Download 4 - Electrical and Computer Engineering

UCONN ECE 5212 092012016 (F. Jain)
L4
1. Modulators using Optical transitions in quantum wells
2.Photon absorption or emission involves electronic transitions
(Absorption is dependent on probability of transition and density of states;
density of states depend on the physical structure of the absorbing layer if it
is thick (>10-15nm), or thin).
In terms of thin, we need to know if it comprise of quantum wells, multiple
quantum wells, quantum wires, and quantum dots.
Absorbing or emitting layer is made of direct gap or indirect gap semiconductors.
•Upward transitions involve photon absorption. These transitions are with and
without phonons. In indirect semiconductors, phonon participation is essential to
conserve momentum.
•Downward transitions result in photon emission.
Transitions involve valence band-to-conduction band, band to impurity band
or levels, donor-to-acceptor, and intra-band.
1
Band-to-band transitions could involve free carriers or excitonic transitions.
Optical Modulators using quantum confined Stark Effect
We cover basics of exciton formation in quantum wells and related changes in the
absorption coefficient and index of refraction. This phenomena is called confined
Stark effect.
Photon energy needed to create an exciton is given by
h =
Eg
+
(Ee1 + Ehh1)
Eex
(Bulk Band gap)
(Zero field energy of
(Binding energy
Electron and holes)
of exciton)
Ee1
Eg
Ehh1
If the photon energy is greater than above
equation, free electron and hole pair is created.
Fig. 1(a) E = 0
2
Application of perpendicular Electric Field
When a perpendicular electric field is applied, the potential well tilts. Its slope is
related to the electric field.
Electron wave
function
Hole wave function
Fig. 1(b) Quantum well in the presence of E
•As E increases Ee & Eh decreases.
As a result photon energy at which absorption peak occurs shifts to
lower values (Red shift) .
•The interaction between electron and hole wavefunctions (and thus, the value of
the optical matrix elements and absorption coefficeint) also is reduced as
magnitude E is increased. This is due to the fact that electron, hole wave
functions are displaced with respect to each other. Therefore, the magnitude of
absorption coefficient decreases with increasing electronic filed E.
3
MQW Modulator based on change in absorption due to
quantum confined Stark effect
Fig.3. Responsivity of a MQW diode
(acting as a photodetector or optical
modulator).
4
Change in index of
refraction
5
Phase Modulator:
A light beam signal (pulse) undergoes a phase change as it transfers an electro-optic medium of length ‘L’
  0  
2
 0 
n.L
0
2
2

n.L 
n.L


In linear electro-optic medium
n 
--------------- (1)
1 3
r.n E
2
…………………(2)
  0 
E= Electric field of the RF driver; E=V/d, V=voltage and d
thickness of the layer.
r= linear electro-optic coefficient
n= index
 
2 1
V
. .r.n 3 .
o 2
d

L


r.n 3 .L.E
o
Alternatively ,   o   .
Here, V 
V
V
d .o
L.r.n 3
6
MQWs have quadratic electro-refractive effect.
Figure compares linear and quadratic
variation n as a funciton of field.
Figure shows a Fabry-Perot Cavity
which comprises of MQWs whose
index can be tuned (n) as a funciton
of field.
7
Fabry-Perot Modulator (pp. 178-179)
8
Fabry-Perot Modulator (pp. 180-181,ECE 5212
9
Fabry-Perot Modulator (pp. 180-181)
10
Asymmetric InGaAsGaAs Modulator
11
Mach-Zehnder Modulator
Here an optical beam is split into two using a Y-junction or a
3dB coupler. The two equal beams having ½ Iin optical power.
If one of the beam undergoes a phase change , and
subsequently recombine. The output Io is related as
1

