EE 232: Lightwave Devices

Lecture #12 – Spontaneous emission

Instructor: Seth A. Fortuna

Dept. of Electrical Engineering and Computer Sciences

University of California, Berkeley

2/28/2019

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Two-level system

2E

1E

Spontaneous emission Absorption Stimulated emission

221 2 21 2

2

21 21

1

1

2

2 1

12 21

0

/

/ )( 1

( ) ( )

( )( / )

dNA N B B

dt

dN

dt

N

B

N

B B

N

A

N

= − +

=

−

=

→−

21

21

12

Spontaneous emission rate

Stimulated emission

y

rate

Absorp

l

tion ra

e

te

( )

( )

( ) Spectra photon d nsit

A

B

B

=

=

=

=

(steady state)

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Photon mode density

“cavity”

We place our two-level system in a “cavity”What are the density of photon modes in this cavity?

( )

3

3

2

3

2

2

22

2

2 3

2 3

1Mode density =

1 2

(2 ) /

8

(2

)

)

)

)

(

/

(

(mod

k

e

V

d

V V

dk

dk

c n d

c n

nd

k

k

d

c

=

=

=

=

=

=

k

c c

kn n

d

dk

= =

(factor of 2 accountsfor polarization)

V

2 3

2 3mode

n

c

= (modes/cm3/eV)

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Relations between the coefficients

1(

exp 1

)phn

kT

=

−

Bose-Einstein distribution(photons per state)

2 3

2 3

( ) ) )

( )

1

( (

1

exp

mode ph

n

T

n

c

k

=

−

=

Recall, we also derived

21 21

1 2 12 21

/

/ )( / ) 1( )

(

B

N

A

N B B =

−

These equations must be equal to each other. This isonly possible if:

1 2 expN NkT

=

12 21B B=

21 21 )(modeA B =

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Spontaneous emission

Transition rate fromState 2 to State 1

21 2 21 2

21 2 21 2

1

1

2

2

2

( )

) ( ) ( )

1) )

(

( (

mode mode ph

mode ph

R

B

A N B

N

N

B

N

n N

B n

→

=

= +

+

= +

stimulatedemissionfrom onephoton ineach mode

stimulatedemissionfrom 𝒏𝒑𝒉photons ineach mode

Spontaneous emission can be interpreted as the stimulated emissionfrom a single photon in each optical mode.

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Spontaneous emission in semiconductor

C

V

2

2 0

02ˆ

cv c v

qAH

me − = p

We need to relate the magnitude of the vector potential to the mode density.

(E

)M energy densi y

(1)

)

t

(

mode

mode

V

V

=

=

2 2

0 0

2 2 2

0 0

1EM energy density

2

1

2

n E

n A=

=2

0 2 2

0

2 3

)

2

2 (modeA

n

n

c

=

=

22 2() 1( )

c v

spon cv e h c v

k k

r E fV

H E f

= − − −

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Spontaneous emission in semiconductor

2

22

2 2 3

0 0

2 22

0 2 2

) ( )

(

)2 2

( (1

2

(

)(2

ˆ

2 2)

( )

(1)

1 )

c v

c v

c v

spon cv e h c v

k k

cv e h c v

k k

cv e h c v

k k

r

n

H E f

qH E

f

E fV

nE f

m

C e

fV c

E fEc

= − − −

= − − −

− − −=

p

2 22

, 0 ,2 2( ( ( )ˆ (1 ( ))) )spon bulk cv r bulk g c g v g

nr C e E f E E

cf

= −

− − −p

Spontaneous emission spectrum 3 -1 -1cm s eV

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Spontaneous emission spectrum

GaAs (bulk)T = 300K

18 -3×10n = 2 cm

17 -3×10n = 5 cm

18 -3×10n = 1 cm

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Spontaneous emission spectrum

gE

)(spon r gEr −

( )g

kT

E

sponr e

−−

c vFF −

For

For

Limited by density of states

Limited by filling of states

)(r gE −

)( g

kT

E

e

−−

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C

V

Low-injection

For low-injection (Quasi-Fermi levels are within the bandgap)

* *

* *

( )( )exp

( )( )(1 ) ~ exp

c g g r

c

v

e

hg r

v

Ef

m

E

F E m

kT

m m Ff

kT

−

−

− −

− −−

* * * *( )( ) ( )( )(1 ) exp

e r hc g g r g v

c v

E Ef

F E m m m m Ff

kT

− −

− −−

−

−

( )exp exp exp

( )exp

g g c v

g

c v

k

E

E

E F F

kT kT T

np

kT N N

−=

− =

−

− −

−

, vcN N (effective density of states)

cF

vF

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Radiative rate

2 22

, 0 ,2 2

( )( ( xˆ ) e p)

g

spon bulk cv r bulk g

c v

n npr

c k

EC e

T NE

N

−

=

− −

p

, 0) (( )rad spon bulkr npR d B = =Radiative rate

-3 -1cm s

10

10

10

13

14

13

10

10

10

3.

G

2

9

aAs 2.0

InP 1.2

1

10

3.2 10

0

S

GaN 2.2

GaP

i

Ge .8

−

−

−

−

−

−

3 -1

0 (cm s )BMaterial

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Quantum well

18 -3×10n = 2 cm

17 -3×10n = 5 cm

18 -3×10n = 1 cm

Note: Linewidth effects are ignored

2 22

2

, 0 ,2 2, ,

| |ˆ ( ) ( )(1 (1

)en en en en

spon QW hm r QW hm c hm hm

h

cv v

hz n mlh

nr e

c LC I H E f E f E

= −

− − − p

InGaAs/InP quantum well6nm thick, T=300K

1 1C HH−

1 1C LH−

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Light emitting diodes

“yellow-gap”

Krames et al. Journal of Display Technology, Vol. 3, No. 2, June 2007

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Source: Compound Semiconductor

Thin-film flip chip flip chip vertically-injected thin film

Source: Lumileds