Lecture #10 Absorption in quantum wellsee232/sp19/lectures... · cv ÖÖ Ö Ö Ö Ö Ö Ö 1 Ö 2...
Transcript of Lecture #10 Absorption in quantum wellsee232/sp19/lectures... · cv ÖÖ Ö Ö Ö Ö Ö Ö 1 Ö 2...
EE 232: Lightwave Devices
Lecture #10 – Absorption in quantum
wells
Instructor: Seth A. Fortuna
Dept. of Electrical Engineering and Computer Sciences
University of California, Berkeley
2/28/2019
2Fortuna – E3S Seminar
Semiconductor quantum well
cE
gE
vE
1eE
2eE
2hE
1hE
3hE
E
( )vg E
( )cg E
2
*
em
*
2
2 em
*
2
3 hm
*
2
2 hm
2
*
hm
*
2)( ) (c g
een
n
mg E H E E E
= − −
2
*
( ) ( )h
m
v hm
mg E H E E
= −
3Fortuna – E3S Seminar
Absorption coefficient
( )2
0ˆ ( )
2| |
tz
cv e h v c
k k
e E f fECV
− − −= p
2
2 2
2 2(2 )2
(2 (2) )t
t t t t
k
tkd k dk dkA A A
k
− − −
→ = =
2 2
*
2 2
*
2 2
* * * *
2 2 * * 2
* 2 2 *
2
2
1 1 1 where
2
2Let
2 2
h
e en
h
h
tg
e
thm
en thm
r
t tr rt
r e
r
e
h
r
kE E E
m
kE E
m
kE E E
m m m m
k dkm E mE k
m dE m E
h
= + +
= −
− = + = +
= → = =
Note that here,
cv c v =p p
and are bloch statesvc
4Fortuna – E3S Seminar
Absorption coefficient
2
2 2
*
2
*
,22
0
)
)
2 2(22
(2 (2
( ) ( )
)t
t t t t t
r
D
k
rr
d k dk dkA A A
mA dE
mA H E dE
k
E dE
k
A H
− − −
− −
→ = =
=
= =
( )
( )
2
0
2
0 ,2
2
0 ,2
2
0 ,2
,
,
ˆ ( )
1ˆ (
|
2| |
| | ( ) ( ) ( )
| ( ) )
)
|
1ˆ
(
( ( )
| ( )1
ˆ)
tz
en
r D hm
z
en en en
r D hm hm c hm
z
cv e h v c
k k
cv v c
n m
cv v
n
en
z
m
c r D hv m
C E f fV
C H E E f E f E dE
C H E f E E f E E
C H E f
e E
e EL
eL
eL
−
=
+
− − −
− − −
= − −
= = − =
−=
p
p
p
p,
( ) ( )n
e
v
n en
hm c h
m
mE E f E E = − = − −
5Fortuna – E3S Seminar
Absorption with inclusion of other valence bands
2
0 , ,2
2
0 , ,2
,
,
,
,
| | ( ) ( ) ( )
| | ( ) ( ) ( )
1ˆ( )
1ˆ( )
en en en
hh r hh D hm hm c hm
z
en en en
lh r lh D h
cv hh v
n m
cv lh v
n m
m hm c hm
z
C H E f E E f E E
eC H E f E E f E E
eL
L
− − −
−
= = − =
= = = − −−
p
p
Valence band dependent Valence band dependent
( ) lh hh = + Total absorption coefficientis the summation of absorptionbetween the conduction and eachvalence band.
