Page 23 - Electrical Properties of Materials

6 The electron as a particle

What can we say about the direction of E H ? Well, we have taken meticu-
lous care to find the correct direction. Once the polarity of the applied voltage
and the direction of the magnetic field are chosen, the electric field is well and
truly defined. So if we put into our measuring apparatus one conductor after
the other, the measured transverse voltage should always have the same po-
larity. Yes . . . the logic seems unassailable. Unfortunately, the experimental
facts do not conform. For some conductors and semiconductors the measured
transverse voltage is in the other direction.
How could we account for the different sign? One possible way of explain-
ing the phenomenon is to say that in certain conductors (and semiconductors)
electricity is carried by positively charged particles. Where do they come from?
We shall discuss this problem in more detail some time later; for the moment
just accept that mobile positive particles may exist in a solid. They bear the
unpretentious name ‘holes’.
To incorporate holes in our model is not at all difficult. There are now two
species of charge carriers bouncing around, which you may imagine as a mix-
ture of two gases. Take good care that the net charge density is zero, and the
new model is ready. It is actually quite a good model. Whenever you come
across a new phenomenon, try this model first. It might work.
Returning to the Hall effect, you may now appreciate that the experimental
determination of R H is of considerable importance. If only one type of carrier
is present, the measurement will give us immediately the sign and the density
of the carrier. If both carriers are simultaneously present it still gives useful
information but the physics is a little more complicated (see Exercises 1.7
and 1.8).
In our previous example we took a typical metal where conduction takes
–1
–3
place by electrons only, and we got a drift velocity of 5 × 10 ms .For a
The corresponding electric field in magnetic field of 1 T the transverse electric field is
a semiconductor is considerably –3 –1
E H = Bv =5 × 10 Vm . (1.21)
higher because of the higher mo-
bilities.
1.5 Electromagnetic waves in solids
So far as the propagation of electromagnetic waves is concerned, our model
works very well indeed. All we need to assume is that our holes and electrons
obey the equations of motion, and when they move, they give rise to fields in
accordance with Maxwell’s theory of electrodynamics.
It is perfectly simple to take holes into account, but the equations, with
holes included, would be considerably longer, so we shall confine our attention
to electrons.
We could start immediately with the equation of motion for electrons, but
let us first review what you already know about wave propagation in a me-
dium characterized by the constants permeability, μ, dielectric constant, , and
conductivity, σ (it will not be a waste of time).
First of all we shall need Maxwell’s equations:
1 ∂E E E
∇ × B = J + , (1.22)
μ ∂t
∂B
∇ × E E E =– . (1.23)
∂t