Interplay of Electron Phonon Coupling Dissipative Phonon Bath and Electron Electron Interaction in a Triangular Quantum-Dot Trimer

TL;DR

The paper analyzes nonequilibrium charge transport through a triangular quantum-dot trimer (TMT) on a substrate, incorporating electron–phonon coupling via the Lang–Firsov transformation, substrate-induced phonon damping via a Caldeira–Leggett bath, and on-site Coulomb interaction treated in mean-field. An extended Anderson-Holstein-Caldeira-Leggett Hamiltonian is solved with nonequilibrium Green’s functions to compute the spectral function $A(E)$, steady-state current $J$, and differential conductance maps, revealing how polaronic renormalization, phonon sidebands, and dissipation compete with Coulomb blockade. Renormalized parameters such as $\tilde{t}=t_{ ext{eff}}$ and $\tilde{\epsilon}_i$, along with phonon dressing, shape the transport spectra and lead to phenomena like negative differential resistance. The findings provide a microscopic framework for tunable vibronic and correlation-driven transport in molecular transistors and nanoscale interferometers, with implications for design of phonon-assisted devices.

Abstract

Nonequilibrium charge transport through a trimer molecular transistor composed of three quantum dots arranged in a triangular geometry, which is placed on a substrate, has been studied in the presence of electron electron and electron phonon interactions. The entire system is described by an extended Anderson Holstein Caldeira Leggett Hamiltonian, in which the Caldeira Leggett term accounts for phonon damping arising from the coupling between the molecular vibrations and the substrate phonon bath. The electron phonon interaction is treated nonperturbatively using the Lang Firsov canonical transformation, while the electron electron interaction is incorporated at the mean field level. Keldysh nonequilibrium Greens function framework is used to study the transport properties, allowing us to calculate the spectral function, tunneling current, ad differential conductance of the trimer molecular transistor. The formalism enables systematic evaluation of the effects of Coulomb interaction, electron phonon coupling, and dissipation on the devices electronic transport characteristics.

Figures (6)

• Figure 1: Schematic representation of the trimer molecular transistor (TMT). The central region consists of three quantum dots (QDs) forming a triangular configuration, coupled to the source (S) and drain (D) electrodes. The inter-dot tunneling amplitudes are denoted by $t_1$, $t_2$, and $t_3$, while $eV_g$ and $eV_b$ represent the gate and bias voltages, respectively.
• Figure 2: Spectral function $A(E)$ as a function of energy $E$ for different values of electron--phonon coupling strength $\lambda$ and dissipation parameter $\gamma$.
• Figure 3: Current $J(eV_b)$ and differential conductance $G = dJ/d(eV_b)$ as functions of the bias voltage $eV_b$. (a) Results for $U = 0$. (b) Results for $U \neq 0$, showing the influence of electron--electron and electron--phonon interactions.
• Figure 4: (a–d) Current $J$ as a function of interaction parameters. (a) Variation of $J$ with on-site Coulomb interaction $U$ for different dissipation strengths $\gamma$. (b) $J(U)$ for different electron–phonon couplings $\lambda$ at fixed $\gamma$. (c) Dependence of $J$ on the phonon damping parameter $\gamma$ for several $\lambda$ values. (d) Dependence of $J$ on inter-dot tunneling $t_2$ for different $\lambda$. The data illustrate the interplay between Coulomb interaction, electron–phonon coupling, and dissipative effects in determining the transport characteristics of the TMT device.
• Figure 5: (a–d) Self-consistent dot occupation $n_d$ as a function of interaction parameters in the trimer molecular transistor (TMT). (a) Dependence of $n_d$ on on-site Coulomb interaction $U$ for different dissipation strengths $\gamma$. (b) $n_d(U)$ for several electron–phonon couplings $\lambda$ at fixed $\gamma$. (c) Variation of $n_d$ with the phonon damping parameter $\gamma$ for different $\lambda$ values. (d) Dependence of $n_d$ on inter-dot tunneling amplitude $t_2$ for various $\lambda$. The results illustrate how Coulomb repulsion, electron–phonon coupling, and dissipation collectively influence the steady-state charge distribution within the TMT device.