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Optimization of Photodetector Thickness in
Vertically-Integrated Image Sensors
Orit Skorka, Dan Sirbu, and Dileepan Joseph
University of Alberta, Canada
Outline
 Motivation
 Problem statement
 Photodetector model
 Mathematical method
 Optimization of thickness
 Conclusions
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Motivation
 Image sensors are required to have high SNR, high
dynamic range, high resolution, and high frame rate
 All these features may not be achievable with current
planar technologies (CCD and CMOS)
 There has been an increased interest in fabricating
image sensors in which the photodetectors are
vertically integrated with the CMOS circuitry
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Motivation
Example –Vertically-integrated CMOS image sensor
fabricated using flip-chip assembly
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Motivation
Advantages of Vertical-Integration
 The photodetectors and the electronics can be optimized
independently of each other in VI-CMOS image sensors.
 There are more degrees of freedom in the photodetector
design.
 Material – no longer restricted to c-Si.
 Photodetector thickness – the vertical dimension can
now be controlled.
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Problem Statement
 Simplified 1D photodetector structure
 Semiconductor layer thickness – l
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Problem Statement
 What is the optimal semiconductor thickness, lopt,
for the photodetector to have a maximum contrast?
 Contrast  
J ph
J dk

J ( 0  0)  J ( 0  0)
J ( 0  0)
Initial Hypothesis
 If the semiconductor is made too thin then very little light is
absorbed, which implies low photocurrent
 If the semiconductor is made too thick then most photo-
generated charge carriers recombine on their way to the
contact, which also implies low photocurrent
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Photodetector Model
 Three-resistor system
 The illumination decays exponentially in the semiconductor
 The absorbed photons generate extra electron-hole pairs
(EHPs), or excess carriers, that improve conductivity
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Photodetector Model
Charge carrier equations
 Poisson’s equation – relates electric potential or electric
field to concentration of charge carriers
 Continuity equations – ensures charge carriers are
neither created nor destroyed at any point
 Drift-diffusion equations – describe the current density
as a sum of a drift current, which arises from existance
of electric field, and diffusion current, which arises from
concentration gradients
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Photodetector Model
Charge carrier equations
 Poisson’s equation:

dE ( z ) q
 p( z )  n( z )  N D ( z )  N Z ( z )
dz

 Continuity equations:
1 dJ p ( z )
1 dJ n ( z )

 g ( z)  r ( z)
q dz
q dz
 Drift-diffusion equations:
dp( z )
dz
dn( z )
J n ( z )  q n n( z ) E ( z )  qDn
dz
J p ( z )  q p p( z ) E ( z )  qD p
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
Photodetector Model
Boundary conditions
 Kirchoff’s voltage law – forces the sum of the voltage drops
over the three resistors to equal the applied potential, Vab.
 Kirchoff’s current law – forces the sum of the hole current
and the electron current in the semiconductor to equal the
current drawn from the power supply.
 Charge neutrality – as electrons and holes are generated and
recombined in pairs, the semiconductor must remain neutral,
assuming initial charge neutrality.
 Generation-recombination balance – in the steady state,
every EHP generation must be offset by an EHP recombination
(perhaps elsewhere) in the semiconductor.
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Photodetector Model
Boundary conditions

 Kirchoff’s voltage law: Vab  J  ( R  R )  0 E ( z )dz
'
a
 Kirchoff’s current law:
 Charge neutrality:
1 
J   J p ( z )  J n ( z ) dz
 0
  p ( z )  n( z )  N

0
'
b

D
 Generation-recombination balance:
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
( z )  N A ( z ) dz  0
 g ( z )  r ( z )dz  0

0
Photodetector Model
 EHP boundary conditions are often defined through
the “recombination velocity” of charge carriers at the
boundaries. It relates the concentration of charge
carriers and current density on the boundaries.
 We have defined EHP boundary conditions by charge
neutrality and generation-recombination balance,
which are easier to interpret. These conditions, fewer
than those used in the literature, prove sufficient to
solve the problem without inconsistency.
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Mathematical Method
 The solution process is based on the mean value of
the different variables in the semiconductor, and on
the deviation of the local quantity from its mean
value. The mean is computed over the length l.
Example – Hole (p) and electron (n) concentration
p( z )  p   p ( z )
n( z )  n   n ( z )
local concentration
mean value
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local perturbation
Mathematical Method
Analytical solution
 Is derived by assuming p(z) = 0 and n(z) = 0 only.
 A systematic way was derived to solve all the remaining
variables and, thereby, to find J = f (0) for any 0.
 However, the solution does not satisfy all equations.
Numerical solution
 Is based on an iterative finite-differences method.
 It shows the equations are consistent and complete.
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Mathematical Method
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Analytical
Numerical
Poisson’s
equation


Continuity
equations


Drift-diff.
equations


K.’s voltage
law (KVL)


K.’s current
law (KCL)


Charge
neutrality


Gen.-rec.
balance


Optimization of Thickness
Our initial hypothesis proved wrong
 If the semiconductor is too thin, very little light is
absorbed. However, the electric field becomes very
strong for a constant applied voltage.
 According to the model, the contrast is low at this
end because the resistance of the semiconductor is
low in comparison to the contact resistances. The
device is dominated by the contacts.
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Optimization of Thickness
 In the absence of contact resistances, the contrast is
maximal at l equals zero, which is nonsensical.
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Optimization of Thickness
Our initial hypothesis proved wrong
 If the semiconductor is too thick, there is more EHP
recombination in the device, as predicted. However,
the total generation rate of EHPs also increases with
semiconductor thickness.
 According to the model, the contrast is low here
because the mean value of excess charge carriers per
unit volume decreases with device length and, thus,
the mean photoconductivity also decreases.
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Optimization of Thickness
 Total EHP generation rate increases with semiconductor
thickness. However, mean excess carriers, and hence
conductivity and contrast decrease as l grows large.
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Optimization of Thickness
 Optimal thickness was found analytically and numerically
 See the paper for material (a-Si:H) and other parameters
 Simulation results: lopt = 400 nm for maximum contrast
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Conclusions
 We presented a new approach to solve semiconductor
charge carrier equations in 1D photodetectors.
 Our boundary conditions used Kirchoff’s laws, charge
neutrality, and generation-recombination balance.
 Our approach was based on mean values of variables and
on deviations of local values from the mean values.
 The method was used for optimization of photodetector
thickness in a vertically-integrated image sensor.
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Acknowledgments
The authors gratefully acknowledge the support of
 Alberta Ingenuity
 The Natural Sciences and Engineering Research Council
(NSERC) of Canada
 The Mary Louise Imrie Graduate Student Award,
University of Alberta
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