Part V Appendices

Appendix A

Quantum Efficiency of Photodetectors

Each of the photodetector structures illustrated in Figure 5.1 can be analyzed relatively easily. Here we provide an analysis for each device, and derive equations for the quantum efficiency as a function of the geometrical and metallurgical parameters of the devices. The simplifying assumptions made in all derivations in the following sections are:

• Abrupt junctions with rectangular depletion regions.

• One dimensional current flow. This would not be true for minimum size devices, where vertical and horizontal dimensions are comparable.

• No high-level injection. This becomes important for very high intensity appli­cations, for example for furnaces or welding inspection.

• No degeneration in highly doped diffusion regions.

• No recombination in depletion regions.

• No surface recombination. This parameter is particularly important for lateral devices and for photo gates, where there is a significant number of active carri­ers close to the surface. In vertical devices, the processes which determine the characteristics of the device depend mainly on the parameters of bulk semicon­ductor.

• No surface reflectance.

• No diffusion in the bulk substrate. This is important for near infra-red detectors, as most of the carrier generation happens close to the bulk substrate.

There are also some other assumptions made for each device which will be explained individually when treating each device.

Appendix A. Quantum Efficiency of Photodetectors

In order to improve the consistency between the simulation results from the derived equations and real measured data, the above parameters should be taken into account. However, the derived equations can still provide a good insight into the device opera­tion, and illustrate the effect of different parameters on the quantum efficiency. More­over, there are no accurate data available for the physical and metallurgical parameters in most processes, and hence these will be ignored here. In the extreme case one can use device simulation software to numerically derive the device characteristics.

A.1 Quantum Efficiency of a Vertical Junction Diode

For the structure shown in Figure A.I, the photocurrent is composed of two compo­nents: the drift current due to the drift of holes and electrons in the depletion region, and the diffusion current due to the diffusion of carriers outside the depletion region.

P-well Xj

Xepi

Figure A.I: The structure of a junction photodetector. Xi is the metallurgical junction depth, W is the width of the depletion region, and xepi is the thickness of the epitaxial layer.

The drift current in the depletion region is:

l ",j+",,,

Jdrift = -q G(x)dx "'j-"'p

(A. I)

246

Vision Chips Part V. Appendix

where G{x) is the carrier generation rate for an incident photon flux, 1).)0, in a semi­conductor with an absorption coefficient of a, and is given by

(A.2)

Hence

1. ;J. -a{x· -x )(1 -aW) drift = q~oe 3 p - e (A.3)

xn and xp are the extent of the depletion region in the nand p sides of the junction and are given by

xn = (A.4)

xp =

where Vr is the reverse bias voltage applied to the junction, and Vo is the built-in potential of the junction, and is given by

Vo = kT In NAND q n~

(A.S)

The diffusion component of the current can be found from the diffusion equation:

D a2pr - Pn - PnO + G{x) = 0 in the N-substrate Pax Tp

(A.6)

where Dn and Dp are the diffusion coefficients of the minority carriers, Tp and Tn are the lifetime of excess carriers, and PnO and npO are the eqUilibrium minority carrier densities. The above equation can be solved under the boundary conditions Pnl"="ePi = 0, Pnl.,=.,;+., .. = 0, npl.,=o = npO, and npl.,=.,;_.,p = 0 to obtain

Pn{x) = PnO + Aei; + Be -i; + Ce-ax (A.7)

247

Appendix A. Quantum Efflciency of Photodetectors

where Lp and Ln are the diffusion lengths of excess carriers, and

( (x· - x »)

D ~ F e-a ("; - ",1- e - 'Ln' +n"o

2 . h (Xj - xp) - sm Ln

The diffusion current can be expressed as:

248

(A.8)

Vision Chips Part V. Appendix

which can be simplified as:

!2:£. 1 - coshKp !2:£. e-O:Xepi q Ln PnO sinhK + q Ln C sinhK p p p p

CD -o:(x, + xn) ( _ COShKk ) +q pe J 0: Lpsinh p

!2:n. 1 - cosh K n + !2:n. F 1 +q Ln npO sinh Kn q Ln sinh K -qFDne-O:(Xj - xn) (0: + co~hK:k '\

Ln smh .,;) (AIO)

The parameters Dn , Dp , Tn, and Tp can be derived from the following empirical formulas for silicon, as a function of impurity densities

r. - 1 p - 7.8 x 10 13 N D + 1.8 x 10 '&1 Nb

_ kT ( 370) Dp - q 370 + 1 + 1.563 x 10 IS N D

r. - 1 n - 3.45 x 10 12 N A + 9.5 x 10 32 N1

Dn= k[ (232+ 1+1.12~~~0 17NA)

(All)

The total current Jopt is the sum of the drift and diffusion currents.

J opt = Jdri/t + Jdi/ / (AI2)

The above equations may be simplified for single-sided and shallow junctions to pro­vide a better understanding of the effect of different parameters on the photoresponse of the device, but we keep them in their general form. The measured absorption coef­ficients for silicon are shown in Figure A.2. Typical parameters of a powell-substrate and a diffusion-well silicon junctions are shown in Table AI. The simulated quantum efficiency, Jopt!iJ!o, for these devices is plotted in Figure A.3. The quantum efficiency of the diffusion-substrate junction is higher than the other two structures, and also spans over a wider spectrum.

