Waves and Optics

1
In this chapter we will study some basic optical
spectra of solids. After studying this chapter,
we are able to classify solids materials as
metals, semiconductors, or insulators according
to their optical behaviors
5.1 Optical Magnitudes and Dielectric
Constants In the previous chapters, we have
examined the fundamentals of the most
spectroscopic techniques, as well as the
instrumentation needed in order to employ these
techniques. Now it is time to invoke some models
in order to predict and interpret the optical
properties of a given solid materials. Our first
task is to connect measurable optical magnitudes
with the dielectric constant, which describes the
response of a given material to an applied
electric field. This electric field is created by
the electromagnetic wave propagating into the
solid. In the optical range (the wavelength
range 200 nm to 3000 nm) is much larger than the
interatomic distance in solids. This fact enables
us to consider a solid as a continuous medium for
the purpose of describing the propagation of the
electromagnetic radiation and then to use
2
a classical description. Considering, for
simplicity, isotropy in the medium, the spatial
and temporal dependence for the electric field of
an electromagnetic wave with an angular frequency
?, propagating along the z direction, can be
written as where k2p/?, is the module of the
light wave vector in vacuum, E0 is the electric
field amplitude at z 0, and N is the complex
refractive index, defined as The real part of
this number is the normal refractive index n
c/v (c and v being the speed of light in vacuum
and in the medium, respectively). The imaginary
part of the complex refractive index, ?, is
called the extinction coefficient. It is
necessary to recall that both magnitudes, n and
?, are dependent on the frequency (wavelength) of
the propagating wave. Taking in account Eq. (5.2)
and that k ?/c, Eq. (5.1) can be rewritten as
(5.1)
(5.2)
3
(5.3)
so that a nonzero extinction coefficient ? leads
to an exponential attenuation of the wave in the
material. As the intensity of the light wave, I,
is proportional to the square of the electric
field module (I ?E2 EE), we write an
expression for the intensity attenuation along
the propagating direction z where I0 is the
intensity of the incident light at z0. Now,
comparing this equation to the Lambert-Beer law
(see Eq. (2.4)), we obtain the relation between
the absorption coefficient a, which is directly
measurable from an absorption spectrum and the
extinction coefficient ? Let us now relate the
refractive index and the extinction coefficient
with the relative dielectric constant of the
solid material. Assuming a nonmagnetic solid
(5.4)
(5.5)
4
(relative magnetic permeability µ1) we know that
, whereeis the relative dielectric
constant of the material. Thus, as eis also a
complex magnitude, , we can
write and this equation can be solved for the
real and imaginary terms as follows Inversely,
we can obtain n and ? as functions of the
relative dielectric constants The
reflectivity, R, of a solid can also be
determined after establishing the boundary
conditions for the electromagnetic radiation at
the interface
(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
5
between the solid and the vacuum. In the simple
case of a solid in a vacuum, and considering
normal incidence of light, the reflectivity can
be derived from basic optics text as Therefore,
the optical magnitudes n, ? or a, and R can be
obtained from the dielectric constant, e. In the
next section, we will develop a simple model to
predict the frequency dependence of the relative
dielectric constants e1 and e2 of a given
material. At that point, we will be able to
determine the measurable optical magnitudes at
any particular frequency if the relative
dielectric constants are known at that
frequency. EXAMPLE The complex refractive index
of germanium at 400 nm is N 4.141 i2.215. At
this wavelength, determine (a) the optical
density for a sample of thickness 1 mm and (b)
the reflectivity at normal incidence.
(5.11)
6

• According to Eq. (5.2), ? 2.215 and, taking
  into account that ? 2pc/?, the absorption
  coefficient is given by Eq. (5.5)
• and, according to Eq. (2.10), the optical density
  for a 1 mm sample is

This is indeed a very large value to be measured
by an optical spectrophotometer. (b) From Eq.
(5.2), we know that n 4.141 and so, using Eq.
(5.11), the reflectivity at normal incidence is
7

• We will now mention some general aspects
  regarding the reflectivity of metals,
  semiconductors, and insulators, in the different
  spectra regions of the optical range (UV, 50-350
  nm visible, 350-700 nm and near infrared
  700-3000 nm) by considering three relevant
  solids
• Al (a typical metal) has a reflectivity that
  varies from about 0 to 90 in the UV, while it is
  highly reflecting in the visible and near
  infrared.