(heavy hole)
(light hole)
6Fortuna – E3S Seminar
Fermi factor
]
1( )
1 exp[ /( )e
e
c
c
fF k
EE T−
=+ ]
1)
1 exp(
( ) /[h
h
v
v
fF kE
ET−
=+
We need a change of variables from 2 2
*,
2 r
e h
kE E E
m→ =
Because of the delta function,*2 2
* 2
2( )
2
en enrhm hm
r
kE E
mkE
m − → = −= =
( )
2 2
*
*
*
2g en
e
en rg n
e
e
e hm
Ek
E
Em
m
mE
E
E
= + +
= −+ +
2 2
*
*
*
2
( )
hm
en rh
h
h
h
h
m m
E
mE
kE
m
Em
= −
− −
=
( ) * * ) / )
1( )
1 exp (c en
g en hm r e c
fE m mE TE F k
= + − −
+ +
* *
1)
1 exp ((
( ) ) /v en
hm r h vhm
fE m m F kTE
= + − − −
7Fortuna – E3S Seminar
Optical matrix element
0
02ˆ ˆ
op
cv c v
ie
m
qAH e
−
=
k r
p
)( ( )ci
cc n
eu z
A
=k r
r )( ( )vi
v v m
eu g z
A
=k r
r
Bloch states
Optical matrix element
periodicwith lattice
envelope function
0
0
)ˆ ( ( ) ( (2
ˆ) )vopc
ii i
mcv vc n
ee eu z
qAH e u g z
mA A
−=
k rk r k r
r p r
* * 30
0
( ( ) ( ( )2
ˆ) )op vc
ii i
c n mv
A ee eu z u g z d
A
qe
m A
− −=
k rk r k r
r p r r
8Fortuna – E3S Seminar
Optical matrix element
* * 30
0
3 3( * *0
0
3( *0
0
(0
)
0
)
ˆ
2
( ( ) (
2
ˆ) )
)ˆ )(
( )2
( ( ) ( (
( )ˆ
ˆ
(
)
)
)
2
opc
c v op
c v p
c
v
o
ii i
cv c n m
i
n m
V
c
i
cv n m
i
cv
v
v
V
ee eu z u g z d
mA A
qA d de
e
z g z u um A
qA de z g z
m A
e
qAe
m
qAH
e i
e
−
−
−
−
+ +
+ +
+
−=
−
−=
−=
−=
k rk r k r
k k k r
k k k r
k k
r p r r
r rr r
rp
p , ,ˆ ˆ ˆ( ) *
*0
0
0
)
,
0,
( ) ( )
( ) ( )2
ˆ
ˆ2
v op t op z
z
c v
z
c v
xx yy i zz
n m
L
cv n m
L
en
m
A
cv h
z g z d
qAz g z dz
m
d de
A
e
e IqA
m
++
−
−=
k k
k k
k k
x yz
p
pNote that here,
cv c vu u=p p
and u are bloch functionsvcu
9Fortuna – E3S Seminar
Overlap integral
0
,
*0ˆ ( ) ( )2
ˆc v
z
cv n m
L
cv
qAz g z
mH e dz
−= k kp
*( ) ( )
z
en
hm n m
L
I z g z dz=
Infinite barrier well (approximation)
1( )z2 ( )z
1( )g z
2 ( )g z
3( )g z
1( )z2 ( )z
1( )g z
2 ( )g z
3( )g z
22
→
11
→
31
→
22
→
11
→
cE
vE
cE
vE
0 for n,m different parity
~ 1 for n=m
~ 0 for n m and n,m same parity
en
hmI
=
Finite barrier well
0 for n m
1 for n=m
en
hmI
=
10Fortuna – E3S Seminar
Bloch functions 𝒖𝒄 and 𝒖𝒗
xuyu
zu
suConduction bandBasis function
Valence bandBasis functions
11Fortuna – E3S Seminar
Bloch functions 𝒖𝒄 and 𝒖𝒗
( ) ( )
1 12 ) 2 )
1 1) )
1 1
2 2
( (6
3
6
( (3
hh x y hh x y
lh x y z lh x y z
so x y z so x y z
u iu u u u
u u u u u u u
u u u
u i
i u i
i u u u u ui
= − + = −
= − + − = − +
= − + + = − −
sc s cuiu uu i= =Conduction bandBloch functions
Valence bandBloch functions
Note: bar denotes spin-downReference: Chuang 4.2, Coldren App 8
Near the bandedge the electron/hole wavevector is primarily directed in the z-direction
ˆc v z zk= =k k
Below are the Bloch functions for electron wavevector in the z-direction as derived from Kane’s 𝑘 ⋅ 𝑝 model for the band structure
wellbarrier
barrier
z
12Fortuna – E3S Seminar
Polarization dependent matrix element
Let’s calculate for the conduction band to heavy-hole band transition 2
ˆcve p
( )
ˆ ˆ
ˆ
ˆ ˆ ˆ ˆ
ˆ
1ˆ
2
cv c v
c v
c v
c x y v
x x y
z
s y z
e e u u
e u i u
z
e
xe u i x
yy z u
u u
e iuiu u
= −
= − +
=
+
+
=
= +