A.2 Quantum Efficiency of a Lateral Junction Diode

The structure of a lateral photodiode is shown in Figure AA. For analysis purposes, a few simplifying assumptions are made. Firstly, only the area between the two diffu­sion regions is assumed to be exposed to light. Otherwise, there will be a large contri­bution from the vertical bipolar component formed by p-diffusionln-weIVp-substrate,

249

Appendix A. Quantum Efficiency of Photodetectors

Absorption Coefficient v. Wavelength (Silicon) 10)~----r-----~----'-----~----~-----r----~----~

102 ::t·, :;" ..

E -< 101 '. ~,

c: <> '0 to:: .....

100 2:1 (,) c 0

'':: e- 10.1 :il

.&> «

10.2

10') 200 300 400 500 600 700 800 900 1000

Wavelength (run)

Figure A.2: Measured absorption coefficient of silicon.

Table A.1: Typical parameters of silicon junctions in a 2JLm p-well standard process. ni = 1.45 x 1010 and Vbias = 0 volt.

Diode structure Xj Xepi ND NA Ln Lp p.m p.m 1/cm3 1/cm3 p.m p.m

p-well-substrate 2.25 10-15 5.07 x 10'" 2.22 X 10"" 199.6 694 n-diff-p-well 0.47 2.25 2.22 x 10"" 1 x 10~u 199.6 0.71 p-diff-substrate 0.47 10-15 1 x lO~u 4.37 x 10"0 446.8 0.289

whereas in reality, the photogenerated electron-hole pairs will diffuse to other areas. Secondly, it is assumed that the effective depth of the device is only Yj, as again there will be some currents diffusing through other areas.

The diffusion equations in the P+ and N-well are

D a2P2n - Pn - PnO + G{y) = 0 in the N-well Pax Tp

(A. 13)

in the P+

250

Vision Chips Part Y. Appendix

0.9

Diff-Well 0.8 , - -'- Diff-8ub

"- Well-8ub / , , ,

0.7 , \

.",. "- , 0.6 I ,

'; \

I" I 0.5 ,.

,.' II ,

0.4 r , i

, \

I \ ~

0.3 \

I \

! \

\ 0.2 "-

"- , " 0.1

_.-=:&.1 0 200 300 400 500 600 700 600 900 1000 1100

Figure A.3: Simulated quantum efficiency versus wavelength for three differentjunc­tion diodes in a 2Jtm process.

By taking into account the boundaryconditionsPnlz~z; = 0 and Pn Iz=z" = O. we will have

(A. 14)

251

Appendix A. Quantum Efflciency of Photodetectors

The drift current is simply

hri/t = -q~oG(y)(xn + xp) ~ -q~OG(y)xn (A. 15)

The total current can be obtained by integrating the addition of the drift and diffu­sion components across the depth and width of the device.

Jtotal = loy; [A-(B+qxn)~oG(Y)ldy = AYj+(B+qxn)~o{e-ayj -I} (A.16)

where

(A. 17)

(A.18)

Figure A.5 shows the simulation result of this structure for a typical 2pm process. As expected, there is a large blue response because all the carriers generated close to the surface are absorbed by the device. The poor response at longer wavelengths is due to the fact that we have considered the contribution of those carriers which are up to y - j deep into the device, which is very shallow. This structure can be combined with the vertical photodiode, by exposing all sides of the diode to light.

A.3 Quantum Efficiency of a Vertical Bipolar tran­sistor

The structure of a vertical bipolar transistor is shown in Figure A.6. It is assumed that only the flat area is exposed to light, as otherwise, there will be some contribution from the vertical walls of the emitter-base and base-collector junctions.

We can write the diffusion equation in the three regions as:

Dne a:n~e _ npe; npeO + G(x) = 0 in the P-Emitter '1x ne

D b a P~b - Pnb - PnbO + G(X) = 0 in the N-Base (A.19) p ~x 'Tpb

D a n~c _ npc - npco + G(x) = 0 in the P-Collector nc ax 'Tnc

252

Vision Chips Part Y. Appendix

Yj

Figure A.4: The structure of a lateral junction diode in an N-Well CMOS process.

The boundary conditions are:

in emitter

in base (A.20)

253

Appendix A. Quantum Efficiency of Photodetectors

0.7.-----.-------.-------.------.-------.-----.

0.6

0.5

0.4

0.3

0.2

0.1

Dill-Well Diff-Sub Well-Sub

oL-______ L-______ L-______ L-______ L-==~~~ ____ ~

o 200 400 600 800 1000 1200

Figure A.5: Simulation result of the lateral photodiode in a 2JLm CMOS process.

The diffusion equations can be solved as follows.

npe(x) = npeO + Aet + Be -t + Ce-ax x X

Pnb(X) = PnbO + Eer,; + Fe -r,; + He-ax (A.21)

npc(x) = npco + Ket + Me -t + Re-ax

(A22)

(A23)

(A24)

254

Vision Chips Part V. Appendix

Xje

Figure A.6: The structure of a vertical bipolar detector in an N-Well CMOS process.