• Si (a typical semiconductor) is highly absorbing
  in the UV, and partially reflecting and absorbing
  in the visible and near infrared.
• SiO2 (a typical insulator) has a strong
  absorption rise (called the fundamental
  absorption edge) in the UV and it is transparent
  in the visible.
• 5.2 The Lorentz Oscillator
• In the previous section we have demonstrated how
  the measurable optical magnitudes are related to
  the dielectric constants (e1 and e2). Now we need
  to establish how these dielectric constants
  depend on the frequency of the

8
incoming electromagnetic radiation. Then, we will
be able to predict the particular shape of
optical absorption and reflectivity spectra of
any material. Obviously, we need to start from
microscopic (classic and quantum) models. These
models require some knowledge about the nature of
the interatomic bonding forces in our solid and
whether or not the valence electrons are free to
move inside the solid. In metals, valence
electrons are conduction electrons, so they are
free to move inside the solid. On the contrary,
valence electrons in insulators are located
around fixed site for instance, in an ionic
solid they are bound to specific ions.
Semiconductors can be regarded as an intermediate
case between metals and insulators valence
electrons can be of both types, free or
bound. The most simple, but general, model to
describe the interaction of optical radiation
with solids is a classic model, due to Lorentz,
in which it is assumed that the valence electrons
are bound to specific atoms in the solid by
harmonic forces. These harmonic forces are the
Coulomb forces that tend
9
to restore the valence electrons into specific
orbits around the atomic nuclei. Therefore, the
solid is considered as a collection of atomic
oscillators, as shown in the following figure.
Each such an atomic oscillator has a
characteristic natural frequency. If we excite
one of these atomic oscillator with its natural
frequency (the resonance frequency), a resonant
process will be produced. From the quantum
viewpoint, these frequencies correspond to those
needed to produce valence band to conduction band
transitions. In the first approach we consider
only a unique resonant frequency, ?0 in other
words, the solid consists of a collection of
equivalent atomic oscillators.
In this approach, ?0 would correspond to the gap
frequency. This model of atomic oscillators, in
which we assume bound valence electrons, is also
perfectly valid for metals, except that in this
case we must set ?0 0.
10
Let us now analyze the interaction of a light
wave with our collection of oscillators with
frequency ?0. In this case, the general motion of
a valence electron bound to a nucleus is a damped
oscillator, which is forced by the oscillating
electric field of the light wave. This atomic
oscillator is called a Lorentz oscillator. The
motion of such a valence electron is then
described by the following differential
equation
(5.12)
where me and e are the electron mass and charge,
respectively, and r is the electrons position
with respect to the equilibrium point. The
harmonic term me?0r represents a Hookes law
restoring force acting on the valence electrons.
Obviously, this term vanishes for metals as
valence electrons becomes free. The damping term
in Eq. (5.12), meGdr/dt, represents a friction
force due to various scattering processes
experienced by the valence electrons, where G is
the damping rate. In solids, it is typically
related to loss of energy due to the excitation
of phonons. The term
11
-eEloc is the force acting on the valence
electron due to the oscillating electric field of
the light, where the subscript loc indicates a
local field. Assuming that this field oscillates
as the solution of Eq. (5.12) is also
time dependent and is given by where r is a
complex number because of a phase shift between r
and Eloc caused by the nonvanishing damping
term. Due to the oscillating nature of the local
electric field, Eloc, an oscillating dipole
moment p -er is induced. Taking into account
that p ßEloc, ßbeing the atomic polarizability,
and using Eq. (5.13), we obtain
(5.13)
(5.14)
12
Once again, it should be noted that the
polarizability is a complex number because of the
damping term. At this point, we can relate the
atomic polarizability ßwith the dielectric
constant e if we make certain assumptions about
the local electric field. Remember that e D/E,
where D E 4pP is the electric displacement
vector. In the units of CGS, we obtain where N
is the density of atoms. The real and imaginary
parts of the dielectric constants are given by
(5.15)
(5.16)
13
(5.17)
In the general case of more than one valence
electron per atom, and allowing for the
possibility of different resonance frequencies
instead of a unique frequency ?0, expression
(5.15) can be generalized to where Nj is the
density of valence electrons bound with a
resonance frequency ?j and is
the total density of valence electrons. The
corresponding quantum mechanical expression of
e(?) in Eq. (5.18) is similar except for the
quantity Nj, which is replaced by Nfj . However,
the physical meaning of some terms are quite
different ?j represents the frequency
corresponding to a transition between two
electronic states of the atom separated by an
energy , and fj is a dimensionless
quantity which
(5.18)
14
is called the oscillator strength related to the
quantum probability for this transition. It
satisfies . We are now able to
understand the response of our solid to an
electromagnetic field oscillating at frequency ?.
