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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Handout 29
Optical Transitions in Solids, Optical Gain, and
Semiconductor Lasers
In this lecture you will learn:
• Electronphoton Hamiltonian in solids
• Optical transition matrix elements
• Optical absorption coefficients
• Stimulated absorption and stimulated emission
• Optical gain in semiconductors
• Semiconductor heterostructure lasers
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Interactions Between Light and Solids
The basic interactions between light and solids cover a wide variety of topics
that can include:
• Interband electronic transitions in solids
• Intraband electronic transitions and intersubband electronic transitions
• Plasmons and plasmonpolaritons
• Surface plasmons
• Excitons and excitonpolaritons
• Phonon and phononpolaritons
• Nonlinear optics
• Quantum optics
• Optical spintronics
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Fermi’s Golden Rule: A Review
Now suppose a time dependent externally applied potential is added to the
Hamiltonian:
t
i
t
i
o
e
V
e
V
H
H
ω
↓
−
↑
+
+
=
ˆ
ˆ
ˆ
ˆ
Consider a Hamiltonian with the following eigenstates and eigenenergies:
{
integer
ˆ
=
=
m
E
H
m
m
m
o
ψ
Suppose at time
t
= 0 an electron was in some initial state
k
:
( )
p
t
=
=
0
Fermi’s golden rule
tells that the rate at which the electron absorbs energy
from
the timedependent potential and makes a transition to some higher energy level is
given by:
h
()
( )
δ
π
h
h
−
−
∑
=
↑
↑
p
m
p
m
m
E
E
V
p
W
2
ˆ
2
The rate at which the electron gives away energy
to the timedependent
potential and makes a transition to some lower energy level is given by:
h
( )
h
h
+
−
∑
=
↓
↓
p
m
p
m
m
E
E
V
p
W
2
ˆ
2
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical Transitions in Solids: Energy and Momentum Conservation
k
r
h
For an electron to absorb energy from a photon
energy conservation
implies:
( ) ( )
h
r
r
+
=
i
n
f
m
k
E
k
E
Final
energy
Initial
energy
Photon
energy
Momentum conservation
implies:
q
k
k
i
f
r
h
r
h
r
h
+
=
Initial
momentum
Final
momentum
Photon
momentum
Intraband photonic transitions are not possible:
For parabolic bands, it can be shown that intraband optical transitions cannot satisfy
both energy and momentum conservation and are therefore not possible
Intraband
Intraband
Interband
Note that the momentum conservation principle is stated in terms of the crystal
momentum of the electrons. This principle will be derived later.
E
3
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Electromagnetic Wave Basics
Consider an electromagnetic wave passing through a solid with electric field
given by:
()
( )
t
r
q
E
n
t
r
E
o
ω
−
−
=
r
r
r
r
.
sin
ˆ
,
The
vector potential
associated with the field is:
t
t
r
A
t
r
E
∂
∂
−
=
,
,
r
r
r
r
t
r
q
A
n
t
r
q
E
n
t
r
A
o
o
−
=
−
=
⇒
r
r
r
r
r
r
.
cos
ˆ
.
cos
ˆ
,
The
power per unit area
carried by the field is given by the Poynting vector:
() ()
η
2
ˆ
2
ˆ
,
,
,
2
2
2
o
o
A
q
E
q
t
r
H
t
r
E
t
r
S
P
=
=
×
=
=
r
r
r
r
r
r
r
n
o
o
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This note was uploaded on 10/21/2009 for the course ECE 4070 taught by Professor Rana during the Spring '08 term at Cornell University (Engineering School).
 Spring '08
 RANA

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