(A.25)

F = p,.bo(e+Y (e W -1 +2e+X )_H(e-Q(-;.+z,..le+Y _e-Q(z;c-a,.cl e+x e e

(A.26)

and

X - Xje +Xne - Lb

Y _ Xjc -Xnc - Lb (A.27)

255

Appendix A. Quantum Efficiency of Photodetectors

U - Xjc + xpc - Lc (A.30)

The diffusion component of the emitter and collector currents can be expressed as:

J . - - D an~e(X) I D apll(x) I dtff,E - q e X X=Xje-Xpe + q b X X=Xje+Xne

Jdiff,e = -qDc an~~x) IX=Xje+xpe + qDb apa~x) IX=Xjc-Zne

(A.31)

The drift components can be simply obtained_by integrating the amount of gener­ated electron-hole pairs in the depletion regions.

Jdrift = Jdepletion -qG(x)dx Jdrift,E = _q(>o(e-O:(Xje + xne) _ e-O:(Xje - Xpe)) (A.32)

Jdrift,e = +q(>o(e-O:(XjC + xpc) _ e-O:(Xjc - Xnc))

As the base of this device is floating, the collector and emitter currents should be equal. The only variable parameter, which is unknown, is VEB. The value of VEB

for which Ie = IE can be found using numerical methods. Figure A.7 shows the quantum efficiency of a typical parasitic PNP transistor in a 2p.m process. The large gain is simply due to the current gain of the bipolar transistor, which is larger than one. As expected the response is relatively flat over the visible spectrum, which is due to the presence of two junctions at two different depths in the device.

A.4 Quantum Efficiency of a Lateral Bipolar Pho­todetector

The structure of a lateral bipolar device is shown in Figure A.8. The simplifying assumptions are similar to those made in Section A.2, except that exposure to light is assumed to be only in the area between the emitter and collector diffusions, and the depletion regions.

Note that in Equation A.33, x denotes the horizontal axis and y the vertical axis. Also yjis the depth of the collector/emitter junctions.

The diffusion equations in the three regions can be written as:

D a2n~e _ npe - npeO = 0 ne ax Tne in P-Emitter

D a2p~b _ pnb - pnbO + G(y) = 0 in N-Base pb ax Tpb

(A.33)

in P-Collector

The diffusion length in the collector and emitter regions is very short. Therefore, we make another simplifying assumption, in that these junctions extend to four times

256

Vision Chips Part V. Appendix

Quantum efficiency of a silicon vertical bipolar transistor 250r-------r-------,-------,-------,-------,-------,

200

~ Iii 150 'u :; E :> C ~ 100 o

50

OL-------~------~------~--------~------~~~---J o 200 400 600 800 1000 1200 Wavelength (nm)

Figure A.7: Simulated quantum efficiency of a vertical bipolar transistor in a 2pm CMOS process. Note that the quantum efficiency is greater than "1", due to the current gain of the transistor.

the diffusion length in these regions. We set the origin at 4Le before the start of the emitter junction, to be able to reuse the derivations for the vertical bipolar transistor.

The boundary conditions for the three regions are:

in emitter { npelx=o = 0 - ~ npelx=Xie-Xpe -npe(e T -1)

{ ~

in base Pnblx=Xie+Xne = PnbO(e VT - 1) (A. 34) Pnblx=Xic-Xnc = -PnbO

in collector { npCIX=Xic+Xpc = -npco

npCIX=Xic+4Lc = 0 257

Appendix A. Quantum Efficiency of Photodetectors

Yj

Figure A.S: The structure of a lateral bipolar detector in an N-Well CMOS process.

By solving the equation we have.

t: -t: npe(x) = npeO + Ae e + Be e

Pnb(X) = PnbO + G(y)Tb + Eeib + Fe-ib (A.35)

t; -t; npc(x) = npco + Ke c + Me c

(A.36)

(A.37)

258

Vision Chips Part V. Appendix

Z _ Xje - Xpe - Le Xje = 4Le (A.38)

~

) (P ()) (e-Y (e VT - 1) + 2e-x ) (A.39) E = (PnbO + G(y Tpb)El = nbO + G y Tpb +X-y -X+Y e -e

X - Xje + Xne - Lb

Y _ Xje - Xne - Lb (A.41)

U _ Xje +xpe - Le

W _ Xje + 4Le - Le (A.42)

(A.43)

(2 -w -u) K _ -npeo e - e - e IV+U _e+W U (A.44)

The diffusion currents at the collector and emitter are:

(A.45)

The drift currents in the depletion regions of the collector and emitter junctions are:

Jdri/t,E(Y) = -qG(y)xne Jdri/t,c(Y) = -qG(y)xne

(A.46)

The emitter and collector currents can be obtained by integrating the corresponding drift and diffusion components of each current over the range [y = 0 to Y = Yj]. Notice that the current density is per unit width of the device, and hence it should be divided by the junction depth Y j to yield a current density per unit area. The simulation result for a PNP device with minimum diffusion spacing (3)') in a 2J.tm CMOS process is shown in Figure A.9. The general shape of the quantum efficiency is very similar to that of a lateral photodiode (Figure A.5).

