Derivation of open boundary conditions for Non-Equilibrium Green's Function (NEGF)

Question

It is widely claimed that the Non-Equilibrium Green's Function (NEGF) equations for the study of quantum transport have been derived from the many-body perturbation theory (MBPT). Yet the bridge between the two is unclear. The form of equations that we work with in the popular NEGF framework, as described by Prof Supriyo Datta (abs, pdf), is much different from the form of equation we see in classical texts on Non-equilibrium field theories.

My question is regarding the derivation of boundary conditions for open systems. The influence of the contacts is folded into the device as a self energy. This is done as follows.

$$H = H_0 + H_R + H_L$$

In the above equation the total Hamiltonian of the system is described in three parts. $H_0$ represents the Unperturbed Hamiltonian of the closed system under experimentation which our system. Now the the left and right contacts are described by Hamiltonians $H_L$ and $H_R$ respectively. The retarded Green's function is calculated as

$$G^R(E) = (EI - H_0 - \Sigma)^{-1}$$

Here $\Sigma$ is the contact self energy attributed to the left and the right contact. This self energy is calculated using the contact Hamiltonians. The contact Hamiltonians are used to calculate the surface Green's functions and then the ansatz for the Bloch waves is used to calculate this self energy.

$$\Sigma_{Lc} = e^{ik\Delta}g^r_{Lc}$$

(the exponential factor is coming from the ansatz used and $g_R$ is the surface Green's function).

For my problem I seek to work in a different basis of states. And, to be able to work all the equations to formulate the Dyson's equation,s I need to follow the derivation of these simplified equations from MBPT itself. Is there any rigorous derivation of these contact self energies? How are these equations derived from the basics of Keldysh formalism? Can you please mention some sources or a detailed method?

This post imported from StackExchange Physics at 2015-12-07 13:19 (UTC), posted by SE-user Praanshu Goyal

Answer 1

Non-Equilibrium Green's Function equations for the study of quantum transport were derived using the Keldysh formalism, e.g., in the first and last reference of this post. Looking up these papers in scholar.google.com and browsing through the citing papers should give you a good entrance to the literature.