Optical properties of semiconductors ppt

Optical Properties of Semiconductors
Tewodros Adaro
July 9, 2021
Tewodros Adaro Optical Properties of Semiconductors

Table of content
Table of content
The complex refractive index of a solid
Reflectivity
Transmission through a thin slab
The Drude theory of conductivity

Maxwell’s equations
In order to understand how light interacts with a semiconductor,
we need to say a few words about light propagation in a given
medium. Consider a medium which has both bound electrons and
free electrons. The propagation of light in this medium is described
by Maxwell’s equations. The Maxwell’s equations can be written in
a form which from the very beginning distinguishes a conducting
medium from a non conducting medium, by writing:
∇X
−
→
E = −
∂
−
→
B
∂t
(1)
∇X
−
→
H = σ(ω)
−
→
E +
∂
−
→
B
∂t
(2)
∇.
−
→
D = ρ (3)
∇.
−
→
B = 0 (4)
where σ(ω) is the complex frequency dependent conductivity of
the medium with a density of ρ mobile charges,

−
→
E ,
−
→
D ,
−
→
H ,
−
→
B are the electric field, displacement, magnetic field and
magnetic flux respectively. We are mainly interested in neutral
media, so we shall put ρ = 0 and assume that the relative
permittivity εb of a medium with bound charges in
−
→
D = εoεb
−
→
E is
time independent and
−
→
D = εo
−
→
E +
−
→
P is the bound polarization
vector which gives the electric dipole moment per unit volume and
εo is the permittivity of free space. We also assume that the
medium is not magnetic so that
−
→
B = µµo
−
→
H , µ = 1 the
permeability of free space. Using the fact that the velocity of light
in free space is c2 = (µoεo)−1 one can combine Eq. ( 1 ) and Eq.
( 2 ) by taking the “curl” (or “rot” ) of Eq. ( 1 ) to give the wave
equation for an EM wave as:
∇2−
→
E =
1
c2
(εb
∂2−
→
E
∂t2
+
σ
εo
∂
−
→
E
∂t
) (5)
As we will see, this equation describes a traveling wave that can be
solved by assuming that the electric field of the light is of the form:

−
→
E =
−
→
Eo exp(i(
−
→
k .−
→
r − ωt)) (6)
The substitution of Eq. (6 ) into Eq. (5 ) then gives rise to the
requirement that to be a solution, the length of the vector
−
→
k (the
wavevector) must satisfy the complex equation:
k =
ω
c
(εb +
iσ
εoω
)
1
2 (7)
since the wavevector k = ko =
ω
c
in free space, we can interpret
the square root factor in Eq. ( 7 ) as the complex refractive index
of the material N
N = (εb +
iσ
εoω
)
1
2 (8)

We recall that in this representation,εb refers to the (relative)
bound electron permittivity, and is itself normally a complex
quantity. This is why some authors prefer to work with a total
relative complex permittivity ( εt(ω) = εb +
iσ
εoω
) and define
−
→
D = εoεt
−
→
E which includes both the complex free and the
complex bound electron permittivities. In the notation that we
have chosen, the conductivity of the medium is made explicit, and
σ(ω) is the complex frequency dependent conductivity of the
system, the real part of which is the AC conductivity or, with
geometry factor (area /length), the “conductance” of the system.
The imaginary part then corresponds to ωC where C is the
capacitance. Indeed if we separate the bound electron permittivity
into real εr and imaginary parts εi we have
N2 = εr + i(εi +
σ
εoω
) (9)

The free electron permittivity is now by definition:
εf (ω) =
σ
εoω
(10)
We can now rewrite the complex refractive index and complex
wavevector as:
N = n + ik (11)
k =
nω
c
+
ikω
c
= Nko (12)
The imaginary part of Eq. ( 11 ) acquires physical significance as
soon as we substitute Eq. ( 12 ) back into the wave solution Eq. (
6 ) and for simplicity assume propagation in the z-direction only,
then we have:

−
→
E =
−
→
Eo exp[iω(
nz
c
− t)] exp(−
ωkz
c
) (13)
for
−
→
Eo = Ex
o
−
→
x the corresponding Hy
o is given by Hy
o = N
r
εo
µo
Ex
o
where we also have from Eq. ( 9 ) and σ = σr + iσi
n2
− k2
= εr −
σi
ωεo
(14)
2nk = εi +
σi
ωεo
(15)
The medium has modified the electromagnetic wave or photon, in
two ways. It has changed the velocity of propagation from c to
c
n
,
and it has given rise to damping. The damping is due to the
imaginary part of k and is caused by the absorption of
electromagnetic energy in the medium. From Eq. ( 14 and 15 ) it
follows that one principal source of absorption is the conductivity
term.

