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EE232 Lecture 14-1 Prof. Ming Wu
EE 232 Lightwave DevicesLecture 14: Quantum Well andStrained Quantum Well Laser
Reading: Chuang, Sec. 10.3-10.4(There is also a good discussion in Coldren, Appendix 11)
Instructor: Ming C. Wu
University of California, BerkeleyElectrical Engineering and Computer Sciences Dept.
EE232 Lecture 14-2 Prof. Ming Wu
Quantum Well Gain1.4 1.6 1.8 2 2.2 2.4 2.61-
0.5-
0
0.5
1
f_g xi 0.05eV, ( )f_g xi 0.1eV, ( )f_g xi 0.15eV, ( )f_g xi 0.2eV, ( )f_g xi 0.25eV, ( )f_g xi 0.3eV, ( )f_g xi 0.35eV, ( )f_g xi 0.4eV, ( )f_g xi 0.45eV, ( )
xieV
1.4 1.6 1.8 2 2.2 2.4 2.6
1- 106´
0
1 106´g xi 0.05eV, ( )g xi 0.1eV, ( )g xi 0.15eV, ( )g xi 0.2eV, ( )g xi 0.25eV, ( )g xi 0.3eV, ( )g xi 0.35eV, ( )g xi 0.4eV, ( )g xi 0.45eV, ( )
xieV
QW Material Gain:
g(!ω) =C0 e! ⋅P!"cv
2
ρr2d (!ω) fg (!ω)
C0 =πe2
nrcε0m02ω
e! ⋅P!"cv
2
≈m0
6Ep
ρr2d (E) =
mr*
π!2LzH (E − Een )
n=1
∞
∑
2
EE232 Lecture 14-3 Prof. Ming Wu
Advantages of Quantum Well Lasers(1) Low threshold current density:Compare fundamental material property
Transparency current density
10 nmSince ~100 nm
10%(
bulkbulk trtr active
QWQW trtr z
QWbulk QW tr ztr tr bulk
tr active
qNJ d
qNJ L
J LN NJ d
t
t
®
=
=
» Þ =
2) Higher differential gain Larger bandwidth:
Resonance frequency:
(3) Lower chirp:Smaller wavelength shift when the laser is directly modulated
gR
p
v aS gaN
wt
®
¶= µ =
¶
EE232 Lecture 14-4 Prof. Ming Wu
Transparency Carrier Concentration in Bulk
3/23/2*3/2
2 2
At transparency: or
Let
Electron concentration:
4 422 3 3
Hole concentration:
C V C V
C C V V
C C V V
B B
C C
e B n CC
B
V
F F E EF E F E
F E F Ek T k T
F E
m k T F EN Nk T
F E
pp p p
- = -- = -
- -D = =
>
æ öæ ö -= = × Dç ÷ç ÷
è ø è ø>
Q
hQ 1
3/2*
2 2
3/2*3/2
*
* *0 0
17 3
22
4 Solve 3
For GaAs ( 0.067 , 0.5 )
2.15, 9 10 cm
V V
B
h
F Ek Th B
V
h
e
e h
m k TP e N e
mN P em
m m m mN
pp
p
--
-D
-D
-
æ ö= =ç ÷
è ø
æ ö= Þ D = Þ Dç ÷
è ø= =
D = = ´
h
EC
EV
CB
VB
FC
FV
Transparency Condition(Bernard-Duraffourg Inversion Condition)
C V gF F F ED = - =
3
EE232 Lecture 14-5 Prof. Ming Wu
Transparency Carrier Concentration in QWAt transparency: FC − FV = Ee1 − Eh1 or FC − Ee1 = FV − Eh1
Let Δ =FC − Ee1kBT
=FV − Eh1kBT
Electron concentration: ∵FC > Ee1
N =me*kBT
π!2Lz
FC − Ee1kBT
!
