A nonstandard way to deal with it Assumes confidence with Basic S olid S tate Physics Crystals Electrons Holes directindirect Band Gap Their foundation on the ID: 635356
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Slide1
Semiconductors and Light
A non-standard way to deal with
it
Assumes
confidence
with
Basic
S
olid
S
tate
Physics
:
Crystals
Electrons
,
Holes
,
direct-indirect
Band Gap
Their
foundation
on the
Blochโs
Theorem
p
n
junction
DC
characteristics
Basic light-
matter
interaction
theory
Quantum-
mechanic
perturbation
theory
Spontaneous
and
stimulated
emission
,
absorptionSlide2
Semiconductors and Light
Does
NOT
aim
to be complete
Aims
to
build
up a self-
consistent
view
of:
Electron,
Hole
and
Photon
densities
Their
link with
measurable
quantities
: I, V, P
Their
call for
technological
solutionsSlide3
Semiconductors and Light
It
gives
a new
insight
on:
Threshold
current
Gain and
loss
coefficients
The
theoretical
foundation
of
known
empirical
formulas
It
should
be
compared
with
literature
.
Coldren
LA, Corzine SW,
Maลกanoviฤ
ML.
Diode lasers and photonic integrated circuits
, Wiley series in microwave and optical engineering.
Jhon
Wiley & Sons, Inc., Hoboken, New Jersey; Second Edition 2012
.
J.T.
Verdeyen
.
Laser Electronics
. Third Edition, Prentice Hall, 1995. Slide4
Semiconductors and Light
Absorption
Emission
Transparency
Refraction
Partial
reflection
Easily
explained
by
Eg:
Absorbption
Transparency
E
g
n
Refraction
Partial
ReflectionSlide5Slide6
Diamond
has
the
closest
lattice
spacing
It
is
Mechanically
Hard
Electrically
InsulatingOptically Transparent
Optically RefractiveSlide7
โฆ
but
Light
Emission
is
another
tale
Fermi Golden
Rule
(holds for any system
)
Momentum
Selection Rule
(specific for crystals)
High
probability
Low
probability
Si excluded from light emitting devices
Recombination
rate
is no more
It
must
obey
the
selection
rulesSlide8
Something
may
be
anticipated
about
light
emission in direct gap semiconductors
Many
electrons
Many
holes
Many
photons
No
electrons
No
holesNo photons
Few
electrons
Few holes
Few photons
h
๐
E
g
๐
๐
=
photon
density
Typical
40kT
Typical
โkTSlide9
How
t
o
get
many
electrons
and many holes
together?
By
injecting
a forward current into a pn
junction
But
in an
ideal diode
, they meet inside the depletion layer without recombining
.Then recombine, separately, entering the neutral regionsSlide10
The
practical
solution
is
to force e-h
recombination
inside the depletion layer introducing
a thin layer at
smaller bandap between the p and n
regions.
In a
well designed
device, a forward bias V will cause a current I to flow and an
optical power POUT
to leave the structure.
This is a Light Emitting Diode
= LEDSlide11
But
something
more
may
be
anticipated about light emission
in direct gap semiconductors
๐
n
๐
p
More
empty
than
filled
states
More
filled
than
empty
states
More
filled
than
empty
states
More
empty
than
filled
states
Possibility
for
population
inversion
More top-down (
emitting
)
than
bottom-up (
absorbing
)
transitions
stimulated
by light.
GAIN = Light
Amplification
The LED
achieves
L
ight
A
mplification
by
S
timulated
E
mission
of
R
adiation
This
is
a LASER DIODESlide12
Form qualitative to quantitative:
we
should
:
List
all
mechanisms
involving photons and electrons and
holesBalance them
Bring optical properties properties inside the world of
diodes:Spectrum
๐๐, gain g, loss
ฮฑ, optical power
POUT
Include concurring phenomena non involving photons
Find lumped equations for V, I and POUT
Bring
diode properties
properties inside the world of lasers:Current I, Voltage V
The starting point will be a Rate Equation for photonsSlide13
Our
program
:
Consider
a Double
Heterostructure
ideal
diode
made of direct gap semiconductor
where all
recombinations are radiativeand happens only
inside the central active
layer
qV
p
n
depletion layer
active layer
Inside
that
layer
all
electron,
holes
and
photon
densities
are
uniform
We
will
look first for the
electro-optical
characteristics
of
such
ideal
diode
a
nd
later
on,
we
will
include
Non-radiative
recombination
inside the
layer
Other
recombinations
and
currents
outside
the
layerSlide14
A
quick
recall
of the
Einsteinโs
treatment of Black Body
radiation
(1905)
Search for the spectral
power density
u๐ at
equilibrium Fermiโs Golden rule not
yet discovered
Classical results from Thermodynamics
Stefanโs
Law
Wienโs
(
Displacement
) Law
Rayleigh
and Jeansโ
Ultraviolet
Catastrophe . But
good for low ๐.
No Fermi-
Dirac
or Bose-Einstein
statistics
available
:
only
Boltzmann
. No
exclusion
principle
No quanta.
