1

EE232 Lecture 14-1 Prof. Ming Wu

EE 232 Lightwave DevicesLecture 14: Quantum Well andStrained Quantum Well Laser

Reading: Chuang, Sec. 10.3-­10.4(There is also a good discussion in Coldren, Appendix 11)

Instructor: Ming C. Wu

University of California, BerkeleyElectrical Engineering and Computer Sciences Dept.

EE232 Lecture 14-2 Prof. Ming Wu

Quantum Well Gain1.4 1.6 1.8 2 2.2 2.4 2.61−

0.5−

0

0.5

1

f_g xi 0.05eV, ( )f_g xi 0.1eV, ( )f_g xi 0.15eV, ( )f_g xi 0.2eV, ( )f_g xi 0.25eV, ( )f_g xi 0.3eV, ( )f_g xi 0.35eV, ( )f_g xi 0.4eV, ( )f_g xi 0.45eV, ( )

xieV

1.4 1.6 1.8 2 2.2 2.4 2.6

1− 106×

0

1 106×g xi 0.05eV, ( )g xi 0.1eV, ( )g xi 0.15eV, ( )g xi 0.2eV, ( )g xi 0.25eV, ( )g xi 0.3eV, ( )g xi 0.35eV, ( )g xi 0.4eV, ( )g xi 0.45eV, ( )

xieV

QW Material Gain:

g(!ω) =C0 e! ⋅P!"cv

2

ρr2d (!ω) fg (!ω)

C0 =πe2

nrcε0m02ω

e! ⋅P!"cv

2

≈m0

6Ep

ρr2d (E) =

mr*

π!2LzH (E − Een )

n=1

∞

∑

2

EE232 Lecture 14-3 Prof. Ming Wu

Advantages of Quantum Well Lasers(1) Low threshold current density:Compare fundamental material property

Transparency current density

10 nmSince ~100 nm

10%(

bulkbulk trtr active

QWQW trtr z

QWbulk QW tr ztr tr bulk

tr active

qNJ d

qNJ L

J LN NJ d

τ

τ

→

=

=

≈ ⇒ =

2) Higher differential gain Larger bandwidth:

Resonance frequency:

(3) Lower chirp:Smaller wavelength shift when the laser is directly modulated

gR

p

v aS gaN

ωτ

→

∂= ∝ =∂

EE232 Lecture 14-4 Prof. Ming Wu

Transparency Carrier Concentration in Bulk

3/23/2*3/2

2 2

At transparency: or

Let

Electron concentration:

4 422 3 3

Hole concentration:

C V C V

C C V V

C C V V

B B

C C

e B n CC

B

V

F F E EF E F E

F E F Ek T k T

F E

m k T F EN Nk T

F E

ππ π π

− = −− = −

− −Δ = =

>

⎛ ⎞⎛ ⎞ −= = ⋅ Δ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

>

Q

hQ 1

3/2*

2 2

3/2*3/2

*

* *0 0

17 3

22

4 Solve 3

For GaAs ( 0.067 , 0.5 )

2.15, 9 10 cm

V V

B

h

F Ek Th B

V

h

e

e h

m k TP e N e

mN P em

m m m mN

ππ

π

−−

−Δ

−Δ

−

⎛ ⎞= =⎜ ⎟

⎝ ⎠

⎛ ⎞= ⇒ Δ = ⇒ Δ⎜ ⎟

⎝ ⎠= =

Δ = = ×

h

EC

EV

CB

VB

FC

FV

Transparency Condition(Bernard-Duraffourg Inversion Condition)

C V gF F F EΔ = − =

3

EE232 Lecture 14-5 Prof. Ming Wu

Transparency Carrier Concentration in QWAt transparency: FC − FV = Ee1 − Eh1 or FC − Ee1 = FV − Eh1

Let Δ =FC − Ee1kBT

=FV − Eh1kBT

Electron concentration: ∵FC > Ee1

N =me*kBT

π!2Lz

FC − Ee1kBT

!

"##

$

%&&= NC

2d ⋅ Δ

Hole concentration: ∵FV > Eh1

P = mh*kBT

π!2Lze−FV −Eh1kBT = NV

2de−Δ

N = P ⇒ Δ =mh*

me*e−Δ ⇒ Solve Δ

For GaAs (me* = 0.067m0 ,mh

* = 0.5m0 )

Δ =1.56, N = NC2d ⋅ Δ =1018 cm−3

Note: N is independent of Lz

EC

EV

CB

VB

FC

FV

Transparency Condition(Bernard-Duraffourg Inversion Condition)

C V gF F F EΔ = − =

EE232 Lecture 14-6 Prof. Ming Wu

Reduction of Lasing Threshold Current Density by Lowering Valence Band Effective Mass

• Yablonovitch, E.;; Kane, E., "Reduction of lasing threshold current density by the lowering of valence band effective mass," Lightwave Technology, Journal of , vol.4, no.5, pp. 504-­506, May 1986

• Yablonovitch, E.;; Kane, E.O., "Band structure engineering of semiconductor lasers for optical communications," Lightwave Technology, Journal of , vol.6, no.8, pp.1292-­1299, Aug 1988

1 1

Bernard-Duraffourg Condition:

C V e hF F E Eω− ≥ ≥ −h

* *

Ordinary Semiconductor6

High transparency carrier concentration

h em m≈ * *

Ideal Semiconductor

Low transparency carrier concentration

h em m≈

4

EE232 Lecture 14-7 Prof. Ming Wu

Bernard-­Duraffourg Condition in Quantum Well

Bernard-Duraffourg Condition:FC − FV = E − Eh1(a) mh

* >me* (as in most semiconductors)

FV > Eh1FC ≫ Ee1

Ntr = ρe2d (FC − Ee1) =

me*

π!2Lz(FC − Ee1)

Large Ntr → High threshold current

(b) mh* =me

* (Ideal semiconductor)

FV = Eh1FC = Ee1

Ntr =me

*

π!2LzfC (E)dE

Ee1

∞

∫ is low

EE232 Lecture 14-8 Prof. Ming Wu

Transparency Carrier Concentration for Ordinary Semiconductor

( )

1

1

* *1 1

*

2

*

20

*

2 0

*

2

*0

17 3

(b) Ideal Semiconductor

=

1

11

1

ln(1 )

ln 2

For =0.067

4.6 10

e

e B

h e V h C e

etr E E

z E k T

B ex

z

xB e

z

B e

z

e

tr

m m F E F E

mN dEL

ek Tm dx

L e

k Tm eL

k TmL

m m

N cm

π

π

π

π

∞

−

∞

∞−

−

= ⇒ =

=+

=+

= − +

=

≈ ×

∫

∫

h

h

h

h1 1

Transparency Condition:

C V e hF F E E− = −

5

EE232 Lecture 14-9 Prof. Ming Wu

Transparency Carrier Concentration for Ordinary Semiconductor

1 1

Transparency Condition:

C V e hF F E E− = −

*2

1 2

*2

2

*

*

* *

*

(a) Ordinary Semiconductor

( )

To estimate , note that

For 6 (in 1.55 m laser), 1.43

1.43

B B

B

d etr e C e

z

k T k Td B hV

z

k T e

B h

h e

B

B etr

mN F ELN P

k TmP N e eL

mN P ek T m

m mk Tk TmN

ρπ

π

µ

π

−Δ −Δ

−Δ

= − = Δ

Δ =

= =

Δ= ⇒ =

≈Δ =

=

h

h

h2zL

EE232 Lecture 14-10 Prof. Ming Wu

Effective Mass Asymmetry Penalty

2 3 2 3

1.43 2ln 2

Threshold current density reduction is more than a factor of 2:

: Shockley-Read-Hall nonradiative recombination lifetime

OrdinarytrIdealtr

th nonrad rad Auger

th

NN

J J J JJ NAN BN CN BN CNqd

J

ττ

= =

= + +

= + + = + +

3

is greatly reduced when is lowered

(1) is reduced by 8x(2) C is also reduced due to band structure change by strain

Auger N

N

6

EE232 Lecture 14-11 Prof. Ming Wu

Bandgap-­vs-­Lattice Constant of Common III-­V Semiconductors

Compressive Strain

Tensile Strain Lattice Matched

In0.53Ga0.47As

EE232 Lecture 14-12 Prof. Ming Wu

Qualitative Band Energy Shifts Under Strain

C

HH, LH

C

SO

SO SO

HH, LHHH

LH

CCC

HH, LHLHHH

SO SO

Compressive Strain

Tensile Strain

CEδ

Pε−

Qε+

Qε−

Hydrostatic Strain

BiaxialStrain

BiaxialStrain

Hydrostatic Strain

7

EE232 Lecture 14-13 Prof. Ming Wu

Strain and Stress

0

0

0

12

11

12 11

( )

: lattice constant of InP0 : compressive strain0 : tensile strain

2

: Compliance Tensor

0.5

xx yy

zz

ij

a a xa

a

CC

CC C

ε ε ε

εε

ε ε ε⊥

−= = =

<⎧⎨ >⎩

= = −

≈

11 12 12

12 11 12

12 12 11

12 12 11

12

11

Biaxial stress:

0 0

2

xx xx

yy yy

zz zz

xx yy

zz

xx yy zz

zz

C C CC C CC C C

C C CCC

σ εσ εσ ε

σ σ σσ

ε ε ε

ε ε

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

= =

=⇒ + + =

= −

EE232 Lecture 14-14 Prof. Ming Wu

Band Edge Shift

12

11

12

11

12

11

( )

( ) 2 1

( ) 2 1

( ) 1 22

: hydrostatic potential :

C g C

HH

LH

C C xx yy zz C

V xx yy zz V

xx yyzz

C V

E E x EE P QE P Q

CE a aC

CP a aC

CQ b bC

a a ab

ε ε

ε ε

ε

ε

δ

δ ε ε ε ε

ε ε ε ε

ε εε ε

= +

= − −= − +

⎛ ⎞= + + = −⎜ ⎟

⎝ ⎠⎛ ⎞

= − + + = − −⎜ ⎟⎝ ⎠

+ ⎛ ⎞= − − = − +⎜ ⎟

⎝ ⎠= −

shear potential

8

EE232 Lecture 14-15 Prof. Ming Wu

Strain Parameters in III-­V(Coldren, p.535)

EE232 Lecture 14-16 Prof. Ming Wu

Band-­Edge Profile and SubbandDispersion