259

Appendix A. Quantum Efficiency of Photodetectors

Jdri/t,E = -q<)OXne(1- e-aYj ) Jdri/t,C = -q<)oxnc(l- e-aYj )

JdifJ,E =

Quantum efficiency of a lateral bipolar transistor 140

120

100

~ c CD 80 '0

== CD

E ~ 60 c <II ::::I 0

40

20

0 0 200 400 600 800

Wavelength (nm)

(A.47)

1000 1200

Figure A.9: Simulated quantum efficiency of a vertical bipolar transistor in a 2J.Lm CMOS process.

260

Vision Chips Part V. Appendix

A.5 Mixed structures

The simulation result~ for the lateral and vertical devices obtained in the previous sections indicate that vertical devices have a relatively flat response over the visual spectrum, while lateral devices have a better blue response. The lateral and vertical devices can be combined in a simple fashion to form new structures. For the pho­todiode structures all that is required is to make the exposure window opening large enough for the edges of the diode to be exposed. Figure A.ID illustrates the mixed devices.

a b

Figure A.tO: a) A mixed lateral and vertical photodiode. b) A mixed lateral and vertical bipolar transistor.

A.6 Quantum Efficiency of a Photogate

The structure of a photogate is shown in Figure A.ll. A photogate is nothing but a MOS capacitor exposed to light. A photogate operates by integrating the photogener­ated carriers in the potential well, which is created by applying a large voltage to the gate. A simple assumption made here is that the depth of the depletion region is small. One can verify this using the following equation.

(A.48)

261

Appendix A. Quantum Efficiency of Photodetectors

In a 2/Lm process the typical values for Xd are less than O.5/Lm. Therefore, it is reason­able to assume that all the charges filling the potential well are diffusing from areas outside the depletion region.

One important drawback of photo gates is that they have very poor blue response because the gate material absorbs this part of the spectrum. In new processes the gate is silicided, which blocks most parts of the visual spectrum, and hence the silicide layer should be masked out from the areas above the photogate. Another solution is to make several windows in the gate so that light can pass through. Even with polysilicon gates it is recommended to use windowed gates for the photogate devices.

The spectral response of the photogate is obtained by solving the diffusion equation in the substrate area. Notice that a photogate works in a reset-and-integrate mode. During the reset cycle, charges are emptied from the potential well, and during the integration cycle, diffusion of photogenerated currents fills up the potential well.

(A.49)

X=r; y = X£Si

<.PI is the photon flux at the surface of the silicon. If we assume that the gate material is polysilicon and has the same absorption coefficient as silicon, we will have:

(A.50)

The simulated spectral response of the photogate is shown in Figure A.12. The device has a better response for the red part of the spectrum, and the blue response is significantly lower, when the effect of the absorption of the gate is considered.

262

Vision Chips Part V. Appendix

Xepi

Figure A.ll: Structure of a photogate device in an N-Well CMOS process.

263

Appendix A. Quantum Efficiency of Photodetectors

Spectral response of a photogate structure

0.9

0.8 -With gate - - Without gate

0.7

--, ~ 0.6

/' /' ,

" I , 0 , Q. III I \

~0.5 I \

~ I \

'0 I \

\ 8,0.4 I en I \

I \ 0.3 ,

I ,

I , " 0.2 >-

I

0.1 I

0 200 300 400 500 600 700 800 900 1000 1100

Wavelength (nm)

Figure A.12: Spectral response of the photogate showing the effect of the gate ab­sorption in the reduction of quantum efficiency.

264

Appendix B

Analysis of Second-Order Resistive Networks

B.1 Stability

The stability criteria for a general N-th order linear resistive network has been derived elegantly by [Matsumoto et al. 93]. However, only the conditions under which the network is stable are addressed, and hence here we present an alternative method based on the analysis of the poles of the network. This method, in addition to giving conditions for stability, yields the location of the poles, and therefore can be used in determining the shape of the convolution kernel of the specific network. Also this method is more suited for a computer synthesis of a resistive network with the desired kernel function.

For the second-order resistive network shown in Figure B.t the Z-transform can be written as:

x(n)

H(z)

= -g2y(n - 2) - gly(n - 1) + (go + 2g1 + 2g2)y(n) -gly(n + 1) - g2y(n + 2)

1 2 --z _ 11

- 4 U 3 1 + 2u + 2v 2 U 1 z + -z - z + -z + v v v

u =!l1. v = fl1. go go

(B.t)

We have chosen u and v as variables because they represent a more physically com­prehensible space for understanding the effect of the resistor values on stability. The poles of the system can be easily found.

-A± A2-4 PI, P2 , P3 ,P4 = --=-=-='--J.r":"::""--= A = u ± u + 4v)2 + 4v

v P1P2 = 1 and P3P4 = 1

(B.2)

Appendix B. Analysis of Second-Order Resistive Networks

x(n-2) ! x(n-l) ! g2

x(n)! x(n+l) ! g2

Figure B.1: Second-order resistive network.

