EE 232: Lightwave Devices Lecture #16 – Steady state LED and LASER characteristics - Part 1 Instructor: Seth A. Fortuna Dept. of Electrical Engineering and Computer Sciences University of California, Berkeley 4/2/2019
EE 232: Lightwave Devices
Lecture #16 – Steady state LED and
LASER characteristics - Part 1
Instructor: Seth A. Fortuna
Dept. of Electrical Engineering and Computer Sciences
University of California, Berkeley
4/2/2019
2Fortuna – E3S Seminar
Light emitting diode at steady state
active
P-type
N-type
contact
contact
cE
vE
vF
cF
d
Rn
t−=
−J
carrier recombination
At steady state: i RJ
qd =i R
dn J
dt qd −
We neglect current spreading in the lateral direction and assume quasi-Fermi levelis constant throughout active region.
Rate equation for active region(ignoring generation term)
d is activeregion thickness
injectionefficiency
3Fortuna – E3S Seminar
Carrier recombination
mSRH s sti Aug rp eR RR RR + + +=
Leaky Bathtub analogy
At steady state,we must continuouslypump the activeregion to maintaina certain carrier density
spR
J qd
N
Shockley-Reed-Hall recombination
Spontaneousemission
Stimulatedemission
Augerrecombination
Coldren et al. Diode Lasers and Photonic Integrated Circuits.
4Fortuna – E3S Seminar
Spontaneous emission power
uSRH si A gerp
JR R
qdR += +
Let SRH s
sp sp
sp
A er ip ug
R R
R R J qdR
==
+ +
Spontaneous emission power (P
= (Photon energy) (Emission rate) (Active region volume)
)
sp
sp sp act
sp i
sp i
sp IQE
P
I
V
J qd
q
R
V
P
I
q
=
=
=
=
For constant internal quantumefficiency (IQE), there is a linearrelationship between power andcurrent. However, in general,IQE does change with current
5Fortuna – E3S Seminar
ABC approximation
2 3
Augei rSRH sp
JR R R
qd
An Bn Cn
+
+ +
= +
2
2
2 3
sp
IQE i sp i i
I
SRH sp Aug
i
er
QE
R Bn
R R An Bn Cn
Bn
A C
R
Bn n
= =+ +
=+
+ +=
+
Low drive current2A Bn Cn +
1/2
~ constantIQE
sp
n J
P J
2Bn A Cn +
Moderate drive current
2
IQE
sp
n J
J
P J
1/3
1/3
2/3
~ IQE
sp
n J
J
P J
−
2Cn A Bn +
High drive current
6Fortuna – E3S Seminar
ABC approximation
2J
J
2/3
J
SRH Sp. Em. AugerEx. InGaAs LED
Note: At high current density, other loss mechanisms beside Auger may become important.See for example, controversy surrounding “droop” in GaN LED efficiency at high current.
7Fortuna – E3S Seminar
LED external quantum efficiency
c
active
P-type
N-type
0
))(2 si
()
1(
4n
c
c
dd
TT
d
= =
Most emitted light will become trappedin the high-index semiconductor.Only light with angle smaller thancritical angle can escape.
0
2
1 2
0
1(0
4
1 4
4 (
)(2 sin )
(2 sin1)
( )
)
1
c
c
c
T
n
n
n n
d
d
− −
=+
+
1sin ~air
c c
n
n n →=
sp IQE c ext IPq q
I
= =
Coupling efficiency Externally collected power
External quantumefficiency
8Fortuna – E3S Seminar
Semiconductor laser at steady state
contact
active
P-type
N-type
d
LFacetmirror
Facetmirror
x
y
x
z
Edge-emitting ridge laser
P-type
N-type
contact
contact
active
N-type
insulator
(Area) (Area)actcav
cav cav
V d dV
V V == →
photon density in lasing modeS
carrier density in active regionn
group velocitygv
fraction of sp. em. into lasing modesp
photon lifetimep
gaing
cavity volumecavV
active region volumeactV
9Fortuna – E3S Seminar
Rate equations for photons and carriers
sp sp stim
p
dS SR R
dt
+= − +
Rate of photonsleaving cavity
Sp. em. rateinto cavitymode
Stimulatedemission
Hi sti rSR sp m Auge
dn JR R R
d qR
t d= − −− −
Shockley-Reed-Hall recombination
Spontaneousemission
Stimulatedemission
Augerrecombination
Rate equation for carriers in active region
Rate equation for photons in cavity
( )rSRH sp stim Auge
i
qdJ R R R R
+ + +=(steady state)
( ) ( )1sp sp p gR gS v −= (steady state)Note, recall:
stim gR v gS=
10Fortuna – E3S Seminar
Below threshold – Spontaneous emission
( ) ( )
( )
1sp sp p g
p sp sp
p sp i sp
S
d
vR g
R
J q
=
−
=
Recall, 1 1
( )th m i p
g p g m i
gv v
= →+
+= =
( )
sp sp i
g m i
J
v qdS
=
+
sp
sp
i
R
J qd
=From before,
This is the number of spontaneouslyemitted photons in the cavity mode
Then,
11Fortuna – E3S Seminar
)
Spontaneous emission power (P
= (Photon energy) (Rate of loss from mirror) (Photon den lsity
)
(Cavity vo ume)
sp
)) )(( ( )
)
(
(
sp g m cav
sp i
g m cav
g m i
sp
P v S V
J
v qdv V
=
+
=
Note:
(Area) (Area)actcav
cav cav
V d dV
V V == →
This is the spontaneous emission power into the cavity mode
m
spm
sp i sp
i
P Iq
=+
Below threshold – Spontaneous emission
12Fortuna – E3S Seminar
At threshold and above threshold
)( ( ( ))SRH sp Augth
i th th e hr tRJ
n R n R nqd
+= +
Laser threshold condition is reached when gain precisely balances cavity loss.Stimulated emission will still be small with respect to spontaneous emission.
