Electro-absorption Modulators
EE232, Spring 2001
Pei-Cheng Ku
April 24 & 26, 2001
Optical Modulation

Direct modulation on semiconductor lasers:

Output frequency drifts




carrier induced (chirp)
temperature variation due to carrier modulation
Limited modulation depth (don’t want to turn off laser)
External modulation


Electro-optical modulation (low efficiency)
Electroabsorption (EA) modulation (smaller modulation
bandwidth)
Electroabsorption (EA) Modulator


EA modulator is a semiconductor device with the
same structure as the laser diode.
In laser diodes, we inject large enough current to
achieve stimulated emission. While in EA
modulator, we apply electric field (reverse bias) to
change the absorption spectrum. No carriers are
injected into the active region. However, carriers
are generated due to absorption of light.
Advantages of EA modulators





Zero biasing voltage
Low driving voltage
Low/negative chirp
high bandwidth
Integrated with DFB
Schematics of an EA modulator
From textbook, figure 13.10.

Waveguide type is more popular.
Outline of our discussion



Know how absorption spectrum changes with
applied electric field (effective-mass
approximation)
See how extinction ratio of modulation can be
enhanced in the quantum well (quantum confined
Stark effect). Excitons will be introduced.
We will follow the textbook (chapter 13). A
recommended reference book is “Introduction to
Semiconductor Optics” by N. Peyghambarian et al.
(1993).
Physics behind EA Modulators

How absorption spectrum in semiconductors can be
changed?

Physical model: effective-mass equation






Single-particle representation
Two-particel representation
Coulomb interaction between electrons and holes: Excitons
Electric field effect: Franz-Keldysh effect
Coulomb+Electric field: DC Stark effect
Coulomb+Electric field+QW: QCSE
Absorption Spectrum Change under
Applied Electric Field

Franz-Keldysh Effect


DC Stark Effect


Neglect Coulomb interaction between electrons and
holes.
Franz-Keldysh effect plus Coulomb interaction between
electrons and holes (excitons).
Quantum confined Start effect (QCSE)


DC Stark Effect in quantum wells
Excitons been confined in quantum well. Stark effect
enhanced.
Effective mass approximation

Electron moving in semiconductor crystal with periodic potential V (r) under
the influence of a potential U (r).
 2 2




V
(
r
)

U
(
r
)
crystal
 2m
 (r )  E(r )
0



Effective mass approximation: if the potential is slowly varying compared to
the period of the lattice, the above equation can be replaced by effective-mass
equation as follows (section 4.4.1 and appendix B):
 2 2




U
(
r
)
 2m*
f (r )  E  Enk (k  0)f (r )


(r )  f (r ) unk (r )
that is the potential causes the envelope of the fast-varying Bloch function to
change. Without potential U (r), the envelope f (r) reduces to plane wave
 2k 2
exp( ik  r ) and E 
2 m*
Wavefunction for electrons moving in
periodic potential

For example, at the valence bandedge. The
wavefunction is p-like (L = 3/2 in sp3 orbitals).
uvk (r )
f (r )  exp( ik  r )
 (r )  f (r )u (r )
Form of U (r)

Franz-Keldysh Effect
U ( r )  eF  r


DC Stark Effect (slow-varying approximation only
holds for Wannier excitons)
e2
U (re )  eF  re 
4r (re  rh )
Quantum confined Start effect (QCSE)
e2
U (re )  eF  re 
 Ve (re )
4r (re  rh )
Effective mass approximation in
Two-Particle Representation


Since U (r) is in general a function of both electron and hole, it’s more
convenient to combine two single-particle equations of motion for
electron and hole together.
Effective-mass equation for electrons:
 2 2




U
(
r
)
e e fe (re )  Ee  Een ( k  0)fe (re )
 2m* e
e


Effective-mass equation for holes:
 2 2




U
(
r
)
h h fh (rh )  Eh  Ehn ( k  0)fh (rh )
 2m* h
h


Add them together 
 2 2 2 2






U
(
r
)