X(n+2)!

g2

This system is noncausal and therefore in general it may not have a unique impulse response, if sufficient constraints are not applied to the system. In causal systems the constraint that the input and output sequence are "zero" for all left-values (or past values for time sequences), guarantees a unique impulse response for an LTI (linear time-invariant) system. In our case, which is a spatial linear noncausal system, the additional constraint is that the system should be symmetrical. In other words the poles of the system are present in pairs such that PnlPn2 = 1. In fact from the Z­transfonn of a general Nth order network given by

H(z) = +N 1

L akzk

k=-N n

ao =90 +2L9n 1

(B.3)

it can be seen that if Pnl is a pole of the system, Pn2 = 1/ Pnl would be another pole of the system.

Under these conditions the system would be unstable when the poles of the system are on the unit circle in the Z domain. For the second-order network the poles derived in Equation B.2 can be analyzed as follows.

1. u > 0, v > 0 (Region 1)

266

In this case A would be real and

A2 _ 4 = 2u2 + 8uv + 4v ±4~~J(u + 4v)2 + 4v > 0 (B.4)

It can be seen that all the poles are real and therefore in this region the network is stable, and the impulse response will consist of two exponential decay functions.

Vision Chips Part V. Appendix

2. u < 0, v > 0 (Region 2) In this case still A is real. However, by checking the expression A2 - 4 one can see that

1 ~u< --

4 (B.5)

or A2 - 4 < O. The system will have two imaginary and two real poles. The magnitude of the imaginary poles will be

Hence the network would be unstable for all u < -!. 3. v < 0 and (u + 4V)2 + 4v < 0 (Region 3)

In this case A will be complex, and v' A 2 - 4 will also be a complex number.

(B.6)

Therefore, the poles of the network will be

.1 X + J X2 + y2 .1-X + J X2 + y2 -A ± ±V 2 + tV 2 (B.7)

Pn= ____ ~L_ ____________ ~-----------------L-

Note that the first and the second ± sign are independent. The magnitude of the poles after some simplification will be

There are no values for which the magnitude of poles becomes" I", and there­fore the network is stable in this region. The impulse response will consist of two exponentially modulated sinusoidal signals.

It is worth noting that (u + 4V)2 + 4v = 0 represents a parabola which lies in

the quarter-plane v < 0, u > -i. 267

Appendix B. Analysis of Second-Order Resistive Networks

v

-1/4

Figure B.2: Stability regions of the second-order resistive network.

4. v < 0, (u + 4v)2 + 4v > ° and u < -! (Regions 4,5, and 6)

268

This region is a part of the quarter-plane [v < 0, u > -!] which is outside the

parabola (u + 4v)2 + 4v = 0. Three regions, 4,5, and 6, can be recognized. We have

(u + 4v)2 + 4v = 0 (B.9)

R . 4' h . db ° -(2u + 1) + J4u+1 4 eglOn IS c aractenze y > v > 8 ' u > - v + r-;c. -(2u+1)+V4u+1 /

v -4v > 0, regIOn 5 by ° > v >8 ' -1 4 < u < r-;c. -(2u + 1) - V4u + 1

-4v - v -4v, and regIOn 6 by v <8 ,u > -1/4. In all these regions, A is real, and the stability depends on the sign of A 2 - 4. If the

Vision Chips Part V. Appendix

sign is negative, the network will have poles on the unit circle and therefore the network becomes unstable. Otherwise the network will be stable.

A2 - 4> 0 =} (A - 2)(A + 2) > 0 (B.IO)

Note that v < O. When u > 0 (in region 4) we have

u+4v> yf-4v > 0 =} (u+4v) > J(U+4V)2 +4v

....... (u + 4v) - v( u + 4v)2 + 4v 0

.....- 2v < (B.ll)

This means that A + 2 < 0, and therefore A2 - 4 is positive in this region. As a result the network is stable.

In region 5, u < 0, and it is rather easy to see that we have

u - 4v < 0 =} u - 4v - sqrt(u + 4V)2 + 4v < 0

=} (u - 4v) - V;~ + 4V)2 + 4v > 0 (B.12)

Therefore, A - 2 > 0, which means that A2 - 4 is positive in this region. Hence all the poles of the system are real and the system is stable.

In the region below the parabola (region 6) we prove that A2 - 4 < O. For this purpose we show that A - 2 < 0 and A + 2 > O. In order to have A - 2 < 0, we should have u - 4v ± J(u + 4v)2 + 4v > O. This can easily be shown as in this region we have u - 4v > O. Therefore we obtain

u < -! =} (u - 4v)2 > (u + 4v)2 + 4v

=} u - 4v - J (u + 4v)2 + 4v > 0 (B.13)

To show that A+2 > 0, u+4v± J(u + 4v)2 + 4v < 0, or (u +4V)2 - [(u + 4v)2 + 4v] > O. Which is true in this region. Therefore, we have proven that A2 - 4 < O. Hence all the poles of the system lie on the unit circle, and the network is unstable.