At threshold
Above threshold and under steady state operation, the gain must “clamp” at the threshold gain. If this were note the case, the fields in the cavity wouldgrow without bound (thus not achieving steady state). This also implies thatthe Fermi-level and current density must clamp at the threshold current density.
Above threshold
for
for
th th
th th
g g I I
n n I I
13Fortuna – E3S Seminar
Gain and carrier density clamping
1
sp sp
p g
R
gS
v
−=
Recall,
Now, let’s derive a stimulated emission lifetime.
1stim g stim
stim g
n ngS
v gR v
S
=
= →
=
We see that there is a negative feedback loop that does not allow the gain to increasebeyond the gain threshold. As current is increased and gain approaches the threshold gain, the photon density increases dramatically thus significantly reducing the stimulated emission lifetime. Any additional current gets immediately “used up” by stimulated emission not allowing the carrier density or gain to increase.
The laser at threshold is similar to a filled bathtub. Any additional water spills over the side.Likewise any additional injected carriers will “spill out” as stimulated emission and willnot increase the carrier density.
spR
J qd
stimR
Coldren et al. Diode Lasers and Photonic Integrated Circuits.
14Fortuna – E3S Seminar
Above threshold
( ( () )
)
)
(
i th th th stim
thi i g th
i
SRH sp Au
t
ge
hg th
r
Jn R n R n
qd
JJv g S
qd qd
v g Sq
R
d
R
J J
−=
=
+ += +
+
From the carrier density rate equation
Plug into the photon density rate equation
( )
sp sp g th g th
p
i thp
v
d
v
S
S
J
R g S g S
q
J
+=
−=
)(mstim g m i th
m i
P v S I Iq
= = −
+
15Fortuna – E3S Seminar
Above threshold (L-I curve)
Current
Pow
er
Spontaneous emissionclamped above threshold
thI
Above threshold, emitted power is dominated by stimulated emission.Spontaneous emission power clamps at the threshold power.
16Fortuna – E3S Seminar
Above threshold (L-I curve)
log(Current)
log(P
ow
er)
thI
Plotted on log-log scale the L-I curve has an “S”-like shape.
17Fortuna – E3S Seminar
Above threshold (L-I curve)
Coldren et al. Diode Lasers and Photonic Integrated Circuits.
18Fortuna – E3S Seminar
Differential quantum efficiency
edI
q dP
=
We define the external differential quantum efficiency of the laser as
This is the probability of externally collecting a stimulated photon for an injectedelectron-hole pair.
ln(1/ )
ln(1/ )
me i i
m i i
R
L R
+ +
= =
By plotting the inverse of the external differential quantum efficiency as a function of cavity length we can determine cavity loss and injection efficiency from the slope.
1 1
ln(1/ )
i
ie i
LR
= +
Cavity length (L)
1
e−
1
i−
ln(1/ )
i
i Rm
=
19Fortuna – E3S Seminar
Summary
Light emitting diode
)( SRH sact Aug r
i
p eI RV RRq
= + + sp IQE c IP
q
=
Semiconductor laser
( )act Auger
i
ms s
i
SRH
p i s
m
sp
p p
qR R
P I
I V
q
R
=
+
+
+
=
Below threshold:
Above threshold:
(Spontaneous emissionpower into lasing mode)
)(
)(
thSRH sp stimact Auger n n
i
mstim i th
m i
I Rq
R R R
P I
V
Iq
=
+ + +
= −+
=
(Stimulated emission power into lasing mode)