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Featured Webinar: How to
Simulate and Optimize CMOS
Image Sensors
Lumerical Solutions, Inc.
http://www.lumerical.com
For More Information
More information on simulation CMOS image sensors on our
Knowledge Base:
http://docs.lumerical.com/en/fdtd/cmos.html
Dr. James Pond
CTO
jpond.support@lumerical.com
Dr. Guilin Sun
Senior R&D Scientist
gsun.support@lumerical.com
Chris Kopetski
Director of Technical Services
ckopetski.support@lumerical.com
Dr. Mitsunori Kawano
Technical Sales Engineer
mkawano.support@lumerical.com
Check out other webinars:
http://www.lumerical.com/support/webinar_schedule.html
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Outline
 Introduction
 Trends in image sensor design
 Impact on simulation
 General simulation considerations
 Simulation steps




Parameterization of your design
FDTD Simulation
Analysis of results
Optimization
 Broadband simulations
 Questions and Answers
http://www.lumerical.com
Introduction
We simulate light interacting with wavelength scale structures
MODE Solutions
FDTD Solutions
12 mm
20 mm
6 mm
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CMOS image sensor simulation
What are the components of a pixel?
 Micro-lens
 Color filters
 Vias and
interconnects
 Silicon
 Transistors and
collection electronics
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CMOS image sensor simulation
Why simulate?
 Simulation gives the opportunity to cheaply and
quickly test ideas, optimize designs and solve
problems


Expensive and time-consuming to build prototypes
Design optimization is challenging and results are not
always intuitive
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CMOS image sensor simulation
What do we want to calculate?
 The quantum efficiency, QE

The ratio of collected electrons to incoming photons
 The optical efficiency, OE
The ratio of generated electrons to incoming photons
 Equal to the absorbed optical power in the Si over the incident power,
assuming that absorption in the Si can only come from exciting an
electron-hole

 More advanced

Spectral cross talk
• Color matrix coefficients


Point spread functions (spatial cross talk)
And more...
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CMOS image sensor trends
Pixel Size (mm)
 Pixel sizes continue to decrease
 Pixel size stays about 20x the current technology node
Year
Source: Advanced Image Sensor Technology, Dr. Albert Theuwissen
http://www.lumerical.com
Impact on simulation
 Ray tracing works well at 5 mm pixels sizes (visible light)
 Starts to break down around 3 mm (visible light)
Source: Hirigoyen et al., “FDTD-based optical simulations methodology
for CMOS image sensor pixels architecture and process optimization”,
PROCEEDINGS- SPIE THE INTERNATIONAL SOCIETY FOR OPTICAL
ENGINEERING, 6816, 2008, [6816 08]
http://www.lumerical.com
Impact on simulation
 At 1.75 mm, visible light
Angular response
Spot at Si surface
for (1) ray tracing
and (2) FDTD
Solutions
Source: Hirigoyen et al., “FDTD-based optical simulations methodology
for CMOS image sensor pixels architecture and process optimization”,
PROCEEDINGS- SPIE THE INTERNATIONAL SOCIETY FOR OPTICAL
ENGINEERING, 6816, 2008, [6816 08]
http://www.lumerical.com
General simulation considerations
 Wave optics vs ray optics
 Simulation methodology
 Using focussed beams
 Using plane waves
 How to obtain unpolarized results
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Wave optics
 At the wavelength scale, some intuitive
questions may not make sense
 Example: “Which lens did the photon pass through
before creating an electron-hole?”
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Wave optics
 An electron-hole created here
is a result of the interference
pattern created by the photon
(a wave) passing through all
lenses at once
 Ray optics does not take these
effects into account and is one
reason for the breakdown at
small pixel sizes
 In this example, the incident
light is green, and is blocked
by the red and blue filters
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Simulation methodology
 We want to calculate the Optical Efficiency, OE
 We need to consider the entire optical system
and illumination conditions
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Simulation methodology
N 

2
Euniform   Ei
i 1
1
OEuniform 
N
Lens or lens system
2
Point source
N
 OEi
Microlen
s array
i 1
Incoherent sum
Uniform
illumination
=
1
2
3
…
N
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Simulation methodology
 The direct approach
 Model a large number of gaussian beams
• The parameters are determined by the lens system
 Shift the position of the beam across the image sensor
 Sum |E|2 or the OE incoherently
 Perfect for looking at spatial PSF
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Point spread function


There is not an obvious definition of a PSF in a digital system
Here is one way
 Fully illuminate one pixel
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Point spread function

32 simulations
 16 beam positions
 2 polarizations per position

Could use these results to reconstruct other illumination conditions
 Structure is locally periodic
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Point spread function
 Incident light is green
 Can see cross talk to other green pixels
 Also spectral cross talk to red and blue pixels
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Simulation methodology
 Another approach for uniform illumination
 An incoherent sum of focussed beams for uniform
illumination is mathematically equivalent to an
incoherent sum of all plane waves supported by the
lens system
 This is the most commonly used approach to optimize
the OE and determine optimal micro-lens shifts
 See Jérôme Vaillant, Axel Crocherie, Flavien Hirigoyen, Adam Cadien, and James
Pond, "Uniform illumination and rigorous electromagnetic simulations applied to
CMOS image sensors," Opt. Express 15, 5494-5503 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5494
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Simulation methodology
 Uniform illumination
z
Objective lens
y
 2   2

