lecture #8 optical transition matrix elementee232/sp19/lectures... · of the transition matrix...
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EE 232: Lightwave Devices
Lecture #8 – Optical transition matrix
element
Instructor: Seth A. Fortuna
Dept. of Electrical Engineering and Computer Sciences
University of California, Berkeley
2/25/2019
2Fortuna – E3S Seminar
Optical transition matrix element
0
0
ˆ2
ˆop
c v
i
cv
qA eH e
m
−
=
k r
p
)(ci
cc
eu
V
=k r
r )(v
v v
ie
uV
=k r
r
Bloch states
Optical transition matrix element
periodicwith lattice
envelope function
0
0
( (2
ˆ ˆ) )opc v
ii i
cv c v
ee eu u
mV V
qAH e
−=
k rk r k r
r p r
* 30
0
ˆ( ) )(2
opc vii i
vc
ee eu
Au d
q
Ve
mV
− −=
k rk r k r
r p r r
3Fortuna – E3S Seminar
Optical transition matrix element
* 30
0
ˆ ˆ) )( (2
op
c v
v
i ii
cc v
e eu e u d
m V
qAe
VH
− −
= k r k r
k rr p r r
)3
( * *0
0
( ( ( ()ˆ ) )2
( ) )v pc oi
c v vv c
qA de u u u u
m Ve i k
− + + − −
−=
k k k r rr r r r
3*0
0
( ( (ˆ ) ) )2
c v vopii i i
c vv ve kqA d
u e e i u e u em V
−
− = − −
k rk r k r k r rr r r
3*0
0
[ˆ ) ] )( (2
p vocii
c v
iqA du e e i u e
m Ve
−
−= −
k rk r k r rr r
4Fortuna – E3S Seminar
Bloch functions 𝒖𝒄 and 𝒖𝒗
Atomic levels
s
p
BandsMolecular levels
p anti-bonding
p bonding
s anti-bonding
s bonding
conductions-“like”
valencep-“like”
gE
xp ypzp
s
5Fortuna – E3S Seminar
Bloch functions 𝒖𝒄 and 𝒖𝒗
xpyp
zp
s
~v y zxu p p p + +
~ cu sConduction band
Valence band
6Fortuna – E3S Seminar
Optical transition matrix element
~ ~
0
0
c x
x y
c v
v y z
z
u s u p p p
s p s p s p
u u
+ +
= = =
→ =
0
)3
( * *0ˆ ˆ ) )( ( ( (( ) ) )2
v opci
cv c v vv c
qA de u u u u
m VH e i k
− + + =
−− −
k k k r rr r r r
0
Envelope functionVaries slowly over unit cell of the crystal
Varies rapidlyover unit cell of the crystal
3)( *0
0
ˆ )( ) )( (2
v opci
vc
qA de
Viu u
me
− + +−= −
k k k r rr r
3*0
0
)( (( (2
ˆ ) ) )c v
qA dF u u
m Ve i=
− −
rr r r
7Fortuna – E3S Seminar
Optical transition matrix element
* 3) )( ) )( ( (c vF diu u − r r r r
* )( ) )
( (c
i
G
vu u
e
i
C
−
= G r
G
r r is periodic and can be represented by a Fourier serieswhere 𝑮 are the reciprocal lattice vectors
3
( ) 3
( ) 3
)
)
)
(
(
(
i
G
i
G
i
G
F e d
F e d
F C
C
e d
C
+
+
=
+=
G r
G
G r R
R G
G r R
R G
r r
r R r
R r
R
3
V
d r3d
R
r
V
8Fortuna – E3S Seminar
Optical transition matrix element
* 3 3
33
33
33 *
( ( ( (
(
(
) )( ) ) )
)
)
) )( ) )
(
( (
i
c G
i
v
V
V
v
G
G
V
i
c
F u u d F C e d
dF C e d
dF d C e
id
F d u
i
u
−
=
−
=
=
=
G r
R G
G r
G
G r
G
r r r r R r
rr r
rr r
rr r r r
3 3*)(0
0
ˆ ˆ )( ) )( (2
v opci