U
(
r
)
e e
h h  (re , rh )  E  E g  (re , rh )
 2 m* e 2 m* h
e
h


where  (re , rh )   B(k e , k h )fe,k e (re )fh ,k h (rh ) and E  Eh  Ee
k e ,k h
original wavefunct ion is (re , rh ) 
 B(k , k
e
k e ,k h
h
)fe,k e (re )fh ,k h (rh )uc ,k e (re )uv ,k h (rh )
Effective mass approximation in
Two-Particle Representation (cont.)
 (re , rh ) 
 B(k , k
e
h
)fe ,k e (re )fh ,k h (rh )
k e ,k h
can also be written in the expansion of plane wave modes of ke and kh as follows.
eik e re eik h rh
 (re , rh )   A( k e , k h )
V
V
k e ,k h

In the path of generalizing single-particle effective mass equation to
two-particle effective mass equation, we didn’t make new
approximations. The new envelope wavefunction (which is a linear
combination of the single-particle envelope functions for electrons and
holes) still has to satisfy the slowly-varying approximation. This will be
valid if the potential seen by electrons can be smoothly transferred to
the potential profile seen by holes.
Validity of Slowly-Varying Approximation
in two-particle representation

Franz-Keldysh Effect
U (re , rh )  eF  (re  rh )

e  |e|
For 5V applied voltage, there is 500kV/cm electric field across 0.1mm active
region. The variation of electric field across a lattice site is 500kV/cm x 5.6Å ~
0.028V which gives an energy variation of 0.028eV for an electron.
For an electron moving in periodic atomic potential, it sees a Coulombic
attraction from the nucleus.
V (r)  
e2
40 r
(Note,  0 instead of  r has been used.)
(Also, one valence electron is available from each atom. It sees " screened" nucleus.)

The potential energy variation across a lattice site is around V(a) ~ 2.569eV
>> 0.028eV!
Excitons

The effective-mass equation for an electron-hole pair is:
 2 2 2 2






U
(
r
)

U
(
r
)
 (re , rh )  E  E g  (re , rh )
e
h
e
e
h
h
 2 m*

*
2mh
e



 
1  1
 e2
where U e (re )  U h (rh )  


2  N e N h i , j  4r | re,i  rh , j |  
(remember to put a factor 1/2 because of pair interactio n)
We have to sum over all possible interactions between electrons and
holes. But if the carrier density is low enough, we can neglect the
contribution from the rest of the electrons and holes and consider only
bare Coulomb interaction, that is:
 e2
U e (re )  U h (rh ) 
4r | re  rh |
Effective-mass equation for Excitons

With the bare Coulomb interaction between electron and hole, we have
the effective-mass equation for an exciton as follows.
 2 2 2 2

e2





 2m* e 2m* h 4 | r  r |(re , rh )  E  Eg (re , rh )
e
h
r
e
h 


This equation is formally identical to a hydrogen atom problem except
the masses for the electron and proton are replaced by the effective
masses for the electrons and holes, respectively.
Any central-force problem (i.e. the potential is a function of relative
coordinate between two particles) can be separated into center-ofmass motion and relative motion.
Relative coordinate : r  re  rh
 (re , rh )   (R, r )
me*re  mh*rh
Center of mass : R 
me*  mh*
Center-of-mass motion and Relative
motion

Effective-mass equation in two-particle representation for
any central-force problem can be rewritten as follows.
 2 2

2 2
 2 M R  2m r  U (r )(R, r )  ( E  Eg )(R, r )
r



Center-of-mass motion is a free-running solution:
eiKR
 (R, r ) 
 (r )
V
 2 2




U
(
r
)
 2m r
 (r )   (r )
r


2 K 2
  E  Eg 
2M
Solutions for Excitons


Same as hydrogen atoms (3D or 2D)
Exciton Bohr Radius and Rydberg energy
4r  2
aB 
mr e2
mr e4
Ry  2
2 (4r )2
Energy levels: bound states and continuum
For bound states (3D) : En  
For bound states (2D) : En  

Ry
n2
Ry
( n  1 / 2) 2
Ionization energy for ground state exciton in 2D case is 4 times larger
than it is in 3D case. Therefore in QW, it’s easier to form bound
exciton states.
Energy Diagram of Excitons
Electron-hole energy E
n=3
n=2
continuum
n=1
In GaAs,
E1=4.2meV (3D)
or
E1=16.8meV (2D)
K
Wavefunction for Excitons (3D)


