Page 120 - Electrical Properties of Materials

102 The band theory of solids

for example, an early triumph of engineering to turn the energy of steam
into a steam engine but, having accomplished the feat, most people could
comprehend that the expanding steam moved a piston, which was connected
to a wheel, etc. It needed perhaps a little more abstract thought to appreciate
Watt’s invention of the separate condenser, but even then any intelligent man
willing to devote half an hour of his time to the problem of heat exchange
could realize the advantages. Alas, these times have gone. No longer can a
layman hope to understand the working principles of complex mechanisms,
and this is particularly true in electronics. And most unfortunately, it is true not
only for the layman. Even electronic engineers find it hard nowadays to follow
the phenomena in an electronic device. Engineers may nowadays be expected
to reach for the keyboard of a computer at the slightest provocation, but the
fundamental equations are still far too complicated for a direct numerical
attack. We need models. The models need not be simple ones, but they should
be comprehensive and valid under a wide range of conditions. They have
to serve as a basis for intuition. Such a model and the concurrent physical
picture are provided by the band theory of solids. It may be said without undue
exaggeration that the spectacular advance in solid-state electronic devices in
the second half of the twentieth century owed its existence to the power and
simplicity of the band theory of solids.
Well, after this rather lengthy introduction, let us see what this theory is
about. There are several elementary derivations, each one giving a slightly dif-
ferent physical picture. Since our aim is a thorough understanding of the basic
ideas involved it is probably best to show you the three approaches I know.

7.2 The Kronig–Penney model
This model is historically the first (1930) and is concerned with the solution
of Schrödinger’s equation, assuming a certain potential distribution inside the
solid. According to the free-electron model, the potential inside the solid is uni-
form; the Kronig–Penney model, goes one step further by taking into account
the variation of potential due to the presence of immobile lattice ions.
If we consider a one-dimensional case for simplicity, the potential energy
of an electron is shown in Fig. 7.1. The highest potential is halfway between
the ions, and the potential tends to minus infinity as the position of the ions is
approached. This potential distribution is still very complicated for a straight-
The ions are located at x =0, a, forward mathematical solution. We shall, therefore, replace it by a simpler one,
2a, ... etc. The potential wells are which still displays the essential features of the function, namely (i) it has the
separated from each other by po- same period as the lattice; (ii) the potential is lower in the vicinity of the lattice
tential barriers of height V 0 , and ion and higher between the ions. The potential distribution chosen is shown
width w. in Fig. 7.2.
The solution of the time-independent Schrödinger equation

2
2
d ψ
+{E – V(x)}ψ =0 (7.1)
2m dx 2
for the above chosen potential distribution is not too difficult. We can solve it
for the V(x)= V 0 /2 and V(x)=–V 0 /2 regions separately, match the solutions
at the boundaries, and take good care that the solution is periodic. It is all