I 0  .I in .1  cos   I in . cos 2
2
2
The transmitt ance, T(  )  cos 2

2
Mach-Zehnder Modulator comprises of two
waveguides which are fed by one common source at
the input (left side). When the phase shift is 180, the
out put is zero. Hence, the applied RF voltage across
the waveguide modulates the input light.
12
Band to band transitions in nanostructures
• Band to band free carrier transitions.
• Band to band transitions involving exciton
formation.
• Light emission and absorption.
13
Energy bands
Fig. 8 and 9. p.119
Energy
Energy
Energy Gap Eg
Energy Gap Eg
k vector
k vector
14
E-K diagram of an indirect
semiconductor
E-K diagram of an direct
semiconductor
V(R)
V0
2b<a
-b
0
Energy bands
a
a+b
 2  2
sinh( b ). sin( a )  cosh( b ). cos( a )  cos( k )( a  b )
2
(Ref. F. Wang, “Introduction to Solid State Electronics”,
Elsevier, North-Holland)
Fig. 13(b) and Fig. 13c. p.123-124
15
Absorption and emission of photons
Conduction
Band
Heavy Hole
Band
 So
Light Hole
Band
 (h ) 
Fig. 2 p.138
Pmo probability that a transition
has occurred from an initial state
‘0’ to a final state ‘m’ after a
radiation of intensity I(h) is ON
for the duration t.
PCV (t )
I ( )Vt
1)
(Transition Probability)
here, (h) is the joint density of states in a volume V (the number of energy levels separated
by an energy
E g h m0

2)
PCV (t )   Pm0 (t )  (h )dm0
0
The joint density of states is expressed as Eq. 2B)
 (h )  V
3
4
2
h  E g 12
(
2
m
)
r
3
h
o
16
Time dependent Schrodinger Equation p.68-69
17
18
Infinite Potential Well energy Levels
Solve Schrodinger Equation
d 2 2m
 2 ( E  V ( x ))  0
dx 2


h
2
In a region where V(x) = 0, using boundary condition that (x=0)=0, (x=L)=0. Plot the (x) in the 0-L region.
V(x) = 0
0
L
19
d 2 2m
 2 ( E  V ( x ))  0
2
dx

In the region x = 0 to x = L, V(x) = 0
d 2
2mE
 2 
2
dx

k x2 
Let
  Ae
 ik x x
2 mE
2
 Be
ik x x
(2)
The solution depends on boundary conditions.
Equation 2 can be written as D coskxx + C sinkxx.
At x=0,   0  D  0 , so D= 0
(3)
  C * sin k x x
nx = 1,2,3…
(4)
Using the boundary condition k x Lx  n x
2 2
2 2
2
(x=L) = 0, we get from Eq. 3
n

h
nx

x
nx
2mE
E


n
kx 

2
2
sin kxL = 0, L = Lx
2
2
m
L
8
mL

L
x
20
Page 52-53: Using periodic boundary condition (x+L) = (x),
we get a different solution:
d 2 2m
dx 2

2
We need to have
k x2 
( E  V ( x ))  0
kx=
2n x
Lx
---(6)
2 mE
2
  Ae ik x  Be ik x
x
x
The solution depends on boundary conditions. It satisfies boundary
condition when Eq.6 is satisfied.
If we write a three-dimensional potential well, the problem is not that much different
ky =
kZ =
2n y
n y= 1,2,3,…
Ly
2n z
Lz
,
n ( x, y, z )  n (r )  ASin (
n z = 1,2,3,….
2n z  z
2n y  y
2n x  x
) Sin (
) Sin (
)
Lx
Ly
LZ
…(7)
The allowed kx, ky, kz values form a grid. The cell size for each allowed state in k-space is
2 2 2 (2 ) 3



