1

EE232 Lecture 7-1 Prof. Ming Wu

EE  232  Lightwave  DevicesLecture  7:  Absorption  and  Gain  Coefficient,Bernard-­Duraffourg Inversion  Condition,  

Lineshape

Instructor:  Ming  C.  Wu

University  of  California,  BerkeleyElectrical  Engineering  and  Computer  Sciences  Dept.

EE232 Lecture 7-2 Prof. Ming Wu

Absorption  Coefficient

When CB and VB are partially filled:

Absorption Condition: VB is occupied, CB is empty

Absorption probability = fV (E1) 1− fC (E2 )( )

R12 (!ω) = 2V

2π!Hba

'2δ(Eb − Ea − !ω)

!

"#

$

%&

k∑ fV 1− fC( )

Emission Condition: CB is occupied, VB is empty

Emission probability = fC (E2 ) 1− fV (E1)( )

R21(!ω) = 2V

2π!Hba

'2δ(Eb − Ea − !ω)

!

"#

$

%&

k∑ fC 1− fV( )

CB

VB

hn

E2

E1

CB

VB

hn

E2

E1

( )12 1V CR f fµ -

( )21 1C VR f fµ -

2

EE232 Lecture 7-3 Prof. Ming Wu

Absorption  CoefficientNet absorption rate:

R(!ω) = 2V

2π!Hba' 2

δ(Eb − Ea − !ω)"

#$

%

&'

k∑ fV 1− fC( )− fC 1− fV( )"

#%&

=2π!Hba' 2 2dk

!

(2π )3∫ δ(Eg +!2k 2

2mr*− !ω) fV (k)− fC (k)"# %&

=2π!Hba' 2 2dk

!

(2π )3∫ δ(Eg +!2k 2

2mr*− !ω) fV (k)− fC (k)"# %&

=2π!Hba' 2

ρr (!ω − Eg ) fV − !ω − Eg( )mr*

mh*

*

+,,

-

.//− fC Eg + !ω − Eg( )mr

*

me*

*

+,,

-

.//

"

#$$

%

&''

α(!ω) =C0 e! ⋅P!"cv

2

ρr (!ω − Eg ) − fg (!ω − Eg )"# %&

Fermi Inversion Factor :

fg (!ω − Eg ) = fC Eg + !ω − Eg( )mr*

me*

*

+,,

-

.//− fV − !ω − Eg( )mr

*

mh*

*

+,,

-

.//

EE232 Lecture 7-4 Prof. Ming Wu

1 1.2 1.4 1.6 1.8

1-

0

1

f_g hvi 10K, 1.5eV, ( )f_g hvi 150K, 1.5eV, ( )f_g hvi 300K, 1.5eV, ( )

hvieV

fg (!ω − Eg ) = fC E2( )− fV E1( )

E2 = Eg + !ω − Eg( )mr*

me*

E1 = − !ω − Eg( )mr*

mh*

!

"

##

$

##

And k1 = k2

fC E2( ) = 1

1+ eE2−FCkBT

fV E1( ) = 1

1+ eE1−FVkBT

fg (!ω − Eg ) = eE1−FVkBT − e

E2−FCkBT

1+ eE2−FCkBT

!

"

##

$

%

&&

1+ eE1−FVkBT

!

"

##

$

%

&&

Fermi  Inversion  Factor

CB

VB

E2

E1

502 .)( >EfC

501 .)( <EfV

502 .)( <EfC

501 .)( >EfV

FC

FV

T=  10  KT=  150  KT=  300  K

( )g gf Ew -h

C VF F FD = -

wh

3

EE232 Lecture 7-5 Prof. Ming Wu

α0 (!ω) =C0 e! ⋅P!"cv

2

ρr (!ω − Eg )

α(!ω) =α0 (!ω) − fg (!ω − Eg )!" #$

g(!ω) = −α(!ω)

g(!ω) =α0 (!ω) fg (!ω − Eg )

Optical  Gain  Coefficient

1 1.2 1.4 1.6 1.80

1 1044´

2 1044´

3 1044´

4 1044´

5 1044´

rr hvi( )

hvieV

1 1.2 1.4 1.6 1.81.5-

0

1.5

f_g hvi 1K, 1.55eV, ( )f_g hvi 77K, 1.55eV, ( )f_g hvi 300K, 1.55eV, ( )

hvieV

1 1.2 1.4 1.6 1.81.5- 106´

0

1.5 106´

a0 hvi( ) f_g hvi 1K, 1.55eV, ( )×

a0 hvi( ) f_g hvi 77K, 1.55eV, ( )×

a0 hvi( ) f_g hvi 300K, 1.55eV, ( )×

a0 hvi( )a0 hvi( )-

hvieV

JointDOS

FermiInversionFactor

Gain

Absorption

T=      1  KT=    77KT  =  300K

EE232 Lecture 7-6 Prof. Ming Wu

0( ) ( ) ( )

Bernard-Duraffourg Inversion Condition

Spectral Range of Gain

g g

C V g

g C V

g f E

F F F E

E F F F

w a w w

w

= -

D = - >

< < D = -

h h h

h

1 1.2 1.4 1.6 1.80

1 1044´

2 1044´

3 1044´

4 1044´

5 1044´

rr hvi( )

hvieV

1 1.2 1.4 1.6 1.81.5-

0

1.5

f_g hvi 10K, 1.5eV, ( )f_g hvi 10K, 1.55eV, ( )f_g hvi 10K, 1.6eV, ( )

hvieV

1 1.2 1.4 1.6 1.81.5- 106´

0

1.5 106´

a0 hvi( ) f_g hvi 10K, 1.5eV, ( )×

a0 hvi( ) f_g hvi 10K, 1.55eV, ( )×

a0 hvi( ) f_g hvi 10K, 1.6eV, ( )×

a0 hvi( )a0 hvi( )-

hvieV

Optical  Gain  Coefficient  versus  Bias

JointDOS

FermiInversionFactor

Gain

Absorption

ΔF=  1.5eVΔF=  1.55eVΔF=  1.6eV

4

EE232 Lecture 7-7 Prof. Ming Wu

Lineshape Broadening

g(!ω) =C0 e! ⋅P!"cv

2

dE∫ ρr (E)δ(E = !ω − Eg ) fg (E)

Lineshape broadening due to intraband scattering of electrons→ Replace delta function by a Lorentzian lineshape function:

δ(E = !ω − Eg )→ L E − (!ω − Eg )( ) = 1π

! / τ inE − (!ω − Eg )( )

2+ ! / τ in( )

2

FWHM = Γ = 2!τ in

τ in : intraband relaxation time ~ 0.1ps

g(!ω) =C0 e! ⋅P!"cv

2

dE∫ ρr (E)1π

! / τ inE − (!ω − Eg )( )

2+ ! / τ in( )

2

!

"

###

$

%

&&&fg (E)

EE232 Lecture 7-8 Prof. Ming Wu

Lineshape Functions  and  Gain  Spectra

Source:  Coldren &  Corzine,  p.133