"##
$
%&&= NC
2d ⋅ Δ
Hole concentration: ∵FV > Eh1
P = mh*kBT
π!2Lze−FV −Eh1kBT = NV
2de−Δ
N = P ⇒ Δ =mh*
me*e−Δ ⇒ Solve Δ
For GaAs (me* = 0.067m0 ,mh
* = 0.5m0 )
Δ =1.56, N = NC2d ⋅ Δ =1018 cm−3
Note: N is independent of Lz
EC
EV
CB
VB
FC
FV
Transparency Condition(Bernard-Duraffourg Inversion Condition)
C V gF F F ED = - =
EE232 Lecture 14-6 Prof. Ming Wu
Reduction of Lasing Threshold Current Density by Lowering Valence Band Effective Mass
• Yablonovitch, E.;; Kane, E., "Reduction of lasing threshold current density by the lowering of valence band effective mass," Lightwave Technology, Journal of , vol.4, no.5, pp. 504-506, May 1986
• Yablonovitch, E.;; Kane, E.O., "Band structure engineering of semiconductor lasers for optical communications," Lightwave Technology, Journal of , vol.6, no.8, pp.1292-1299, Aug 1988
1 1
Bernard-Duraffourg Condition:
C V e hF F E Ew- ³ ³ -h
* *
Ordinary Semiconductor6
High transparency carrier concentration
h em m» * *
Ideal Semiconductor
Low transparency carrier concentration
h em m»
4
EE232 Lecture 14-7 Prof. Ming Wu
Bernard-Duraffourg Condition in Quantum Well
Bernard-Duraffourg Condition:FC − FV = E − Eh1(a) mh
* >me* (as in most semiconductors)
FV > Eh1FC ≫ Ee1
Ntr = ρe2d (FC − Ee1) =
me*
π!2Lz(FC − Ee1)
Large Ntr → High threshold current
(b) mh* =me
* (Ideal semiconductor)
FV = Eh1FC = Ee1
Ntr =me
*
π!2LzfC (E)dE
Ee1
∞
∫ is low
EE232 Lecture 14-8 Prof. Ming Wu
Transparency Carrier Concentration for Ordinary Semiconductor
( )
1
1
* *1 1
*
2
*
20
*
2 0
*
2
*0
17 3
(b) Ideal Semiconductor
=
1
11
1
ln(1 )
ln 2
For =0.067
4.6 10
e
e B
h e V h C e
etr E E
z E k T
B ex
z
xB e
z
B e
z
e
tr
m m F E F E
mN dEL
ek Tm dx
L e
k Tm eL
k TmL
m m
N cm
p
p
p
p
¥
-
¥
¥-
-
= Þ =
=
+
=+
= - +
=
» ´
ò
ò
h
h
h
h1 1
Transparency Condition:
C V e hF F E E- = -
5
EE232 Lecture 14-9 Prof. Ming Wu
Transparency Carrier Concentration for Ordinary Semiconductor
1 1
Transparency Condition:
C V e hF F E E- = -
*2
1 2
*2
2
*
*
* *
*
(a) Ordinary Semiconductor
( )
To estimate , note that
For 6 (in 1.55 m laser), 1.43
1.43
B B
B
d etr e C e
z
k T k Td B hV
z
k T e
B h
h e
B
B etr
mN F ELN P
k TmP N e eL
mN P ek T m
m mk Tk TmN
rp
p
µ
p
-D -D
-D
= - = D
D =
= =
D= Þ =
»D =
=
h
h
h2zL
EE232 Lecture 14-10 Prof. Ming Wu
Effective Mass Asymmetry Penalty
2 3 2 3
1.43 2ln 2
Threshold current density reduction is more than a factor of 2:
: Shockley-Read-Hall nonradiative recombination lifetime
OrdinarytrIdealtr
th nonrad rad Auger
th
NN
J J J JJ NAN BN CN BN CNqd
J
tt
= =
= + +
= + + = + +
3
is greatly reduced when is lowered
(1) is reduced by 8x(2) C is also reduced due to band structure change by strain
Auger N
N
6
EE232 Lecture 14-11 Prof. Ming Wu
Bandgap-vs-Lattice Constant of Common III-V Semiconductors
Compressive Strain
Tensile Strain Lattice Matched
In0.53Ga0.47As
EE232 Lecture 14-12 Prof. Ming Wu
Qualitative Band Energy Shifts Under Strain
C
HH, LH
C
SO
SO SO
HH, LHHH
LH
CCC
HH, LHLHHH
SO SO
Compressive Strain
Tensile Strain
CEd
Pe-
Qe+
Qe-
Hydrostatic Strain
BiaxialStrain
BiaxialStrain
Hydrostatic Strain
7
EE232 Lecture 14-13 Prof. Ming Wu
Strain and Stress
0
0
0
12
11
12 11
( )
: lattice constant of InP0 : compressive strain0 : tensile strain
2
: Compliance Tensor
0.5
xx yy
zz
ij
a a xa
a
CC
CC C
e e e
ee
e e e^
-= = =
<ìí >î
= = -
»
11 12 12
12 11 12
12 12 11
12 12 11
12
11
Biaxial stress:
0 0
2
xx xx
yy yy
zz zz
xx yy
zz
xx yy zz
zz
C C CC C CC C C
C C CCC
s es es e
s s s
se e e
e e
é ù é ù é ùê ú ê ú ê ú=ê ú ê ú ê úê ú ê ú ê úë û ë û ë û
= =
=Þ + + =
= -
EE232 Lecture 14-14 Prof. Ming Wu
Band Edge Shift
12
11
12
11
12
11
( )
( ) 2 1
( ) 2 1
( ) 1 22
: hydrostatic potential :
C g C
HH
LH
C C xx yy zz C
V xx yy zz V
xx yyzz
C V
E E x EE P QE P Q
CE a aC
CP a aC
CQ b bC
a a ab
e e
e e
e
e
d
d e e e e
e e e e
e ee e
= +
= - -
= - +
æ ö= + + = -ç ÷
è øæ ö
= - + + = - -ç ÷è ø
+ æ ö= - - = - +ç ÷
è ø= -
shear potential
8
EE232 Lecture 14-15 Prof. Ming Wu
Strain Parameters in III-V(Coldren, p.535)
EE232 Lecture 14-16 Prof. Ming Wu
Band-Edge Profile and SubbandDispersion