In Chapter 1, I used the generic structure shown in Fig. 11.1.1a (see Section 11.1.1) to focus and motivate this book. We spent Chapters 2 through 7 understanding how to write down a Hamiltonian matrix [H0] for the active region of the transistor structure whose eigenvalues describe the allowed energy levels (see Fig. 11.1.1b). In Chapter 8, I introduced the broadening [Г1] and [Г2] arising from the connection to the source and drain contacts. In Chapter 9, I introduced the concepts needed to model the flow of electrons, neglecting phase-breaking processes. In Chapter 10 we discussed the nature and meaning of phase-breaking processes, and how the resulting inflow and outflow of electrons is incorporated into a transport model. We now have the full “machinery” needed to describe dissipative quantum transport within the self-consistent field model (discussed in Chapter 3) which treats each electron as an independent particle moving in an average potential U due to the other electrons. I should mention, however, that this independent electron model misses what are referred to as “strong correlation effects” (see Section 1.5) which are still poorly understood. To what extent such effects can be incorporated into this model remains to be explored (see Appendix, Section A.5).

My purpose in this chapter is to summarize the machinery we have developed (Section 11.1) and illustrate how it is applied to concrete problems. I believe these examples will be useful as a starting point for readers who wish to use it to solve other problems of their own.

A tale of many electrons

The twentieth century has witnessed some of the greatest revolutions in the history of science: in physics we have gone from classical to quantum mechanics, our views of space and time have forever been changed by the theory of relativity, a comprehensive microscopic theory of matter has emerged, the invention of solid state electronics has ushered in the information age. Through all these changes one thing has always kept the center stage: the electron. Discovered by J. J. Thomson in 1897, the electron was the first elementary particle to be clearly identified. Although the discovery took place in an entirely classical context, further investigation of the properties of the electron soon led the way into the new world of quantum mechanics. In particular, the Davisson–Germer electron diffraction experiment (1927) established the wave-like properties of matter particles; the discovery of the half-integer spin (Goudsmit and Uhlenbeck, 1925) and the related statistical properties of the electron (Pauli, 1925; Fermi, 1925; Dirac, 1929) laid the foundation for the understanding of the atomic structure; Dirac's relativistic treatment of the electron (Dirac, 1928) created a new branch of theoretical physics: quantum field theory.

It was clear from the beginning that electrons are a pervasive component of matter. An electrical current in a metal wire is nothing but a flow of many electrons: this is the basic assumption of the classic Drude–Lorentz model of electrical conduction in metals, which was proposed as early as 1900. In 1911, however, Kamerlingh Onnes astonished the world (and himself) by showing that a metal (Hg) cooled below liquid helium temperature lost all electrical resistance and became a perfect conductor.