+
+ + −
p p
p p p
p p p
( )
22 2
ˆ ˆ
1ˆ
2
f
1 3ˆ
2
or
2
cv x x y
c b
s
sv x xu
iu
e x
x iu u
x u M
=
=
+
= =
−p p
p p
( )
22 2
ˆ ˆ
1ˆ
2
f
1 3ˆ
2
or
2
cv x y
c
y
b
s
s y yv u
iu
e y
y iu u
y u M
=
=
+
= =
−p p
p p
13Fortuna – E3S Seminar
Polarization dependent matrix element
Let’s calculate for the conduction band to heavy-hole band transition 2
ˆcve p
( )
ˆ ˆ
ˆ
ˆ ˆ ˆ ˆ
ˆ
1ˆ
2
cv c v
c v
c v
c x y v
x x y
z
s y z
e e u u
e u i u
z
e
xe u i x
yy z u
u u
e iuiu u
= −
= − +
=
+
+
=
= +
+
+ + −
p p
p p p
p p p
( )
2
for
0
ˆ ˆ
1ˆ 0
2
ˆ
cv z x y
cv
s
e z
z iu
z
iuu
=
+ =
− =
=
p p
p
Light with polarization in the z-directionwill not cause a transition betweenthe conduction and heavy hole band!(Reminder: this is at the bandedge)
14Fortuna – E3S Seminar
Polarization dependent matrix element
Let’s calculate for the conduction band to light-hole band transition 2
ˆcve p
)ˆ ˆ1
62(zcv x zs x yye e iu u i uu += + − + −p p p p
22 2
for
12 )
1
6
ˆ ˆ
ˆ (6
1ˆ
2
cv x zs
v
x
s x x
y
c b
i
u
u
e x
x iu u u
x u M
−
=
=
=
+ −
=
p p
p p22 2
for
12 )
1
6
ˆ ˆ
ˆ (6
1ˆ
2
cv x zs
v
y
s y y
y
c b
i
u
u
e y
y iu u u
y u M
−
=
=
=
+ −
=
p p
p p
22 2
f
1
ˆ
2
o ˆ
ˆ (6
ˆ
r
2
2 )
3
cv x zs z
s z z
y
cv b
i
z
u
e
z iu u u
z u u M
−
=
=
= =
+ −p p
p p
15Fortuna – E3S Seminar
Momentum matrix element (bandedge)
TE polarization ˆ ˆ ˆ ˆ or e x e y= =
2 2
2 2
ˆ
1
3
2
ˆ2
c hh b
c lh bMe
Me −
− =
=p
p
(heavy hole, bandedge)
(light hole, bandedge)
TM polarization ˆ ˆe z=
2
2 2
ˆ 0
ˆ 2
c hh
c lh b
e
e M
−
−
=
=
p
p
(heavy hole, bandedge)
(light hole, bandedge)
16Fortuna – E3S Seminar
Momentum matrix element (general)
2ˆ
cve p can also be calculated “away from the bandedge” (i.e. 𝑘𝑡 ≠ 0)
TE polarization ˆ ˆ ˆ ˆ or e x e y= =
2 2 2
2 2 2
ˆ
4
3(1 co )
5 3ˆ cos
4
s4
c hh b
c lh b
M
M
e
e
−
−
= +
=
−
p
p
(heavy hole)
(light hole)
TM polarization ˆ ˆe z=
2 2 2
2 2 2
3ˆ sin
2
1ˆ (1 3cos )
2
c hh b
c lh b
Me
Me
−
−
+
=
=
p
p
(heavy hole)
(light hole)
2
2 2
*
cos
2
en
ten
e
k
m
E
E
=
+
Note:
17Fortuna – E3S Seminar
Momentum matrix element (general)
Relative magnitude of 𝑀𝑏2 for conduction to heavy hole
and light hole transitions.
2ˆ
cve p can also be calculated “away from the bandedge” (i.e. 𝑘𝑡 ≠ 0)
Source: Zory. Quantum Well Lasers
18Fortuna – E3S Seminar
Summary
*
,2 2
rr D
m
=
0 2
2
0 0
Cn
q
c m
=
22
0 ,2
, ,
| | ( ) ( ) ( )1
ˆ( ) cv v
hh lh n m
m
en en en en
hm r D hm h c hm
z
C I H E f E E f E EeL
= = − =− − − p
( ) * *x / ]
1( )
1 e p (c en
g en hm r e c
fE E m m F kTE
= + −
+
+ −
* *
1)
1 exp ((
( ) ) /v en
hm r h vhm
fE m m F kTE
= + − − −
*( ) ( )
z
en
hm n m
L
I z g z dz= cv c vu u=p p
and u are bloch functionsvcu
19Fortuna – E3S Seminar
Calculated absorption spectrum
InP/InGaAs quantum well (𝐿𝑧= 11nm) T=10K
E1-HH1E1-LH1
E2-HH2 E2-LH2
TE bandedge matrix elements are used
20Fortuna – E3S Seminar
Comparison with experimental data
Source: Klingshirn. Semiconductor Optics.
simple absorption model
Our simple absorption model does not include excitonic effectsor transitions to unbound states.