1

EE232 Lecture 6-1 Prof. Ming Wu

EE 232 Lightwave Devices Lecture 6: Electron-Photon Interaction,

Optical Matrix Element, Absorption Coefficient

Instructor: Ming C. Wu

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

EE232 Lecture 6-2 Prof. Ming Wu

Vector Potential Maxwell's Equations

∇× E!"= −

∂B!"

∂t

∇×H!"!= J!"+∂D!"

∂t∇iD!"= ρ

∇iB!"= 0

%

&

''''

(

''''

D!"= εE!"

and B!"= µH!"!

Vector potential A!"

B!"=∇× A

!"

⇒∇iB!"=∇i ∇× A

!"( ) = 0

Vector potential is not unique:

A '!"!= A!"+∇ξ→∇× A '

!"!= B!"

∇× E!"= −

∂B!"

∂t= −

∂∂t∇× A!"=∇× −

∂A!"

∂t

%

&''

(

)**

E!"= −

∂A!"

∂t−∇φ

Wave Equations:

∇2A!"−µε

∂2A!"

∂t2%

&''

(

)**−∇ ∇iA

!"+µε

∂φ∂t

%

&'

(

)*= −µJ

!"

∇2φ +∂∂t

∇iA!"

( ) = − ρε

+

,

--

.

--

Wave Equations are invariant under "Gauge Transformation"

A '!"!= A!"+∇ξ

φ ' = φ − ∂ξ∂t

+

,-

.-

⇒One can choose the value of ∇iA!"

2

EE232 Lecture 6-3 Prof. Ming Wu

Lorenz and Coulomb Gauges

Lorenz Gauge:

Set ∇iA!"+µε

∂φ∂t

= 0

∇2A!"−µε

∂2A!"

∂t2= −µJ!"

∇2φ −µε∂2φ∂t2

= −ρε

$

%&&

'&&

Coulomb Gauge

Set ∇iA!"= 0

∇2A!"−µε

∂2A!"

∂t2$

%&&

'

())−∇ µε

∂φ∂t

$

%&

'

()= −µJ

!"

∇2φ = −ρε

*

+

,,

-

,,

For optical field, ρ = 0, φ=0

E!"= −

∂A!"

∂t

EE232 Lecture 6-4 Prof. Ming Wu

Electron-Photon Interaction Hamiltonian in the absence of light

H0 =P2

2m0+V (r!)

Generalized momentum of a charged particle in an electromagnetic field

P!"→ P!"− eA!"

( )Note: ∂P

!"

∂t= Force→ ∂P

!"

∂t− e ∂A!"

∂t

$

%&&

'

())=

∂P!"

∂t− eE!"$

%&&

'

())

(modification represents EM fields' ability to accelerate or decelerate electrons)

H = 12m0

P!"− eA!"

( )2+V (r!)

=P2

2m0−e

2m0P!"⋅ A!"+ A!"⋅P!"

( )+ e2A2

2m0+V (r!)

•  For a brief review of Hamiltonian classical mechanics, see http://en.wikipedia.org/wiki/Hamiltonian_mechanics#Using_Hamilton.27s_equations

H ' = − e2m0

P!"⋅ A!"+ A!"⋅P!"

( )+ e2A2

2m0

H ' ≈ − e2m0

P!"⋅ A!"+ A!"⋅P!"

( )P!"⋅ A!"

( )φ = −i!∇⋅ A!"φ

= −i! ∇⋅ A!"

( )φ + A!"⋅∇φ%

&'()*

= −i!A!"⋅∇φ under Coulomb Gauge

= A!"⋅P!"

( )φH ' = − e

m0A!"⋅P!"

3

EE232 Lecture 6-5 Prof. Ming Wu

Optical Matrix Element H ' = − e

m0A!"⋅P!"

A!"= e#A0 cos k

!op ⋅ r!−ωt( ) = e! A02 e

ik!op⋅r!−iωt + e!

A02e−i k!op⋅r!+iωt ; e! is optical polarization

H ' = −eA02m0

eik!"!op⋅r"−iωt e! ⋅P

"#≈ −

eA02m0

e−iωt e! ⋅P"#

; (Dipole or long-wavelength approx)

Hba' = b −

eA02m0

e! ⋅P"#$

%&&

'

()) a = −

eA02m0

e! ⋅ b P"#a = −

eA02m0

e! ⋅P"#ba

Alternative form:

P!"=m0

ddtr!=m0

1i!r",H0

*+

,-=m0i!r"H0 −H0 r

!( )

b P!"a =

m0i!b r"H0 −H0 r

!( ) a = m0i! Ea − Eb( ) b r

!a =

m0i!!ω b r

!a

Hba' = −

em0A!"$

%&&

'

())⋅ −iωm0( ) b r

!a = −eE

"!⋅ r!ba

EE232 Lecture 6-6 Prof. Ming Wu

Typical Values of Optical Matrix Element

Hba' = −

eA02m0

e! ⋅P"#cv = −eE

"#⋅ r#ba = −eiω

A02e! ⋅ r#ba ⇒ r

#ba =

P"#cv

m0ωThe optical matrix element is often expressed in Ep :

e! ⋅P"#cv

2

=m06Ep

Typical values of Ep (Table K.2 on p.709 of Chuang textbook)

