202 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 49, NO. 1, JANUARY 2002

Fig. 4. f –I characteristics of the lateral NPN and PNP devices.

The base resistance of this device is higher as compared to the verticalbipolar transistor as well as the lateral BJT reported earlier [7]. Thehigher base resistance is because of the lateral base contact as well asno silicide used in the process. But the lateral BJT reported earlier hastop base contact, and silicide has been used to minimize the base resis-tance [7]. The values of other basic parameters such as Re, Rc, and Vaof the lateral NPN are 290, 9.7 k, and 13 V, respectively. The valuesof the junction capacitancesCje, Cjc, andCjs of the lateral NPN are0.29 fF, 0.28 fF, and 2.9 fF, respectively. A noise figure of 2.3 dB at2 GHz withVCE = 2 V was obtained for the lateral NPN. The highfrequency performance of the lateral bipolar devices also depends onthe collector width. The collector width of the lateral bipolar devicesis approximately 1�m. The high frequency performance can be fur-ther improved by reducing the base and collector widths, and also byusing silicide to further reduce the base resistance. The further improve-ment in device performance was predicted using 2-D device simulatorMEDICI [9]. Simulation results show that by narrowing the base widthto 0.08�m and collector width to 0.5�m, the lateral NPN can achievea fmax of 47 GHz.

CMOS transistors with a channel length of 0.5�m and a channelwidth of 5�m were characterized. The threshold voltages of the NMOSand PMOS devices are+0.8 V and�0:80.8 V, respectively.

IV. CONCLUSION

In summary, simple and high performance TFSOI lateral comple-mentary BJT structures were implemented in a TFSOI CMOS processwith minimized base and external base linkage regions (for minimizedoverall base resistance). The experimental results showed that thisTFSOI C-BiCMOS technology is very promising for high-levelintegration of RF mixed-signal circuits for wireless communicationapplications.

ACKNOWLEDGMENT

The authors would like to thank the fabrication staffs of the Micro-fabrication Facility at the HKUST for their help in the processing andconstant support.

REFERENCES

[1] R. Reedy, J. Cable, and D. Kelly, “Single chip wireless systems usingSOI,” in IEEE Int. SOI Conf., 1999, pp. 8–11.

[2] H. Ammo, H. Ejiri, S. Kanematsu, H. Kikuchi, M. Yano, and H.Miwa, “A complementary BiCMOS technology for low power wirelesstelecommunication applications,” inProc. ESSDERC, 1999, pp.444–447.

[3] S. Parke, F. Assaderaghi, J. Chen, J. King, C. Hu, and P. K. Ko, “Aversatile, SOI BiCMOS technology with complementary lateral BJT’s,”in IEEE IEDM Tech. Dig., 1992, pp. 453–456.

[4] G. G. Shahidiet al., “A novel high-performance lateral bipolar on SOI,”in IEEE IEDM Tech. Dig., 1991, pp. 663–666.

[5] W. M. Huang, K. Klein, M. Grimaldi, M. Racanelli, S. Ramaswami, J.Tsao, J. Foerstner, and B. Y. Hwang, “TFSOI BiCMOS technology forlow power applications,” inIEEE IEDM Tech. Dig., 1993, pp. 449–452.

[6] T. Shino, K. Inoh, T. Yamada, H. Nii, S. Kawanaka, T. Fuse, M. Yoshimi,Y. Katsumata, S. Watanable, and J. matsunaga, “A 31 GHzf lateralBJT on SOI self-aligned external base formation technology,” inIEEEIEDM Tech. Dig., 1998, pp. 953–956.

[7] H. Nii, T. Yamada, K. Inoh, T. Shino, S. Kawanaka, M. Yoshimi, andY. Katsumata, “A novel lateral bipolar transistor with 67 GHzf onthin-film SOI for RF analog applications,”IEEE Trans. Electron De-vices, vol. 47, pp. 1536–1541, July 2000.

[8] TSUPREM User’s Manual, 1999. ver. 4.[9] MEDICI User’s Manual, 1999. ver. 4.1.

The PSD Transfer Function

Michiel de Bakker, Piet W. Verbeek, Gijs K. Steenvoorden, andIan T. Young

Abstract—This article describes the dynamic behavior of position sen-sitive detectors. Earlier work has described the PSD response to changesin light intensity. Here, we investigate the PSD response to a change oflight spot position. We call the linear filter describing this response the PSDtransfer function (PTF).

Index Terms—Position sensitive detectors, transfer functions.

