Page 99 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
• Basic quantum mechanics;
• The semiconductor electronic system, and the band structure of sili-
con.
Chapter We start with the free electron, introducing the Schrödinger equation.
Roadmap Next, the electron is bound by a variety of potentials. The hydrogen atom
represents the simplest quantum-mechanical model of an atom, with
good predictive qualities. This leads up to the periodic potential, a model
for the periodically-placed crystal atom potentials. This model naturally
leads to the concepts of a forbidden band, and band splitting. The chapter
wraps up with and expression for the effective mass of the bound elec-
trons.
In this chapter we take some space to explain the necessary mathematical
instruments and physical concepts to understand the quantum nature of
phenomena described in the book. You are strongly encouraged to care-
fully study the mathematical manipulations of wavefunctions, operators
and all the objects described in this chapter. Moreover, you should freely
work with all the objects and practise to manipulate them. Most of the
time you will gain a deeper understanding, and of course there is no risk
of doing any harm to them. The worst that can happen is that you might
not find any meaningful physical interpretation for what you did. For a
detailed understanding of specific topics special literature is given in the
references.

3.1 Quantum Mechanics of Single Electrons

Quantum mechanics describes the fundamental properties of electrons
[3.1], [3.2]. It tells us that both particle-like and wave-like behavior is
possible. In a semiconductor both types of behavior are observable. The
particle concept turns out to be an excellent description for a wide range
of classical phenomena and the most common applications. Modern
devices on the nanometer scale instead demonstrate the quantum nature

96 Semiconductors for Micro and Nanosystem Technology