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.