1 review problem allowed modes b region of possible oscillations a)find the photon lifetime of the...
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1Review Problem
22 0.98
2
d
2
d
d
60cmf
1
2 0.90 23 0.98
1 2 0.98
Scattering Loss at
lens =3% per surface
T T
0
gl
1T
2T
0
0
0 ( )
/ 2F c d
1 ... M
allowed modes
- Loss per unit of length
- Gain per unit of length
B
Region of possible oscillations
1/ 2
a) Find the photon lifetime of the passive cavity; that is, with 0() = 0.
b) What is the passive cavity Q?
c) What is the free spectral range of the cavity?
d) What is the minimum gain coefficient 0(0) necessary to sustain oscillations in this cavity?
e) If the gain coefficient 0(0) were 2 x 10-2 cm-1 (at line center) and the line shape of the transition was approximated by a Lorentzian with h = 1.5 GHz, then how many TEM0,0,q modes are above the threshold?
f) What is the stimulated emission cross section (at line center)?
g) What is the absorption cross section?
h) The characteristic beam parameter z0 is related to the dimensions of the cavity and the focal length of the lens by
1) Where is z = 0 in the cavity?
2) Find a formula for the resonant frequency of the TEMm,p,q mode.
3) What is the difference in resonant frequencies (in MHz) of the TEM 0,0,q and the TEM1,0,q modes?
5 121
2
1
50 cm
30cm
0.8μm
1.5GHz
10 sec
1
2
g
h
d
l
A
J
J
1/ 22
1/ 200
33 1
4
w dz df
f
2Introduction to Optical Electronics
Quantum (Photon) Optics (Ch 12)
Resonators (Ch 10)
Electromagnetic Optics (Ch 5)
Wave Optics (Ch 2 & 3)
Ray Optics (Ch 1)
Photons & Atoms (Ch 13)
Laser Amplifiers (Ch 14)
Lasers (Ch 15) Photons in Semiconductors (Ch 16)
Semiconductor Photon Detectors (Ch 18)
Semiconductor Photon Sources (Ch 17)
Optics Physics Optoelectronics
3Semiconductors
201
212
1
2
1R
2R 1iW
E1
E2
h21
k
E
Ec
Ev
h
Photons & Atoms
• Distinct Energy Levels
• Probability: Boltzmann
• Gain: Population Inversion
Photons in Semiconductors
• Energy Bands
• Probability: Fermi-Dirac
• Gain: Quasi-Fermi Energies Efc – Efv > Eg
21
21
E1
E2
4Review of Quantum Mechanics
• Free Electron Theory of Solids
• Free Electrons are Waves, (r,t)
• Obey Schrodingers’ Equation
– Time-Independent Schrodinger’s Equation
22( , ) ( , )
2j r t r t
m
22 ( ) ( )
2r E r
m
5Free-Electron’s Energy Spectrum
E
E dE Number of states: ?EN dE
Number of states: ?
