-
Fundamentals of Optical Science
–
OSE 5312 Fall 2003
Tuesday, September 9, 2003
Some comments on the
applicability of the Lorentz model to real
materials:
(a) Insulators
The Lorentz model works surprisingly
well, provided we remember that real
materials correspond to a collection of
Lorentz oscillators with different
frequencies. The outer, or valence,
electrons predominantly determine the
characteristics of the optical
properties a solid. In an ionically
–
bonded material,
e.g. alkali-halides such as KCl, the
valence electrons are quite strongly localized at
the negative ion (for KCl, this would
be the Cl atom), and hence the optical
spectrum contains some atomic-like
features, with many resonances. As the
valence electrons are tightly bound,
the resonance frequency is high so that these
materials may have a transparency range
that extends far into the uv. This can be
seen in the reflectance spectrum for
KCl shown below (taken from Wooten, Ch. 3.)
For these types of materials, the
external field and the local field can be quite
different and it is not trivial to
calculate the local field. For this reason, the
Lorentz
model does not give
quantitatively accurate results for ionic
materials.
(ii) Doped
Insulators
1
Doped insulators, for example ions in
glass, behave somewhat like the ions would
in a gas, except that the locally
strong electric fields of the host materials may
distort the spectrum slightly. The
figure below shows the absorption of
Nd
3+
ions
in a
glass host material.
Usually, the
absorption of the dopant material is in a region
of transparency of the
host so that we
can approximate the polarization as a
superposition of polarizations
due to
the host and dopant material. For the case of a
single resonant absorption
line, we may
write:
P
tot
?
P
host
?
P
dopant
2
?
?<
/p>
?
p
?
?
0
?
?
host
?
2
2
?
?
?
?
i
?
?
?
0
< br>?
?
?
?
E
?
?
where
?
host
is assumed
to be real and constant. Hence;
?
r
(
?
)
?
1
?
?
host
?
?
p
< br>2
2
2
?
0
?
?
?
i
?
?
Often, we label 1 +
?
host
as the “high frequency dielectric
constant”,
?
?
,
so that:
?
r
(
?
)
?
?
?
?
?
p
2
2
2
?
0
?
?<
/p>
?
i
?
?
.
2
The
static dielectric constant, defined as
?
st
=
?
(
?
=0)
is therefore given by setting
?
= 0 in the above
expression, so that:
Hence, the static dielectric function
of a material is affected by dopants, even
though the resonant frequency for the
dopant is far away from
?
=
0.
?
st
?
?<
/p>
?
?
?
p
2
2
?
0
.
(iii) Semiconductors:
Semiconductors are covalently bonded
materials where the electrons are evenly
shared between neighboring atoms.
(Some insulators are covalently bonded, too.)
This means that the electrons are
smeared out into broader bands and that their
resonance frequencies are lower than
for ionically bonded materials. Usually these
materials can be described by a single
energy gap and single broad absorption band
above the energy gap. The example of
Silicon is shown below:
3
Estimation of
?
r
(
?
=0) for Si:
Noting that the
reflectance of Si rises sharply at about 3 eV, we
may take this as an
estimate for
?
0
. Hence
?
0
?
(3 x 1.6 x10
-19<
/p>
)/
?
= 4.53 x
10
15
rad/s.
Now,
?
r
< br>(
?
)
?
1
?
?
p
2
0
2
2
?
?
?
?
i
?
?
, so that
?
r
(
0
)
?
1
?
?<
/p>
p
?
0
2
2
, so that if we can
determine
?
p
, we can
estimate
?
(0). Now
< br>?
p
?
Ne
2
/
?
0
m
, and since each Si
atom
has 4 valence electrons, N =
4N
Si
?
4x 2
?
10
28
m
-3
. This gives an estimate
of
?
p
?
1.6 x
10
16
rad/s (corresponding
to about 10.5 eV) and hence
?
(0)
?
14. This
is compared to a measured
value for
?
(0) of 12, so the
approximations are
reasonable. Note
that Si appears as a grayish reflector throughout
the visible
spectrum. (1.7 ~ 3.2 eV)
(iv) Metals:
- Drude theory of optical properties of
metals.
We can extend the Lorentz model
to metals, in which case, since the electrons are
unbound or
frequency,
?
0
2
= K/m is also zero. This is known as
the “Drude” model.
The
equation of motion is then:
,
m
which has solution;
2
?
?
r
(
t
)
?
t
2
?
m
?
?
?
< br>r
(
t
)
?
t
?
?
?
e
E
(
t
)
?
e
r
(
t
)
?
m
< br>?
?
E
(
?
)
2
?
?
i
?
?
?
and hence
?
(
?
)
is given by,
?
(
?
)
?
?
?
p
2
2
?
?
i
?
?
< br>
where once again the
plasma frequency is defined by
?
p
2
=
Ne
2
/
?
0
m
. Hence,
4
or,
?
r
'
(
?
)
?
1
?
?
p
2
?
'
(
< br>?
)
?
?
?
2
p
1
?
?
?
2
2
,
?
(
?
)
?
?
2
p
?
/
?
?
?
?
2
2
1
?<
/p>
2
?
?
2
,
?
r
(
?
)
?
< br>?
p
2
?
/
?
?
2
?
?
2
Now, in a metal, the
damping term
?
is just the
electron collision rate, which is just
the inverse of the mean electron
collision time,
?
