Quantum theory of dispersion of light ppt

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Quantum theory of disperstion of light
Tewodros Adaro
July 13, 2021
Tewodros Adaro Quantum theory of disperstion of light

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Table of Contents
The bound and valence electron contributions to the
permittivity
Time dependent perturbation theory
Real transitions and absorption of light
The permittivity of a semiconductor
The effect of bound electrons on the low frequency optical
properties

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The bound and valence electron contributions to the
permittivity
The bound and valence electron contributions to the permittivity
Consider now the influence of bound electrons on the optical
properties.
When bound charges are subject to an electric field, they will
also be displaced, but not freely, and not to ”infinity”, as the
frequency tends to zero.
For bound electrons, the external field is only a small
perturbation, which gives rise to polarization of the bonds and
orbits, and we can apply methods of quantum mechanical
perturbation theory.

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We consider therefore the effect of the time dependent external
field as a an additional new term in the total energy or
Hamiltonian of the system:
V(t) = −q
−
→
E .−
→
r (1)
The wave function is also written as
Ψn(−
→
r , t) = Φn(r)e
−iEnt
} (2)
with energy eigenvalues En. In the presence of the perturbation,
the electrons are no longer in their stationary sates, but can now
admix with other, higher lying excited states, and change their
orbital configurations, and in principle also undergo transitions into
these excited states.

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The change of spatial configuration is just what polarization is
in the classical sense,
And the transition into excited states is what we call
absorption of energy from the light beam.
We shall now see how polarization and absorption can be
computed in quantum mechanics.
We do this by assuming
System was in its ground state g for t < 0
The effect of the perturbation applied at t = 0, is to generate
a new electronic configuration which is a superposition of the
ground state and all the other excited states of the system.

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The new wave function is a solution of the time dependent
Schrödinger equation in the presence of the coupling term
described in Eq. (1).
So we can write for t > 0:
Ψ(−
→
r , t) = Φge
−iEgt
~ + Σn̸=gCn(t)Φne
−iEnt
~ (3)
where g denotes the ground state and n the excited states.
The next step is to determine the new admixture coeﬀicients
Cn(t).
We do this by substituting Eq. (3) into the time dependent
Schrödinger equation (Eq. (4).

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i~
∂Ψ(−
→
r , t)
∂t
= HΨ(−
→
r , t) (4)
On one side we take the derivative with respect to time to obtain:
i~
∂Ψ
∂t
= EgΦge
−iEgt
~ +Σn̸=gEnCnΦne
−iEnt
~ +Σn̸=gi~
∂Cn
∂t
Φne
−iEnt
~ (5)
on the other side of the Schrödinger equation we have
(Ho+V(t))Ψ(−
→
r , t) = EgΦge
−iEgt
~ +Σn̸=gEnCnΦne
−iEnt
~ −q−
→
r .
−
→
Eo(eiωt
+e−iωt
)Ψ(−
→
r , t)
(6)

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We now equate Eq.(5) and Eq.(6), and cancel the common
terms.
This leaves the last terms of the right-hand-side of Eq.(5) and
Eq.(6) as equal to each other.
Now we multiply the new equation on both sides with Φ∗
j e
iEjt
~
and integrate over space.
This operation eliminates all orthogonal terms, because we are
using the fact that states belonging to different eigenvalues
are orthogonal to each other We also drop all terms which
involve the product of the perturbation V(t) and a
coeﬀicientCl(t) because such terms are necessarily of second
order or above in the strength of the perturbation.

