|Q&A: NEGF, Lectures 18-24||Q&A: NEGF, Lectures 18-24||Q&A: NEGF, Lectures18-24|
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The NEGF (Non-Equilibrium Green’s Function) method is regarded by many as an esoteric tool for specialists, but we believe it should be part of the standard training of science and engineering students.
** NEGF is also discussed in my other BOOKS
Chapter 8 of Electronic Transport of Mesoscopic Systems, Cambridge (1995) —> ETMS
Chapters 8-11 of Quantum Transport: Atom to Transistor, Cambridge (2005) —> QTAT
Table of Contents
- Does the book teach the “real NEGF” or “NEGF-lite for dummies”? (Updated 9/20/13)
- When applying NEGF to a physical structure, do you “dissect the Hamiltonian into disjoint parts”? (Updated 9/20/13)
- Does NEGF agree with Lanaduer and Boltzmann in the appropriate limits? (Updated 9/20/13)
- How can we teach NEGF without requiring advanced quantum statistical mechanics? (Updated 5/2/12)
The standard approach to NEGF is based on many-body perturbation theory (MBPT) which requires advanced quantum statistical mechanics. We use relatively elementary arguments, but in the end we obtain the same basic NEGF equations, namely,
So I would argue that what we teach is indeed THE “real NEGF.”
See the last item below for a fuller discussion of Eqs.19a,b and how we teach NEGF.
When applying NEGF to a physical structure, do you “dissect the Hamiltonian into disjoint parts”? (Updated 9/20/13)
Not really. We solve the NEGF equations (Eqs.(19.1), (19.2)) with appropriate spatial boundary conditions set by the contacts, as sketched below.
The contacts are assumed to have constant electrochemical potentials, based on the argument that ideally they are infinitely more wide than the channel, and hence can carry the same current with only an infinitesimal slope in the potential.
Designating a region as a contact implies that there is a significant increase in the number of conducting channels due to an increase in width and/or in the density of states.
When analyzing a given structure, it is important to scrutinize the results carefully, to make sure that the assumed contacts are indeed functioning as nearly ideal reservoirs.
For example, below is a figure excerpted from a paper written 20+ years ago
showing a wide-narrow-wide structure (Fig.27a of paper).
One could impose the contact boundary conditions at the wide-narrow interfaces. Instead we chose to impose it inside the wide region as shown. There was no significant change in the current, because there is negligible drop in the electrochemical potential across the wide regions (Fig.27b of paper).
Note the sharp drops in the electrochemical potential at the narrow-wide interfaces due to the interface resistance, discovered by Sharvin and Imry. Note also the small drops where the wide regions meet the contacts. The electrostatic potential is obtained by convolving the electrochemical potential with a screening function as explained in the paper.
All these concepts were summarized in Chapter 2 of my 1995 book * ETMS , and are also discussed in this book.
This paper ended by saying that “the main contribution of this work lies in putting the ideas (of Landauer, Buttiker, Imry and others) on a rigorous quantitative footing using a quantum kinetic approach.”
No, this was one of a series of papers exploring the basic conceptual issues of mesoscopic transport using NEGF. A later paper used the same approach to establish the spatial variation of the energy dissipation separating the Joule heat from the Peltier component. See
Yes, with small dephasing, it matches the scattering theory for coherent transport, also called the Landauer-Buttiker formalism.
With sufficient dephasing, it reproduces results from the semiclassical Boltzmann method, thus providing a bridge from quantum to semiclassical transport.
Many examples are included in the book essentially as homework problems that even beginning graduate students can reproduce. Below is one such example showing the variation of the local electrochemical potential across a barrier (1D version of the example in 1.2.2)
Note how the NEGF result for coherent transport shows oscillations around the Boltzmann result, which damp out on including dephasing.
The NEGF method was established in the 1960’s through the classic work of Schwinger, Kadanoff, Baym, Keldysh and others. But the viewpoint we have developed and advocated is somewhat different and unique.
See for example Lectures 18-23 from Lessons from Nanoelectronics (LNE), World Scientific (2012) * Download Lectures 19-20.
Our central equations (see Lecture 19 of LNE)
are essentially the same as Eqs.(75)-(77) of L. V. Keldysh, Sov. Phys. JETP, vol. 20, p. 1018 (1965), which is one of the seminal founding papers on the NEGF method that obtained these equations using many-body perturbation theory (MBPT) and established the “diagram technique for non-equilibrium processes” for calculating the self-energy functions denoted by the Greek letter sigma in Eqs.(19.1) and (19.2).
Much of the literature is based on the original MBPT-based approach and this makes it inaccessible to those unfamiliar with advanced quantum statistical mechanics.
We obtain Eqs.(19.1) and (19.2) directly from a one-electron Schrodinger Equation using relatively elementary arguments and use them to discuss many problems of great interest like quantized conductance, (integer) quantum Hall effect, Anderson localization, resonant tunneling and spin transport.
All these problems involve quantum transport but do not require a systematic treatment of many-body effects. On the other hand it goes beyond purely coherent transport allowing us to include phase-breaking interactions (both momentum-relaxing and momentum-conserving) within a self-consistent Born approximation.
Eqs.(19.1) and (19.2) provide a unified framework for such problems, spanning a wide range of materials and phenomena all the way from molecular to ballistic to diffusive transport and has been widely adopted by the nanoelectronics community for device analysis and design.
* The answer is NO, if we associate NEGF with the MBPT used to obtain the sigma’s appearing in (19.1) and (19.2).
* The answer is YES, if we associate NEGF with (19.1) and (19.2) irrespective of how the sigma’s are obtained.
Which answer we choose is clearly a matter of perspective, but the second viewpoint seems more in keeping with semiclassical transport theory, where the (steady-state) Boltzmann approach is identified with the equation:
and NOT with the evaluation of the scattering operator Sop which is analogous to the sigma’s in (19.1) and (19.2). Indeed Boltzmann himself was unaware of the Fermi’s golden rule commonly used nowadays to evaluate the Sop appearing in the equation bearing his name.
Similarly, totally new approaches for evaluating the sigma’s have been and will be developed as we apply NEGF to newer problems.
In other words, we feel that the scope and utility of Eqs.(19.1) and (19.2) transcends the MBPT-based approach originally used to derive it.
It teaches us how to combine quantum dynamics with “contacts”, much as Boltzmann taught us how to combine classical dynamics with “contacts”, using the word “contacts” in a broad figurative sense to denote all kinds of irreversible processes.
We feel that this should be a part of the training of all science and engineering students so that they can apply it effectively to a wide variety of basic and applied problems that require “connecting contacts to the Schrodinger equation“.