Page 35 - Electrical Properties of Materials

18 The electron as a particle

to the light line, i.e. its velocity tends to the velocity of light. The wave is of
Metal Air course not a pure electromagnetic wave, which would always propagate with
the velocity of light. It is a combination of a plasma wave and an electromag-
netic wave. One might call it a hybrid wave. In fact the modern term is much
Amplitude more pompous. It is called a bulk plasma wave or, even worse, a bulk plasmon
polariton. The word ‘polariton’ is attached to it to signify that it is a hybrid
wave. But why a bulk plasmon polariton? Because there is another variety as
well, called a surface plasma wave or surface plasmon polariton. Such a wave,
needless to say, is called a surface wave because it sticks to a surface. What
kind of surface? The best example of such a wave, and the one relevant here,
is a wave at the interface of a metal and a dielectric, say air. If the wave sticks
Fig. 1.9 to the surface, its amplitude must decline in both directions, both in the metal
A surface wave may exist at a and in air, as shown schematically in Fig. 1.9. One could say that it is the elec-
metal–air boundary. The amplitude of tric field that acts as the glue, sticking to charges in the metal as illustrated in
thewaveishighest at thesurface, Fig. 1.10. We shall not derive the dispersion equation here. It is a fairly long
from where it declines exponentially derivation. We just give here the equation itself, which is quite simple:
in both directions.

ε eff 1/2
k = ω(ε 0 μ 0 ) 1/2 , (1.72)
1+ ε eff
where μ 0 =4π × 10 –7 Hm –1 is the free-space permeability and

(ε 0 μ 0 ) –1/2 = c. (1.73)

For propagation, k must be real. This occurs when

ε eff < –1. (1.74)

Conveniently, as discussed in the previous section, the effective dielectric con-
stant of a metal is negative below the plasma frequency. The limit is when
ε eff = –1. Below this frequency (see eqn (1.53)), ε eff declines further so that
eqn (1.72) always yields a real value and, consequently, a surface wave can al-
ways exist. Substituting eqn (1.53) into (1.72), we find the dispersion equation.
The corresponding dispersion curve is shown in Fig. 1.8. The wave is what one
calls a slow wave since it is to the right of the light line, having a phase velocity
always below that of light.

Metal E

H
.
Fig. 1.10
Electric field lines for a surface
plasma wave in the vicinity of a
Dielectric
metal–air boundary.