Volume EdE
6Fermi Gas
0E
0T K
fE E fE
10 f E
7Fermi Gas
0E
0T K
fE E fE
10 f E
( ) /
1
1f BE E k Tf Ee
8Band Theory of Solids
1n
2n
3n
infn
antibondbond
1 Hydrogen Atom 2 Hydrogen Atoms 3 Hydrogen Atoms N Hydrogen Atoms
N levels
N levels
N levels
9Energy BandsConduction & Valence
Ele
ctro
n E
ner
gy E
ConductionBand
ValenceBand
Bandgap energy Eg
Electron
Hole
En
erg
y Bandgap Energy • Eg=1.11 eV (Si)• Eg=1.42 eV (GaAs)
Conduction Band
Valence Band
10
Direct & Indirect Semiconductor Bandgaps
Ec
Ev
Ec
Ev
Eg=1.11 eV
Silicon (Si)
Eg=1.42 eV
[111] [100]k
E
[111] [100]k
EGallium Arsenide (GaAs)
k
E
k
E
cE
cE
11Semiconducting Materials
II III IV V VIAluminum (Al) Silicon (Si) Phosphorus (P) Sulfur (S)
Zinc (Zn) Gallium (Ga) Germanium (Ge) Arsenic (As) Selenium (Se)
Cadmium (Cd) Indium (In) Antimony (Sb) Tellurium (Te)
Mercury (Hg)
Al
Ga
In
P
As
Sb
Al
Ga As
Ga
In
P
As
x
1-x
x
1-x
y
1-y
Binary Ternary QuaternaryElemental
Si
Ga
12Lattice Constants
13Density of Statesnear the band edges
3/ 2
1/ 2
2 3
2( ) ,
2c
c c c
mE E E E E
3/ 2
1/ 2
2 3
2( ) ,
2v
v v v
mE E E E E
E
k
E
Ec
Ev
Eg
d
Ec
Ev
E
Allowed energy levels(at all k)
Ec
Ev
Density of states
( )v E
( )c E
14Semiconductor’s Density of States
Light HoleHeavy Hole
Electrons
3/ 2
1/ 2
2 3
2( ) ,
2c
c c c
mE E E E E
3/ 2
1/ 2
2 3
2( ) ,
2v
v v v
mE E E E E
15
Fermi-Dirac Distribution f(E)
Ec
Ev
EfEg Ef
Ec
Ev
Ef
Ec
Ev
0.50 1 0.50 1
f(E)
T > 0 KT = 0 KE E
1-f(E)
f(E)
1( )
exp 1f
B
f EE E
k T
16
Carrier Concentration (n & p)
Ec
Ev
EfEg
( )p E
( )n E
( ) ( ) ( )c c
c
E E
n n E dE E f E dE
( ) ( ) 1 ( )v vE E
vp p E dE E f E dE
( )f E 1 ( )f E
17
n- and p-type Semiconductors
Ef
0 1 f(E)Carrier
concentration
Ec
Ev
Ef
Carrierconcentration
Ec
Ev
( )p E
( )p E
( )n E
( )n E
0 1 f(E)
ED
EA
E
E
Donor level
Acceptor level
18Exercise 16.1-2Exponential Approximation of the Fermi Function
When ,f BE E k T the Fermi function f(E) may be approximated by an exponential
function. Similarly, when , 1 ( )f BE E k T f E may be approximated by an
exponential function. These conditions apply when the Fermi level lies within the bandgap, but away from its edges by an energy of at least several times Bk T (at room
temperature 0.026 eVBk T whereas 1.11eVgE in Si and 1.42 eV in GaAs). Using
these approximations, which apply for both intrinsic and doped semiconductors, show that (15.1-8) gives
exp
exp
exp
c fc
B
f vv
B
gc v
B
E En N
k T
E Ep N
k T
Enp N N
k T
where 3/ 222 2 /c c BN m k T h and 3/ 222 2 /v v BN m k T h . Verify that if fE is closer
to the conduction band and , then v cm m n p whereas if it is closer to the valence band,
then p n
19
Semiconductors
Density of StatesProbability of Occupation
Concentration of Carriers
Concentration of Carriers
(Approximation)
3/ 2
1/ 2
2 3
2( )
2c
c c
mE E E
cE E
3/ 2
1/ 2
2 3
2( )
2v
v v
mE E E
vE E
( ) /
1( )
e 1f BE E k Tf E
1 ( )f E
( ) ( )c
c
E
n E f E dE
( ) 1 ( )vE
vp E f E dE
exp c fc
B
E En N
k T
3/ 222 2 /c c BN m k T h
if f BE E k T
exp f vv
B
E Ep N
k T
3/ 222 2 /v v BN m k T h
if f BE E k T
Law of Mass Action:2in p n
/1/ 2where the intrinsic carrier conce (ntration )is g BE k T
i c vn N N e
20
Quasi-Equilibrium Carrier Concentrations
Ec
Ev
Efc
Eg
( )p E
( )n E
Efv Efv
Ec
Ev
22/32 2/3
/
22 /32 2/3
/
1for : where 3
21
1for : 1 1 where 3
21
fc B
fv B
fc cE E k Tc
fv vE E k Tv
n E f E E E nme
p E f E E E pme
21Exercise 16.1-3Determination of the Quasi-Fermi Levels Given
the Electron and Hole Concentrations
(a) Given the concentrations of electrons n and holes p in a semiconductor at T = 0 K, use (15.1-7) and (15.1-8) to show that the quasi-Fermi levels are
22/32 2/3
22/32 2/3
32
32
fc cc
fv vv
E E nm
E E pm
(b) Show that these equations are approximately applicable at an arbitrary
temperature T if n and p are sufficiently large so that fc c BE E k T and
v fv BE E k T , i.e., if the quasi-Fermi levels lie deeply within the conduction
and valence bands.