For the sake of simplicity, we return to the use
of expressions (5.16) and (5.17), related to a
solid made of single-electron classical atoms,
and to only one resonant frequency ?0, related to
the band gap.
Using these expressions, we have displayed the
dependencies of e1 and e2 on the incident photon
energy. At the same time, we can use (5.9),
(5.10), and (5.11) to obtain the spectral
behavior of the measurable magnitudes n, ?, and
R.
15
Inspection of Figures (a) and (b) shown in last
slide shows four significant spectral
regions. Spectral region I corresponds to
illuminating frequencies that are much lower than
the resonance frequency that is ?ltlt?0. According
to Eqs. (5.17) and (5.8), in this region e2
2n? 0 and then ? 0. Also, in accordance with
Eq. (5.7), e1 n2 ?2 n2 gt1. We can see in the
figure (b) that this spectral region is
characterized by a high transparency weak
absorption and relatively low reflectivity. Spectr
al region II corresponds to frequencies close to
the resonance frequency, ??0. In this region, e1
reaches a maximum, and then decreases with the
frequency of light in a certain energy range and
increases again. The mentioned decrease is a
manifestation of an anomalous dispersion in this
spectral region. This anomalous dispersion is
accompanied by an appreciable increase in the
reflectivity and a strong absorption peak. In the
region III, ? gtgt ?0 . The reflectivity is high
while the extinction
16
coefficient ?(?) shows a rapid decrease with
increasing frequency. Finally, the onset of
spectral region IV is defined by e1 0, and
hence for frequencies ?gt ?p, where ?p is the
called the plasma frequency. This frequency
indicates a region of very high transparency (?0
and R0). Letting e10 in Eq. (5.16) and assuming
?gtgt?0gtgtG, the plasma frequency is given (in CGS
units) by 5.3 Metals We will now analyze the
general optical behavior of a metal using the
simple Lorentz model developed in the previous
section. Assuming that the restoring force on the
valence electrons is equal to zero, these
electrons become free and we can consider that ?0
0 in Eq. (5.12). This is the so-called Drude
model, which was proposed by P. Drude in 1900. We
will show how this
(5.19)
17
model successfully explains a number of important
optical properties, such as the fact that metals
are excellent reflectors in the visible while
they become transparent in the ultraviolet. The
previous assumption, ?0 0, can be also justified
from the quantum point of view, as the most
relevant transitions in metals take place within
a band, usually the conduction band. The energy
levels within a band of a solid are separated by
energies of about 10-27 eV, and so in the optical
range (photon energies of some eV) the validity
of the assumption for the transitions between
these so-closely spaced levels is evident. 5.3.1
Ideal Metal To simplify, we begin thinking of an
ideal metal that is, a metal without damping.
Later, we shall discuss how damping forces can
affect the optical properties of this ideal
metal.
18
By setting ?0 0 and G0 in Eqs. (5.16) and
(5.17), we obtain Using expressions (5.9) and
(5.10), the optical magnitudes n and ?can be
obtained as functions of the frequency of the
illuminating light. The following figure shows a
(5.20)
(5.21)
plot of the optical magnitudes as functions of
?/?p. For light frequencies lower than ?p, the
refractive index is zero (n0), while the
extinction coefficient decreases with frequency.
19
When ??p, ?0. For frequencies higher than the
plasma frequency, the extinction coefficient
remains equal to zero, while the refractive index
rises with increasing frequency n2 1-(?p/?)2
toward a limit n1. The following figure shows
the optical spectra expected for our ideal metal,
calculated from expressions (5.5) and (5.11).