In general, the poles of an Nth order network can be found using numerical tech­niques, and the stability can be checked by looking at the position of the poles with respect to the unit circle.

B.2 Impulse Response

The impulse response of the resistive network can be found relatively easily, once the poles of the network have been obtained from Equation B.2. It should be noted that although the system has four poles, only two poles determine the impulse response in the stable regions. This is because the system is noncausal, and we have applied the

269

Appendix B. Analysis of Second-Order Resistive Networks

constraint that the impulse response is symmetrical. There are two pairs of poles such that

P1P2 = 1 P3P4 = 1

(B.l4)

One of the poles in each pair is associated with the left-hand impulse response, and the other one with the right-hand response. In the stable region, as the condition in Equation B.l4 holds, we only need to consider the poles with a magnitude less that 1, in order to determine the impulse response. The transfer function obtained in Equation B.l can be rewritten as

H(z)

1 2 --z V

4 U 3 1 + 2u + 2v 2 U 1 z + -z - z + -z + v

+

where

Pl,P2 = ----,~--

P3, P4 = ----,~--

(P4 - P3)(A1 - A 2 )

Z P4 }

(B.1S)

(B.l6)

1. Regions 1 and 2:

270

In these regions Al and A2 are both real. It is easy to see that one pair of poles (PbP2) are alway negative and the other always positive. Note that both poles in a pair have the same sign.

What is more interesting, is the relative magnitude of the poles. In region 1 the magnitude of the positive pole is larger than that of the negative pole, and vice versa in region 2. On the Y-axis, where u = 0, both poles have the same magnitude but opposite signs. The insets a, b, c, d, and e in Figure B.3, clearly show the manner in which the kernel function changes in regions 1 and 2. The oscillating component comes from the negative pole.

Vision Chips Part V. Appendix

2. Region 3: In this region the poles are complex, and as we derived in Equation B.7

A ( .1 X + V X2 + Y2 .1-X + V X2 + Y2)

- 1 ± +y 2 +ty 2

P2,P1 = -------l--------------~2~------------------~

( Jx + y'X2 + y2 .J-x + y'X2 + Y2) (B.17) -A2 ± - + t

2 2 P4, P3 = -------l--------------~2~------------------~

A A - u±ivl(u+4v)2+4vl 1, 2 - 2v

It can be seen that P1 = p,i, and P2 = pj. Therefore, if for example P1 is the pole within the unit circle, P4 will also be inside the unit circle. The shape of the kernel function is an exponentially modulated sinusoid.

3. Regions 4 and 5:

In region 4 both poles are real and positive. However, the coefficient of one of the exponential term of the impulse response, associated with one of the poles, is negative, while the other is positive. Hence, the kernel function is the difference of two exponentials (DoE). The DoE has been used in several vision chips to approximate the difference of Gaussians (DoG).

In region 5 both poles are real and negative. Again the coefficient of one of the exponential terms in the impulse response is positive and the other one is negative.

Figure B.3 illustrates some of the kernel functions in different regions of stability. Apart from plot (a) in the figure, which lies on the boundary of the stability region, the rest of the curves belong to points within the stability regions. The plots have been obtained by simulating a 127-cell resistive network with the input to all cells being zero, except the middle cell which has an input of "1".

271

Appendix B. Analysis of Second-Order Resistive Networks

u=·0.25 v=l 0

·1

o 50 100

u=-<l.24 v=1

ofF .0.5t::!j

o 50 100

0.4

0.2

u=O v=l

(m)

50 100 u=·0.24 v=O. l

u=·0,24 y=10 0,15

0.1

0.05

u=O v=10

(c)

100

u=l v=10

0'1wd) 0.08

0.06

0.04

0.02

o 50 100 u=10 v=10

o.06[AJe) 0.04

0.02

o 50 100

u=10 v=1

0'1~ o.osUU o o 50 100

u=10 v=O.l

~.:1Fl o.osUU o 50 100 0

u-.0.24 y-·0.05 u=l v~-0.24 u=2 v=·O.99 0 50 100

S8jk) OArn') 2gjil "'1 ">Of 1 a ~2 0

0.1

-5 0 -2 a 50 100 0 50 100 0 50 100 0 o 50 100

Figure B.3: Kernel functions of the second-order resistive network for several sample points. Notice the values on the vertical axis of the plots.

272

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295

INDEX

Index

Astrom, A, 142 3-D VLSI, 123

AAER, see Asynchronous address-event representation

absolute value of difference, 179, 199 absorption coefficient, 247 accommodation, 168 Active Pixel Sensors, 75

correlated double sampling, 77 fixed pattern noise, 77 photocircuits, 76

adaptive retina, 132 ADC, see Analog-to-Digital Conversion aggregation network, 150 Aizawa, K., 188 Analog Delay Elements, 99 Analog Memory Elements, 96 Analog neural networks, 23 Analog versus Digital, 24-27 Analog VLSI, 4, 5, 28, 123

design methodologies, 36 design techniques, 44 framework, 4, 5 technology, 43

Analog-to-Digital Conversion, 56, 78, 102,139,196,225

Andreou, A, 138, 186, 188 ANN, see Artificial Neural Networks anti-blooming, 94 anti-bump circuit, 169 APS, see Active Pixel Sensors Artificial Neural Networks, 28 Artificial Retina, 135, 156

