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.