Electron transport in a one dimensional conductor with inelastic scattering by self-consistent reservoirs

We present an extension of the work of D'Amato and Pastawski on electron transport in a one-dimensional conductor modeled by the tight binding lattice Hamiltonian and in which inelastic scattering is incorporated by connecting each site of the lattice to one-dimensional leads. This model incorporates B\"uttiker's original idea of dephasing probes. Here we consider finite temperatures and study both electrical and heat transport across a chain with applied chemical potential and temperature gradients. Our approach involves quantum Langevin equations and nonequilibrium Green's functions. In the linear response limit we are able to solve the model exactly and obtain expressions for various transport coefficients. Standard linear response relations are shown to be valid. We also explicitly compute the heat dissipation and show that for wires of length $N >> \ell$, where $\ell$ is a coherence length scale, dissipation takes place uniformly along the wire. For $N << \ell$, when transport is ballistic, dissipation is mostly at the contacts. In the intermediate range between Ohmic and ballistic transport we find that the chemical potential profile is linear in the bulk with sharp jumps at the boundaries. These are explained using a simple model where the left and right moving electrons behave as persistent random walkers.