lecture #12 spontaneous emissionee232/sp19/lectures... · 2019. 2. 28. · fortuna –e3s seminar 2...
TRANSCRIPT
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
2Fortuna – E3S Seminar
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)
3Fortuna – E3S Seminar
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)
4Fortuna – E3S Seminar
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 =
5Fortuna – E3S Seminar
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.
6Fortuna – E3S Seminar
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
= − − −
7Fortuna – E3S Seminar
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
8Fortuna – E3S Seminar
Spontaneous emission spectrum
GaAs (bulk)T = 300K
18 -3×10n = 2 cm
17 -3×10n = 5 cm
18 -3×10n = 1 cm
9Fortuna – E3S Seminar
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
−−
10Fortuna – E3S Seminar
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
11Fortuna – E3S Seminar
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
12Fortuna – E3S Seminar
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−
13Fortuna – E3S Seminar
Light emitting diodes
“yellow-gap”
Krames et al. Journal of Display Technology, Vol. 3, No. 2, June 2007
14Fortuna – E3S Seminar
Source: Compound Semiconductor
Thin-film flip chip flip chip vertically-injected thin film
Source: Lumileds