Coulomb potential is spherical symmetric with respect to
the center of mass.
 (r)   nl ( r ) Ylm ( ,f )
Three quantum numbers: n, l, m.
n  1, 2, 3, 
l  0, , n  1
m  l ,  l  1, , 0, , l  1, l

“Accidental degeneracy” for same quantum number n.
Orbitals designated by 1s, 2s, 2p, 3s, 3p, 3d, …
Radial Solution (3D)
1/ 2
 2  ( n  l  1)! 
 r  2 l 1  2r 

 Ln l 

exp  

3
 naB  2n[( n  1)! ] 
 naB 
 naB 
where L2nll1 q  is the associated Laguerre polynomial .
 2r 

 nl ( r )  
 naB 
e.g. 1s ( r ) 
l
3
1
aB3
e
r
aB
probability
1s
0.012
0.01
2s
0.008
2p
0.006
0.004
0.002
2
4
6
8
10
r/aB
Spherical Harmonics Ylm (3D)
| Y00 |
| Y10 |
| Y11 |
| Y20 |
| Y21 |
| Y22 |
Wavefunction Representation for Excitons (3D)
Sitting at hole
reference frame.
P-like exciton
envelope function  (r )
(e.g. n=2)
S-like exciton
envelope function  (r )
(e.g. n=1)
s-like electron (L=1/2) bloch function
p-like hole (L=3/2) bloch function
Wavefunction for Excitons (2D)


Coulomb potential is cylindrical symmetric with respect to
the center of mass.
eimf
 (r)   nm (  )
2
Two quantum numbers: n, m.
n  1, 2, 3, 
m  n  1, , 0, , n  1

“Accidental degeneracy” for same quantum number n.
Radial Solution (2D)
4
 nm ( r ) 
aB
|m|
1/ 2

 

2r
( n  | m | 1)!

 

3
(
n

1
/
2
)
a
(
2
n

1
)
(
n

|
m
|

1
)!

B  


 2|m| 

r
2r




 exp  
Ln |m|1 

 ( n  1 / 2)a B 
 ( n  1 / 2)a B 
where L2n|m||m|1 q  is the associated Laguerre polynomial .
r
1
4 2 r aB
e.g. 10 ( r ) 
e
compared to 1s ( r ) 
e aB
aB
aB3
2D excitons are more tightly bounded.
Validity of Approximations used in the
Calculation of Excitons


Slowly-Varying Potential Profile Approximation:
GaAs/InGaAs/AlGaAs lattice density ~ 5.6Å
Bare Coulomb interaction approximation:
Exciton radius aB for GaAs ~ 140Å
Exciton density:
3D: 8.7e16/cm-3
2D: 8.1e10/cm-2


For EA modulators, no carriers are injected into active region. Carrier
density ~ intrinsic doping level ~ 1e16/cm-3 (bulk) or 1e10/cm-2 (100Å
QW). Bare coulomb interaction is a good approximation.
But for laser applications, carrier density is high: 5e18/cm-3 (bulk) or
5e12/cm-2 (100Å QW). Coulomb interaction between electrons and holes
is screened.
 e2 exp( r )
U (r) 
4s
r
(Yukawa' s potential)
Optical Absorption without U (r)

From Fermi’s golden rule, the absorption coefficient can be
derived as in equation (9.1.23) between two energy
bands:
2
2 ' 2
 ( )   H cv  ( Ec  Ev   )( f v  f c )
V
kv
kc

where matrix element H cv' is calculated as in section 9.3.1
in the case of no external electric field and neglecting the
Coulomb interaction between electrons and holes. The
result is as follows (equation 9.3.11).
H cv'  
eA0
eˆ  pcv kc ,kv
2m0
  (  )  C0
2
2
ˆ
e

