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

+ 2 like - 0 dislike
167 views

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

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverflo$\varnothing$Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.