ee 232 lightwave devices lecture 21: steady state response ...€¦ · ee232 lecture 21-7 prof....

EE232 Lecture 21-1 Prof. Ming Wu

EE 232 Lightwave DevicesLecture 21: Steady State Response of

Semiconductor LEDs and Lasers

Instructor: Ming C. Wu

University of California, BerkeleyElectrical Engineering and Computer Sciences Dept.

Acknowledgment: some lecture materials are provided by Seth Fortuna


Current and photon confinement

Edge-emitting stripe laser

activeP-type

N-type

contact Photon confinement

Index contrast confinesmode in transverse

direction

Oxide confines currentin lateral direction.

Current confinement

lateral direction

transverse direction

oxide

No confinement inlateral direction

No confinement inlateral direction



Edge-emitting ridge laser

active

P-type

N-type

insulator

contact Photon confinement

Index contrast confinesmode in transverse and

lateral direction

Heterostructureconfines current in

transverse direction.

Ridge confines currentin lateral direction.

Current confinement



Photon confinement


lateral direction

Heterostructureand reverse-biasedpn junction confines current in transverse and lateral direction.

Current confinement

active

P-typeN-type N-type

P-type

N-type

P-type

Edge-emitting buried ridge stripe

contact



Surface-emittingoxide aperture laser

Photon confinement

Oxide confines currentin lateral direction.

Current confinement

Heterostructureconfines current

in transverse direction

contact

oxide

DBRmirror

active

P-type

N-type


lateral direction



Single quantum well separateconfinement heterostructure

cE

vE

cE

vE

cE

vE

Multiple quantum well (MQW) separate confinement heterostructure

Graded index separateconfinement heterostructure (GRINSCH)

cE

vEDouble heterostructure


Quantum well laser

active

P-type

N-type

d

L Facetmirror

Facetmirror

contact

contact

separate confinement heterostructure

quantum wells

P-type

N-type

SCH

• Quantum well(s) can only very weakly confine a mode.

• Usually, quantum wells are sandwiched inside another higher bandgap material that can be engineered to improve mode guiding and electron injection into the quantum well(s).

• This type of structure is called a separate confinement heterostructure (SCH).

Posi

tion

Refractiveindex

Posi

tion

Bandgap


Gain in quantum well

w!11ehE

gain

pgPeak gain occurs at the bandedge

Assuming only subband is filled in conduction and valence bands we can write an approximate expression for the peak gain

1 1m 1 1( [ ] [ ])e e

p c h v hg g f fE Ew w= - == ! !

( )11

1

1( ) )1

1 expex /p

(ec h c

g e c

fE E F

E n nkT

w = = - -é ùë û

=+ + -

!

( )11

1

1( ) )p

exp(/1 ex

eh

vv

hvf E F

E p nkT

wù

=+

= = -éë û-

!

( )1e ) / ]ln 1 xp[(c c g en n F E E kT-+ -= *

2e

cz

m kTL

np

=!( )1ln 1 exp[( ]) /vhvp n E F kT= -+

*

2z

hv

m kTL

np

=!

Recall,

The Fermi functions can be written in terms of the carrier density

Then, [ ]m 1 exp( ) exp( )p vcg g n n p n= - - - -


Gain in quantum well

[ ]m 1 exp( ) exp( )p vcg g n n p n= - - - -

The plot of peak gain vs. carrier concentration (red line) is approx. linear on a semi-log plot, therefore a simplerapproximate expression is often used(dashed gray curve).

0 0ln[ / ]pg g n n=

As discussed previously, the current density can be written in terms of a polynomial of the carrier density (e.g. 𝐽 ∝ 𝑛! if spontaneous emission dominates). Therefore, we can write a similar approximateexpression for peak gain in terms of carrier density.

0 0ln[ / ]pg g J J= Note: g0 are not the same in both expressions


Quantum well laser - threshold gain

00

ln ww

Jg gJ

æ öç ÷=è ø

: quantum well threshold gainJ : threshold quantum well current densityw

w

g

1 2

1 12lnw w w in gL R R

aæ ö

G = çè

+ ÷ø

n : number of quantum wells in active region: fraction of mode in quantum well

w

wG

quantum wells

P-type

N-type

SCH

zw

op

Ld

G=G

opticalconfinement

factoreffectivewidth of

optical mode

quantum wellthickness


Threshold current and gain in active region with multiple quantum wells

00

ln w ww w w

w

n Jn g n gn J

æ=

öç ÷è ø

w wn J

ww

ng

2wn =

w wth

i

n JJh

=Note:

threshold currentat device terminals

Then: 0

0

0

0 1 2

0

0 1 2

exp

exp ln

exp l

1 1 12

1 12

n 1

w wth

i

wth i

i w w

wth i

i w w

n J gg

n Jn g L

J

R R

wLn JIn g L R

J

R

h

ah

ah

æ öç ÷è øæ öé ù

÷

=

= ç ÷ê úç ÷G ë ûè øæ öé ùç ÷ê úç G ë ûè

+ø

+

= w: active region width: active region lengthL

ww

ng

1 L

ln()

thJ1wn =

2wn =

3wn =

3wn =

1wn =


Active region optimization

0 1 2

1 l1 n 102

th

w wopt

I LL n g R R

æ öç ÷G

® =è

=¶ ø

¶

Optimize cavity length (L) or a fixed number of quantum wells

Optimize number of quantum wells (nw) for a fixed cavity length

0 1 2

1 l1 1n02

thi

w wopt

I ndn g L R R

aæ öç ÷+

G=

è= ®

ø

¶

For a general cavity,1 2

l1 1 12

nip gv L R R

at

= +æ öç ÷è ø

therefore,

0 pQ w t=

0

0

1opt

w g

ng v Q

wG

=


Additional details on the “ABC” approximation

2 3

SRH sp AugerR R R

An n

R

B Cn

+ +

+

=

+!

The ABC approximation is widely used to estimate recombinationrates in LEDs and lasers (at or below threshold). Although strictly speaking, it is valid only when Boltzmann statistics are valid so some care needs to be applied when using the approximation.

We have already proved that the spontaneous emission rate has ann2 dependence (when Boltzmann statistics apply). See the previous lecture on spontaneous emission.

Let’s look at the Shockley-Reed-Hall and Auger rates.

“ABC” approximation

Recombination rate in LED or Laser at or below threshold:


Shockley-Reed-Hall recombination

cE

vE

TE

The derivation of the SRH rate is foundin many basic semiconductor textbooks.

20 0

1 10 0) )( ( s

SRHnp ds d

nRC n n C p p

n p n nn n

d d dd d- -+ + ++ ++ +

!

0

0

n p

np pn

pn d

dd d=

= +

+

=

,n n n th dsC v Ns=

,p th dsp pC v Ns=

dsn dspand are the electron and hole concentrations when the Fermi level is at the defect state energy

We see that in general we cannot write SRHR An=

But, we can do so if we restrictour analysis to a “low-injection” or“high-injection” regime.

Capture rates

ns

ps

nvpv

Capture cross-sections

Thermal velocity

dsN : defect density


Low-injection and high-injection regimeActive region materials have a background doping due to unintentional dopingimpurities introduced during growth. Let’s assume our active region is unintentionally doped p-type.

p0 ≫ δn Low-injection regime2

0 01 1

0 0( ) )( n n lowp ds ds

SRHn

nR A nn p n n CC n n C pn p n

d d d dd d- -

+ +@

+ + ++=

+!

0n pd ! High-injection regime

hn p

SRH igp

hnn

C CR A n

C Cd

+@ !

We see that we can write so long we stay in one of the two regimes.If the electron capture is the rate-limiting step (for p-type material), then the A coefficient will be identical in both regimes.

SRHR An=


Auger recombinationElectron recombines with hole and gives up excess energy to another carrier instead of releasing a photon. Several different Auger processes are possible (as shown below). Often there is a material-dependent dominant process.


Auger recombination

The CCCH Auger rate is given by

( )( )0 1 2 3

1 30

2

4

2

24

0 2

2 3

1 1

exp exp exp 1

exp)

(

(

f

)

or

Auger

c c v

g

c v

Auger

R C f f f f

F E F E FkT kT kT

E En pkT N N kT

R p Cn p

E

C

n

C

Cn

é ù é ù é ù= ê ú ê ú ê úë û ë û ë ûé ù

= ê úë

= - -

- -

=

-

-

û= =

Very likely that State 4 is emptysince it is well beyond the bandedge


Auger recombination

CH

TE

k

EEnergy and momentum conservationneeds to be simultaneously conserved.This sets a threshold value for E4 whichwe call ET. Materials with small ET willhave large Auger rates since

exp( / )Auger TR E kTµ -

ET is related to the curvature of the bandsthrough

* *

* *

2 e hT g g

e h

m mE aEm m

E+= =

+

the value of a is approximately unity for III-V semiconductors therefore,

exp( / )Auger gR E kTµ -

Auger recombination is higher inlow bandgap materials.

ee 232 lightwave devices lecture 21: steady state response ...€¦ · ee232 lecture 21-7 prof....

Documents