2
Euniform   dk x dk y W (k ) E (k )
 2

OEuniform   dk x dk y W (k ) OE(k )
k
x
W is a property of the lens system and pixel location
A reasonable approximation for the central pixel is
1, k   k0  NA
W 
0, k   k0  NA
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Simulation methodology
 Uniform illumination
z
y
For an edge pixel, W is different.
Again at low NA, a reasonable approximation
can be made from the NA and geometry of the
system
kCRA
k
Objective lens
x
1, k   k CRA  k0  NA
W 
CRA
0, k   k   k0  NA
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Simulation methodology
 Optimizing OE for uniform illumination means
optimizing the integral under the curve
kxCRA
NA
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Simulation methodology
 We often plot the angular
response but the integral is
really over k
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Simulation methodology
 Several options to optimize the optical efficiency
 Simulate only the chief ray angle (CRA) and try to
make the peak efficiency appear at the center of the
integration window
 Simulate the CRA and some points near the edges of
the integration window
 Simulate enough angles to get an accurate integral
• Excellent agreement with experimental results
Source: Hirigoyen et al., “FDTD-based optical simulations methodology for CMOS image
sensor pixels architecture and process optimization”, PROCEEDINGS- SPIE THE
INTERNATIONAL SOCIETY FOR OPTICAL ENGINEERING, 6816, 2008, [6816 08]
http://www.lumerical.com
Unpolarized results
 FDTD simulations have well defined polarization
 Unpolarized results are obtained by averaging the results of 2
orthogonal polarization simulations incoherently
2
E
unpolarized
1

2
2
1

2
2

2
 d E ( )
0


2
 d E1 cos( )  E2 sin( )
0
1  2  2
  E1  E2 

2
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Simulation steps




Parameterization of design
FDTD simulation
Analysis of results
Optimization
 Parameter sweeps
 Optimization algorithms
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Parameterization of your design
 Parameterized is essential for optimization
 CMOS image sensors are complex 3D structures
 Import from GDSII, AFM data
 Use script to reproducibly draw the structure
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Parameterization of your design
 Create your own properties
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Parameterization of your design
 Construct image sensor
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Parameterization of your design
 Parameterization can include position of sources and monitors
 Any group can set properties of the children
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Parameterization of your design
 Essential for
 Reproducibility
 Easy parameter sweeps
 Optimization
 Lumerical’s hierarchical group layout and script
based parameterization makes almost anything
possible
 It is worth the initial investment!
http://www.lumerical.com
Angular response curve
 How to obtain the angular response curve and
spectral cross talk
 How to optimize the microlens shift and other layer
shifts
 Optimization of different parameters, such as lens
radius of curvature
http://www.lumerical.com
Angular response curve




Plane wave source
Bloch boundary conditions
One unit cell
Arbitrary lens shift or layer shift is OK
 Structure is still periodic
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Angular response curve

Calculate response of each pixel by integrating the power into each region at the
surface of the Si
 We will consider 3D optical generation effects later!
 Saves considerable simulation time because we don’t need to simulate the Si volume
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Angular response curve
 Note that by integrating over a particular region
of Si, we are making a first effort at calculating
the QE (photons to collected electrons) of the
device
 We still often refer to this as the Optical Efficiency
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Tip
 Use a coarse mesh for simulations
 Memory scales as dx3
 Simulation time scales as dx4
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Tip
 Use Lumerical’s conformal mesh technology to
get submesh accuracy
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Tip
 You don’t need a small dx and dy in the Si, just dz
 The in-plane wavector is the same in Si as in the upper layers
(Snell’s law)
 Not strictly true when scattering structures are on the Si surface
 But in practice is an excellent approximation
 Do some convergence testing to confirm
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Tip
 Simulation times at mesh accuracy 1
 6 points per wavelength
Machine
Year
Approximate cost (US$)
Parallel processing time
(all cores)
Lenovo laptop, 2 cores
2009
$1500
3:20 minutes
Intel Core i7, 8 cores
2010
$1000
56 seconds
Intel X5550 Worsktation,
2 processors, 16 cores
2009
$3700
43 seconds
AMD Opteron, 4
processors, 32 cores
2010
$3500
15 seconds
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Angular response curve
 Make an analysis group perform the integral
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Angular response curve
 Create a nested sweep
 Polarization (for unpolarized result)
 Source angle
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Angular response curve
 Demonstration
 Use all the computers in your office
 Save, run and load files if you have a cluster
Ideal shape is cosine
Ideal peak efficiency is
50% for green light
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Concurrent computing
 Optimization and parameter sweep require many
simulations
 Send them to many different workstations
 Each workstation can run in distributed computing
mode, using all cores
N computers means you
can get your
optimization or
parameter sweep results
N times faster!
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Optimization
 How to optimize the microlens shift and other
layer shifts
 Calculate a map of shift vs optical efficiency for all
possible CRAs
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Optimization
 Optimization of different parameters
 Example: lens radius of curvature
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Optimization
 For lens radius of curvature (ROC), assume
spherical lens
 Set up a nested parameter sweep
 Use only 2 angles: 0 and 15 degrees
 Plot average OE as a function of ROC
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Optimization
 There is a maximum near 1.3 microns.
 The normal incidence light has 2 maxima, one
near 1.1 microns
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Optimization
 Setup an optimization task
 Actual optimal result is about 1.29 microns
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Loss per unit volume in silicon
 Insert a “Power absorbed” object in the Si
Absorbed power, log scale
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Advanced 3D efficiency analysis
 We can calculate the optical generation rate, G
 The number of generated electrons per unit time per
unit volume
 Assume that all photons are absorbed by exciting an
electron-hole