Lx L y Lz
V
V  L X LY LZ
…(8)
21
Density of States in 3D semiconductor film (pages 53, 54)
Density of states between k and k+dk including spin
N(k)dk =
2
4k
 dk
3
(2 )
k
k+dk
2
…(10)
Density of states between E and E+dE
N(E)dE=2 
4k 2 k

 dE
(2 ) 3 E
E=Ec +
…11
2k 2
2mc
(12)

dE  2 2k  2 k


dk 2me  me 
…(13)
E
N(E)
VB
N(E)dE =
N(E)dE =

2 me 3 / 2 1 / 2
) E dE
2
2
2
1 2mn  3 / 2 1 / 2
(
) E dE
2 2  2
1
(
…14
…15
22
Energy Levels in a Finite Potential Well
AlxGa1-xAs
GaAs
AlxGa1-xAs
∆EC
∆Ec = 0.6∆Eg
∆Ev = 0.4∆Eg
∆Eg
V(z)
0
z
-EG
z
0
∆EV
-EG+∆EV
23
In the barrier region
24
Wavefunction matching at well-barrier boundary
25
Wavefunction matching at well-barrier boundary
26
27
Electrons
EC1 = 72.5 meV
Heavy Holes
EHH1 = 22.3 meV
Light Holes
ELH1 = 52.9 meV
28
kL/2
αL/2
Electrons
Heavy Holes
Light Holes
Where, radius is
EC1 = 72.5 meV
EHH1 = 22.3 meV
ELH1 = 52.9 meV
2m eb  L 
V0  
2

2
Output: Equation (15) and (12), give k,
α’ (or
α). Once k is known, using Equation (4) we get:
E
2k 2
2m ew
The effective width, Leff can be found by:
Leff 
 2 2
E1 is the first energy level.
2me* E
(4)
29
Summary of Schrodinger Equation in finite quantum well region.
 2 2




V(z)

 (x, y, z)  E (x, y, z)
2m



h
2
(1)
In the well region where V(x) = 0, and in the barrier region V(x) = Vo = ∆Ec in conduction band, and ∆Ev in the valence band.
The boundary conditions are:


L 
L 
(5) Continuity of the wavefunction



  z      z  
2 
2 


(6) Continuity of the slope
1 d 
L 
1 d 
L 
 z 

  z   
m eb dz 
2  m ew dz 
2 
AlxGa1-xAs
AlxGa1-xAs
GaAs
V(z)
∆EC
∆Ec = 0.6∆Eg
∆Ev = 0.4∆Eg
L
∆Eg
L
2
0
 (z)  C 2 cos kz  C '2 sin kz
0
z
-EG
z
2
∆EV
-EG+∆EV
2m ew E
k 
2
2
 (z)  C1e

α zL
2

2m eb (V0  E)
α 
2
2
z
L
2
2
m
 L
 k   ew
m eb
 2
2
2m
 αL 
L
   2eb V0  

 2 
2
Eigenvalue equation:
L
α
kL m ew 2
tan

2
m eb L
k
2
2
2
'
2m
 L α L
L

  2eb V0  
k


 

 2  2 
2
2
2
α 
'
m ew

m eb
30
C. 2D density of state (quantum wells)
page 84
Carrier density number of states
2
 dk x dk y  f ( E )
V nz 
Without f(E) we get density of states

(1)
dkt + kt
dk y
dk x
2
2
dk x dk y  


V
V 2 / L x 2 / L y
kt
Quantization due to carrier confinement along the z-axis.
me
 2
 Lz
 H (E  E
n
)
nz
31
Density of states summary
Density of
States N(E)
Plots
3D
2D Well
1D Wire
0-D Box
3-D (bulk)
No confinement
2-D (Quantum well)
1-D of confinement
1-D (Quantum wire)
2-D of confinement
0-D (Quantum dot)
3-D of confinement
N(E)
N(E)
N(E)
E 1 E2
d(E)
E1/2
E1/2
E1 E2
E
3
2
1
1  me 
2