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The orthogonality rule, and the first order perturbation
approximation only leaves one term in the sum of the last term on
the right-hand-side of Eq. (6) which now gives:
i~
∂Cj
∂t
= −
∫
d−
→
r Φ∗
j (−
→
r )q−
→
r
−
→
Eo(eiωt
+ e−iωt
)e
i(Ej−Eg)t
~ Φg(−
→
r ) (7)
this can be integrated to give:
Cj(t) = −q
−
→
Eo.−
→
rjg[
1 − ei(~ω+Ej−Eg)t/~
~ω + (Ej − Eg)
−
1 − e−i(~ω+Ej−Eg)t/~
~ω − (Ej − Eg)
] (8)
where the position matrix element is:
rjg =
∫
d−
→
r Φ∗
j (−
→
r )−
→
r Φg(−
→
r ) (9)

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For simplicity we assume that the wave is polarized in the
x-direction so the first factor reduces to qEx
oxjg. Eq.(9) is,
apart from a factor q, the matrix element of the dipole
moment of the electron, it is a measure of how much the
excited state j has ground state g character mixed into it
when acted on by the position coordinate.
The matrix element of an operator Eq.(9), in this case the
displacement, −
→
rαβ is sometimes also written in the Dirac
notation < α|−
→
r |β >.
The above results now allow us to compute how the applied
field polarizes the bound electron system.

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By definition the induced time dependent dipole moment Px(t) is
given by the charge q times the expectation value of the position
operator:
Px(t) = −q
∫
d−
→
r Ψ∗
(−
→
r , t)xΨ(−
→
r , t) (10)
Substitute the solution from the wave function and keep only the
linear terms in the coeﬀicients immediately gives us:
Px(t) = −Σjq[xgjCj(t)e−iωjt
+ xjgC∗
j eiωjt
] (11)
Px(t) = Σjq2
|xgj|2
[
1
Ejo − ~ω
+
1
Ejo + ~ω
]Ex
o(eiωt
+ e−iωt
) (12)

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From the dipole moment induced by the field we can now deduce
the polarizability in the usual way:
αp(ω) = Σjq2
|xgj|2 2Ejo
E2
jo − (~ω)2
(13)
and by introducing the oscillator strength Fj:
Fj =
2mo
~2
Ejg|xgj|2
(14)
We can rewrite the ground state polarizability in an elegant form:
αp(ω) =
q2
mo
Σj
Fj
ω2
jg − ω2
(15)
with ωjg = (Ej − Eg)/~

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The significance of this expression becomes clear when we note
that the oscillator strengths obey a simple sum rule:
ΣjFj = 1 (16)
This sum rule is important. It is a check of consistency and follows
from two quantum mechanical identities. The momentum position
commutation relation:
xPx − Pxx = i~ (17)
and taking the expectation value of this equation and expanding
over a complete set of intermediate states:
i~ = Σi(xilPli,x − Pil,xxli) (18)
and using an identity from Heisenberg’s equation motion which
reads:
Pij,x = xij(Ej − Ei)mo/i~ (19)

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Substituting Eq. (19) into Eq. (18) gives the sum rule. Now we
know the bound electron polarizability, we can compute the
relative permittivity by considering the polarizability of Nb such
atoms or molecules per unit volume.
ε(ω) = 1 +
Nbq2
εomo
Σj
Fj
ω2
j − ω2
(20)
The sum now runs over the eigenstates of one such elementary
unit, i.e. an atom or a molecule. In the zero frequency limit we
have
ε(0) = 1 + Σj
Fjω2
p
ω2
j
(21)

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and in the high frequency limit when the light energy exceeds all
bound to bound transitions, we recover the corresponding Drude
result:
ε(ω) = 1 −
ω2
p
ω2
(22)
which also implies that close to the plasma frequency, the
permittivity can be negative, and the refractive index purely
imaginary implying perfect reflection.

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2. Real transitions and absorption of light
So far we have not considered what happens when the energy of
the photon matches the energy difference between two bound
levels. From Eq. ( 20), we should expect an infinite response. But
what does this mean?
When we have matching of energies we should expect the
electron to reach the excited state and the photon to be
absorbed.
In order to track such a transition mathematically we go back
to Eq. ( 8) and evaluate the probability that the particle is in
the excited state j at time t having started at t = 0 in the
ground state.

