1

EE232 Lecture 9-1 Prof. Ming Wu

EE 232 Lightwave Devices Lecture 9: Optical Matrix Element,

k-Selection Rule, Quantum Well Gain / Absorption

Instructor: Ming C. Wu

University of California, Berkeley Electrical Engineering and Computer Sciences Dept.

EE232 Lecture 9-2 Prof. Ming Wu

Bloch Function

•  Bloch function –  Electron wavefunction

in a periodic potential can be expressed as a product of a periodic function and a slowly varying plane wave envelop function

Atom

Periodic Potential

a =ψa (r!) = uv (r

!) e

ik!v ⋅r!

V

b =ψb(r!) = uc (r

!) e

ik!c⋅r!

V

2

EE232 Lecture 9-3 Prof. Ming Wu

Optical Matrix Element

Hba' = b −

eA02m0

e! ⋅P"##

$%%

&

'(( a = −

eA02m0

e! ⋅ b P"#a = −

eA02m0

e! ⋅P"#ba

Electron wavefunction can be expressed in Bloch functions:

a =ψa (r!) = uv (r

!) e

ik!v ⋅r!

V; b =ψb(r

!) = uc (r

!) e

ik!c⋅r!

V

Hba' = −

eA02m0

e! ⋅ b P"#a = −

eA02m0

e! ⋅ uc*(r!)∫ e−i k

!v ⋅r!

eik!op⋅r!

−i"∇( )uv (r!)eik!v ⋅r! d 3rV

= −eA02m0

e! ⋅ uc*(r!)∫ e−i k

!c⋅r!

eik!op⋅r!

−i"∇( )uv (r!)+ (−i!k

"v )uv (r!)+

,-.e

ik!v ⋅r! d 3rV

= −eA02m0

e! ⋅ uc*(r!)∫ −i!∇( )uv (r

!) d

3rΩ

e−i k!c⋅r!

eik!op⋅r!

∫ eik!v ⋅r! d 3rV

= −eA02m0

e! ⋅ p"#cv ⋅δk

!c ,k!op+k!v

k-Selection Rule

Matrix element of periodic function over unit cell

Slowly Varying Envelop Approx

EE232 Lecture 9-4 Prof. Ming Wu

Optical Matrix Element for Quantum Wells

a =ψa (r!) = uv (r

!) e

ik!t ⋅ρ!"

Agm(z)

b =ψb(r!) = uc (r

!) e

ik!t '⋅ρ!"

Aφn (z)

Hba' = −

eA0

2m0

e! ⋅ b P!"a

= −eA0

2m0

e! ⋅ p!"cv ⋅δk

!t ,k!t '⋅ Ihmen

Ihmen = φ*

n (z)gm(z)dz−∞

∞

∫

Overlap integral of QW envelop wavefunctionsIhmen = δmn for infinite potential well

Quantum Well Wavefunction

Atomic Wavefunction

Envelop Wavefunction

Slowly Varying Envelop

3

EE232 Lecture 9-5 Prof. Ming Wu

Interband Transitions in Quantum Wells (1)

0V =∞

0x = x L=

xk

yk

E

1eE

2eE

1hE

2hE

Allowed Transition

22ehE

11ehE

ωh

1 21 2

Only transition from m=1 hole subband to n=1 electron subband is allowed

e eh hE Eω< <h

EE232 Lecture 9-6 Prof. Ming Wu

Interband Transitions in Quantum Wells (2)

0V =∞

0x = x L=

xk

yk

E

1eE

2eE

1hE

2hE

Allowed Transition

22ehE

11ehE ωh

!ω > Eh2e2

m=1 → n=1m=2 → n=2

"#$

Since the density of states for each subband is the same, the maximum optical gain is twice of that of a single-band transition

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EE232 Lecture 9-7 Prof. Ming Wu

Optical Gain for Interband Transition in Quantum Wells

g(!ω) =C0 e! ⋅P!"cv

2

ρr2d (!ω) fg (!ω)

ρr2d (E) =

mr*

π!2LzH (E − Ehn

en )n=1

∞

∑

fg (!ω) = fC Een + !ω − Ehnen( )mr

*

me*

!

"##

$

%&&− fV −Ehm − !ω − Ehn

en( )mr*

mh*

!

"##

$

%&&

For more general case:

⇒ g(!ω) =C0 e! ⋅P!"cv

2

Ihmen 2 ρr

2dH (!ω − Ehmen )

m,n∑ fg (!ω)

ρr2d =

mr*

π!2Lz

EE232 Lecture 9-8 Prof. Ming Wu

Quantum Well Gain

1.4 1.6 1.8 2 2.2 2.4 2.61−

0.5−

0

0.5

1

f_g xi 0.05eV, ( )f_g xi 0.1eV, ( )f_g xi 0.15eV, ( )f_g xi 0.2eV, ( )f_g xi 0.25eV, ( )f_g xi 0.3eV, ( )f_g xi 0.35eV, ( )f_g xi 0.4eV, ( )f_g xi 0.45eV, ( )

xieV

1.4 1.6 1.8 2 2.2 2.4 2.6

1− 106×

0

1 106×g xi 0.05eV, ( )g xi 0.1eV, ( )g xi 0.15eV, ( )g xi 0.2eV, ( )g xi 0.25eV, ( )g xi 0.3eV, ( )g xi 0.35eV, ( )g xi 0.4eV, ( )g xi 0.45eV, ( )

xieV

Fermi Inversion

Factor

Gain

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EE232 Lecture 9-9 Prof. Ming Wu

Solving Quasi-Fermi Level in Quantum Well

Electron concentration:

N = dEρe2d (E) fC

n∫ (E)

ρe2d (E) = me

*

π!2LzH (E − Een )

n=1

∞

∑

fCn (E) = 1

1+ eEen+Et−FCkBT

Use dx1+ ex∫ = −ln(1+ e−x )

N =n∑ me

*kBTπ!2Lz

ln 1+ eFC−EenkBT

%

&

''

(

)

**

For large quasi-Fermi Energy:FC ≫ Een

ln 1+ eFC−EenkBT

"

#

$$

%

&

''≈FC − EenkBT

N =n∑ me

*

π!2LzFC − Een( )

For small quasi-Fermi Energy:FC ≪ Een

ln(1+ eFC−EenkBT ) ≈ e

FC−EenkBT = e

−Een−FCkBT

N =me*kBT

π!2Lze−Ee1−FCkBT = NCe

−Ee1−FCkBT