ABSTRACT
This chapter discusses the damping effect on the non-equilibrium transport properties of the single molecular transistor due to the presence of the substrate in ohmic case at zero temperature. It explains the Anderson-Holstein Hamiltonian in combination with the Caldeira-Leggett model, which was proposed by Caldeira and Leggett in 1981 to describe the dissipation of a quantum system due to its interaction with a set of harmonic oscillators. The chapter deals with an extension of the Anderson-Holstein model called the Anderson-Holstein-Caldeira-Leggett model which is useful to describe the quantum dissipative effects in an electronic device at the nanoscale such as a single molecular transistor. The transport properties of a single molecular transistor device have been studied both experimentally and theoretically by several groups. In this transistor device the charge transport takes place through a single molecule which is connected with two metal electrodes that act as source and drain.