2
.
E
2 
2 
2  h 

n
me
2
 h Lz
E

.U E  Enze
E1 E2


n ,e
2

h
E E 
k x2  k y2
2 me
e
Energy level
e
nz
2
E nze 
N(E)
N(E)
h
2 me
2
 n

 Lz
E  E
e
nz
E
1
e
e

.d E  Enze  Eny
 Enx
n , l , k Lx Ly Lz

 E ne, y

1
2


 .F me , Lz ,V0 

ki 
 2 me 
 2 


 h 
Lz L y
2
E e  E nze  E nye 
1
2
E1,2
E
 
h
k x2
2me
n
ai
32
Photon Absorption
Absorption coefficient h depends on type of transitions involving phonons
(indirect) or not involving phonons (direct). Generally, absorption starts when
photon energy is about the band gap. It increases as hv increases above the band
gap Eg. It also depends on effective masses and density of states.(p.140, eqs 1-2)
The absorption coefficient in quantum wires is higher than in wells. It also starts
at higher energy than band gap in bulk materials.
Absorption coefficient is related to rate of emission.
Rate of emission in quantum wells is higher than in bulk layers.
Rate of emission in quantum wires is higher than in quantum wells.
Rate of emission in quantum dots is higher than in quantum wires.
Rate of emission has two components:
1.Spontaneous rate of emission
2.Stimulated rate of emission.
33
 Direct and Indirect Energy Gap Semiconductors
Semiconductors are direct energy gap or indirect gap. Metals do
have not energy gaps. Insulators have above 4.0eV energy gap.
Energy
Energy
Energy Gap Eg
Energy Gap Eg
k vector
E-K diagram of an indirect
semiconductor
k vector
E-K diagram of an direct
semiconductor
Fig. 10b. Energy-wavevector (E-k) diagrams for indirect and direct
semiconductors. Here, wavevector k represents momentum of the particle
(electron in the conduction band and holes in the valence band). Actually
momentum is = (h/2)k =  k
34
Effect of strain on band gap
Ref: W. Huang, 1995 UConn doctoral thesis with F. Jain
1.Under the tensile strain, the light hole band is lifted above the heavy hole, resulting in a smaller band gap.
2.Under a compressive strain the light hole is pushed away from heavy. As a result the effective band gap as well as light
and heavy hole m asses are a function of lattice strain. Generally, the strain is +/- 0.5-1.5%. "+" for tensile and "-" for
compressive.
3.Strain does not change the nature of the band gap. That is, direct band gap materials remain direct gap and the indirect
gap remain indirect.
35
Electrons & Holes
Photons
Phonons
Statistics
F-D & M-B
Bose-Einstein
Bose-Einstein
Velocity
vth ,vn
1/2 mvth2 =3/2 kT
Light c or v = c/nr
nr= index of refraction
Sound
vs = 2,865 meters/s in GaAs
Effective
Mass
mn , mp
(material dependent)
No mass
No mass
Energy
E-k diagram
Eelec=25meV to 1.5eV
ω-k diagram (E=hω)
ω~1015 /s at E~1eV
Ephotons = 1-3eV
ω-k diagram (E=  ω)
ω~5x1013/s at E~30meV
Ephonons = 20-200 meV
Momentum
P=  k
k=2π/λ
λ=2πvelec/ω
momentum: 1000 times
smaller than phonons
and electrons
P=  k
k=2π/λ
λ=2πvs/ω
36
Direct
 = A(h - E g )
p. 143 A=
and
1
2
Indirect p.135 .
 (h ) =


2  4 
1
2
  2e   3 2mv    m 2 E  2
 

 o g 
2
2
 6m0 nr  0 c
  2me 


3
C(h - E g + E p )2
Ep
e kT - 1
+
C(h - E g - E p )2
Ep
1 - e- kT
(phonon absorption) + (phonon emission)
M me mh e2
C=
6  2 h7 mo2 n  o CW
 | Pio|2 Bc
| Pmi |2 Bv 
+

