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EE 232: Lightwave Devices
Lecture #11 – Gain in quantum wells
Instructor: Seth A. Fortuna
Dept. of Electrical Engineering and Computer Sciences
University of California, Berkeley
2/25/2019
2Fortuna – E3S Seminar
Transparency condition for quantum well(estimation)
2
0
( )D
subband
n
n zs
n N f E dEL
= = 2
0
*
1 exp[( ) / ]
Let ( ) /
en
z c
c
m dEN
L E F kT
x E F kT
=+ −
= −
( )2
( )/
( )
*
/
2
*
2
*
[
1
)]
) / ]
ln(1
ln 1 exp[(
g en c
g en c
en x
z
xe
z
en
E E F kT
E E
n
F k
c
z
T
g e
m dxN
L e
m kT
L
m kTN E E kF T
L
x e
−
+
+ −
=+
=
= −
− +
+ −
( )*
21 )ln ex /p ][(h
m hm v
z
m kTP E F kT
L= + −
similarly,
(total carrier density) (carrier density within single subband)
3Fortuna – E3S Seminar
Transparency condition for quantum well(estimation)
If only one subband is filled,
2
*
2
*
ln exp 1
ln exp 1
zc g en
mv h
h
z
e
F nL
kT Em kT
LF kT E
m k
E
pT
− +
− +
=
+
= −
en
c v hmF F E− = (transparency condition)
After several lines of algebra, and assuming quasi-neutrality (n=p), we obtain an equation in terms of transparency carrier density
/ /1tr c vtrn n n n
e e− −
+ =
where,*
2
*
2
z
z
ec
hv
m kTn
L
m kTn
L
=
=
n
18 310 cmtrn −=
L 8nmz =
InGaAs
//
tv
trc
rn
nn
ne
e−
−+
4Fortuna – E3S Seminar
0.5( )vf E
E
1
Gain spectrum (T=0K)
0
Fermi inversion factor
0
1 1C HH−
1 1C LH−
E
cF
vF
1HH
1LH
1 1C HH−
) (
()
(
(
)
)p c v
g
g ff
=
−
−
=
0.5 1( )cf E
1C
1 1C LH−
1
1
e
hEvcF F−
5Fortuna – E3S Seminar
0.5( )vf E
E
Gain spectrum (T=0K)
0
Fermi inversion factor
0
1 1C HH−
1 1C LH−
E
cF
vF
1HH
1LH
1
1 1C HH−
) (
()
(
(
)
)p c v
g
g ff
=
−
−
=
0.5 1( )cf E
1C
1 1C LH−
1
1
e
hEvcF F−
6Fortuna – E3S Seminar
0.5( )vf E
E
1
Gain spectrum (T=0K)
0
Fermi inversion factor
0
1 1C HH−
1 1C LH−
E
cF
vF
1HH
1LH
1 1C HH−
) (
()
(
(
)
)p c v
g
g ff
=
−
−
=
0.5 1( )cf E
1C
1 1C LH−
1
1
e
hEvcF F−
7Fortuna – E3S Seminar
0.5( )vf E
E
1
Gain spectrum (T=0K)
0
Fermi inversion factor
1
1
e
hEvcF F−
0
1 1C HH−
1 1C LH−
E
cF
vF
1C
1HH
1LH
1 1C HH−
) (
()
(
(
)
)p c v
g
g ff
=
−
−
=
0.5 1( )cf E
1 1C LH−
8Fortuna – E3S Seminar
0.5( )vf E
E
1
Gain spectrum (T=0K)
0
Fermi inversion factor
0
1 1C HH−
1 1C LH−
1 1C HH−
1 1C LH−
E
cF
vF
1HH
1LH
) (
()
(
(
)
)p c v
g
g ff
=
−
−
=
0.5 1( )cf E
1C
1
1
e
hEvcF F−
C1-LH1 transition starts contributing to absorption
9Fortuna – E3S Seminar
0.5 1
0.5
( )cf E
( )vf E
E
1
Gain spectrum (T=300K)
0
Fermi inversion factor
0
1 1C HH−
1 1C LH−
E
cF
vF
1HH
1LH
) (
()
(
(
)
)p c v
g
g ff
=
−
−
=
1 1C HH−
1 1C LH−
1C
1
1
e
hEvcF F−
10Fortuna – E3S Seminar
0.5 1
0.5
( )cf E
( )vf E
E
1
Gain spectrum (T=300K)
Fermi inversion factor
0
1 1C HH−
1 1C LH−
E
cF
vF
1HH
1LH
) (
()
(
(
)
)p c v
g
g ff
=
−
−
=
1 1C HH−
1 1C LH−
1C
1
1
e
hEvcF F−
0
11Fortuna – E3S Seminar
0.5 1
0.5
( )cf E
( )vf E
E
1
Gain spectrum (T=300K)
Fermi inversion factor
1
1
e
hEvcF F−
0
1 1C HH−
1 1C LH−
E
cF
vF
1HH
1LH
) (
()
(
(
)
)p c v
g
g ff
=
−
−
=
1 1C HH−
1 1C LH−
1C
0
12Fortuna – E3S Seminar
0.5 1
0.5
( )cf E
( )vf E
E
1
Gain spectrum (T=300K)
Fermi inversion factor
0
1 1C HH−
1 1C LH−
1 1C HH−
1 1C LH−
E
cF
vF
1C
1HH
1LH
1
1
e
hEvcF F−
0
) (
()
(
(
)
)p c v
g
g ff
=
−
−
=
13Fortuna – E3S Seminar
0.5( )vf E
E
1
Gain spectrum (T=300K)
0
Fermi inversion factor
0
1 1C HH−
1 1C LH−
1 1C HH−
1 1C LH−
) (
()
(
(
)
)p c v
g
g ff
=
−
−
=
E
cF
vF
1HH
1LH
0.5 1( )cf E
1C
1
1
e
hEvcF F−
14Fortuna – E3S Seminar
Gain spectrum
Fermi inversion factor
1 1C HH−
1 1C LH−
1 1C HH−
1 1C LH−
) (
()
(
(
)
)p c v
g
g ff
=
−
−
=
1K77K
300K
1K77K
300K
0.85eVF =
InP/In0.53Ga0.47As quantum well(Transverse electric mode)
6 nmzL =
0.85eVF =
Bandgap temperature dependence is ignored here
15Fortuna – E3S Seminar
Gain spectrum
Fermi inversion factor
1 1C HH−
1 1C LH−
1 1C HH−
1 1C LH−
) (
()
(
(
)
)p c v
g
g ff
=
−
−
=
InP/In0.53Ga0.47As quantum well(Transverse electric mode)
6 nmzL =
.
1.0
.
0.
0 9
0
6
8
F
F
F
F
=
=
=
=T=300K
.
1.0
.
0.
0 9
0
6
8
F
F
F
F
=
=
=
=
T=300K
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