FDTD 

Pabs (r ,  ) Psource( ) Pabs (r ,  )
G (r ,  ) 

FDTD


Psource
( )
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Advanced 3D efficiency analysis
 We can make a better effort at calculating the quantum efficiency of
the device
 The number of collected electrons over the number of incident photons
 We assume that all electrons in a given spatial volume are collected
 We simply integrate G over that volume
Example
integration
volume
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Use in subsequent electrical modeling
 Once we can calculate the generation rate, we can use it
as an input to electrical modeling
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Electrical modeling
STMicroelectronics and CMOS image sensors
Three steps to modeling
1. Process modeling
2. Optical modeling (FDTD Solutions)
3. Electrical device modeling
Comparison with
experiment
550nm
Source: A. Crocherie et al., “From
photons to electrons: a complete 3D
simulation flow for CMOS image
sensor”, IEEE 2009 International
Image Sensor Workshop (IISW)
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Broadband simulation
 First time users
 3 simulations for red, green, blue wavelengths
 Can calculate spectral cross talk
• Can make initial calculation of the color matrix
 More advanced users
 Will want to try and include more spectral information
• Initially, in small wavelength bands near red, green and blue
• Eventually can try full bandwidth in one simulation
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Broadband challenge
 FDTD is appealing because we can obtain the
entire spectrum from 1 simulation
 Challenges
 Dispersive material models
 Incident angle changes with wavelength
 Incident beam profile changes with wavelength
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Broadband challenge
 Dispersive materials
 Well-known frequency domain relationship


D( )   ( ) E ( )
 FDTD is a time domain technique: relationship?
t 


D(t )   (t )  E (t )   E (t ) (t  t )dt 
0
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Broadband challenge
 Common solutions are Lorentz or Drude models
 Often insufficient for real materials
 Lumerical’s Multi-Coefficient Model (MCM) can solve for materials
with arbitrary dispersion such as Si, GaAs, or color filters
Silicon
GaAs
Red filter
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Blue filter
Broadband simulation
 Tips
 The angle of incidence changes
with wavelength
• Can easily be corrected in
angular response curve
 You may want to lock the
simulation mesh as you vary
the source bandwidth
 You will likely want to lock
material fits to a particular
wavelength range
 Many fits with large numbers of
coefficients will reduce
numerical stability
• Most issues can be resolved by
carefully controlling the fit
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Broadband simulation
 Broadband simulation can give excellent
agreement with experimental results
Source: Crocherie et al., “Three-dimensional broadband FDTD optical
simulations of CMOS image sensor”, Optical Design and Engineering III,
Proc. of SPIE, 7100, 2008, [71002J]
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Challenges and solutions
Challenge
Solutions and best practices
Wavelength scale devices
Full vectorial 3D Maxwell solver
Simulation methodology
“Think before you simulate”
•Setup simulation methodology to calculate the same results you can obtain experimentally
•Understand how to deal with effects like incoherent and unpolarized light
•Reduce unnecessary computational requirements
•Store only necessary field data
Complex 3D geometries
Parameterize designs
Simulation time
•Use coarse mesh size where possible
•Always for initial simulations
•Do convergence testing of mesh size last!
•Make reasonable approximations
•Example, treat metal as PEC
•Advanced meshing
•Use non-uniform meshing
•Use conformal meshing
•Use distributed parallel computation to take advantage of modern hardware
Broadband simulation
•Time domain gives broadband results
•Highly dispersive materials require multi-pole material models
•Carefully control your models
•Keep material fits constant as you change simulation bandwidths
•Use a fixed mesh as you change simulation bandwidths
Optimization and parameter
sweeps
•Using a global search algorithm that significantly reduces the number of simulations required
•Use concurrent computing to use all your available computer resources optimally
http://www.lumerical.com
For More Information
Dr. James Pond
CTO
jpond.support@lumerical.com
Dr. Guilin Sun
Senior R&D Scientist
gsun.support@lumerical.com
Chris Kopetski
Director of Technical Services
ckopetski.support@lumerical.com
Dr. Mitsunori Kawano
Technical Sales Engineer
mkawano.support@lumerical.com
Sales inquiries
sales@lumerical.com
Sales representatives (other regions)
http://www.lumerical.com/company/representatives.html
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