2
2
 (  E o - h ) (  E m - h ) 
Absorption coefficient h depends on type of transitions involving phonons
(indirect) or not involving phonons (direct). Generally, absorption starts when
photon energy is about the band gap. It increases as h increases above the band
gap Eg. It also depends on effective masses and density of states. (p.140, Eqs 1-2)
The absorption coefficient in quantum wires is higher than in wells. It also starts
at higher energy than band gap in bulk materials.
•Downward transitions result in photon emission. This is called radiative
transition. When there is no photon emission it is called non-radiative transition.
Rate of emission in quantum wells is higher than in bulk layers.
Rate of emission in quantum wires is higher than in quantum wells.
Rate of emission in quantum dots is higher than in quantum wires.
Rate of emission has two components:
1.Spontaneous rate of emission
2.Stimulated rate of emission.
37
Quantum efficiency
Absorption coefficient is related to rate of emission via Van RoosbroeckShockley. This enables obtaining an expression for radiative transition lifetime
tr.
There are ways to compute non-radiative lifetime tnr. The internal quantum
efficiency is expressed as follows: Pr is the probability of a radiative transition.
Pr
1/t r
t
nr
=


q
Pnr  Pr 1 / t nr  1 / t r ( t nr +t r )
Two types of downward transitions.
1. Radiative transitions.
2. Non-radiative transition:
38
Absorption coefficient is related to rate of emission.
8 2 nr2 ( )d
R( )d  2 h / kT
c (e
 1)
Eq. 7 –page 214
R()d is the rate of emission of photons at  within an interval d
Van Roosbroeck - Shockley Relation under equilibrium
8nr2
R   R( )d 
c2
0

 ( ) 2 d
 (e 
h / kT
Eq. 8 –page 214
 1)
Absorption coefficient is related to imaginary component of the complex index of refraction.
Index of refraction nc is complex when there are losses.
nc= nr – i k where extinction coefficient k = α/4, here alpha is the absorption
coefficient.
Also, nc is related to the dielectric constant. nc =
RC  np
RC  anp
=
In non-equilibrium, we have excess electron-hole pairs. Their recombination gives emission
of photons. The non-equilibrium rate of recombination Rc is
or
In equilibrium,
RC  R p0 n  n0 p

R
n0 p0
R  an0 p0
R  n  p


R
n0
p0
n  p; and
n  n0  n
p  p0  p
n0 p 0
ni2
n
1
tr 



R  R(n0  p 0 )  R(n0  p 0 ) R(n0  p 0 )


 n0 p 0 
39
Radiative life time in intrinsic and p-doped semiconductors (p.216)
tr
intrinsic
n
 i
2R
ni2
n
tr 

R p0 R
Nonradiative recombination: Auger Effect and other mechanisms (217)
Nonradiative transitions are processes in which there are no photons emitted. Thus, there are
several models by which energy is dissipated.
Experimental observations are in terms of (1) emission efficiency, (2) carrier lifetime (coupled
with emission kinetics), (3) behavior (or recombination mechanisms response) to temperature
and carrier concentration variations.
Recombination processes which are not associated with an emission of a photon are as follows:
•Auger effect (carrier-carrier interaction)
•Surface recombination, recombination through defects (Shah-Noyce-Shockley)
•Multi phonon emission
and others which may fit with the above proposed criterion.
Since Auger processes involve carrier-carrier interactions, it is
typical to assume that probability of occurrence of such a
process should increase with the carrier concentrations.
t Auger
 Eg 
 