p

cv  ( Ec  Ev   )( f v  f c )
V k
Optical Absorption with U (r)
The interband matrix element now becomes :
H cv'  
eA0
ik r
 final wavefunct ion | e op eˆ  pcv | init wavef unction 
2m0
Recall that the wavefunct ion is (re , rh ) 
 B( k , k
e
h
)fe,ke (re )fh,k h (rh )uc,ke (re )uv ,k h (rh )
k e ,k h
eik e re eik h rh
where the envelope function  (re , rh )   B( k e , k h )fe,k e (re )fh ,k h (rh )   A( k e , k h )
V
V
k e ,k h
k e ,k h
eik e re eik h rh ik opr
 H   A( k e , k h )eˆ  pcv 
e dr
V
V
k e ,k h
'
cv
 eˆ  pcv eiKR  A( k e , k e )  eˆ  pcv V  (r  0)
ke
 absorption spectrum
 ( )  2C0
  dE | eˆ  p
cv
all possible
states
In central force problem, |  (r  0) |2 |  (r  0) |2 .
|2 |  (r  0) |2  ( E   )
Absorption Spectrum Calculation
Summary





Absorption is proportional to the probability that we find
electron and hole at the same “position” (i.e. r=0).
For central force problem, we only need to evaluate the
envelope wavefunction for the relative motion and take the
square of its absolute value.
Note the factor of 2 in the formula of the absorption
spectrum comes from the fact that we consider two
particles at one time.
The delta function takes care of the (joint) density of
states available.
We have to sum over all possible states to get the total
absorption.
Remarks on the first assertion  (r  0)



2
Same “position” actually means electron and hole are in the same
lattice site. What happens in the unit cell is governed by the oscillation
strength pcv which depends on the details of the Bloch functions inside
the lattice site.
This is because  (re , rh ) is an “envelope” function. The true
wavefunction is the product of the envelope function with the Bloch
function.
2
 (r  0) is the probability of finding the electron and hole in the
same lattice site.
Density of States function g (E)

Density of states means total number of degenerate states for a given
energy level. For example in bulk materials, since there’s no restriction
on the motion of the electrons, for a given energy E, the number of
states that satisfy
2
E

2
2
2
(
k

k

k
)
x
y
z
*
2m
is the density of states at that energy.
  dE f ( E )   dE f ( E ) g ( E )
all possible
states
g (E) as a function of dimensionality
2
2
n

d
k   g(E)d E

n 
V k
( 2 )
n  0 : d n k   ( E ) dE
g(E)   ( E )
dk
m  1
 dE
  r2 
dE
d
E
dE
 2  E
dk
dk  2mr 
n  2 : d 2k  2kdk  2kdE
  2  dE
dE   
n  1 : dk  dE
 2m 
n  3 : d 3k  4k 2dk  2  2 r 
  
g(E)  1 / E
g(E)  constant
3/2
E dE
g(E)  E
Diagrammatic representation for g (E)
0D
g(E)
1D
3D
2D
E
For a given energy
range, the number of
carriers necessary to
fill out these density of
states:
3D>2D>1D>0D.
Franz-Keldysh Effect

Franz-Keldysh effect is also a central-force problem.
U ( re , rh )  eF ( re  rh )
e  |e|
  2 d 2


  ( z )  E z ( z )

eFz
2
2
m
dz
 r

1/ 3
change of variable
E 
 2m eF  
Z   r2   z  z 
eF 
   
d 2 ( Z )
 Z ( Z )  0
2
dZ
for Z  0
 ( Z )  Ai( Z ) 
Z
3
 (  Z )  Ai(  Z ) 

 2 3/ 2 
 2 3 / 2 
I
Z

I
 1 / 3  Z 
 1 / 3  3


3


Z
3

 2 3/ 2 
 2 3 / 2 
J
Z

J
 1 / 3  Z 
 1 / 3  3


3


Airy Function Ai (Z)


Z>0: electron-hole energy+Eg < electric field potential
Z<0: electron-hole energy+Eg > electric field potential, i.e.
above bandgap  oscillation wavefunction
Ai(Z)
Smaller period
0.4
0.2
-10
-7.5
-5
-2.5
2.5
-0.2
-0.4
5
Z
Absorption Spectrum of Franz-Keldysh Effect

Absorption spectrum reduce to the familiar square root
dependence of energy when F0.
Exciton Absorption Spectrum

3D Excitons
A
 ( )  2 0 3
2 R y aB
  1    E g 1 
   E g    E g

 2   S3 D 
4  3  



n 
Ry
 n 1 n  R y
 Ry 



where
S3 D ( x ) 