But loss of amplitude can also be caused by the bound electrons
absorbing light energy and getting excited into higher energy levels
in the solid. Bound electron absorption happens at relatively high
frequencies, so that in practice, as we shall see later, the low
frequency damping is mainly due to free charges, and the high
frequency damping mainly due to band to band absorption. Noting
that the energy density is proportional to the square of the electric
field amplitude, we recover the Beer Lambert law:
|E|2
= |Eo|2
exp −αz (16)
α = 2k
ω
c
(17)
where α is the absorption coefficient and measured in units of m−1
in the MKS units as used here.

A word of caution as to the definition of the absorption coefficient.
In the transmission of light through a material, the electric field
amplitude can decay not just because of absorption. The decay
may be due to disorder i.e. scattering, and this is why some
authors prefer to compute the power dissipated per unit length.
The optical power density of the electromagnetic wave in units of
W
m2 is given by the time averaged Poynting vector:
−
→
S =
1
2
Re(
−
→
E X
−
→
H ∗
) =
nc
2
εo(Ex
o )2
exp(−ez)
−
→
Z (18)

Reflectivity
Reflectivity
Before getting on with the evaluation of the complex permittivities
and conductivity, it is convenient to investigate what happens
when photons, or in other words the light beam, are incident onto
a medium with complex refractive index coming from free space.
Consider for simplicity normal incidence as shown in Fig. 1.
Figure: 1.The reflection and transmission process expressed in terms of a
diagram.

Reflectivity
The wavevector k = koz has a z-component only, and is traveling
in the z-direction. We assume that the wave is polarized with its
Ex vector lying in the x-y plane and pointing in the x-direction.
The boundary of the two media is atz = 0, so in the region z > 0,
i.e. in the medium, the EM wave is traveling in one direction only,
and given by:
Ex (t, z) = Eo exp(iω(
Nz
c
− t)) (19)
We are assuming that the medium is thick, so that there is no
back reflected wave from a second interface. In the z < 0 region,
free space, we have both the incoming wave Ei and the reflected
wave Er :
Ex (t, z) = Ei exp(iω(
z
c
− t)) + Er exp(−iω(
z
c
− t)) (20)
The continuity requirement of the electric field at the boundary
z = 0 gives us:
Eo = Ei + Er (21)

Reflectivity
Knowing the electric field allows us to deduce the magnetic field
using Maxwell’s equation so that, for z > 0:
Hy =
−1
ωµo
(Nko)Ex (22)
and then use the continuity condition for H at the boundary, which
gives
Nko = Ei + Er (23)
Note the magnetic field at z = 0 depends on the direction of
propagation. From this pair of equations we can deduce the
relation:
Er
Ei
=
1 − N
1 − N
(24)
The ratio of reflected to incident power is the reflectivity
R = |
Er
Ei
|2 of the medium, and the squared of the absolute value of
Eq.(24) giving:

Reflectivity
R = |
1 − N
1 + N
|2
=
(n − 1)2 + k2
(n + 1)2 + k2
(25)
Thus knowing the complex refractive index as a function of
frequency, allows us to immediately calculate the reflectivity of a
medium. One should note that there is, at this stage, no simple
intuitive way of seeing from Eq.(25) when a medium is highly
reflective or not. One has to calculate the equation. In order to
develop this intuition, we need to go one step forward and actually
derive explicit expressions for the refractive index in limiting
situations of interest. Before that, it is useful and instructive to
also consider the optical transmission and reflection through a slab
of finite thickness d.

If R is the reflectivity, A the absorbance, and T the transmissivity,
for a slab of finite thickness d, we must, by energy conservation,
have R + T + A = 1. In the region z < 0, we have two waves as
before, the incoming and reflected waves Ei and Er1. In region
z > 0, inside the medium, the EM wave now also consists of 2
components, one moving forward as before Et1, and one back
reflected from the second interface Er2. The second interface is at
z = d. The waves Et1 and Er2 are traveling inside the medium and
are therefore simply related via Eq. (6) to the corresponding waves
at z = d, E
0
t1 and E
0
r2 by a e±idkoN phase factor. Outside, we have
the outgoing transmitted wave into free space Et2. The boundary
condition for the electric and magnetic field must be taken at
z = 0 and at z = d and give 4 equations for 4 unknowns
(Er1, Et1, Er2, Et2) and allow an explicit solution of this problem as
before.

The transmissivity T defined as T =|
Et2
Ei
|2 becomes:
T =
(1− | ro1 |2)2e−αd
(1− | ro1 |2 e−αd )2
(26)
(| ro1 |2)2 =|
1 − N
1 + N
|2 where α = 2
ω
c
k is the absorption coefficient
in the medium and | ro1 |2)2 can be recognized to be from Eq.
(25) the reflectivity of the slab if it were very thick. The reflectivity
of the slab R is given by the ratio |
Er1
Ei
|2 and correspondingly:
R =
|ro1|2(1 − e−αd )2
(1 − |ro1|2e−αd )2
(27)

From Eq.(26 ) and Eq.(27) one can now deduce the absorbance
A = 1 − R − T. In the limit of a very thick slab, eαd and R
reduces to the previous expression.