 kT 
3
2
e
 Eg 


 kT 
40
Auger Transitions (p217-219)
1
t
1
t
 Anp  Bn 2
(n type semiconductor)
 Anp  Bp 2
(p type semiconductor)
e
First term:
AnP
h
e
Second Term:
Bp2
h
41
Defect states caused nonradiative transitions
Recombination via lattice defects or inclusions:
Defects (dislocations, grain boundaries) and inclusions produce a continuum of energy states. These can trap
electrons and holes which may recombine via continuum of states. Such a transition would be non-radiative.
The energy states are distributed as:
Nonradiative recombination:
A microscopic defect or inclusion could induce a
deformation of the band structure.
e.g.
E
Deformation due to Local
Trapped Centers
Density of states
States in the Gap
E*
EF
Density of states
due to defects and
inclusions
EF
E*
42
The absorption coefficient relations are in Chapter 3
Transitions involve:
• Valence band-to-conduction band
-Direct band-to-band  = A(h - E )
-Indirect band-to-band (h ) = C(h e- E -+1 E ) + C(h1--Ee - E )
•Band to band involving excitons (h < Eg)
•Band to impurity band or levels,
•Donor level-to-acceptor level, and
•Intra-band or free carrier absorption.
g
1
2
2
g
Ep
kT
p
2
g
Ep
kT
p
Similar transitions in downward direction result in
emission of photons.
43
Density of states in bulk is N(E)dE =
1
2
2
(
2mh  3 / 2 1 / 2
) E dE
2
Spectral width of emitted radiation
The electron concentration ‘n’ in the entire conduction band is given by

n
 N ( E ) f ( E )dE
(EC is the band edge)
Ec
This equation assumes that the bottom of the conduction band is =0.

n   N ( E ) f ( E )dE
0
f FD ( E ) 
1
1 e
( E  E f / kT )

1
1 2mn  3 / 2 1 / 2
( 2 ) E dE
2
1

exp(
E

E
/
kT
)
2


f
0
n
1
1 2m
n  (1 / kT ) 3 / 2 *  1 / 2 * exp[ E f / kT ] * 2 ( 2 n ) 3 / 2
2
2

Ev
p   N ( E )[1  f ( E )]dE

( E f Eg )
 2 m p kT 
p= 2

kT
e

2
 h

3/2
Electron and hole concentration as a function of energy
44
Graphical method to find carrier concentration in bulk or
thick film
(Chapter 2 ECE 4211)
E
E
Ec
dn electron conc.
between E and E+dE
N(E)
E
Ef
dp hole conc. between
E and E+dE
Density of states
Ev
dp
dp = N(E)[1-f(E)]
Here, N(E) is for the
valence band
0
1/2
1
f(E)
(a)
Fig. 28 (a) Density of states, Fermi-Dirac distribution function f(E), and distribution of
electrons and hole in conduction and valence band. The electron concentration increases
as Ef gets closer to the conduction band edge (see Fig. 28b).
45
Graphical method to find carrier concentration in quantum well
E
E
E2e
E
E1e
0=Ec
N(E)
dn electron conc.
between E and E+dE
Ef
dp hole conc. between
E and E+dE
Density of states
Ev =-Eg
dp
E1hh
dp = N(E)[1-f(E)]
Here, N(E) is for the
valence band
E2hh
0
1/2
1
f(E)
(a)
Fig. 28 (a) Density of states, Fermi-Dirac distribution function f(E), and distribution of
electrons and hole in conduction and valence band. The electron concentration increases
as Ef gets closer to the conduction band edge.
46
Spectral width of emitted radiation
Spectral width in quantum well active layer
is smaller than bulk thin film active layer
47
DEVICES based on optical transitions
Emission: LEDs
Lasers
Photodetectors
Absorption: Solar cells
Optical modulators and switches
Optical logic
48
Transitions in Quantum Wires:
The probability of transition Pmo from an energy state "0" to an energy state in the conduction
band "m" is given by an expression (derived in ECE 5212). This is related to the absorption
coefficient . It depends on the nature of the transition.
The gain coefficient g depends on the absorption coefficient  in the following way:
g = -1 fe - fh ).
1
fe 
1 e
fn  1 
Ec  E fn
kT
1
1 e
Ev  E fp
kT
The gain coefficient g can be expressed in terms of absorption coefficient , and Fermi-Dirac
distribution functions fe and fh for electrons and holes, respectively. Here, fe is the probability of
finding an electron at the upper level and fh is the probability of finding a hole at the lower level.
Free carrier transitions: A typical expression for g in semiconducting quantum wires, involving free
electrons and free holes, is given by:
2 e2
(2 meh )1/2
[
| M b |2  |  e (y)  h (y)dy |2
g free (  ) =