v c
V
vc
qA d de u uH
me
Vi
−
+ + −
−=
k k k r r rr r
Plug backinto opticalmatrix element
Non-zero only if c op v c v= + →k k k k k
Note: 1ie =G R
9Fortuna – E3S Seminar
Optical transition matrix element
3*0 0
0 0
, ,ˆ ˆ ˆ)( ) )( (
2 2c v c vvcv c cv
qA qAduH e i u
m me
−−
−
= = k k k k
rr r p
2ˆ
cve p can be evaluated using the k p method where it is shown that
2 02 0
*
)ˆ
(1
26
3
g g
cv b
eg
m E EmM
mE
e+
= = −
+
pΔ is the spin-orbit split-off bandseparation
It is also common to see 𝑀𝑏2 written as
2 0
6b p
mM E=
where 𝐸𝑝 can be experimentally measured GaAs: 𝐸𝑝= 25.7 eV
InP: 𝐸𝑝= 20.7 eV
0
,
22 20ˆ ˆ
2 c vcv cv
qA
mH e
=
k kpcv c vu u=p p
and u are bloch functionsvcu
10Fortuna – E3S Seminar
Absorption coefficient
2
0 )( ) (b r v cM ffC = −
*
2 2
3221
2
rr gE
m
= −
0 2
2
0 0
Cn
q
c m
=
* *
1
1 exp[( )( ) / ]c
g ceg r
fm m FE kTE −
=+ + −
* *( ( ) ) / ]
1
1 exp[v
g r h v
fE m m F kT
=+ − − −
2 0
6b p
mM E=
11Fortuna – E3S Seminar
Simplified two-band 𝒌 ⋅ 𝒑 theory
22
0
) ( ) ( )2
( )( nk n nkEVm
+ − =
r r k r
:
(( ) ) i
nk nku e
n
= k rr rGeneral solution:
band
Plug general solution in Schrodinger’s eqn:
2 2 22
0 0 0
2 2
0
0
0
0 0 0 0
) ( ) ( ) ( )
ˆ ˆ ( ) ( ) ( )
ˆ
ˆ ( ) ( )
(2 2
2
(0)
nk n nk
nk n nk
n n
u E u
H H u E u
H
H
kV
m m
k
u
m
m
E u
m
− + = −
= −
=
+
+
=
r k p r k r
r k r
k p
r r
where:
See Chuang Ch.4Haug and Koch Ch. 3
12Fortuna – E3S Seminar
Simplified two-band 𝒌 ⋅ 𝒑 theory
Second order time-independent perturbation theory:
( )( )
2 2
0
2 2 2
0 0 0
ˆ ˆˆ(0)
2 (0) (0)
(0)2 2 0
( )
2
(0) ( )
nm mnn n nn
m n n m
n nn
m n n m
EE
E
n m m nE
H HkH
m E
k
m m m E E
+= +
+−
+ +−
= +
k
k p k pk p
Consider two states: (0)
(0) 0
e g
hE
E E=
=
22 21
0 *
0
22 21
0 *
0
2 2
2 2
(0)2 2
(0)2 )
2(
)2
1
21
(
)
(h h
g
c
h
cv
e e g
e
v
g
E
E
k km E
m E m
k km
m E
E
mE
−
−
= +
=
=
−
+
+ +
=−
pk
pk
Assuming cubic symmetry(Lots of math details skipped)
(bottom of conduction band)
(top of valence band)
0
13Fortuna – E3S Seminar
Simplified two-band 𝒌 ⋅ 𝒑 theory
2 2
1 1
0 0* * *
0 0
2 22 02
* 2 *
0
2 21 1 11 1
41ˆ
12
cv cv
r e h g g
gcv
cv b
r g r
m E m Em
m EM
m
m mm m
em E m
− −
= + = + + −
= → = =
p p
pp
Reduced mass and bandgap can be measured from absorption coefficient. This allows for empirical determinationof the transition matrix element.
More rigorous treatment includes multiple valence bands (hh,lh, and so) and degeneracy at k = 0 (result given without proof on slide 9). But, this simple approximation derived here gives surprisingly reasonable estimate (within about a factor of 2) of the true matrix element.