22
Electron-Hole Generation & Recombination
Ec
Ev
Generation Recombination
Ec
Ev
Trap
0
1where
( )o
nR
r n p n
23Exercise 16.1-4Electron-Hole Pair Injection in GaAs
Assume that electron-hole pairs are injected into n-type GaAs
0 0( 1.42eV, 0.07 , 0.5 )g c vE m m m m at a rate 23 310 per cm per second.R The
thermal equilibrium concentration of electrons is 16 30 10 .n cm If the recombination
parameter 11 310 / and 300 , determine:r cm s T K (a) The equilibrium concentration of holes 0p .
(b) The recombination lifetime . (c) The steady-state excess concentration n (d) The separation between the quasi-Fermi levels fc fvE E , assuming that 0T K
24How to Handle an Inverted SemiconductorVerdeyen’s Approach
1/ 23/ 2
2 /
21( ) ( ) ( )
2 1fc B
c c c
ccc E E k T
E E E
E Emn n E dE E f E dE dE
e
1/ 23/ 2
2 /
21( ) ( ) 1 ( )
2 1
v v v
fv B
E E Evv
v E E k T
E Emp p E dE E f E dE dE
e
/
/
or
/
fv v B
c fc B
c v
B
E E k T
E E k T
u E E u E E
x u k T
a e
b e
Setting:
25Inverted Semiconductor Example: GaAs
3/ 2 1/ 2
/
2 20
21
2 1/c fc BE E k Te B
x
m k T xn e dx
e b
3/ 2 1/ 2
/
2 20
21
2 1/fv v BE E k Te B
x
m k T xp e dx
e a
1/ 2
2
0
2let
1/ ,x
xI dx
e a b
/
/
fv v B
c fc B
E E k T
E E k T
a e
b e
26Semiconductors
Density of States
Probability of Occupancy
Carrier Concentrations
Law of Mass Action
3/21/2
2 3
3/21/2
2 3
2,
2
2,
2
cc c c
vv v v
mE E E E E
mE E E E E
p
p
/
1
1f BE E k Tf E
e
1v
c
E
c v
E
n E f E dE p E f E dE
p p
2 where exp2
gi i c v
B
En p n n N N
k T
27
Generation, Recombination & Injection
Rate of Recombination
Recombination Lifetime,
Internal Quantum Number
(low concentrations)
0 0 0 0
0
0
where G
and
and
r n p R G r n p
n n n
p p n
0 0
1where
( )
nR
r n p n
0 0
1 where
( )
r r
r nri
nr
r r nr
r r
r r r
r n p
28
Semiconductor Fermi Energy Levels
p n
Ca
rrie
rC
on
cen
tra
tion
Ele
ctro
n E
ne
rgy
Before Contact
p
n
n
p
Position
E0Neutral p Neutral n
After Contact
DepletionLayer
eV0
Ca
rrie
rC
on
cen
tra
tion
Ele
ctro
n E
ne
rgy
p(x)
n(x)
n(x)
p(x)
x
29
Forward-Biased p-n JunctionForward Bias
E0 - ENeutral p Neutral n
eV
Ca
rrie
rC
on
cen
tra
tion
Ele
ctro
n E
ne
rgy
p(x)
Excesselectrons
n(x)
Excess holes
x
+_
e(V0-V)
V
Efc
Efv
n
p
E0Neutral p Neutral n
Neutral
eV0
Ca
rrie
rC
on
cen
tra
tion
Ele
ctro
n E
ne
rgy
p(x)
n(x)
n(x)
p(x)
x
Ef
30Current-Voltage Characteristics of an ideal p-n Junction Diode
is
i
V
i
_
+
V
i p
n_
