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The Solid State 369

Positive ions

U

0

a

Figure 10.39The potential energy of an electron in a periodic array of positive ions.

depleted region in the channel and the fewer the electrons available to carry the cur-

rent. Thus the gate voltage controls the channel current. Very little current passes

through the gate circuit owing to the reverse bias, and the result is an extremely high

input impedance.

Even higher input impedances (up to 10^15 ) together with greater ease of manu-

facture are characteristic of the metal-oxide-semiconductor FET (MOSFET), a FET in

which the semiconductor gate is replaced by a metal film separated from the channel

by an insulating layer of silicon dioxide. The metal film is thus capacitively coupled to

the channel, and its potential controls the drain current through the number of induced

charges in the channel. A MOSFET occupies only a few percent of the area needed for

a junction transistor.

10.8 ENERGY BANDS: ALTERNATIVE ANALYSIS

How the periodicity of a crystal lattice leads to allowed and

forbidden bands

A very different approach can be taken to the origin of energy bands from that de-

scribed in Sec. 10.6. There we saw that bringing together isolated atoms to form a solid

has the effect of broadening their energy levels into bands of allowed electron energies.

Alternatively we can start with the idea that an electron in a crystal moves in a region

of periodically varying potential (Fig. 10.39) rather than one of constant potential. As

a result diffraction effects occur that limit the electron to certain ranges of momenta

that correspond to allowed energy bands. In this way of thinking, the interactions

among the atoms influence the behavior of their valence electrons indirectly through

the crystal lattice these interactions bring about, rather than directly through the atomic

interactions themselves. An intuitive approach will be used here to bring out more

clearly the physics of the situation, instead of a formal treatment based on Schrödinger’s

equation.

The de Broglie wavelength of a free electron of momentum pis

Free electron  (10.17)

Unbound low-energy electrons can travel freely through a crystal since their wave-

lengths are long relative to the lattice spacing a. More energetic electrons, such as those

with the Fermi energy in a metal, have wavelengths comparable with a, and such elec-

trons are diffracted in precisely the same way as x-rays (Sec. 2.6) or electrons in a beam

h

p

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