I. INTRODUCTION

In this paper we derive the response of a position sensitive detector(PSD) to a moving light source. A PSD is a lateral photodiode: due toa nonuniform illumination over the PSD surface two lateral photocur-rents are generated. The ratio of the photocurrents is a measure for thecenter of gravity of the intensity distribution of the illumination. Thiseffect is known as the lateral photo effect, and J. T. Wallmark was oneof the first researchers to recognize the possiblities [1]. G. Lucovskydescribed the theory behind the the lateral photo effect [2].

A model of a PSD is shown in Fig. 1. If the PSD layer is perfectlyhomogeneous, the position of light incidence is [3]:

up = Liright

iright + ileft: (1)

In this article we start with repeating the Lucovsky equation (i.e., thebasic equation describing PSD behavior), and we give the solution for

Manuscript received May 25, 2001; revised August 21, 2001. This work wasbeen supported by the Dutch Technology Foundation STW. The review of thisbrief was arranged bny Editor K. Najafi.

M. de Bakker, P. W. Verbeek and I. T. Young are with the Pattern RecognitionGroup, Faculty of Applied Sciences, Delft University of Technology, NL-2628CJ Delft, The Netherlands (e-mail michiel@ph.tn.tudelft.nl).

G. K. Steenvoorden is with the TNO Institute of Applied Physics, 2600 ADDelft, The Netherlands.

Publisher Item Identifier S 0018-9383(02)00223-X.

0018–9383/02$17.00 © 2002 IEEE

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 49, NO. 1, JANUARY 2002 203

theintensityimpulse response. Next, we show how we model a movinglight source. After deriving the PSD response to a moving light source,we compare our results to Pstar3.0 simulations.

II. THE LUCOVSKY EQUATION

The Lucovsky equation is ([2]–[4]):

�uu �RC

L2�t = �jpn

�

D(2)

where� represents the potential difference between the PSD layer andthe substrate1 , R andC are the PSD resistance and capacitance,Lthe PSD length, andjpn the current density entering the PSD layer atw = 0. In the general situation� andjpn depend ont as well as onu:�(u; t) andjpn(u; t).

III. I NTENSITY IMPULSE RESPONSE

In this section we compute the response of a PSD to an impulse in in-tensity. The starting point is the Lucovsky equation, (2), with boundaryconditions�ju=0;L = VR (an externally applied reverse bias voltage),andt > 0. If we write�0 = ��VR, and substitutejp��(u�up)�(t�t0)for jpn, we can write for (2):

�0uu �RC

L2�0t = �

�

Djp � �(t� t0)�(u� up) (3)

with boundary conditions�0ju=0;L = 0. The solution of (3) can befound by performing a finite Fourier sine transform (FFST) with re-spect tou and a Laplace Transform with respect tot. The theory on theFFST can be found in [7]. The result of this double transform we call~�(n; s):

�n�

L

2

+RC

L2s � ~�(n; s) = �

�

Djp � sin

n�upL

� e�st : (4)

Solving (4) for~�(n; s)=jp and rearrangement of terms leads to

~�(n; s)

jp=

L2

RC

�

Dsin

n�u

L� e�st

(n�)RC

+ s(5)

�0 can be found by performing the double backtransform of (5):

�0

jp=

1

n=1

2�

D

L

(n�)2sin

n�upL

sinn�u

L� gn(t� t0) (6)

with

gn(t) = �2ne�� t

s(t); �2n =(n�)2

RC: (7)

We introduce�2n rather than�n in order to emphasize that the valueis always positive, and proportional ton2. In (7), s(t) is the unit stepfunction. The (lateral) current density at the left contact(u = 0) is(note that�0u = �u):

jpulseleft (up; t) =��u(0; t)

�

=�jpD

1

n=1

2

n�sin

n�upL

gn(t� t0): (8)

The minus sign injpulseleft indicates that the current at the left contactflows in the negativeu direction. In the remainder of this article we use

1The following notation is applied:� = @ �=@u and� = @�=@t.

Fig. 1. Sketch of a position sensitive detector. A p-type layer with resistivity� and implantation depthD has been implanted in an n-type substrate. Incidentillumination leads to two lateral photo currents: one to the left, and one to theright end contact. The center of gravity is determined by (1).

a new coordinate system for the PSD, with the origin in the center ofthe PSD. This is achieved by replacingsin(n�up=L) in (8) by

Sn(up) = sinn�upL

+1

2: (9)

In the new coordinate system we can use symmetry properties of thePSD to compute the photocurrent to the right PSD contact, by replacingup by�up. The impulse responses for the left and right PSD signalsare in the new coordinate system:

jpulseleft (up; t) =�jpD

1

n=1

2

n�Sn(up)gn(t� t0);

jpulseright (up; t) = +jpD

1

n=1

2

n�Sn(�up)gn(t� t0): (10)

Equation (10) also implies that the closer the light source originatesto a contact, the faster the response is.

