202

Fig. 4. f

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

–I

characteristics of the lateral NPN and PNP devices.

The base resistance of this device is higher as compared to the vertical bipolar transistor as well as the lateral BJT reported earlier [7]. The higher base resistance is because of the lateral base contact as well as no silicide used in the process. But the lateral BJT reported earlier has top base contact, and silicide has been used to minimize the base resistance [7]. The values of other basic parameters such as Re, Rc, and Va of the lateral NPN are 290 , 9.7 k , and 13 V, respectively. The values of the junction capacitances Cje , Cjc , and Cjs of the lateral NPN are 0.29 fF, 0.28 fF, and 2.9 fF, respectively. A noise figure of 2.3 dB at 2 GHz with VCE = 2 V was obtained for the lateral NPN. The high frequency performance of the lateral bipolar devices also depends on the collector width. The collector width of the lateral bipolar devices is approximately 1 m. The high frequency performance can be further improved by reducing the base and collector widths, and also by using silicide to further reduce the base resistance. The further improvement in device performance was predicted using 2-D device simulator MEDICI [9]. Simulation results show that by narrowing the base width to 0.08 m and collector width to 0.5 m, the lateral NPN can achieve a fmax of 47 GHz. CMOS transistors with a channel length of 0.5 m and a channel width of 5 m were characterized. The threshold voltages of the NMOS and PMOS devices are +0.8 V and 00:80.8 V, respectively. IV. CONCLUSION In summary, simple and high performance TFSOI lateral complementary BJT structures were implemented in a TFSOI CMOS process with minimized base and external base linkage regions (for minimized overall base resistance). The experimental results showed that this TFSOI C-BiCMOS technology is very promising for high-level integration of RF mixed-signal circuits for wireless communication applications.

[2] H. Ammo, H. Ejiri, S. Kanematsu, H. Kikuchi, M. Yano, and H. Miwa, “A complementary BiCMOS technology for low power wireless telecommunication applications,” in Proc. ESSDERC, 1999, pp. 444–447. [3] S. Parke, F. Assaderaghi, J. Chen, J. King, C. Hu, and P. K. Ko, “A versatile, SOI BiCMOS technology with complementary lateral BJT’s,” in IEEE IEDM Tech. Dig., 1992, pp. 453–456. [4] G. G. Shahidi et 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 for low power applications,” in IEEE IEDM Tech. Dig., 1993, pp. 449–452. [6] T. Shino, K. Inoh, T. Yamada, H. Nii, S. Kawanaka, T. Fuse, M. Yoshimi, lateral Y. Katsumata, S. Watanable, and J. matsunaga, “A 31 GHz f BJT on SOI self-aligned external base formation technology,” in IEEE IEDM Tech. Dig., 1998, pp. 953–956. [7] H. Nii, T. Yamada, K. Inoh, T. Shino, S. Kawanaka, M. Yoshimi, and on Y. Katsumata, “A novel lateral bipolar transistor with 67 GHz f thin-film SOI for RF analog applications,” IEEE Trans. Electron Devices, 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, and Ian T. Young Abstract—This article describes the dynamic behavior of position sensitive detectors. Earlier work has described the PSD response to changes in light intensity. Here, we investigate the PSD response to a change of light spot position. We call the linear filter describing this response the PSD transfer 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 to a nonuniform illumination over the PSD surface two lateral photocurrents are generated. The ratio of the photocurrents is a measure for the center of gravity of the intensity distribution of the illumination. This effect is known as the lateral photo effect, and J. T. Wallmark was one of the first researchers to recognize the possiblities [1]. G. Lucovsky described the theory behind the the lateral photo effect [2]. A model of a PSD is shown in Fig. 1. If the PSD layer is perfectly homogeneous, the position of light incidence is [3]:

up = L

iright : iright + ileft

(1)

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

ACKNOWLEDGMENT The authors would like to thank the fabrication staffs of the Microfabrication Facility at the HKUST for their help in the processing and constant support. REFERENCES [1] R. Reedy, J. Cable, and D. Kelly, “Single chip wireless systems using SOI,” in IEEE Int. SOI Conf., 1999, pp. 8–11.

Manuscript received May 25, 2001; revised August 21, 2001. This work was been supported by the Dutch Technology Foundation STW. The review of this brief was arranged bny Editor K. Najafi. M. de Bakker, P. W. Verbeek and I. T. Young are with the Pattern Recognition Group, Faculty of Applied Sciences, Delft University of Technology, NL-2628 CJ Delft, The Netherlands (e-mail [email protected]). G. K. Steenvoorden is with the TNO Institute of Applied Physics, 2600 AD Delft, 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

the intensity impulse response. Next, we show how we model a moving light 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

0 RC t = 0jpn L2 D

(2)

where represents the potential difference between the PSD layer and the substrate1 , R and C are the PSD resistance and capacitance, L the PSD length, and jpn the current density entering the PSD layer at w = 0. In the general situation and jpn depend on t as well as on u: (u; t) and jpn (u; t). III. INTENSITY IMPULSE RESPONSE In this section we compute the response of a PSD to an impulse in intensity. The starting point is the Lucovsky equation, (2), with boundary conditions ju=0;L = VR (an externally applied reverse bias voltage), and t > 0. If we write 0 = 0VR , and substitute jp 1 (u0up ) (t0t0 ) for jpn , we can write for (2):

0

uu

0 0 RC t = 0 jp 1 (t 0 t ) (u 0 up ) L D

(3)

0

2

with boundary conditions 0 ju=0;L = 0. The solution of (3) can be found by performing a finite Fourier sine transform (FFST) with respect to u and a Laplace Transform with respect to t. The theory on the FFST can be found in [7]. The result of this double transform we call 8~ (n; s):

