Page 20 - Electrical Properties of Materials

The effect of an electric field—conductivity and Ohm’s law 3

The average time between collisions, τ, has many other names; for example,
mean free time, relaxation time, and collision time. Similarly, the average
velocity is often referred to as the mean velocity or drift velocity. We shall
call them ‘collision time’ and ‘drift velocity’, denoting the latter by v D .
The relationship between drift velocity and electric field may be obtained
from eqns (1.3) and (1.5), yielding
e

v D = τ E , (1.6)
m
where the proportionality constant in parentheses is called the ‘mobility’. This
is the only name it has, and it is quite a logical one. The higher the mobility, the more
Assuming now that all electrons drift with their drift velocity, the total num- mobile the electrons.
ber of electrons crossing a plane of unit area per second may be obtained
by multiplying the drift velocity by the density of electrons, N e . Multiplying
further by the charge on the electron we obtain the electric current density

J = N e ev D . (1.7)

Notice that it is only the drift velocity, created by the electric field, that
comes into the expression. The random velocities do not contribute to the
electric current because they average out to zero. ∗ ∗ They give rise, however, to electrical
We can derive similarly the relationship between current density and electric noise in a conductor. Its value is usu-
ally much smaller than the signals we
field from eqns (1.6) and (1.7) in the form
are concerned with so we shall not worry
about it, although some of the most in-
2
N e e τ teresting engineering problems arise just
J = E . (1.8) when signal and noise are comparable.
m
see Section 1.8 on noise.
This is a linear relationship which you may recognize as Ohm’s law
J = σE , (1.9)

where σ is the electrical conductivity. When first learning about electricity you
looked upon σ as a bulk constant; now you can see what it comprises of. We
can write it in the form In metals, incidentally, the mobil-
ities are quite low, about two or-
e ders of magnitude below those of

σ = τ (N e e)
m semiconductors; so their high con-
= μ e (N e e). (1.10) ductivity is due to the high density
of electrons.
That is, we may regard conductivity as the product of two factors, charge
density (N e e) and mobility (μ e ). Thus, we may have high conductivities be-
cause there are lots of electrons around or because they can acquire high drift
velocities, by having high mobilities.
Ohm’s law further implies that σ is a constant, which means that τ must
†
be independent of electric field. From our model so far it is more reasonable † It seems reasonable at this stage to as-
to assume that l, the distance between collisions (usually called the mean free sume that the charge and mass of the
path) in the regularly spaced lattice, rather than τ, is independent of electric electron and the number of electrons
present will be independent of the elec-
field. But l must be related to τ by the relationship,
tric field.
l = τ(v th + v D ). (1.11)