 0 n rm0 c L x l,h L y Lz
  e (z)  h (z)dz |2  (E - E e - E h )-1/2  (E)  L(E)dE( f e + f h - 1)]
49
Excitonic Transitions in Quantum Wires
Excitonic Transitions: This gets modified when the exciton binding energy
in a system is rather large as compared to phonon energies (~kT). In the case
of excitonic transitions, the gain coefficient is:
2 e2
g ex (  ) =
 o nr mocLy Lz
 [| M
b
2
|  2  1/2 | ex (0) |2
l,h
 (| e ( y)h ( y)dy |2 | e ( z )h ( z )dz |2 )   ex  L( E ex )  ( f c + f v - 1)]
50
Laser Modulators
The gain coefficient depends on the current density as well as losses in the cavity or distributed feedback structure.
J=
8 n2r 2 e L x  s
 q c2 [1  e
h  
kT
.g
]
g=
1
1
ln (
) + f + c + d
2L
R1 R2
Threshold current density is obtained once we substitute g from the above equation.
For modulators, we need to know:
1.Absorption and its dependence on electric field (for electroabsorptive modulators)
2.Index of refraction nr and its variation n as a function of electric field (electrorefractive modulators)
3.Change of the direction of polarization, in the case of birefringent modulators.
In the presence of absorption, the dielectric constant  (1-j2) and index of refraction n (nr –j k) are
complex.
However, their real and imaginary parts are related via Kramer Kronig's relation.
See more in the write-up for Stark Effect Modulators.
Lasers and Modulators:
In the case of lasers, we need to find the gain coefficient and its relationship
with fe and fh, which in turn are dependent on the current density (injection
laser) or excitation level of the optical pump (optically pumped lasers).
51
p114 notes
4. Quasi Fermi Levels Efn, Efp, and 
Quantum Wells
me 
m
dE
= 2hh
2
( E  E Fn ) / kT

 Lx E 1  e
 Lx
ND
Quantum Wires


 1 e
dE
( E  E Fn ) / kT
Ehh
m
 2lh
 Lx

1 e
dE

 N D  (4)
Elh


1
 N D 1  f ( E D )  N D 1 
(
E
 E ) / kT
 1  1 e D fn

2
(2me )1/ 2 ( E  Ee ) 1/ 2 dE (2mhh )1/ 2
=
Lx Ly Ee 1  e( E  EFn ) / kT
Lx Ly

( E  E Fn ) / kT






( E  Ehh ) 1/ 2 dE (2mlh )1/ 2
E 1  e( E EFn ) / kT  Lx Ly
hh

( E  Elh ) 1/ 2 dE
E 1  e( E EFn ) / kT  (5)
lh
Quantum Dots
1
L x L y Lz
or 


d ( E  Ee (l ,m ,n ) ) dE
1 e
E e ( l ,m , n )
1
Lx L y L z


Elh ( l ,m ,n )
( E  E Fn ) / kT
=
1
L x L y Lz
d ( E  Elh(l ,m ,n ) ) dE
1  e ( E  EFn ) / kT


E h h ( l , m ,n )
d ( E  Ehh(l ,m ,n ) )dE
1  e ( E  EFn ) / kT
 (6)
52
Laser operating wavelength (p 114)
5. Operating Wavelength :
The operating wavelength of the laser is determined by resonance condition
L=m/2nr
Since many modes generally satisfy this condition, the wavelength for the dominant
mode is obtained by determining which gives the maximum value of the value of the
gain. In addition, the index of refraction, nr, of the active layer is dependent on the
carrier concentration, and knowing its dependence on the current density or gain is
important. For this we need to write the continuity equation.
6. Continuity equation and dependence of index of refraction on injected carrier
concentration
The rate of increase in the carrier concentration in the active layer due to forward current
density J can be expressed as:
dn
J (n  n po ) g m
=


