Lessons from Nanoelectronics

The following sections are included:

• Dedication
• Preface
• Acknowledgments
• List of Available Video Lectures Quantum Transport
• Constants Used in This Book
• Some Symbols Used
• Contents

“Everyone” has a smartphone these days, and each smartphone has more than a billion transistors, making transistors more numerous than anything else we could think of. Even the proverbial ants, I am told, have been vastly outnumbered…

Over a century ago Boltzmann taught us how to combine Newtonian mechanics with entropy-driven processes and the resulting Boltzmann transport equation (BTE) is widely accepted as the cornerstone of semiclassical transport theory. Most of the results we have discussed so far can be (and generally are) obtained from the Boltzmann equation, but the concept of an elastic resistor makes them more transparent by spatially separating force-driven processes in the channel from the entropy-driven processes in the contacts.

In this part of this book I would like to discuss the quantum version of this problem, using the non-equilibrium Green’s function (NEGF) method to combine quantum mechanics described by the Schrödinger equation with “contacts” much as Boltzmann taught us how to combine classical dynamics with “contacts”…

In the last chapter I tried to provide a super-brief but hopefully self-contained introduction to the Hamiltonian matrix H whose eigenvalues tell us the allowed energy levels in the channel. However, H describes an isolated channel and we cannot talk about the steady-state resistance of an isolated channel without bringing in the contacts and the battery connected across it. In this chapter, I will describe the NEGF-based transport model that can be used to model current flow, given H and the Σ’s (Fig. 18.1)…

In the next three chapters we will go through a few examples of increasing complexity which are interesting in their own right but have been chosen primarily as “do it yourself” problems that the reader can use to get familiar with the quantum transport model outlined in the last chapter. The MATLAB codes are all included in Appendix H.

In this chapter, we will use 1D quantum transport models to study an interesting question regarding multiple scatterers or obstacles along a conductor. Are we justified in neglecting all interference effects among them and assuming that electrons diffuse like classical particles as we do in the semiclassical picture?…

As I mentioned, our primary objective in Chapters 19–23 is to help the reader get familiar with the NEGF model through “do it yourself” examples of increasing complexity. The last chapter used 1D examples. In this chapter we look at 2D examples which illustrate one of the key results of mesoscopic physics, namely the observation of conductances that are an integer multiple of the conductance quantum q²/h.

Back in Chapter 17 we used this picture (Fig. 21.1) to summarize our NEGF model in which the channel is described by a Hamiltonian H while the self-energies Σ₁ and Σ₂ describe the exchange of electrons with the physical contacts. Σ₀ describes the interactions with the surroundings which can be viewed as additional conceptual “contacts”…

In the last chapter we have seen how to include inelastic interactions through the self-energy function Σ₀ in the simplest approximation, technically known as the self-consistent Born approximation. This is just a small subset of the interactions considered in the classic work on NEGF which provide clear prescriptions for including any microscopic interaction to any degree of approximation (see for example, Danielewicz, 1984 or Mahan, 1987). In practice, however, it remains numerically challenging to include interactions even in the lowest approximations. But practical issues apart, can the NEGF method model “everything”, at least in principle?

Back in Section 12.2 of Part A, we talked about the concept of spin potentials and how they can be generated in two ways, namely (Fig. 23.1)

• use of magnetic contacts with ordinary channels (Section 12.2.1),
• use of ordinary contacts with spin-momentum locked channels (Section 12.2.2)

In this book (Part B) we have introduced the formalism of quantum transport, while Part A was based on the semiclassical view that treats electrons primarily as particles, invoking its wave nature only to obtain the density of states, D(E) and the number of modes, M(E). The distinction between quantum and classical viewpoints is a much broader topic of general interest and we believe that spin transport provides a natural framework for understanding and exploring at least some aspects of it. Let me try to elaborate with a few thoughts along these lines.

Intel has a website, From Sand to Circuits (http://www.intel.com/about/companyinfo/museum/exhibits/sandtocircuits/index.htm) describing the amazing process that turns grains of sand into the chips that have enabled the modern information age. This book has been about the physics that these “grains of sand” and the associated technology have helped illuminate in the last 30 years, the physics that helped validate the concept of an elastic resistor with a clean separation of entropy-driven processes from the force-driven ones…

This book is based on a set of two online courses originally offered in 2012 on nanoHUB-U and more recently in 2015 on edX. These courses are now available in self-paced format at nanoHUB-U (https://nanohub.org/u) along with many other unique online courses…

The following sections are included:

• Appendix F List of Equations and Figures Cited From Part A
• Appendix G NEGF Equations
  • Self-energy for Contacts
  • Self-energy for Elastic Scatterers in Equilibrium:
  • Self-energy for Inelastic Scatterers
• Appendix H MATLAB Codes Used for Text Figures
  • Chapter 19
    • Fig. 19.2 Transmission through a single point scatterer in a 1D wire
    • Fig. 19.4 Normalized conductance for a wire with M = 1 due to one scatterer
    • Fig. 19.5 Normalized conductance for a wire with M = 1 due to six scatterers
    • Figs. 19.6–19.7 Potential drop across a scatterer calculated from NEGF
    • Figs. 19.8–19.9 Potential drop across two scatterers in series calculated from NEGF
  • Chapter 20
    • Fig. 20.1 Numerically computed transmission as a function of energy
    • Fig. 20.3 Transmission calculated from NEGF for ballistic graphene sheet and CNT
    • Fig. 20.4 Normalized Hall resistance versus B-field for ballistic channel
    • Fig. 20.5 Grayscale plot of local density of states
  • Chapter 22
    • Fig. 22.7, n versus μ, single dot
    • Fig. 22.8, I versus V, single quantum dot
    • Fig. 22.9, n versus μ, double quantum dot
  • Chapter 23
    • Fig. 23.9 Voltage probe signal as the magnetization of the probe is rotated
    • Fig. 23.10 Voltage probe signal due to variation of gate voltage controlled Rashba coefficient
• Appendix I Table of Contents of Part A: Basic Concepts
• Appendix J Available Video Lectures for Part A: Basic Concepts

The following section is included:

• Index