1000 Solved Problems in Modern Physics

5.1 Basic Concepts and Formulae 293

Metals, insulators and semiconductors

Materials are distinguished by the extent to which the valence and conduction bands

are filled by electrons. The bands in solids may be filled, partially or empty. A good

conductor has a conduction band that is approximately half filled or the conduction

band overlaps the next higher band. In this case it is very easy for the valence elec-

tron to be raised to a higher energy level under the application of electric field and

provide electrical conduction.

In an insulator the valency band is completely filled and the energy gap (Eg) with

the conduction gap is large (∼5eV).

In the case of semiconductors the valence band is completely filled, like an insu-

lator. However, the conduction band is empty, so that at room temperature some

of the electrons acquire sufficient energy to be found in the conduction band. Fur-

thermore, the electrons leave behind unfilled “holes” into which other electrons in

the valence band can move in the electrical conduction regime. The excitation of

electron into these holes has the net effect of positive charge carriers aiding the

electrical conduction. Such semiconductors are known as intrinsic semiconductors.

However, with the introduction of certain impurities into a material in a controlled

way, a procedure known as doping conduction is dramatically increased. Such doped

semiconductors are known as extrinsic semiconductors, on which are based numer-

ous semiconductor devices. If the majority charge carriers are electrons, the material

is called an n-type semiconductor and if the holes are the majority charge carrier the

material is called a p-type semiconductor.

The Fermi energyEFlies in the middle of the energy gap.

The mobility of charge carriers is defined as

μ=vd/E (5.15)

The conductivityσhas two contributions, one from the electrons and the other

from the holes.

σ=nneμn+npeμp (5.16)

n=σ/eμ (5.17)

τ=μm/e (5.18)

Superconductivity
Some materials when cooled below a certain temperature, called critical temperature
(Tc), have zero resistance. The material is said to be a superconductor.Tcvaries from
one superconductor to another.
When a superconductor is placed in a magnetic field,Tcdecreases with the
increasingB. WhenBis increased beyond a critical magnetic fieldBc, the super-
conductivity will not take place no matter how low the temperature.
Tc(B)=Tc0(1–B/Bc)^1 /^2 (5.19)
whereTc0is the critical temperature with zero magnetic field, andBis the applied
field.