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From Eq. ( 8) we note that in the expression for Cj(t), there
are two terms, one corresponding to the possibility of
absorption, namely a resonance when ~ω = ~ωj and one
corresponding to emission. For simplicity we keep the
absorption term only so we have:
|Cj(t)|2
∼ |
qxgj
~
Ex
o|2 sin2(ωj − ω)t/2
(ωj − ω)2
(23)
The right hand term or sine function is strongly peaked at ω = ωj
and decays strongly with frequency, it is a well known function of
mathematical physics, and is best analyzed if instead of the
probability,

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we consider the probability per unit time of finding the particle in
the excited state j, that is divide by time t to study
Wgj = |Cj(t)|2/t Dividing the right-hand-side of Eq. (23) by t, and
letting time go to infinity gives us a function which we recognize to
be the well known Dirac delta function:
t −→ ∞ ⇒
sin2(~ωj − ~ω)t/2~
t~2(ωj − ω)2/4
=
2π
~
δ(~ωj − ~ω) (24)
the Dirac delta function δ(x) has the property that:
∫ ∞
−∞
dxδ(x) = 1 (25)
And also as the imaginary part of the fraction:
Im(
1
x − iη
) = πδ(x) (26)
with infinitesimal η

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So basically Eq. (23) contains the statement that the particle can
end up in an excited state if energy is conserved in the long time
limit. Although the Heisenberg uncertainty relation allows energy
not to be conserved at short times, to complete the transition, to
make a temporary admixture real, energy conservation must be
satisfied in the long time limit. We can summarize this result in
the form known as the Fermi golden rule which states that if a
particle is subject to perturbation of the form 2V(r)cos(ωt) then
the probability per unit time of finding it in an eigenstate j given
that it started in g at t = 0 is given by the formula:
Wgj =
2π
~
|
∫
d−
→
r Φ∗
j V(−
→
r )Φg|2
δ(~ω − Ej + Eg) (27)

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Now we can understand the meaning of the resonances in the
permittivity expression Eq. (20). They do indeed indicate
absorption processes, and the way to take care of the singularity is
to introduce the notion of a lifetime. Clearly when excited, the
electron can recombine back down again so it has a finite lifetime
in the excited state, and by Heisenberg uncertainty principle,
because of this time uncertainty, it has a finite energy uncertainty
or energy broadening. There is a broadening associated with each
level j and the lifetime is measured in Hz. The broadening
introduces a complex number in the denominators of
Eq. ( 8) so that the relative permittivity becomes the complex
function (T = 0 K):
εb(ω) = 1 +
Nbq2
εomo
Σj
Fj
ω2
j − ω2 − iωΓj
(28)

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This function has a real and an imaginary part. The imaginary
part, we know, is related to the absorption coeﬀicient, and this
time it is not the joule heating of free electrons as in Drude theory,
but the absorption of photons by bound electrons in the solid. We
are now in the position to write down an expression for the relative
permittivity of the solid including both boundNb and free electrons
nc:
ε(ω) = 1 +
Nbq2
εomo
Σj
Fj
ω2
j − ω2 − iωΓj
−
ncq2
εom∗
(
ωτ − i
ω2τ2 + 1
) (29)
At this stage it is also useful to generalize the bound relative
permittivity to finite temperatures, allowing the light to admix
bound levels up and admix thermally excited levels down in energy,

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to find ( Γij largest of the two widths and fl is the Fermi -Dirac
function):
εb(ω) = 1 +
Nbq2
εomo
Σi̸=j
~|xij|2(fi − fj)(ωj − ωi)
(ωj − ωi)2 − ω2 − iωΓij
(30)

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We can apply these results to a semiconductor. Consider a direct
bandgap semiconductor with no free carriers for the sake of
simplicity. In this case the bound electrons are in the valence band
and the quantum label j becomes a Bloch
−
→
k -state and the
number of orbital Nb/volume falls under the Bloch integral
−
→
k .
The transitions that the light can induce are from valence to
conduction band and involve a negligible momentum of the light
wave.
In direct bandgap materials, or for suﬀiciently high photon energy,
Eq. ( 28 ) means that the permittivity involves to a good
approximation only the vertical
−
→
k -valence to same
−
→
k -conduction
band admixtures. We also assume that the valence band is full and
the conduction band empty so that we have (T = 0K):
εs(ω) ∼ 1 +
q2
εomo
ΣkFk
1
(ωk,c − ωk,ν)2 − ω2
(31)