The free carrier contribution to the complex refractive index
(The Drude theory of conductivity)
consider a nearly free electron gas in a time dependent electric field
to be. In particular, this can be the electric field vector of an
impinging light (EM) wave as considered above.
The Newton’s law for carriers of effective mass m∗ in a time
dependent field Eoe−iωt and subject to the frictional force can be
written as
m∗ d2x
dt2
+ m∗ dx
dt
1
τ
= −qE(t) (28)
The displacement x(t) of the particle is also expected to oscillate
in time and follow the field, so that a solution to this equation
could be x(t) = xoe−iωt.

Substitute this trial function into Eq.(28) and differentiate in time.
The condition that this be a solution to Eq.(28) is that:
−m∗
ω2
xo − m∗ iω
τ
xo = −qEo (29)
which immediately allows us to extract the amplitude xo as:
xo =
qτ
m∗iω
(
1
1 − iωτ
)Eo (30)
When negative charges move against a positive background they
produce a dipole. The polarization density produced by the time
varying field is the next quantity of interest. Thus the polarization
density produced by a density nc of displaced electronic charges is
given by:
P = −ncqx(t) = −
ncq2τ
m∗iω
(
1
1 − iωt
)Eoe−iωt
(31)

from which we can now also deduce the polarizability or optical
susceptibility as the ratio:
αp(ω) =
Pc(t)
Eoe−iωt
(32)
and write:
αp(ω) = −
ncq2τ
m∗iω
(
1
1 − iωt
) (33)
And for the complex conductivity we have, from the current:
−
ncq
dx
dt
Eoe−iωt
= σ(ω) =
ncq2τ
m∗
(
1
1 − iωt
) (34)

From the polarizability, we can deduce the relative permittivity
produced by nearly free electrons, in the usual electrodynamic way
εf = (1 +
αp
εo
)
εf = 1 −
ncq2τ
εom∗iω
(
1
1 − iωt
) (35)
It is convenient and useful to rewrite the relative permittivity in a
form which involves the plasma frequency ωp and rewrite it as:
εf = 1 −
ω2
p
ω2
(
ωτ(ωτ − i)
1 + (ωτ)2
) (36)
ω2
p =
ncq2
m∗εo
(37)

The plasma frequency is the frequency at which the electron gas
would oscillate as a whole if the electrons were collectively
displaced and released from their equilibrium position.This can
happen as follows: the electrons (nc per unit volume) are all
displaced by a field by a distance x. This displacement causes a
polarizationp = ncqx, which produces an electric field and
restoring forceFR = −
nq2x
εo
. The restoring force acting on each
electron is proportional to the displacement, and we thus have
simple harmonic motion with frequency ωp =
s
ncq2
m∗εo
.

Now that we have the permittivity, we can apply it to find out a
bit more about the optical properties of systems with free charge:
metallic systems. Assume that the solid in question is a pure nearly
free electron gas embedded in a jellium. A real metal will have
both free and bound electron contributions, but the free electron
responds strongly, and this term is often dominant.
There are two interesting limits for the refractive index.
First, when ωτ << 1, the second complex term on the
right-hand-side of Eq. (36) dominates and εf (ω)reduces to:
εf (ω) ∼ i
ncq2τ
εom∗ω
(38)
the permittivity is purely imaginary, and the square root of i has an
equal real and imaginary part of cos(π
4 ) and sin(π
4 ) , giving:
n(ω) = [
ncq2τ
2εom∗ω
]
1
2 (39)

and which via Eq.(25) gives rise to a high reflection coefficient for
small frequencies.
Secondly, in the limit that ωτ >> 1, the relative permittivity is
dominated by the real part and reduces to the form:
εf (ω) ∼ (1 −
ω2
p
ω2
) (40)
In this limit the permittivity is purely real, which means that there
is no absorption. It is also negative when the frequency is smaller
than the plasma frequency. This implies that in this region, the
refractive index is purely imaginary and according to Eq. (25) we
have perfect reflectance. Perfect reflectance means that the wave
is not allowed to travel inside the medium. It can just tunnel in a
little and go back out again. The fact that the permittivity can
become less than 1, and even negative, turns out to be one of the
most significant properties of metallic systems. It gives rise to the
phenomenon of surface plasmon excitations at metal dielectric
interfaces and in metal particles.

These are collective charge oscillations which can be excited by
light, are mobile, and absorb the light very efficiently when the
energy momentum conservation laws for their production are
satisfied. Indeed when ε(ω) = 0, a transverse wave can excite a
longitudinal wave. When the frequency is above the plasma
frequency, the permittivity is real and ε < 1, it vanishes at the
plasma frequency. The refractive index in this limit becomes:
n(ω) =
s
(1 −
ω2
p
ω2
) (41)
and gives rise to an unattenuated wave which is part reflected and
part transmitted.

The bulk reflectivity of a metal can be evaluated numerically and is
given by substituting Eq. (36) into Eq. (25). The result is shown
in Fig.2.
Figure: 2. The reflectivity and transmissivity of an electron gas (thin
film).

Thank You

Optical properties of semiconductors ppt

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