Obviously, the Drude model predicts that ideal
metals are 100 reflectors for frequencies below
?p and highly transparent for higher frequencies.
This result is in rather good agreement with the
experimental spectra observed for several metals.
In fact, the plasma frequency ?p defines the
region of
20
transparency of a metal. It is important to
realize that, according to Eq. (5.19), this
frequency only depends on the density of the
conduction electrons N, which is equal to the
density of the metal atoms multiplied by their
valency. This allows us to determine the region
of transparency of a metal provided that N is
known, as in the next example. EXAMPLE Sodium is
a metal with a density of conduction electrons
N2.65 X 1022 cm-3. Determine (a) its plasma
frequency, (b) the wavelength region of
transparency, and (c) the optical density at very
low frequencies for a Na sample of 1 mm
thickness. (a) According to Eq. (5.19),
21
(b) Then metallic sodium will become transparent
for frequencies higher than 9.18 x 1015 Hz. The
corresponding wavelength ?p, called the cutoff
wavelength, is (c) At low frequencies (long
wavelengths), Na is a fully reflector material.
Its optical absorption coefficient is also very
large. It is given by
Thus, at ?0 and according to
ODax/2.303, the corresponding optical density
for a Na sample of 1 mm thickness
22
In the previous example, we have calculated the
plasma frequency for metallic Na from the free
electron density N. In the following table, the
measured cutoff wavelength, ?p, for different
alkali metals are listed together with their free
electron densities. The calculated values for the
cutoff wavelength according to Eq. (5.19) are
also given for comparison.

Thus, in general, metals are good filter for the
UV radiation and good reflector for visible
light.
23
5.3.2. Damping effects If we now consider a
nonvanishing damping term, G?0, Eqs. (5.20) and
(5.21) must be written as The damping term
in a metal is due to the scattering suffered by
the free electrons with atoms and electrons,
which produces the electrical resistivity. Let us
now examine the motion of the free electrons in a
metal just after the driving external local field
is eliminated. Then Eq. (5.12) for the Lorentz
oscillator appears in a simplified form, as

(5.22)
(5.23)
24
(5.24)
which can be written in terms of the free
electron velocity, ?, as The solution of this
differential equation is ??0e-Gt. This solution
indicates that the electron velocity decays
exponentially to zero with a decay time of
t1/Gafter the driving force is eliminated. This
time represents the mean free collision time for
electrons in metals. Typically, this time is
about 10-14 s, and corresponds to a damping
frequency G1014 s-1. This means that the damping
effects will be significant for frequencies 1014
s-1. In the figure shown in next slide, the
reflectivity spectra of aluminum predicted by the
ideal Drude model and the damping model are
plotted to compare with the experimental
reflectivity spectrum of Al.
(5.25)
25
Aluminum has a free electron density of N18.1 x
1022 cm-3 (three valence electrons per Al atom)
and so, according to Eq. (5.19), its plasma
energy is h?p 15.8 eV. Compared with the
experimental spectrum, the calculated spectrum
based on the damping model is only slightly
improved. For getting better theoretical
spectrum, we need to develop more accurate model.
26
5.4 Semiconductors and Insulators Unlike metals,
semiconductors and insulators have bound valence
electrons. This property gives rise to interband
transitions. In this section, we attempt to
understand how the absorption spectrum of a given
material is related to its band structure and, in
particular, to the density of states for the
transitions involved. First of all, let us
recall the physical meaning of band structure in
a solid. Transitions in isolated atoms arise from
a series of states with discrete energy levels.
Consequently, optical transitions between these
levels give rise to sharp absorption and emission
lines at specific resonance frequencies. These
spectra correspond to the case of individual
atoms, or of widely separated atoms (at infinite
distances). As the interatomic distance is
decreased, the individual atomic charge
distributions begin to interact. As a result,
each atomic energy level shifts and split into
(2l1)N molecular energy levels, N being the
number of atoms (or ions) involved in the bonding
and (2l1) being the orbital degenerancy of the
atomic energy level
27
(l being the orbital quantum number). In solids,
atoms hold together at short equilibrium
distances and N is of the order of Avogadros
number. Consequently, each energy level splits
into a high number of closely spaced
levels, giving rise to a continuous band. In any
case, in spite of the fact that optical
transitions in solids are much more complicated
than atomic transitions, some features still
retain the character of the transitions for
individual atoms. The right figure provides a
good qualitative scheme to explain the metallic
or insulator character of a solid.
