Page 33 - Electrical Properties of Materials

16 The electron as a particle
∗ different. The actual value can be measured directly with the aid of cyclotron
The actual value is called, quite reason-
ably, the effective mass.
The charge of the electron is a fun- grateful for small favours.
damental constant in a solid; the
mass of an electron is not.
aid of cyclotron resonance, where by clever means the sign of attenuation
is reversed, turning it into gain. As far as I know none of these devices
reached the ultimate glory of commercial exploitation. If cyclotron resonance
is no good for devices, is it good for something else? Yes, it is an excellent
measurement tool.
It is used as follows: we take a sample, put it in a waveguide and launch
an electromagnetic wave of frequency, ω. Then we apply a magnetic field and
measure the amplitude of the output electromagnetic wave while the strength
of magnetic field is varied. When the output is a minimum, the condition of
cyclotron resonance is satisfied. We know ω so we know ω c ; we know the
value of the magnetic field, B 0 so we can work out the mass of the electron
from the formula
eB 0
m = . (1.64)
ω
But, you would say, what is the point in working out the mass of the elec-
tron? That’s a fundamental constant, isn’t it? Well, it is, but not in the present
context. When we put our electron in a crystal lattice, its mass will appear to be
∗
resonance. So once more, under the pressure of experimental results we have
to modify our model. The bouncing billiard balls have variable mass. Luckily,
the charge of the electron does remain a fundamental constant. We must be
1.7 Plasma waves
Electromagnetic waves are not the only type of waves that can propagate in a
solid. The most prominent ones are sound waves and plasma waves. We know
about sound waves; but what are plasma waves? In their simplest form they are
density waves of charged particles in an electrically neutral medium. So they
exist in a solid that has some mobile carriers. The main difference between this
case and the previously considered electromagnetic case is that now we permit
the accumulation of space charge. At a certain point in space, the local density
of electrons may exceed the local density of positive carriers. Then an electric
field arises, owing to the repulsive forces between these ‘unneutralized’ elec-
trons. The electric field tries to restore the equilibrium of positive and negative
charges. It drives the electrons away from the regions where they accumulated.
The result is, of course, that the electrons overshoot the mark, and some time
later, there will be a deficiency of electrons in the same region. An opposite
electric field is then created which tries to draw back the electrons, etc. This is
the usual case of harmonic oscillation. Thus, as far as an individual electron is
concerned, it performs simple harmonic motion.
If we consider a one-dimensional model again, where everything is the same
in the transverse plane, then the resulting electric field has a longitudinal com-
ponent only. A glance at eqn (1.26), where ∇×E E E is worked out, will convince
you that if the electric field has a z-component, only then ∇×E E E = 0, that is
B = 0. There is no magnetic field present; the interplay is solely between the
charges and the electric field. For this reason these density waves are often
referred to as electrostatic waves.