Page 257 - Electrical Properties of Materials

Acoustic waves 239

where J I is the time-varying ionic current density, equal to
dx T
with ν T = . (10.59)
J I = eN o ν T
dt
One further equation needed is Poisson’s equation, which relates the electric
field to the net charge density,
dE
=–en I . (10.60)
ε ∞
dx
With our usual wave assumption exp[–i(ωt–kx)] we find, after a certain amount
of algebra,
J I = σE, (10.61)
where
2
iωε ∞ ω 2 I 2 e N I
σ = 2 and ω = . (10.62)
I
ω – ω 2 m ε ∞
∗
T
The effective permittivity due to lattice wave interaction with electromagnetic
waves may be worked out from the relationship
J I –iωε ∞ E =(σ –iωε ∞ )E =–iωε eff E, (10.63)
whence
2
2
2
σ ε ∞ (ω – ω – ω )
I
T
ε eff = ε ∞ – = . (10.64)
2
iω ω – ω 2
T
The usual notation is
2
2
2
ω = ω + ω , (10.65)
L T I
and in the usual terminology ω L is the longitudinal optical phonon frequency
and ω T is the transverse optical phonon frequency; these are related to each
other by

2
2
ω = ε ∞ ω . (10.66)
T
L
ε s
The final form for the effective dielectric constant is then
2
2
ε ∞ (ω – ω )
L
ε eff = . (10.67)
2
ω – ω 2
T
It may now be seen that the effective dielectric constant is negative in the range
ω T <ω <ω L . (10.68)
Optical phonons have been of only moderate interest in the past. This ∗ This is a classical phenomenon which
∗
might change in view of the advent of the new subject of metamaterials. In should be described as the optical branch
of acoustic waves. Alas, the quantum-
fact, in Section 15.9 (see Chapter 15) we shall be greatly interested in negative
mechanical term has now been widely
material parameters such as negative permittivity and negative permeability. In accepted.
that section we shall discuss a device in which a negative dielectric constant
due to the mechanism discussed above is used in a novel type of microscope.