## Lessons from Nanoelectronics

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### Teaching NEGF

## Note

In the Second Edition (2017) the book was reorganized to correspond more closely to the two online courses. The 2017 Edition also split the book into Part A (Chapters 1-16) and Part B (Chapters 17-25).

These Q&A were written in 2013 based on the First Edition (2012).

Here are the corresponding chapters in the 2017 Edition (2012 Edition):

Part A: 1-4 (1-4), 5 (15), 6 (5), 7 (8), 8-9 (6-7), 10 (12 and 14), 11 (13), 12 (9), 13-14 (10-11), 15-16 (16-17).

Part B : 17-20 (18-21), 22 (23), 23 (22), 24 (24).

Chapters 21 and 25 of 2017 Edition are new.

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 discussed in my BOOKS

Chapter 8 of Electronic Transport of Mesoscopic Systems, Cambridge (1995) —> ETMS

Chapters 8-11 of Quantum Transport: Atom to Transistor, Cambridge (2005) —> QTAT

Lectures 18-23 of Lessons from Nanoelectronics, World Scientific (2012) —> LNE

## How can we teach NEGF without requiring advanced quantum statistical mechanics?

### Central equations

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. Our central equations (see Lecture 19 of LNE * Download Lectures 19-20)

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.

### Our approach

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.

### But is this the “real NEGF” ?

* 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 S_{op} 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 S_{op} 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.

### Contact-ing Schrodinger

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*“.