IV. M OVING LIGHTSOURCE

In this section we arrive at the goal of this article, i.e., the PSD re-sponse to a moving light source. We introduceup(t), the position of alight spot, and we assume a constant intensity fort > 0. The currentdensity to the left contact can then be described by the convolution ofjpulseleft (up(t)t) with the unit step function

jleft(t) =jpulseleft (up(t); t) � s(t)

=�

jpD

1

n=1

2

n�

t

0

Sn (up(t0))gn(t� t0)dt0:

(11)

In the following, we show specific interest in a light source of whichthe position is modulated with a sine function, and we assume that theamplitude of the oscillations is small with respect to the PSD lengthL(see Fig. 1):

up(t) = uc +�u sin (!t) (12)

204 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 49, NO. 1, JANUARY 2002

Fig. 2. Intensity impulse response computed with (10) for a test PSD withR = 700 k andC = 28 pF. The current densities in the graphs are normalized forj =(D � RC).

whereuc is the central position of the oscillations,�u is the amplitudeand! the angular frequency. Rewriting (11) [using (12)] leads to

jleft(t) = �jp

D�

t

0

1

n=1

2

n�Sn(uc) cos n�

�u

Lsin!t0

+Cn(uc) sin n��u

Lsin!t0 gn(t� t0)dt0 (13)

in whichCn(u) is (compare with (9)):

Cn(u) = cosn�up

L+

1

2: (14)

Next, we linearize the time dependent sine and cosine functions for�u � L, and split the integral:

jleft(t) = �jp

D

1

n=1

2

n�Sn(uc)

t

0

gn(t� t0)dt0

�jp

D

1

m=1

2

n�Cn(uc) � n�

�u

L

t

0

gn(t� t0) sin!t0dt0: (15)

The first term of (15) is the intensity step response (see, e.g., W.P. Connors [4]). Before evaluating the second term we take a closerlook at the intensity impulse response (10). Concerning the sum overn, we can divide the intensity impulse response in two domains (seeFig. 2(a) and (b): domainA: t 2 [0; "] and domainB: t 2 [";1], seealso [5] and [6]). What we find is that on domainA, the summationovern has to be carried out up ton!1 to obtain the correct value.For domainB (t > "), the summation converges for a low summationlimit we call N .

The summation limitN and the deadtime" depend onR andC aswell as on the positionuc. The closer the light source moves to thecontact, the shorter" becomes, and the higher the required summationlimit N is. However, the" andN that are required foru values closeto a contact, are also sufficient for all points further away. In short,what we are going to do, is take the" andN corresponding toup =�0:4L (i.e., close to the left contact) and use this for all positionsup 2

[�0:4L; 0:4L]. For the PSD from Fig. 2, we find forup = �0:4L(from numerical computations)N = 40 and" = 5 ns.

Next, we compute the second integral of (15) by integrating up tot � ":

t�"

0

gn(t� t0) sin!t0dt0 =e�� "gn(t� ") � sin!(t� ")

�e�� "

gn(t) � sin!t: (16)

In the last step we used the fact that"� !�1. Next, we rewrite (15) to

jleft(t) = jstep

left (uc; t) + jsineleft (uc; t) (17)

wherejstepleft (uc; t) is the intensity step response foru = uc and it de-scribes the switching on of the light source att = 0. In (17),jsineleft (uc; t)describes the PSD response to the sinusoidal movement of the lightsource [using (16) and taking the sum up toN ]:

jsineleft (uc; t) = 2

jp

D

�u

L

N

n=1

Cn(uc) � e�� "

gn(t) � sin!t: (18)

From (18) we learn thatjsineleft can be described as a sum of linearfilters,gn(t), operating on the sinusoidal movement of the light source,�u sin!t.