GaAs: Ep = 25.7 eV The corresponding r#ba ~ 0.4 nm

AlAs: Ep = 21.1 eV

InAs: Ep = 22.2 eV

InP: Ep = 20.7 eV

GaP: Ep = 22.2 eV

4

EE232 Lecture 6-7 Prof. Ming Wu

Fermi’s Golden Rule

2' 2

2'

Two-level system, monochromatic light:

2 1 1 1( ) Unit: [ J ]J-s J

Final state with density of states , monochromatic light:

2 ( ) ( )

Unit:

ba ba b a

b

ba ba b a b b b

W H E Es

W H E E E dE

πδ ω

ρ

πδ ω ρ

= − − =

= − −∫

hh

hh

23 3

2'

23 3

1 1 1 1[ J J ]J-s J cm -J s-cm

Two-level system, non-monochromatic light:

2 ( ) ( ) ( )

1 1 1 1 Unit: [ J J ]J-s J cm -J s-cm

where ( ) : number of photons per unit volume per energy

ba ba b aW H E E P d

P

πδ ω ω ω

ω

=

= − − ⋅

=

∫ h h hh

h interval

bE

aE

hv

bE

aE

hv

bE

aE

hv

EE232 Lecture 6-8 Prof. Ming Wu

Two-Level System in Non-Monochromatic EM Field

2 2' '

/

1 2 2( ) 2 ( )

1 Average number of photons per state (Bose-Einstein distribution)1

To calculate density of states for photons: satisfies pe

op

k B

ba ba b a ph ba ph bak

ph h k T

ik r

W H E E n H n N EV

ne

e

ω

π πδ ω

⋅

= − − ⋅ =

=−

∑

r r

hh h

( )2

3

riodic boundary condition

dispersion relation (~ E-k relation for electrons)

Number of states with photon energy E per unit volume, per energy interval

2 4 8( )2

kr

ba

ba b a k

kcn

k dkN E E EV

L

ω

πδ ω

π

=

= ⋅ − − =⎛ ⎞⎜ ⎟⎝ ⎠

∫ h ( )

( )

2

3

3 2

3 3

2

8

r k rk

r baba

n n dc c

n EN Eh c

ωπω

π

π

⎛ ⎞⎜ ⎟⎝ ⎠

=

∫

5

EE232 Lecture 6-9 Prof. Ming Wu

Joint Density of States for Semiconductor

2 2 2 2

* *

2 2

* *

2 2

*

The electron states associated with optical transition

are related by conservation of mementum:

2 2

1 1( )2

2Joint densit

b a

b C a Ve h

b a C Ve h

gr

k k kk kE E E Em m

kE E E Em m

kEm

ω

≈ ≈

= + = −

⎛ ⎞− = − + +⎜ ⎟

⎝ ⎠

= +

r r r

h h

h

hh

3/2*

2 2

y of states for the pair of electron states:

21( )2

rr g g

mE Eρ ω ωπ

⎛ ⎞− = −⎜ ⎟

⎝ ⎠h h

h

CB

VB

hν

Eb

Ea

Ec

EV

EE232 Lecture 6-10 Prof. Ming Wu

Absorption Coefficient CB completely full, VB completely empty, total upward transition at !ω:

R0 (!ω) =2V

2π!Hba' 2δ(Eb − Ea − !ω)

!

"#

$

%&

k∑ = 2π

!Hba' 2 2dk

!

(2π )3∫ δ(Eg +!2k 2

2mr*− !ω)

R0 (!ω) =2π!Hba' 2 ρr (!ω − Eg ) unit: [

1m3s

]

Absorption coefficient

α0 (!ω) =R(!ω)

photon flux=

R(!ω)ε0εrE0

2

2cnr

1!ω

=R(!ω)

ε0nrω2A0

2c2!ω

unit: [ 1m

]

Hba' = −

eA02m0

e! ⋅P!"cv

α0 (!ω) =C0 e! ⋅P!"cv

2

ρr (!ω − Eg )

C0 =πe2

nrcε0m02ω

≈ 7×109 unit: m2

kg

!

"#

$

%&

hν

Eb

Ea

6

EE232 Lecture 6-11 Prof. Ming Wu

Absorption Coefficient for GaAs

α0 (!ω) =C0 e! ⋅P!"cv

2

ρr (!ω − Eg )

nr 3.5:= ε0 8.854 10 12−⋅Fm

:= h_bar 1.055 10 34−×m2 kg⋅s

= 1eV 1.6 10 19−× J=

Eg 1.42eV:= ωEgh_bar

:= ω 2.154 1015×1s

=

C0π q2⋅

m02 ω⋅ ε0⋅ c⋅ nr⋅:= C0 4.842 109×

m2

kg=

Ep 25.7eV:= m06Ep⋅ 6.243 10 49−× kg J⋅=

m_r0.067 0.5⋅0.067 0.5+

m0⋅:=

ρr hv( )1

2π 22 m_r⋅

h_bar2⎛

⎝

⎞

⎠

3

2⋅ hv Eg−⋅:= ρr 1.43eV( ) 6.102 1043×

1

m3 J⋅=

α0 hv( ) C0m06

⋅ Ep⋅ ρr hv( )⋅:= α0 1.43eV( ) 1.845 105×1m

= α0 1.43eV( ) 1.845 103×1cm

=