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where the Bloch sum over the occupied states is normalized by the
volume and defined as:
Σkν = Nb (32)
with Nb denoting the effective number of bound eigenstates per
unit volume. At ω = 0, the largest contributions in this sum are
from the band edge states, so the denominator can be replaced by
the bandgap Eg/~ and the oscillator strength for the vertical band
to band transition F−
→
k
is to a good approximation reducible under
the sum to give the total valence band electron density and
therefore the expression:
q2
moεo
ΣkFk ∼
Nbq2
εomo
= (ωb
p)2
(33)
εs(0) ∼ 1 + [
~ωb
p
Eg
]2
(34)

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where ωb
pis the effective bound electron plasma frequency and can
be obtained by comparison with experiment. It should be roughly a
factor
Eg
EB,ν
(EB,ν is the valence band width) smaller than the
absolute valence band plasma frequency. This expression is valid
for the low frequency permittivity of a semiconductor of energy gap
Eg. Given that a bandgap can typically be∼ 31014Hz, we see that
the low frequency limit can go a long way. So in the range
0 ∼ 1011Hz for example, the zero frequency form is quite
adequate, and for a doped semiconductor, the bound valence band
contribution can be combined with the free electron contribution.

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At finite temperature, the above expression is still a good
approximation in a wider gap semiconductor, but the full
generalization for finite temperature, substituting for the oscillator
strength, and including the broadening is in fact:
εs(ω) ∼ 1 +
q2
~εo
Σk|xkc,kν|2 (ω − ωkc + ωkν) − iγ
(ω − ωk,c + ωk,ν)2 + γ2
[f(Ekν) − f(Ekc)]
(35)
where the sum is now over the k index normalized per unit volume.
The xposition matrix element has to be evaluated using the
valence and conduction band Bloch functions. Fortunately and to
a good approximation, this matrix element can be calculated using
Kane theory to give us the result [Rosencher and Vinter 2002]:
|xkvkc|2
= |
∫
d−
→
r ψ∗
c (
−
→
K )Xψc(
−
→
K )|2
=
1
3
~2
E2
g
Ep
mo
(36)

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where Ep is the Kane parameter and a number which varies only
slightly between 20 and 25eV in most semiconductors.
This powerful last equation now allows us to compute the
permittivity for most situations of interest in semiconductor
physics. All we need for Eq. ( 35 ) is the density of band states
which as we know is usually well described in the nearly free
electron approximation.

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properties
properties
We have seen that bound electrons usually contribute frequency
dependence to the permittivity only at high frequencies. When we
consider both free and bound carriers we must go back and see
how one affects the other. One of the important consequences
ofεbon the free carrier response is in the regime ωτ ≫ 1 discussed
previously for free carriers only. The combined permittivity in this
regime is approximately real, but the bound electron contribution is
significant, so that the refractive index now becomes
n(ω) = [(1 + εb)(1 −
ω2
p
ω2(1 + εb)
)]1/2
(37)
or as is the notation of some other authors one can also replace:
ε(∞) = 1 + εb (38)

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properties
One can think of Eq. (38) as a renormalization of the plasma
frequency of the electrons to
ω2
p −→
ω2
p
(1+εb) This is a real effect because the electrons are now
oscillating in a medium in which the electric field of the restoring
force, is screened by the permittivity of the bound carriers.
For example GaAs: εb = 13.1 , Si: εb = 11.9 , C: εb = 5.7.
From Eq. (32), it follows that the large bandgap materials are
expected to have the lower permittivity, and this is in general
observed.

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Thank you
Thank you

Quantum theory of dispersion of light ppt

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