V. PSD TRANSFERFUNCTION

In this section we investigate what the impact of the PSD responseto a moving light source is to the position we determine using thePSD equation, (1). We define~uPSD as the position we determine if wesimply insertiright(t) andileft(t) in the PSD equation (see the modelof Fig. 1):

~uPSD(t) =L

2

iright(t)� ileft(t)

iright(t) + ileft(t): (19)

Equation (19) is adjusted for the new coordinate system. We regardthe situationt! 1, wherejstep has reached its final value:

limt!1

jstepright(t) + j

stepleft (t) =

jp

D

2ucL

limt!1

jstepright(t)� j

stepleft (t) =

jp

D: (20)

In the forthcoming step, we make use of the distributive property ofthe linear filtersgn(t), together with the following relations:

Cn(uc) +Cn(�uc) =2 cosn�uc

L� cos

1

2n�

Cn(uc)�Cn(�uc) =� 2 sinn�uc

L� sin

1

2n�: (21)

Next, we rewrite the sum of the “sine parts”

limt!1

jsineright(t) + j

sineleft (t) =

jp

D

4�u

Lgcos(uc; t) � sin!t (22)

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 49, NO. 1, JANUARY 2002 205

Fig. 3. Performance of the PTF (for a PSD withL = 20 mm,R = 700 k andC = 28 pF): computation of~u for a sudden change in position att = 0 (u(t) = u + 0:2 � s(t)).

and the difference

limt!1

jsineright(t)� jsineleft (t) = �jpD

4�u

Lgsin(uc; t) � sin!t (23)

with:

gcos(uc; t) =

N

n=1

cosn�ucL� cos

1

2n� � e�� "gn(t)

gsin(uc; t) =

N

n=1

sinn�ucL� sin

1

2n� � e�� "gn(t): (24)

If we apply this analysis to (19), we find for~uPSD(t) (remember thatiright / jright andileft / �jleft):

~uPSD(t) =L

2

2uL�

4�uLgcos(uc; t) � sin!t

1� 4�uLgsin(uc; t) � sin!t

: (25)

Since we assumed that�u � L, we can linearize the denominator(omitting the termO(�u=L)2):

~uPSD(t) � uc � 2gcos(t) ��u sin!t+ 4ucLgsin(t)

��u sin!t = gPSD(uc; t) � up(t): (26)

In (26),gPSD(t) is the PSD Transfer Function:

gPSD(uc; t) = �2gcos(uc; t) + 4ucLgsin(uc; t): (27)

VI. RESULTS

In this section we compare the results of the PTF to Pstar simula-tions ([8]). In the Pstar simulations a fine grain RC transmission line ismodeled. Light source movement is modeled by changing the positionof a current source along the RC transmission line.

Fig. 3 shows the results for~uPSD(t) for both the PTF [computedusing (26)] as well as the Pstar simulations at three different positionon the PSD. It can be seen that the maximum deviation is around 3�m.

VII. CONCLUSIONS

In this brief, we show that the response of a PSD (1) to changes inposition of the light source can be described as a linear filter operatingon the “real” movement of the light source [see (26)]. This filter isintroduced as the PSD transfer function (PTF) [gPSD(u; t) in (27)],and depends on theu-position along the PSD. The assumptions madein the analysis are that the total light intensity remains constant, andthat the change in position is small with respect to the PSD lengthL.PSD response computations for a sudden change in position show thatthe PTF analysis is in agreement with Pstar simulations.

206 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 49, NO. 1, JANUARY 2002

REFERENCES

[1] J. T. Wallmark, “A new semiconductor photocell using lateral photoef-fect ,” Proc. IRE, pp. 474–483, 1957.

[2] G. Lucovsky, “Photoeffects in nonuniformly irradiatedp–n junctions,”J. Appl. Phys., pp. 1088–1095, 1960.

[3] D. J. W. Noorlag, “Lateral-photoeffect position sensitive detectors,”Ph.D. thesis, Delft Univ. Technol., Delft, The Netherlands, 1982.

[4] W. P. Connors, “Lateral photodetector operating in the fully reverse-biased mode,”IEEE Trans. Electron Devices, vol. ED-19, pp. 591–596,1971.

[5] C. A. Klein and R. W. Bierig, “Pulse-response characteristics ofposition-sensitive photodetectors,”IEEE Trans. Electron Devices, vol.ED-22, pp. 532–537, 1974.

[6] C. Narayananet al., “Position dependance of the transient response ofa position-sensitive detector under periodic pulsed light modulation,”IEEE Trans. Electron Devices, vol. 41, pp. 1688–1694, 1993.

[7] R. V. Churchill and J. W. Brown,Fourier Series and Boundary ValueProblems, 4th ed. New York: McGraw-Hill, 1987.

[8] “Philips EDT/Analogue Simulation,” inPstar User Guide for Pstar3.0. Eindhoven, The Netherlands: Philips Electronics, 1994.