal leads acting as a source and a drain. The system is modeled by the AH Hamiltonian with a linear Caldeira-Leggett term to include the linear coupling between the substrate and the molecule which describes the damping effect. We have calculated the spectral function, tunneling current and differential conductance of the SMT system using the Keldysh Green function method. We have also analyzed the effect of el-ph interaction and the damping effect due to the substrate. We have shown that the el-ph interaction induces polaronic effects that renormalize the SMT parameters and shift the peak of the spectral function towards red and also make them sharper. The spectral function also develops side bands whose heights decrease and the widths broaden with increasing damping rate. As λ increases, the tunneling current is found to decrease smoothly but rapidly to zero at a critical value of λ. The el-ph interaction also sharpens the conductivity peaks and gives rise to new satellite peaks that originate because of phonon-assisted tunneling transport. It is also shown that the local el-el interaction causes a reduction in both the tunneling current and the differential conductance. Due to the damping effect of the substrate, the effective phonon frequency of the molecule oscillator decreases as a result of which the tunneling current increases. In the presence of el-ph and el-el interactions and damping, the differential conductance exhibits an interesting behavior. The spin-orbit interaction may also have an interesting effect on the tunneling current and the differential conductance. This issue is presently under investigation and the results will be published in due course. Which will be the subject matter of a future investigated.