ECE

 561-­‐002
Frequency   Variation  in  Materials
Prof.  Schamiloglu
Spring  2016
Consider the effect of ac electric fields on a dielectric. The polarization electric field discussed
earlier now has a frequency dependence given as

P(ω) = ϵ0 χe (ω)Ea (ω),

where the electrical susceptiblity is a complex quantity given as

χe (ω) = χ′e (ω) − jχ′′e (ω)

and can be due to all three polarization mechanisms, depending on the complexity of the
medium. It should be noted that, in general,

χ′e (−ω) = χ′e (w),

and
χ′′e (−ω) = −χ′′e (w).
 The real and imaginary parts of the complex susceptibilities are plotted in Fig. 5.1. This
sketch does not represent one material, but rather a conglomeration of various materials.
Most of these measurements were performed by von Hippel at MIT in the early 1950’s.
We know that the dielectric constant is related to the susceptibility through

ϵ̇ = 1 + χ̇e .

We would therefore expect that the dielectric constant should exhibit a similar frequency
dependence for its real and imaginary components. Figure 5.2 presents plots of the real
and imagninary parts of the dielectric constant for two sets of parameters. The important
points to consider from this figure is that peaks are observed for ϵ′′r at various frequencies
called resonant frequencies. In other words, the material achieves its most lossy state at the
resonant frequency. Some materials would have multiple resonant frequencies. When one is
away from the resonant frequency, the curves for |ϵ̇r | possess a positive slope. This is called
normal dispersion since it is the most common. However, in the vicinity of the resonances,
|ϵ̇r | exhibits a negative slope. This is called anomalous dispersion, although there is nothing
truly unusual of this.
 As an example of the frequency behavior of the real part of the index of refraction and the
absorption coefficient, consider the familiar substance, liquid water. The index of refraction
and the absorption coefficient over a wide frequency range is plotted in Fig. 5.3, which was
reproduced from Jackson’s book. The frequency dependence of these quantities is shown
on a log-log plot spanning 20 decades in frequency and 11 decades in absorption. At very
low frequencies, the index of refraction is about 9, caused by the partial orientation of the
permanent dipole moments of the polar molecules. Above 10 GHz, the curve falls smoothly
to the structure in the infrared region. In the visible, the index of refraction is constant at
about 1.34. Above about 6 × 1015 Hz (hν ≃ 25 eV), data are not available.
The absorption coefficient has a more dramatic dependence on frequency. At frequencies
below about 0.1 GHz the absorption is very small. As the frequency increases to about
100 GHz, the absorption coefficient rises rapidly. This corresponds to an attenuation length
of about 100 µm. This is the well-known microwave absorption by water vapor. It is this
reason that terminated the trend during World War II towards achieving better resolution
by going to shorter wavelengths. In the infrared region, absorption bands that are associated
with vibrational modes of the molecule causes the coefficient to peak. Then the absorption
coefficient falls abrubtly and steadily over 7 decades in a narrow frequency band of 4 − 8 ×
1014 Hz. This is the dramatic absorption window in the visible region. This transparency of
water is related to the basic energy level structure of the atoms and molecules. In the very
far ultraviolet the absorption peak reaches its maximum at a frequency 5 × 1015 Hz (21 eV).
This is precisely the plasmon energy h̄ωp , corresponding to a collective excitation of all the
electrons in the molecule.
Figure  1.  Real  and  imaginary  parts  of   the  electric   susceptibility   as   a   function   of  
frequency  (from   Balanis).
Figure  2.  Frequency  variations   of   the  real   and  imaginary   parts  of  the  complex  
dielectric   constant   for  various  parameters   (from   Balanis).
Figure  3.  Index  of  refraction  and  absorption   coefficent of  liquid   water   (from  
Jackson).  Real  part.
Figure  4.  Index  of  refraction  and  absorption   coefficent of  liquid   water   (from  
Jackson).  Imaginary  part.
Kramers-Kronig Relations
A fact that was not stressed earlier is that in our simplistic model of the bulk dielectric, in
which we assumed the dielectric is comprised of Ne dipole moments per unit volume, the
material was assumed to possess one resonant frequency. In general, however, an atom may
possess many natural frequecies. This can be modeled by assuming that there are many
independent oscillators comprising the material. Therefore, assuming that the material is
comprised of p independent oscillators,
p
P  Ne (Q2 /m)
= .
E s=1
(ω 2 − ω 2 ) + jω(d/m)
s

This is the defining relationship for the complex permittivity, and we obtain
p
 Ne (Q2 /m)
ϵ̇ = ϵ − jϵ = ϵ0 +
′ ′′
.
s=1
(ωs2 − ω 2 ) + jω(d/m)

Let us remove the permittivity of free space and write an expression for the complex dielectric
constant:
p
 Ne (Q2 /m)
ϵ̇r = ϵr − jϵr = 1 +
′ ′′
.
s=1
(ωs2 − ω 2 ) + jω(d/m)

Once again, we can obtain expressions for the real and imaginary components of the complex
dielectric constant by multiplying the numerator and denominator of this expression by the
complex conjugate of the denominator. The resulting expressions are
p
 (Ne Q2 /ϵ0 m)(ω 2 − ω 2 )
ϵ′r =1+ s
,
s=1
(ωs2 − ω 2 )2 + (ωd/m)2

and
Ne Q2  
p
 (ωd/m)
ϵ′′r = .
s=1
ϵ0 m (ωs2 − ω 2 )2 + (ωd/m)2
 There is an important relationship between the real and the imaginary parts of the
complex dielectric constant, given by:
 ∞
1 ϵ′′r (ω ′ )
ϵ′r (ω) = 1+ P dω ′ ,
π −∞ ω′ − ω

and  ∞
−1 ϵ′r (ω ′ ) − 1
ϵ′′r (ω) = P dω ′ .
π −∞ ω′ − ω

The letter P before the integrand denotes principal part. These relations, or the ones written
below, are called the Kramers-Kronig relations, or dispersion relations. They were first
derived by H. A. Kramers (1927) and R. de L. Kronig (1926) independently. The symmetry
property of the complex susceptibilty, mentioned earlier, shows that ϵ′r is an even function
of ω, whereas ϵ′′r is an odd function. The integrals above can be transformed to span only
positive frequencies, resulting in
 ∞
2 ω ′ ϵ′′r (ω ′ )
ϵ′r (ω) =1+ P dω ′ ,
π 0 ω ′2 − ω 2

and  ∞
−2ω ϵ′r (ω ′ ) − 1
ϵ′′r (ω) = P dω ′ .
π 0 ω ′2 − ω 2

(This is the form of the equations presented in Balanis.) In writing these expressions, we
have assumed that ϵ̇(ω) was regular at ω = 0.
The Kramers-Kronig relations are very general. They follow from the assumption of a
causal relationship between the polarization and electric fields. Empirical knowledge of ϵ′′r
from absorption experiments allows the calculation of ϵ′r from the integrals above.
In addition to the Kramers-Kronig relations, there are other simple relations that allow
the calculation of the real and imaginary parts of the complex relative permittivity for many
materials as a function of frequency. These are obtained using the Debye equation, though
we will not mention this any further.