28
For atomic (gas) sodium (Na), the electronic
configuration is 1s22s22p63s1, leading to filled
electronic energy levels 1s, 2s, and 2p, while
the 3s level is half-filled. The other excited
levels, 3p, 4s , are empty. In the solid state
(the left-hand side in the figure in last slide),
these atomic levels are shifted and split into
energy bands bands 1s, 2s, and 2p are fully
occupied, while the 3s (l0) band, the conduction
band, is half-filled, so that a large number of
(N(2l1)/2N/2) of empty 3s excited levels are
still available. As a result, electrons are
easily excited into empty levels by an applied
electric field, and so become free electrons.
This aspect confers the typical metallic
character to solid sodium. The case of sodium
chloride (NaCl) (the right-hand side in the
figure in last slide), a typical insulator, is
quite different. The electronic configuration of
Cl atoms is 1s22s22p63s23p5. However, NaCl is an
ionic crystal and its crystal structure is
constituted by Na and Cl- ions. This is because,
at the short equilibrium distances in the NaCl
solid, each Na atoms prefers to transfer its 3s
electron to a 3p level of a Cl atom. This process
leads to the
29
empty of the 3s band of Na ions , whereas the 3p
band of Cl- ions is fully occupied (notice that
the 3p band is mixed with the 3s Cl- band). The
energy difference between the top of the 3s3p
(Cl-) band ( the valence band) and the bottom of
the 3s (Na) band (the conduction band) is about
8 eV and is known as the band-gap energy.
Consequently, NaCl is an insulating material.
From the optical viewpoint, the appearance of
this large band gap gives rise to a transparency
region in the visible and in the ultraviolet up
to a wavelength of about 155 nm. The band gap
also results in a continuous absorption spectrum
for photonic energies higher than 8 eV. This
spectrum is due to interband transitions from the
3s3p (Cl-) states to excited states of the 3s
band (Na) and states of other higher energy
bands. Thus, the energy gap gives the region of
transparency for a nonmetallic solids.
The right table gives the energy-gap values and
corresponding wavelengths for some typical
semiconductors
30
and insulators. The small energy-gap values ( 1
eV) for Ge, Si, and GaAs, explain the
semiconductor character observed at room
temperature for these solids, as sufficient
valence electrons can be thermally excited across
the band gap. These small energy-gap values also
explain their opaque aspect in the visible
spectral region. On the other hand, the large
energy-gap values of ZnO, diamond, and LiF make
these crystals highly transparent in the visible.

The right figure (a) shows the band structure of
silicon around the energy gap, Eg1.14 eV. This
band structure is responsible for several
features in the absorption spectrum (Fig. (b)). A
very weak rise (not apparent in the figure. It is
indicates by a big arrow.) at about 1.14
31
eV was observed. The two main peaks E1 and E2 are
related to the higher transitions.

• 5.5 The Spectral Shape of the Fundamental
  Absorption
• In the absorption spectrum, the spectral region
  in the vicinity of the energy gap (i.e., around
  1.14 eV for Si) is usually called fundamental
  absorption edge and it is shown by a rapid rise
  in the absorption coefficient. The fundamental
  absorption edge provides very useful information
  on the band structure around the energy gap. In
  this section, we will examine the spectral shape
  of the fundamental absorption edge and its
  relationship with the band structure.