( I sp  I )  (7)
dt qd
tn
E
dn
J ( n ) g m
=


( I sp  I av )  (8)
dt qd t n
E
53
Examples of Quantum dot lasers
54
Why quantum well, wire and dot lasers, modulators and solar cells?
•
•
•
•
Quantum Dot Lasers:
Low threshold current density and improved modulation rate.
Temperature insensitive threshold current density in quantum dot lasers.
Quantum Dot Modulators:
High field dependent Absorption coefficient (α ~160,000 cm-1) : Ultra-compact intensity modulator
Large electric field-dependent index of refraction change (Δn/n~ 0.1-0.2): Phase or Mach-Zhender Modulators
Radiative lifetime τr ~ 14.5 fs (a significant reduction from 100-200fs).
Quantum Dot Solar Cells: High absorption coefficent enables very thin films as absorbers.
Excitonic effects require use of pseudomorphic cladded nanocrystals (quantum dots ZnCdSe-ZnMgSSe, InGaN-AlGaN)
Table I Computed threshold current density (Jth) as a function of dot size in for InGaN/AlGaN Quantum Dot Lasers (p.11)
Quantum Dot Size
q
and Jth 100100100Å
50 5050Å
q
Defect Status
Negligible
Dislocations
(ideal)
Traps
Nt=2.9x1017cm-
Jth
A/ cm2
=418nm
0.9
76
0.0068 10,118
q
Jth
A/ cm2
=405nm
0.9
0.0068
58
7,693
3 53535Å
q
Jth
A/ cm2
=391nm
0.9
0.0068
54
7,147
3
(Dislocations
=1x1010 cm-2)
Excitonic
Enhancement
(in presence of
dislocations)
0.049
1,404
0.17
304
(Ref. F. Jan and W. Huang, J. Appl. Phys. 85, pp. 2706-2712, March 1999).
0.358
136
55
Quantum Confined Stark Effect (QCSE) in
Conventional Field Effect Structure (FES)
•
Quantum Confined Stark Effect:
•
Under electric fields, large shifts in optical absorption and
change in refractive index are observed.
•
Fig 17. Resulting absorption shift via QCSE.[6]
Result of decreases wave function interaction. (electron
states to lower energies, holes to higher energies)
Fig 18. (a) refractive index increase (higher absorption) in
MQW. (b) Refractive index shift increases.
QD Modulator Modeling
1.
Modeling of II-VI (ZnSe-CdSe) QDs
•
Electro-Absorption
•
•
Greater binding energies = more stable excitons
(smaller QD advantageous)
Electro refraction
•
•
Index changes occur singular for large QDs
Previously modeled index changes up to .021
Fig 18. Absorption change of 30Å QDs under varying field.[13]
Modeling Hierarchy
1.
2.
3.
4.
5.
Fig 19. Refractive index change of 30Å QDs for varying bias Voltages. [13]
Calculation of elastic Strain in the superlattice (mismatch)
and calculate new band gap and band offset.
Create Hamiltonian to solve Schrodinger's equation for
wavefunction.
Calculate imaginary part of dielectric constant (exciton
transitions included)
Calculate real part of the dielectric constant using KramerKronig relations
Calculate index of refraction and absorption coefficients. [13]
Mathematical Representation of QCSE
Index of Refraction
Under Electric Field, Excitonic binding
energies decrease
0.03
0.03
0.02
0.02
0.01
0.01
0.00
-0.01 700
-0.01
-0.02
-0.02
-0.03
50kV/cm
750
800
850
900
Wavelength (nm)
Fig 20. Refractive index of 5nm QD with 50kV/cm and 100kV/cm E field.