Asynchronous address-event represen-tation,61

AVLSI, see Analog VLSI

Bair, w., 167 Bernard, T., 135 BiCMOS, 34, 151 biharmonic equation, 18, 131 bit-serial processor, 139 Boahen, K., 138, 186, 188 Boolean artificial retina, 135 brightness change constancy equation,

9 Bugeye, 207 Bugeye chips, 208 Bugeye I, 171 Bugeye II, 209-222

architecture, 210 photodetector, 212 reconfigurability, 211 spatial smoothing, 215 TCD,216 testability, 210

Bugeye V, 223-229 architecture, 223 interfacing, 224 motion detection, 228 photocircuits, 225

bump circuit, 182,201

carrier generation rate, 247 carrierlifetime, 247 CCD, 34,175 CCD retina, 143 CCD/CMOS, 34,155,177,184,199 CDS, see correlated double sampling

Vision Chips

cellular neural networks, 28, 80, 190 Centroid Computation, 150 centroid computation, 187 channel length modulation-orA, 212 Charge coupled devices, 34 charge integration photocircuit, 75, 143,

225 charge integration read-out, 102 Charge mode, 38 charge redistribution network, 175 Chiu, C-F., 152 CLM-orA, 212, 217 CMOS, 32 CNN, see cellular neural networks, see

cellular neural networks column parallel, 189, 190 compression sensor, 188, 189 computational devices, 32 conditional replenishment, 188 Contrast Enhancement

biological models, 18 circuits, 94 comparison, 19 computational models, 15 models, 13

correlated double sampling, 77 correlation based motion detector, 169 correlation circuit, 134, 182 CSEM,195 CurrentMode,232 Current mode, 37

dark current, 67 Data read-out, 60 delay line-based motion detection, 176 Delbriick, T., 73, 168, 169,201 depletion region, 245-247, 256, 259,

261,262 DeWeerth, S.P., 150, 187 Difference of Exponentials, 17, 271 Difference of Gaussians, 131, 271 diffusion current, 247, 256 diffusion equation, 247 diffusion length, 248

INDEX

diffusive network, 138, 182 Digital Noise, 124-126 direction-of-heading sensor, 182 directionally selective motion detection,

172,186 Division by spatial average, 16,214 drift current, 246, 256 Dron, L., 175 DSMD, see directionally selective mo­

tion detection dynamic range, 25

elementary motion detector, 11, 169, 179,186

EMD, see elementary motion detector Erten, G., 201 Espejo, S., 190 Etienne-Cummings, R., 194

ferroelectric liquid crystal, 160 finite detector size, 50 fixed pattern noise, 77 FLC, see ferroelectric liquid crystal floating gate MOSFET, 133 focus of expansion, 182, 184 focusing chip, 168 FOE, see focus of expansion Forchheimer, R., 142 Foveated sensors, 53, 143, 145 FPN, see fixed pattern noise FUGA,145

GaAs, 36, 160 Gabor functions, 17 Gauss theorem, 179 Gaussian filter, 138 Gaussian filtering, 131 Gilbert multiplier, 134, 155, 169 Gottardi, M., 177

Hakkaranien,I.M., 199 Hamamoto, T., 189 Harris, I., 94, 147 HEMT,36 hexagonal network, 138, 169

297

INDEX

hexagonal network, 138, 169 Hexagonal Tessellation, 50 Horiuchi, T., 176

IBIDEM, 145 image compression, 188, 189 IMEC,145 Indiveri, G., 179, 182 insect vision, 171, 173,207 Integrator, 99 Interconnection, 123-124

2-D, 124 3-D, 124 density, 123

Keast, C.L., 155 Kobayashi, H., 138 Koch, C., 167 Kramer, J., 179

lamina, 171 Laplacian of Gaussian, 16, 131, 167 lateral bipolar, 256 Lateral inhibition, 18-19, 153 lateral photodiode, 249 LED, 160 Lee, H.S., 199 light emitting diodes, 160 light-modulated MaS, 239 Linear lateral inhibition, 18, 96 Linear-polar mapping, 53, 145 LLI, see Linear lateral inhibition lobula,171 Localization Computation, 150 LoG, see Laplacian of Gaussian log-domain integrator, 100 Log-polar mapping, 53, 145 Lyon, R., 164

Mahowald, M., 132, 133 MAPP, see Matrix Array Picture Pro-

cessor Matrix Array Picture Processor, 140 McQuirk, L, 184 Mead, C., 7,23,132,133,164,165

298

medulla, 171 Meitzler, R., 170, 186 MESFET,36 metal-semiconductor-metal, 160 mexican hat, 131 Mismatch, 109-123

electrical variations, 110 geometrical variations, 110 in CLM-OTA, 218 in current mirrors, 112 in MNC network, 226 in networks, 113 in translinear circuits, 112 modeling, 11 0 sources, 110

