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)