Transcript Design, Modeling and Simulation of Optoelectronic Devices

Design, Modeling and Simulation of
Optoelectronic Devices
A course on:
Device Physics Processes
Governing Equations
Solution Techniques
Result Interpretations
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Course Outline
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Introduction
Optical equations
Material model I – single electron band structure
Material model II – optical gain and refractive index
Carrier transport and thermal diffusion equations
Solution techniques
Design, modeling and simulation examples
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Semiconductor lasers
Electro-absorption modulators
Semiconductor optical amplifiers
Super-luminescent light emitting diodes
Integrated optoelectronic devices
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Introduction - Motivation
• Increased complexity in component design to meet the
enhanced performance on demand
• Monolithic integration for cost effectiveness - similarity
to the development of electronic integrated circuits
• Maturity of fabrication technologies
• Better understanding on device physics
• Maturity of numerical techniques
– which leads to the recent rapid progress on the
computer-aided design, modeling and simulation of
optoelectronic devices
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Introduction - Motivation
Conventional Approach
New idea
New idea
Computer-aided
design and modeling
“Back of envelop”
design
Simulation
Experiment (costly)
Works?
Works?
No (very likely)
Yes
No (very likely)
Experiment (costly)
Yes
End
Works?
Effective Approach
No (less likely)
Yes
End
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Introduction - Physics Processes in
Optoelectronic Device
Potential and carrier distribution
Band structure
Bias I(t) or V(t)
(The Poisson and continuity
equations)
(The
SchrÖdinger
equation)
Ambient
temperature T(t)
Temperature distribution
(The thermal diffusion equation)
Saturation and
detuning
Output
Material gain
and refractive
index change
(The
Heisenberg
equation)
Optical field distribution
(The Maxwell equations)
Recombinations
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Introduction - Description of Physics Processes
• From external bias to electron and hole generation –
carrier transport model (Maxwell’s equations in its quasistatic electric field form for bulk, or Heisenberg equation
for low-dimension materials such as QW and QD)
• From electron and hole recombination to optical gain
generation – semiconductor material model (SchrÖdinger
equation)
• From optical gain to photon generation and propagation
– Maxwell’s equations (in its full dynamic form)
• Unwanted accompanying thermal process – thermal
diffusion model
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Introduction - Course Organization
• Lectures:
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Optical equations: 8 hours
Material model I: 8 hours
Material model II: 4 hours
Carrier transport and thermal diffusion: 2 hours
Solution techniques: 6 hours
Design, modeling and simulation examples: 4 hours
Total: 32 hours
• Textbook:
– Optoelectronic Devices: Design, Modeling and Simulation, X. Li,
Cambridge University Press
• Assessment:
– Working groups of 3-5 people
– Minor project on modeling (governing equation extraction for given
components) 40%
– Major project on simulation (problem solving on the extracted governing
equation) 60%
– Open-book, take-home (open-discussion) for minor and major projects
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Optical Equations 1
Maxwell’s equations – A historical review for a
better understanding
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Electrostatic Field
• Coulomb’s law

F
e1e2 
er
4 0 r122
1
E
F
1 e1
1 Ze

e

e
2 r
2 r
e2 4 0 r12
4 0 r
• Feature 1: why inversely proportional to the distance
square?
• Implication – flux conservation in a 3D space
• Hence we have the Gauss’ law (electric)
 
Ze
E

d
s

s
4 0
1   Ze
1
e

d
s


s r 2 r
0
0


E 
0

V
dV
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Electrostatic Field
• Feature 2: centered force
• Implication – zero-curled field (swirl free)

rB
rA
  Ze rB 1   Ze rB 1
Ze rB 1
Ze 1 1
E  dl 
e

d
l

dr

dr

(  )
2 r
2
2



r
r
r
A
A
A
4 0 r
4 0 r
4 0 r
4 0 rA rB
 
 E  dl  0
l

 E  0
• Hence we can introduce the scalar potential to obtain the
Poisson’s equation

E  
 2  

0
• Advantages of using scalar potential instead of vectorial
field: 1. only single variable is involved; 2. with both
features in the electrostatic field captured.
• Disadvantage: PDE order is raised
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Electrostatic Field
• Summary: the electrostatic field is divergence driven,
curl free, and fully described by the Poisson’s equation.
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What happens for the dielectric media in the
electrostatic field?
• Methodology to treat the dielectric media:
• Dipole forms inside the dielectric media if we move this media into
an electrostatic field generated by a single charge;
• Dipole generates a new (extra) electrostatic field which can be
calculated by the Poisson’s equation;
• Equivalent this extra electrostatic field to a field generated by
another (equivalent) single charge;
• Sum up the electrostatic fields generated by the two charges;
• Using the linear superposition theory, we can treat the dielectric
media (with many dipoles) in any electrostatic field (formed by an
arbitrary charge distribution).
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What happens for the conductive media in the
electrostatic field?
• Inside the conductive media, the electrostatic field is
zero (due to the motion of free electrons, which must
distribute in such a way that makes the field generated
by the redistribution cancelled out with the original field
applied to this conductive media).
• Consequently, inside the conductive media, the scalar
potential is identical everywhere.
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What happens if the charge moves?
• Current forms as