2 / x
is Sommerfeld enhancemen t factor.
1  e  2 / x
2D Excitons
A0
 ( ) 
2R y aB2
 
   E g

   E g 
1
1




4



S
 
2D 
3
2 


( n  1 / 2) 
 n 1 ( n  1 / 2)  R y
 R y 
where
S2 D ( x ) 
2
1 e
 2 / x
is Sommerfeld enhancemen t factor.
Exciton Absorption Spectra (schematics)
Without damping

With damping
2D and 3D exciton absorption spectra with zero/finite
linewidth. The spectra is very sensitive to the temperature.
Excitonic effect at different temperature
T
Bulk GaAs
Absorption peak blueshifts
because the bandgap becomes
larger at lower temperature.
Quantum well (quasi 2D)
Sommerfeld Enhancement

Even without excitonic peaks, bandedge
absorption is enhanced due to Coulombic
interaction between electrons and holes.
Lots of bound states near
the onset of continnum sum
together to give Sommerfeld
enhancement.
Quantum Confined Stark Effect*


e2
 He  Hh 
 (re , rh )  ( E  E g ) (re , rh )


4

r

r
r
e
h


 2 2
 2 2
where H e 
e  | e | F  re  Ve H h 
h  | e | F  rh  Vh
*
*
2me
2mh
Quantum well potential

Because the potential of the quantum well breaks the spherical
symmetry, it’s not a central force problem anymore! This symmetry
breaking, unfortunately, forces us to solve this problem numerically.
However, we’ll try to factorize this equation and get more physical
insights.
*D. A. B. Miller et al., Phys. Rev. B., 32, 1043 (1985)
Factorization of Schrödinger equation for QCSE
Define center of mass and relative coordinate s in xy plane :
me*ρe  mh*ρ h
ρ  ρe  ρ h
and
Rt 
M
*
*
where ρ  xxˆ  yyˆ and M  me  mh .
The envelope function  (re ,rh ) can also be factorized as
eiK t R t
 (re ,rh ) 
G (ρ, ze , zh )
A
because the potential function doesn' t depend on center of mass coordinate .
Now the original Schroeding er equation can be rewritten as follows.
 2 2


2 d2
2 d 2
e2
 2 K t2 
 


 Ve ( ze )  Vh ( zh )G (ρ, ze , zh )   E  E g 

G (ρ, ze , zh )
*
2
*
2
2
2
2
m
2
m
d
z
2
m
d
z
2
M
4r   z


r
e
e
h
h


Solutions to QCSE



Consider solutions inside the quantum well. For given z, we can see
that the problem reduces to 2D-exciton problem. For given , this is a
quantum well problem. Therefore we can expect the solution to this
equation can be expanded in terms of the product of 2D-exciton
envelope function and the quantum well wavefunction. We then
expect that this expansion should only need a few terms to give a
good approximation.
Physically, because of the carrier confinement due to the quantum
well, we should expect most of the excitons have their electron and
hole either both in the quantum well or both outside the quantum well.
As we have seen before, 2D exciton has 4 times larger ionization
energy and therefore is much more stable than excitons outside the
quantum well.
Absorption Spectrum for QCSE

Exciton absorption
peak red-shifts
with increasing
electric field
strength.
Absorption peak shifts quadratically with
applied electric field


If the quantum well “only perturbs” the solution a little bit, the absorption peak
position shift can be calculated from time-independent perturbation theory.
The energy shift, for example for the ground state exciton in infinite quantum
well, due to the electric field is:
| eh, j | z | eh, k |2
E ( F )  E (0)  eF  eh1 | z | eh1  e F 

( 0)
( 0)
Ek  E j
j k
e1
h1
e1
h1
2
2
The first order term   eh1 | z | eh1   0
 E ( F )  E (0)  C1 (m  m ) e F L / 
e1
h1
e1
h1
*
e
*
h
2
2 4
eff
2
Potential profile has
reflection symmetry along
z-direction. Thus exciton
wavefunction is either even
or odd with respect to z=0.
Absorption peak shifts quadratically with applied electric field.
Design considerations for EA modulators