, i.e.
?
=
?
-1
. Hence,
?
r
'
(
?
)
?
1
?
?
p
?
2
< br>2
2
2
1
?
?
?
,
?
r
(
?
)
?
?
p
?
?
?
1
?
?
?
2
2
2
?
The collision rate can be
quite rapid - tens of femtoseconds. But for
optical
frequencies, (e.g. for
?
= 500 nm,
?
=
2
?
c/
?
= 3.8x10
15
rad/s)
(
??
)
2
>> 1
. Under
this
approximation, we find:
?
r
'
(
?
)
?
1
?
?
p
?
2
2
,
< br>?
r
(
?
)
?
?
p<
/p>
3
2
This
approximation may break down in the far-infrared
spectral region, where
damping may be
significant. Note that damping is absolutely
necessary to have an
imaginary part of
?
(
?
)
or
?
r
(
?
).
It is
useful to look at some plots of
?
r
(
?
),
n(
?
),
?
(
?
)
and R(
?
). These are plotted
on the next page for
?
p
= 10 and for
?
?
0
or
?
= 0.5. In the limit of
no damping,
the n = 0 and R =1 for 0 <
?
<
?
p
. Above
?
p
,
?
is zero and the
reflectance drops as
n rises from zero
to unity. Note that even for
?
r
” = 0,
?
and hence
?
is not zero.
Introducing some damping causes R to be
< 1 and the reflectance drop at
?
p
is less
severe. The behavior of
?
r
, n and
?
is consistent with what we
now expect for a
Lorentz oscillator
with
?
0
= 0.
?
?
?
?
p
?
?
3
2
.
5
Clearly, the sharp edge in
the reflectance seen at the plasma frequency can
be
expected to be the predominant
spectral feature in the optical properties of
metals.
20
?
= 0.0005
Re
?
(
?
)
20
?
= 0.5
Re
?
(
?<
/p>
)
Im
?
(
?
)
0
0
Im
?
(
?
)
0
0
20
20
0
2
4
?
6
8
10
< br>0
2
4
?
6
8
10
3
3
n
(
?
)<
/p>
?
(
?
)
R
(
?
)
2
n
(
?
)
?
(
?
)
R
(
?
)
2
1
1
0<
/p>
5
?
10
15<
/p>
0
5
?
10
15
1
?
1
0
1
00
0
3
1
00
0
0
4
4
1
?
1
0
3
1
?
1
0
1
00
1
00
2
1
0
Re
?
(
?
)
Im<
/p>
?
(
?
)
1
n
(
?
)
Re
?
(
?
)
?
(
< br>?
)
Im
?
(
?
)
n
g
(
?
)
1<
/p>
0
1
0
0.1<
/p>
n
(
?
)
?
(
?
)
n
g
(
?
)
n
g
(
?
)
?
1
0.1
0.0
1
3
.00
?
1
1
1
0
0.0
1
?
1
n
g
(
?
)
3
1<
/p>
?
1
0
4
2
1
?
1
0
0
0
5
?
1
0
1
5
1
5
5
.00
0
01
?
1
1
0
0
1
5
?
2
1
0
3
?
1
5
1
5
4
0
4
5
6
The last plots show the real and
imaginary parts of the dielectric constant on a
log
scale. It is interesting to note
that only the real part of
?
indicates notable behavior
around the
plasma frequency. One can not see evidence of the
plasma frequency
by looking at the
imaginary part of
?
alone,
yet both n(
?
) and
?
(
?
)
clearly show
evidence of the plasma
frequency.
Optical absorption in low electron
density materials
–
Semiconductors:
Recalling that the
absorption coefficient is given by
?
(
?
) =
2k
0
?
= 2
??
(
?
)/c =
??
r
”
(
?
)/cn(
?
).
Now for very high frequencies, or for low electron
densities, as
may be found in doped
semiconductors,
?
p
2
<<
?
2
,
n(
?
) ~ 1 so that,
?
(
?
)
?
?
c
?
(
?
)
?
?
p
?
c
?
2
2
?
?
?
2<
/p>
2
c
?
p
,
where
?
p
is the
wavelength corresponding to the plasma frequency.
The
?
2
dependence of
?
is commonly seen in semiconductors, where dopant
densities are
typically in the range of
10
16
to
10
19
cm
-3
as compared to ~
10
22
cm
-3
in metals.
This absorption is commonly referred to
as free-carrier absorption.
Tin-doped Indium Oxide (ITO) a
transparent conducting electrode
material
ITO is a
semiconducting material that gives quite high
electrical
conductivity, yet is
transparent in the visible. It is particularly
useful in low-
current applications,
such as liquid crystal displays. This is achieved
by
having a material with low electron
density, but those electrons should be
highly mobile, which means they travel
through the material with relatively
few collisions. By choosing the right
density of tin doping, ITO can be
highly effective. Below, we show the
real and imaginary part of
?
for ITO
from a paper by Hamberg and
Granqvist, Journal of Applied Physics,
Volume 60, Issue 11, 1986, Pages
R123-R159. The plasma frequency,
dependent of the Sn density, is
typically around 0.7 eV, which corresponds
to
?
~1.7
?
m. Due to this, and
the free carrier absorption described above,
ITO is not as useful in the near
infrared (
?
> 1
?
m) as it is in the visible.
7
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