Mixed mode, 40-41 MNC, see multiplicative noise cancel­

lation MNC Circuit, 234 MNCSI, see Multiplicative Noise Can­

cellation and Shunting Inhi­bition

Motion Detection computational algorithms, 8 correlating, 164 feature based, 10 intensity based, 9

MSM, see metal-semiconductor-metal MSV, see multi-scale veto multi-scale veto, 175 multi-sensitivity photodetector, 152, 192 multiplicative lateral inhibition, 193 multiplicative noise

cancelling, 214 source of, 212

multiplicative noise cancellation, 174, 207,212

Multiplicative Noise Cancellation and Shunting Inhibition, 207,231-241

NCP, 135 Near Sensor Image Processing, 142 negative impedance converter, 139

Vision Chips

negative resistors, 138 neighborhood combinatorial processing,

135 Neural Networks, 3

interconnection, 28 learning, 30 neurons, 29 weight, 30

NIC, see negative impedance converter Nilson, C.D., 153 NMOS, 164 noise, 26 Nonlinear lateral inhibition, 19 NSIP, see Near Sensor Image Process­

ing

optical flow, 8, 165 Optical mouse, 164, 195 Optical Neurochip, 160 Optical neurochip, 159 Orientation detection, 146 outlier, 9, 92

parallel ADC, 139 PASIC, see Processor ADC and Sen-

sor Integrated Circuit PCHI, see Processor per chip PeOL, see Processor per column Photocircuits, 68

charge integration, 75 current amplifier, 74 logarithmic compression, 69 logarithmic with adaptation, 72 logarithmic with buffer, 71 logarithmic with feedback, 70

Photodetectors, 65, 245 dark current, 67 derivation of quantum efficiency,

245 mixed structures, 261 multi-sensitivity, 152 quantum efficiency, 66

photogate, 261 photon flux, 247 physics of computation, 32

INDEX

pixel parallel, 55 PPIX, see Processor per pixel Processor ADC and Sensor Integrated

Circuit, 139 Processor per chip, 55 Processor per column, 55, 189 Processor per pixel, 55 pulse code modulation, 26 pulse mode motion detector, 185

Quantum Efficiency, 66, 245 of lateral bipolar, 256 of lateral photodiode, 249 of vertical bipolar, 252 of vertical photodiode, 246 photogate, 261

Radon transform, 192 Random Access, 62 Receptive Field, 151 receptive field, 19,239 Reciprocal Lattice, 50 Regularization theory, 18 Reichardt motion detector, 8, 11, 163,

170, 185, 194 Resistive Fuse, 94, 147 resistive network, 152

impulse response, 269 Resistive Networks, 81

nonlinear, 92 resistive fuse, 94 saturating, 94 spatial response, 84 stability, 82

resistive networks, 265 stability, 265

SAER, see Synchronous address-event representation

Sarpeshkar, R., 185 Saturating Resistor, 94 Scanning, 60, 101 Second-Order resistive networks, 265 Shunting inhibition, 19, 96, 153, 193,

207

299

INDEX

shunting inhibition, 231 silicided, 262 silicon retina, 131-133,137, 152 SLM, see spatial light modulators smoothing network, 138 Sobel edge detector, 151 Sodini, C.G., 155 Solar Illumination Monitoring, 158 spatial aliasing, 50 spatial light modulators, 160 Spatial Smoothing

circuits, 78-94 spatial smoothing, 212 Spatio-Temporal Processing, 96-99 Standley, DL, 146 stereo correspondence, 201 Stereo Matching, 133 stereo matching, 201 substrate controlled MOS, 234 substrate coupling, 124 substrate node, 236 Subtraction from spatial average, 15 surface reconstruction, 18 Synchronous address-event representa-

tion,61,189 Syrzycki, M., 151

Tanner, J., 164, 165 TCD, see temporal contrast detector TDOFM, see temporal domain optical

flow measurement template model, 13, 171,209 temporal contrast detector, 124, 209,

216 temporal domain optical flow measure­

ment, 194 Tessellation, 49-54

biological plausibility, 53 foveated, 53 hexagonal, 50 reciprocal lattice, 50 rectangular, 50 remapping, 53

Test Patterns, 107

300

testing, 101 threshold voltage mismatch, 110 time-to-crash sensor, 179 transconductance mismatch, 110

variable sensitivity photodetector, 156, 160

velocity tuned m.otion sensor, 169 Venier, P., 158 vertical bipolar, 252 Vision Chips

advantages, 1 architectures, 49 design flow, 47 disadvantages, 2 history, 3 requirements, 31 synthesis, 47 test patterns, 107 testing, 101-107

Visualization, 103 Voltage mode, 37 Voronoi Cell, 50 Voronoi Distance, 52 VSD, see variable sensitivity photode­

tector VSP, see variable sensitivity photode­

tector VSPD, see variable sensitivity photode­

tector

Ward, V., lSI Windowing, 62 winner-takes-all, 169, 176, 179, 183,

201 Wodnicki R., 145 WTA, see winner-takes-all Wu, C-Y., 152

Yang, W., 177 Yu, T.C.B., 160

zero-crossing, 167, 183, 186