Page 19 - Electrical Properties of Materials

2 The electron as a particle

interatomic spaces, colliding occasionally with lattice atoms. We may even go
further with this analogy and claim that in equilibrium the electrons follow the
same statistical distribution as gas molecules (that is, the Maxwell–Boltzmann
distribution) which depends strongly on the temperature of the system. The
1
average kinetic energy of each degree of freedom is then k B T where T is
2
absolute temperature and k B is Boltzmann’s constant. So we may say that the
∗ We shall see later that this is not so for mean thermal velocity of electrons is given by the formula ∗
metals but it is nearly true for conduction
electrons in semiconductors. 1 2 3
th
2 mv = k B T (1.1)
2
v th is the thermal velocity, and m is because particles moving in three dimensions have three degrees of freedom.
the mass of the electron. We shall now calculate some observable quantities on the basis of this
simplest model and see how the results compare with experiment. The success
of this simple model is somewhat surprising, but we shall see as we proceed
that viewing a solid, or at least a metal, as a fixed lattice of positive ions held
together by a jelly-like mass of electrons approximates well to the modern view
of the electronic structure of solids. Some books discuss mechanical properties
in terms of dislocations that can move and spread; the solid is then pictured as a
fixed distribution of negative charge in which the lattice ions can move. These
views are almost identical; only the external stimuli are different.

1.2 The effect of an electric field—conductivity and Ohm’s law

Suppose a potential difference U is applied between the two ends of a solid
length L. Then an electric field

U
E = (1.2)
L
is present at every point in the solid, causing an acceleration
e
a = E . (1.3)
m
Thus, the electrons, in addition to their random velocities, will acquire a ve-
locity in the direction of the electric field. We may assume that this directed
velocity is completely lost after each collision, because an electron is much
lighter than a lattice atom. Thus, only the part of this velocity that is picked up
in between collisions counts. If we write τ for the average time between two
collisions, the final velocity of the electron will be aτ and the average velocity

1
v average = aτ. (1.4)
2
This is simple enough but not quite correct. We should not use the average
time between collisions to calculate the average velocity but the actual times
and then the average. The correct derivation is fairly lengthy, but all it gives
†
† See, for example, W. Shockley, Elec- is a factor of 2. Numerical factors like 2 or 3 or π are generally not worth
trons and holes in semiconductors,D. worrying about in simple models, but just to agree with the formulae generally
van Nostrand, New York, 1950, pp. quoted in the literature, we shall incorporate that factor 2, and use
191–5.
v average = aτ. (1.5)