Page 21 - Electrical Properties of Materials

4 The electron as a particle

Since v D varies with electric field, τ must also vary with the field unless
v th v D . (1.12)

As Ohm’s law is accurately true for most metals, this inequality should hold.
In a typical metal μ e =5 × The thermal velocity at room temperature according to eqn (1.1) (which
–1 –1
–3
2
10 m V s , which gives a actually gives too low a value for metals) is
–3
drift velocity v D of 5 × 10 ms –1
1/2
–1
for an electric field of 1 V m . 3k B T ~ 5 –1
v th = = 10 ms . (1.13)
m
Thus, there will be a constant relationship between current and electric field
8 ∗
∗ This is less true for semiconductors as accurate to about 1 part in 10 .
they violate Ohm’s law at high electric This important consideration can be emphasized in another way. Let us draw
fields.
the graph (Fig. 1.2) of the distribution of particles in velocity space, that is with
rectilinear axes representing velocities in three dimensions, v x , v y , v z . With no
electric field present, the distribution is spherically symmetric about the origin.
v x The surface of a sphere of radius v th represents all electrons moving in all
possible directions with that r.m.s. speed. When a field is applied along the
x-axis (say), the distribution is minutely perturbed (the electrons acquire some
additional velocity in the direction of the x-axis) so that its centre shifts from
8
(0, 0, 0) to about (v th /10 ,0,0).
v –1 8 –2
th Taking copper, a field of 1 V m causes a current density of 10 Am .Itis
quite remarkable that a current density of this magnitude can be achieved with
v x an almost negligible perturbation of the electron velocity distribution.
v y 1.3 The hydrodynamic model of electron flow

Fig. 1.2 By considering the flow of a charged fluid, a sophisticated model may be de-
Distributions of electrons in velocity veloped. We shall use it only in its crudest form, which does not give much of
space. a physical picture but leads quickly to the desired result.
The equation of motion for an electron is
dv
m = eE . (1.14)
dt
If we now assume that the electron moves in a viscous medium, then the
forces trying to change the momentum will be resisted. We may account for
this by adding a ‘momentum-destroying’ term, proportional to v. Taking the
ζ may be regarded here as a meas- proportionality constant as ζ eqn (1.14) modifies to
ure of the viscosity of the medium.

dv
m + ζv = eE . (1.15)
dt
In the limit, when viscosity dominates, the term dv/dt becomes negligible,
resulting in the equation

mvζ = eE , (1.16)
which gives for the velocity of the electron
e 1
v = E . (1.17)
m ζ