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EE232 Lecture 3-1 Prof. Ming Wu
EE 232 Lightwave DevicesLecture 3: Basic Semiconductor Physics
and Optical Processes
Instructor: Ming C. Wu
University of California, BerkeleyElectrical Engineering and Computer Sciences Dept.
EE232 Lecture 3-2 Prof. Ming Wu
Optical Properties of Semiconductors
• Optical transitions– Absorption: exciting an electron to a higher energy level by absorbing a photon
– Emission: electron relaxing to a lower energy state by emitting a photon
EC
EV
Absorption Emission
InterbandTransition
EA
Impurity-to-BandTransition
Intraband Transition(Free-Carrier Absorption)
2
EE232 Lecture 3-3 Prof. Ming Wu
Band-to-Band Transition
• Since most electrons and holes are near the band-edges, the photon energy of band-to-band (or interband) transition is approximately equal to the bandgap energy:
• The optical wavelength of band-to-band transition can be approximated by
λ =cν=hcEg
≈1.24Eg
λ : wavelength in µmEg : energy bandgap in eV
hv = Eg
EE232 Lecture 3-4 Prof. Ming Wu
Energy Band Diagram in Real Space and k-Space
EC
VB
E
x
E
k
CB
EV
Real Space K-Space
* 212e C eE E m v= +
* 212h V hE E m v= -
Ee = EC +!2k 2
2me*
Eh = EV −!2k 2
2mh*
Effective Mass Approximation
!k =me*ve
Momentum:
3
EE232 Lecture 3-5 Prof. Ming Wu
Band-to-Band Transition
CB
VB
hn
CB
VB
hn
CB
VB
hn hn
Absorption SpontaneousEmission
StimulatedEmission
Photodetectors;;Solar Cells LED Optical Amplifiers;;
Semiconductor Lasers
EE232 Lecture 3-6 Prof. Ming Wu
Conservation of Energy and Momentum
• Conditions for optical absorption and emission:– Conservation of energy
– Conservation of momentum
CB
VB
hn
E2
E1
kk1
k2
E
E2 −E1 = hν
( ) ( )
2 1
2 1
2 1
2, ~
2~
~ 0.5 ~1
h
h
k k k
k ka
k
a nm m
k k
n
n
p
pll µ
- =
<<
Þ =LatticeConstant
Optical transitions are “vertical” lines
4
EE232 Lecture 3-7 Prof. Ming Wu
Direct vs Indirect Bandgaps
• Direct bandgap materials– CB minimum and VB maximum occur at the same k
– Examples• GaAs, InP, InGaAsP• (AlxGa1-x)As, x < 0.45
• Indirect bandgap materials– CB minimum and VB maximum occur at different k
– Example• Si, Ge• (AlxGa1-x)As, x > 0.45
– Not “optically active”
CB
VB
hn
CB
VB
hn X
Phonon
EE232 Lecture 3-8 Prof. Ming Wu
Absorption Coefficient
• Light intensity decays exponentially in semiconductor:
• Direct bandgap semiconductor has a sharp absorption edge
• Si absorbs photons with hv > Eg = 1.1 eV, but the absorption coefficient is small– Sufficient for CCD
• At higher energy (~ 3 eV), absorption coefficient of Si becomes large again, due to direct bandgap transition to higher CB
xeIxI a-= 0)(
5
EE232 Lecture 3-9 Prof. Ming Wu
Review of Semiconductor Physics
Electron and hole concentrations:
( ) ( )
( ) ( )
Fermi-Dirac distributions:1( )
1 exp
1( )1 exp
: electron quasi-Fermi level : hole quasi-Fermi l
C
V
n eE
E
p h
nn
B
pp
B
n
p
n f E E dE
p f E E dE
f EE Fk T
f EF Ek T
FF
r
r
¥
-¥
=
=
=æ ö-
+ ç ÷è ø
=-æ ö
+ ç ÷è ø
ò
ò
evel
EE232 Lecture 3-10 Prof. Ming Wu
Electron/Hole Density of States (1)
• Electron wave with wavevector k
• Periodic boundary conditions
• An electron state is defined by
• Number of electron states between k and k + Dk in k-space per unit volume
xL
zL
yL
eik!⋅r!
eik!⋅r!
= eik!⋅ r!+Lx x!
( ) = eik!⋅ r!+Ly y!"
( ) = eik!⋅ r!+Lz z!
( )
kx ,ky ,kz( )↑↓= m 2πLx,n 2πLy,l 2πLz
#
$%%
&
'((↑↓
2 2
2
2 4 ( )2 2 2 k
x y z
k dk k dk k dkV
L L L
p rp p p p× = =
k
k k+D
xk
yk
zk
Spin Up and Down
6
EE232 Lecture 3-11 Prof. Ming Wu
Electron/Hole Density of States (2)
• Number of electron states between E and E + DE per unit volume
• Likewise, hole density of states
E = EC +!2k 2
2me*⇒ dE = !
2kme*dk
k 2
π 2dk = me
*
!2π 22me
* E − EC( )!
dE = ρe (E)dE
ρe (E) =12π 2
2me*
!2!
"##
$
%&&
3/2
E − EC
ρh (E) =12π 2
2mh*
!2!
"##
$
%&&
3/2
EV − E
EE232 Lecture 3-12 Prof. Ming Wu
Electron and Hole Concentrations
n = fn (E)ρe (E)dEEC
∞
∫ =1
1+ expE − FnkBT
!
"#
$
%&
12π 2
2me*
!2!
"##
$
%&&
2
E − EC dEEC
∞
∫
n = NC ⋅F1/2Fn − ECkBT
!
"##
$
%&&
NC = 2πme
*kBT2π 2!2
!
"##
$
%&&
3/2
p=NV ⋅F1/2EV − FpkBT
!
"##
$
%&&
NV = 2πmh
*kBT2π 2!2
!
"##
$
%&&
3/2
( )0
Fermi-Dirac Integral
1 ( 1) 1
Gamma Function
3( )2 2
j
j x
xF dxj e hh
p
¥
-=G + +
G =
ò
7
EE232 Lecture 3-13 Prof. Ming Wu
Approximation of Electron/Hole Concentration
Fj η( ) = 1Γ( j +1)
x j
1+ ex−η0
∞
∫ dx ≈ eη when η <<1
43η3
π
&
'(
)
*+
1/2
when η >>1
,
-..
/..
When Fn << EC (Boltzmann approximation)
n ≈ NC ⋅e−EC−FnkBT
When Fn >> EC (Degenerate)
n ≈ NC ⋅4 Fn − EC
kBT
&
'(
)
*+
3/2
3 π
Fn − ECkBT
C
nN
Boltzmann Approx
Degenerate
Quasi-FermiLevel Cross Band Edge