Planck
(1901)
not
sure
of
their
existence
. Einstein
going
to
explain
the
phototelectric
effect
on the
same
year
(1905)Slide15
E
2
g
2
Level 2
Level 1
2-level
system
with N
0
particles
E
nergy
Density
of
states
Population
E
1
g
1
Rate of
spontaneous
2โ1 transitions
Rate of
stimulated
2โ1
transitions
Rate of
stimulated
1โ
2
transitions
Coefficients
A and B can
depend
on
๐
but
not
on T
At
equilibrium
2โ1 = 1 โ
2Slide16
At high temperature:
Do
not
change
with T
Go to
unity
Increase
with T
4
Do
not
include T
Wienโs
(
Displacement
) LawSlide17
Must be (
Rayleigh
and Jeans):
Planckโs
LawSlide18Slide19Slide20
Top-down
optical
transitions
proportional
to
Bottom-up
optical
transitions
proportional
toSlide21Slide22Slide23
The
diode
at
equilibrium
must
give
back the Black Body formula
Conduction
-to-
valence
spontaneous
transitions (e-h recombination,
photon emission)
Conduction-to-
valence stimulated transitions(e-h recombination, photon emission)
Valence-to-conduction stimulated transitions (e-h generation,
photon absorption)
At
equilibrium
V=0
As
for
Einsteinโs
treatment:Slide24
Out of
equilibrium
,
rates
can be
not
constant
, Vโ 0 and an escape term must be introduced
, that must vanish at
equilibrium
In the steady state:
w
e
can solve
for
Using the
previous
forms
Slide25
where
h
๐
E
g
๐
๐
=
photon
density
๐
0๐
B
may
be
and
๐
0
๐
is
a slow
function
of h
๐.
We
can
safely
assume for
both
:
B and
๐
0
๐
are
constant
Is
the joint
density
of
states
for
electrons
and
holes
.
It
is
null
for
a
nd
is
linear
in for
thick
layers
and
constant
for
thin
layers
(the
normal
case)
๐
C
is the average permanence time of radiation inside the active layer. It is a function of
๐
because of refraction and resonances. But we start keeping it constant.Slide26
Let
us
suppose
E
g
= 1
eV
and qV=0.5 eV
h
๐
E
g
๐
๐
=
photon
density
calculated
Expected
(qualitative)
Linear scale
Thick
layer case. Linear
vertical scale. Pay attention to the
values
in the
abscissaSlide27
Let
us
now
change
qV:
Logarithmic
scale
Thick
layer case. Log vertical scale.
Pay attention
to the values of qVSlide28
Thin
layer
case. Log
vertical
scale
.
Pay
attention to the values of
qV
Logarithmic
scale
At
qV
=1.02
eV
Something happens when
qV approaches EgSlide29
For
possible
emission
,
we
have
For
e
xponentials
at
the denominator are
extremely
smallSlide30
The
total
optical
power
i
ncreases
unlimited
Infinite
energy
is
required
to
further increase qV.
But the denominator vanishes
as
Voltage
clamps
at a threshold value
t
hat
is
the minimum
among
the
ones
allowed
For a
thin
layer
this
isSlide31
What
is
it
the happening?
R
sp
R
st
R
abs
R
esc
qV
th
E
g
Spontaneous
emission
and
absorption
dominate
o
ver stimulated emission. It is
the LED regime.
Stimulated
emission
balances
absorption
.
It
the
the
transparency
condition
.
Stimulated
emission
overcomes
absorption
.
Light
starts
to be
amplified
(super-
radiance
)
Spontaneous
emission
is
blocked
. Voltage
is
calmped
.
Stimulated
emission
dominates
.
It
is
the
LASER regime
.Slide32
P
TOT
qV
The more the
bathtube
is
filled
, the
higher
is
the output
flux
But
when
the
edge
is
reached
, the
flux
can
increase
without
increasing
the water
levelSlide33
Current in the ideal
laser
diode
The
current
is
q
times the net recombination rate.
For the
ideal LED/Laser diode, where recombination
is always radiative
h
๐
E
g
๐
๐
=
photon
density
Integration
factor
.Slide34
Shockley
region
(LED)
Laser
region
V<
V
th
: LED
range
V=
V
th
: Laser
range
Equation of an
ideal
diode
with
saturation
current
=
This
is
the DC transfer
function
I(V) of an
ideal
laser
diode
.Slide35
Optical
Power
in the
ideal
laser
diode
Optical
Power
=
Photon Density x
Photon Energy x Volume / Lifetime
But
:
Collection
efficiency
(
coupling+conversion
+โฆ)
I
ph
P
OUTSlide36
measured
Optical
Power
in the
real
laser
diode
I
ph
is
not
the
only
current
For V<
Vth non-radiative recombination dominates
A non-radiative current Inr exists
The
total
current
is made of I= Inr
+ IphAs
qV clamps at qV
th
,
I
nr
stops
, and
I
ph
grows
alone
A
threshold
current
I
th
defines
the
transiton
In
practise
:
qV
th
I
th
Quantum
Efficiency
:Slide37
Threshold
currentSlide38
Collected
escaping
power
Total
escaping
power
Conversion
efficiency
(
often
omitted
)
My opinion: a
big
mistake.
It must go to unity for I>
Ith
f
or
any
V,I
f
or V>
V
th
(
that
is
I>
I
th
)
What
do
textbooks
tell
?Slide39
Lumped
equations
for the DC regime
I(V), P
OUT
(V) โ
direct
substitution
P
OUT
(I) โ eliminate
It
comes
out a
huge
formula,
but it
is
analytic under the form I(
POUT)calculates and embeds the threshold condition
deals with the sub-thresholdregimeSlide40