• If a solid is illuminated with radiation of
  frequency ??g (Eg h?g), the radiation will
  become partially absorbed. The absorption
  coefficient at this frequency is proportional to
• the probability Pij of a transition from the
  initial state i to the final state f
• the density of electrons in the initial state ni
  and the density of available

32
final states nf, including all of the states
separated by an energy equal to h?. So, we can
write
(5.26)
where A is a factor to allow agreement with the
absorption coefficient units. Before we deal with
the spectral shape of a(?), let us examine the
selection rule imposed by the conservation of
momentum. For a given transition, where ki and
Kf are the wave vector of initial and final
electronic states, and kp is the wave vector of
the incoming photon. Since the magnitude of the
wave vector for a typical visible photon (?600
nm) is kp 2p/?105 cm-1, while for a typical
electron in a crystal, ki,f p/a 108 cm-1, we
find that kp ltlt ki,f , so that the selection rule
(5.27) becomes
(5.27)
(5.28)
33
This important selection rule indicates that
interband transitions must preserve the wave
vector. Transitions that preserve the wave vector
(such as those marked by vertical arrows in the
following figure) are called direct transitions,
and they are easily observed in materials where
the top point in the valence band has the same
wave vector as the bottom point in the conduction
band. These materials are called direct gap
materials.
In materials with a band structure such as that
sketched in Fig. (b), the bottom point in the
conduction band has a quite different wave vector
from that of the top point in the valence band.
These materials are called indirect gap
materials. Transitions at the gap photon energy
are not allowed by the rule given in
34
Eq. (5.28), but they are still possible with the
participation of lattice phonons. These
transitions are called indirect transitions. The
momentum conservation rule for indirect
transitions can be written as where k? denotes
the wave vector of the phonons involved. The
sign in Eq. (5.29) indicates that indirect
transitions can occur by absorbing () or by
emitting phonons. In the first case, the crystal
is illuminated with photons of energy EEg-E?,
while in the second case the crystal is
illuminated with photons of energy EEgE?, (E?
being the energy of the phonon involved). Indirect
transitions are much weaker than direct
transitions, because the latter do not require
the participation of phonons. Hereafter, we will
deal with the spectral shape expected for both
direct and indirect transitions. 5.5.1 The
Absorption Edge for Direct Transitions For a
direct allowed absorption edge, its absorption
coefficient can be written as
(5.29)
35
Several III-V semiconductors, such as InP,
GaAs, InAs, show direct absorption edge
transitions. The next example shows the analysis
of the fundamental absorption edge for InAs.
(5.30)
EXAMPLE The right figure (a) shows the
dependence of the absorption coefficient versus
the photon energy for indium arsenide. (a)
Determine whether or not InAs a direct-gap
semiconductors. (b) Estimate the band-gap energy.
(c) If an InAs sample of 1 mm thickness is
illuminated by a laser of 1W
36
at a wavelength of 2 µm, determine the laser
power of the beam after it passes through the
sample assuming that the loss of light is only
due to optical absorption. (a) To determine
whether or not InAs is a direct gap material, we
calculate the square of the absorption
coefficient a2 versus the photon energy from the
absorption spectrum given in the figure (a). This
plot is shown in the figure (b) in last slide a
linear dependence a2 ? (h?-h?g) is clearly
observed, in agreement with Eq. (5.30).
Consequently, we can say that InAs is a direct
gap semiconductor. (b) The band gap value of InAs
can be directly obtained by extrapolating the
linear dependence in the figure (b) to zero,
which leads to Eg h?g 0.35 eV. (c) According to
the figure (a) in last slide, the absorption
coefficient at 2 µm (0.62 eV) is 106 m-1. Thus ,
we can obtain the attenuation of the laser
intensity by using Eq. (2.4)
37
So that the power is completely attenuated by
absorption. 5.5.2 The Absorption Edge for
Indirect Transitions For indirect gap materials,
the absorption coefficient in the vicinity of h?g
is given by where the term ? indicates whether
a phonon of frequency ? is absorbed or
emitted. The general shape of the absorption edge
for an indirect gap material has been sketched in
the figure (a) shown in next slide. In this
figure (a plot of a1/2 versus ?), two different
linear regimes are clearly observable. The
straight line at lower frequencies shows an
absorption threshold at a frequency of ?1?g-?,
which corresponds to a process involving
absorption of phonons of energy h?. The second
straight line intersects the
(5.30)
38
the frequency axis at ?2?g? and corresponds to
a process in which phonons of energy h? are
emitted. Because of the involvement of phonons in
indirect transitions, one expects that the
absorption spectrum of indirect gap materials
must be substantially
influenced by temperature changes, shown in the
right figure (b). It is not difficult to
understand this point since the number of phonons
depends strongly on temperature.