+
V
exp 1sB
eVi i
k T
31PIN Diodein Thermal Equilibrium
p i n
Depletion layer
Electric Field
Ec
Ec
Electronenergy
-
+x
Fixed-chargedensity
x
Electric-field magnitude
32
Photon Absorption & Emission Mechanics
Eg=1.42 eV
Ec
Ev
Band-to-Band Transitions
EA = 0.088 eV Eg=0.66 eV
Acceptor-LevelTransition
Free-CarrierTransition
33Absorption
c E
v E
Intensity
34Stimulated Emission
c E
v E
35Absorption Phenomenon
36Band-To-Band Photon Interactions
E
Ec
Ev
E1
E2
k
h
k
h
k
h h
h
ih
i
h
ih
Absorption SpontaneousEmission
StimulatedEmission
37Optical Joint Density of States
3/ 2
1/ 2
2 3
2( ) ,
2c
c c c
mE E E E E
3/ 2
1/ 2
2 3
2( ) ,
2v
v v v
mE E E E E
2 2( ) ( )d E dE
22
3/ 21/ 22
2
( ) ( )
2,g g
dEE
d
mh E h E
2 ( )rc g
c
mE E h E
m
1 2( )rv g
v
mE E h E E h
m
Eg
h
38Band-To-Band Photon Interactions
Photon AbsorptionIndirect-gap Semiconductor
Photon EmissionIndirect-gap Semiconductor
k
Photon
Phonon
k
Thermalization
PhotonAbsorption
h
39Exercise 16.2-1
Requirement for the Photon Emission Rate to Exceed the Absorption Rate
(a) For a semiconductor in thermal equilibrium, show that ( )ef is always smaller
than ( )af so that the rate of photon emission cannot exceed the rate of photon
absorption.
(b) For a semiconductor in quasi-equilibrium fc fvE E , with Radiative transitions
occurring between a conduction-band state of energy 2E and a valence-band state
of energy 1E with the same k, show that emission is more likely than absorption if
the separations between the quasi-Fermi levels is larger than the photon energy, i.e., if
fc fE E h
What does this condition imply about the locations of fcE relative cE and fE ,
relative to E ?
40Spontaneous Emission Spectral Densityin Thermal Equilibrium
Eg . . . . .h
Eg
rsp()
kBT
1/ 2
0( ) exp ,gg
sp gB
h Eh Er D h E
k T
3/ 2
0 2
2where exp gr
r B
EmD
k T
41
0 1 2
0.5x104
104
Absorption Coefficientin Thermal Equilibrium
h- Eg
()
(cm
-1)
1/ 2
1 1 2( ) ( ) ( )gD h E f E f E
3/ 2 2
1 2
2where r
r
mD
h
1
( ) ( ) ( )exp / 1
c v
f B
f E f E f EE E k T
42Exercise 16.2-2Wavelength of Maximum Band-to-Band Absorption
Use
2 3/ 2
1/ 2
2
2 1rg
r
c mh E
h
(15.2-28)
to determine the (free-space) wavelength p at which the absorption coefficient of a
semiconductor in thermal equilibrium is maximum. Calculate the values of p for GaAs.
Note that this result applies only to absorption by direct band-to-band transitions.