J  v
• The total number of charges must be conserved, hence
the current flow through the surface of a closed region
equals to the reduction of the charge density rate inside
the closed region, i.e., (carrier continuity equation)
 
 ( Ze)

J

d
s




s
V t dV
t


J  
t
• The current is driven by the Coulomb force, hence the
Ohm’s law holds


J  E
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Static Motion (DC Current)
• Inside the current flow region, the carrier continuity leads to a
continuous current flow

 J  0
• Outside the current flow region, the field is still electrostatic as it is
formed by the constant charge distribution inside the DC current flow
region.
• Therefore, the electric field originated from a DC current is still swirl
free, i.e.,

 E  0
• It can be mapped to the electrostatic field inside a dielectric media if we
view J as D, σ as ε0εr, and E the same. (Known as the electrostatic
imitation, this mapping is widely used for field measurement, we can
always take the field measuring of a DC current source rather than a
charge distribution, or vice versa, once a proper mapping between the
DC current source and the charge distribution is established.)
• What is the new effect of the charge motion then?
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Static Motion (DC Current)
• Outside of the current flow region, if there is another charge
stream in a static motion (i.e., a DC current), this charge
stream “feels” a new force that is described by the BiotSavert’s law in a form analogous to the Coulomb’s law,
except for the two involved scalar charges must be replaced
by the two involved vectorial DC currents:
 0
F
4

l1 l2

 
I1dl1  ( I 2 dl2  er )
r122

   0 I 2 dl 2  er  0

  B( r ) 
2

l
2
4
4
r12
I1dl1 

F

l1
' 
Idl  er r '
l' | r  r ' |2
• A swirl is generated, we name it the magnetic flux. Since this
swirl acts on the moving charge only, without any effect to
the stay still charge, it differs from the electric field.
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Static Motion (DC Current)
• Therefore, the magnetic interaction between two moving
charges is a reflection of a purely “derivative” effect;
• Whereas the electric interaction between two charges,
regardless of their status, in motion or at rest, is a reflection
of a “static” effect.
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Magneto-static Field
• Feature 1: the flux is a closed vectorial flow
• Implication – the flux is continuous in a 3D space,
known as zero-diverged (source/drain free)
• Hence we can introduce the vector potential and the
Gauss’ law (magneto) holds
' 
   0 Idl  er r '  0
B( r ) 
 ' 2 
'

l
4 | r  r |
4


B   A
 
J  er r '
0
V | r  r ' |2 dV '    4

J
V | r  r ' |dV '

B  0
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Magneto-static Field
• Feature 2: non-centered force, otherwise, the magnetostatic flux has neither divergence nor curl, which makes
it zero everywhere according to the Helmholtz’s theorem;
the flux is, again, inversely proportional to the distance
square
• Implication – flux conservation in a 3D space
• Hence we have the (derivative form of) Ampere’s law





2
  B      A  (  A)   A  0   2 0
4


J
V | r  r ' |dV '  0 J
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Magneto-static Field
• Summary: the magneto-static flux is divergence free,
curl driven, and fully described by the vectorial
Poisson’s equation


 A   0 J
2
• However, unlike the scalar potential introduced in the
electrostatic field, the vectorial potential introduced in
the magneto-static field is not so popular as it doesn’t
have much pronounced advantages compared to the
flux description. One advantage it bears, however, is
that, unlike in the flux description where a flux
component is usually a mix up of different source
component, the vectorial potential component and the
source component has one-to-one correspondence.
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Summary on the Electro- and Magneto- Static
Fields
Field
tester
Stay still
charge
Charge
in static
motion
Stay still charge source

     
D  


  E  0 or E  




D  E
D  E
    0
 J  0


  E  0 or E  




J  E
J  E
Charge source in static motion
(DC current source)

     
D  


  E  0 or E  




D  E
D  E
 

  [(1 /  )  A]  J
B  0

 

  H  J or B    A


 
B  H
H  B/
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Summary on the Electro- and Magneto- Static
Fields
Field
tester
Stay still
charge
Charge
in static
motion
Stay still charge source
Stay-still charge field: flux
conserves (Gauss’ law) and
centered-force field (swirl-free)
in 3D; either flux/field (D, E) or
potential (φ) description
Field inside the DC current flow
region: charge conserves (the
charge continuity equation) and
centered-force field (swirl-free)
in 3D; either flux/field (J, E) or
potential (φ) description
Charge source in static motion
(DC current source)
DC current field: flux conserves
(Gauss’ law) and centered-force field
(swirl-free) in 3D; either flux/field
(D, E) or potential (φ) description
DC current to DC current field: flux
continues (divergence-free) and
conserves (Ampere’s law) in 3D;
either flux/field (B, H) or potential
(A) description
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