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

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?

Anyway, there are at least two methods to derive it (as far as I know). One popular method is the equation of motion'' method which formulates the contour Dyson equation first and then uses Langreth rules to obtain similar recursive equations for real-time greens functions. This is detailed in the book Quantum Kinetics in Transport and Optics of Semiconductors. The other method I know is that you can work with the nonequilibrium path-integral formalism, where the Grassman variables of contacts can be integrated out, leading to the effective action of the device region. Then your second equation can be recognized immediately.