Operation principle
Contrast ratio
Insertion loss
Modulation efficiency
Chirp considerations and optimization
Packaging and Integration
EA Modulators use QCSE

By applying electric field parallel to the quantum well
growth direction, the absorption can be changed
dramatically at the desired wavelength.
Contrast Ratio
Ron / off
Pout (Von  0) e  (0) L

  (V ) L
Pout (Voff  V ) e
Ron / off (dB)  10 log( Ron / off )  4.343   (V )   (0)L


The larger the contrast ratio, the better it is.
Contrast ratio can be made as large as possible by
increasing the length of the modulator. But
propagation loss then becomes an issue.
Insertion Loss
Pin  Pout (V  0)
 ( 0 ) L
Loss 
 1 e
Pin


The longer the modulator is, the larger the
insertion loss. Therefore, contrast ratio should not
be increased by lengthen the device too much.
Waveguide structure also needs careful design to
increase the coupling efficiency from the singlemode-fiber output.
Modulation Efficiency

Modulation efficiency quantifies how much voltage do we need to
modulate the optical signal.
Ron / off
V


 (V )   (0)L

 4.343
 4.343
V
F
Smaller detuning will increase the modulation efficiency. However, it
also results in a larger insertion loss.
Chirp
dRe[ n] / dN
4 dn ' / dN
e  

dIm[ n] / dN
 dg / dN
(1   e )
f FW HM 
Rsp
Np
2


Fundamental linewidth
Usually, the light to the input of the EA modulator comes directly from
the output of a laser diode (e.g. DFB). This light signal is almost single
frequency but will have a finite linewidth due to the frequency noise
inside the laser diodes. This variation of frequency is called “chirp”.
Chirp results from spontaneous emission and carrier induced phase
fluctuation from the coupling between imaginary and real parts of the
refractive index (Kramers-Kronig relation).
Chirp generated from Direct Modulation



Recall that the generalized rate equation for laser diodes we derived in
the class:
dS
 v g ( g   ) S  Rsp
dt
Neglect spontaneous emission
dN
J
N

  v g gS
dt qd 
df 1
  dS 
 ( g   )v g e  e     chirp
dt 2
2 S  dt 
The amount of chirp generated from direct modulation depends on e
and the signal waveform (note that S is the photon number inside the
cavity and the output photon number is S (1-R) where R is the
reflectivity of the output facet).
Large-signal modulation: blue-shift at the beginning of the pulse and
red-shift at the tail.
Disadvantage of Chirp



In optical fiber, the dispersion is anomalous at 1.55mm, that is the
refractive index of the fiber increases with the wavelength (cf. the
regular optical glass in which the blue color sees higher refractive
index than the red color does). The speed of light in a media is
inverse proportional to the refractive index. Therefore in fiber, longer
wavelength signal travels in a slower speed.
Because of this property, if the signal pulse has higher frequency
component (positive chirp) at the beginning of the pulse and lower
frequency (negative chirp) at the tail, the pulse will experience
broadening during the propagation through the fiber.
If we can have opposite frequency chirp for the signal pulse, we can
compensate the dispersion in the fiber.
Chirp generated in EA Modulator
 
 e  dS 
 
2 S  dt 
dRe[ n] / dN
4 dn' / dN 4 dn' / dN
e  


dIm[ n] / dN
 dg / dN
 d ( ) / dN



We have loss instead of gain. With the increase of carrier density
inside the active region due to the absorption of signal, we expect the
linewidth enhancement factor to become negative if generated
carriers can be removed from the cavity fast enough (i.e. no
saturation of absorption or equivalently speaking no gain is generated
by the small amount of inversion carriers).
This will give us positive chirp when signal has negative slope
(corresponding to the tail).
Aside: In semiconductor optical amplifier, the chirp has opposite sign
to laser diodes if gain is saturated.
Negative Chirp in EA Modulator

Chirp is a strong function of wavelength and bias
40GHz Modulator
Long device length for easy packaging
short modulator section to reduce capacitance
low driving voltage of 2.8V
low insertion loss of 8dB
Integrated DFB-EA Transmitter

10Gb/s module, Ith = 20mA, Pmax = 4mW @80mA ,
extinction ratio = 15dB for -2.5V.