0

n L

2

+

RC s L2

1 8~ (n; s) = 0

jp D

1 sin

nup L

1 e0st

:

(4)

~ (n; s)=jp and rearrangement of terms leads to Solving (4) for 8 8~ (n; s) = jp

L2 RC

D

sin nu e0st L (n ) RC + s

1

=

1 2 n=1

sin(

Sn (up ) = sin n up L

nup nu ( )2 sin L sin L 1 gn (t 0 t0 )

L D n

pulse jright (up ; t) = +

(6)

+ 21

)

:

(9)

In the new coordinate system we can use symmetry properties of the PSD to compute the photocurrent to the right PSD contact, by replacing up by 0up . The impulse responses for the left and right PSD signals are in the new coordinate system:

(5)

1

jp 2 S (u )g (t 0 t0 ); D n=1 n n p n

1

jp 2 S (0up )gn (t 0 t0 ): D n=1 n n

(10)

Equation (10) also implies that the closer the light source originates to a contact, the faster the response is. IV. MOVING LIGHTSOURCE

with

( ) = 2n e0 t s(t);

2

gn t

n

) = (n RC

2

:

(7)

2

We introduce n rather than n in order to emphasize that the value is always positive, and proportional to n2 . In (7), s(t) is the unit step function. The (lateral) current density at the left contact (u = 0) is (note that u0 = u ): pulse

jleft

(up ; t) = 0 u (0 ; t) =0

jp D

1 2 sin nup gn (t 0 t0 ):

n=1 n

following notation is applied:

In this section we arrive at the goal of this article, i.e., the PSD response to a moving light source. We introduce up t , the position of a light spot, and we assume a constant intensity for t > . The current density to the left contact can then be described by the convolution of pulse jleft up t t with the unit step function

()

L

[email protected]

[email protected] and

= @[email protected]

0

( ()) pulse jleft (t) =jleft (up (t); t) 3 s(t) 1 t jp = 0 D n2 Sn (up (t0 )) gn (t 0 t0 )dt0 : n=1

0

(11) (8)

pulse The minus sign in jleft indicates that the current at the left contact flows in the negative u direction. In the remainder of this article we use

1The

a new coordinate system for the PSD, with the origin in the center of the PSD. This is achieved by replacing nup =L in (8) by

pulse jleft (up; t) = 0

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

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

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

up (t) = uc + 1u 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 with R

j =(D 1 RC ).

1

where uc is the central position of the oscillations, u is the amplitude and ! the angular frequency. Rewriting (11) [using (12)] leads to

j jleft (t) = 0 p D

1

t

1 2 S (u )cos n n c

0 n=1

+Cn (uc )sin in which

n

1u sin !t0 n gn (t 0 t0 )dt0 (13)

L

Cn(u) is (compare with (9)): Cn(u) = cos n uLp + 21 1 2 S (u ) n n c

j jleft (t) = 0 p D n=1

1

t 0

:

(14)

gn (t 0 t0 )dt0

2 Cn (uc ) 1 n 1u 0 jDp n L m=1

gn (t 0 t0 )sin !t0 dt0 : (15)

0

[0 ]

0

(

[

= 40

)sin

=

e0

=

step where jleft uc ; t is the intensity step response for u uc and it desine scribes the switching on of the light source at t . In (17), jleft uc ; t describes the PSD response to the sinusoidal movement of the light source [using (16) and taking the sum up to N ]:

=0

sine jleft (uc ; t) = 2 jDp 1Lu Cn (uc ) 1 e0 " gn (t) 3 sin !t: n=1

(

)

(18)

sine From (18) we learn that jleft can be described as a sum of linear filters, gn t , operating on the sinusoidal movement of the light source, u !t.

1 sin

()

=5

= = 04

( ) sin ( " g (t) 3 sin !t: n

In this section we investigate what the impact of the PSD response to a moving light source is to the position we determine using the PSD equation, (1). We define uPSD as the position we determine if we simply insert iright t and ileft t in the PSD equation (see the model of Fig. 1):

)

(16)

~ ()

()

u~PSD (t) =

]

)

04 [ 04 04 ]

)

(17)

V. PSD TRANSFER FUNCTION

t

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 closer look at the intensity impulse response (10). Concerning the sum over n, we can divide the intensity impulse response in two domains (see Fig. 2(a) and (b): domain A: t 2 ; " and domain B: t 2 "; 1 , see also [5] and [6]). What we find is that on domain A, the summation over n has to be carried out up to n ! 1 to obtain the correct value. For domain B t > " , the summation converges for a low summation limit we call N . The summation limit N and the deadtime " depend on R and C as well as on the position uc . The closer the light source moves to the contact, the shorter " becomes, and the higher the required summation limit N is. However, the " and N that are required for u values close to a contact, are also sufficient for all points further away. In short, what we are going to do, is take the " and N corresponding to up 0 : L (i.e., close to the left contact) and use this for all positions up 2 0 : L; : L . For the PSD from Fig. 2, we find for up 0 : L (from numerical computations) N and " ns. Next, we compute the second integral of (15) by integrating up to t 0 ": t0" gn t 0 t0 !t0 dt0 e0 " gn t 0 " 3 ! t 0 "

(

step sine jleft (t) = jleft (uc ; t) + jleft (uc ; t)

N

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

1

In the last step we used the fact that " ! 01 . Next, we rewrite (15) to

(

L

1u sin !t0

= 700 k and C = 28 pF. The current densities in the graphs are normalized for

L iright (t) 0 ileft (t) 2 iright(t) + ileft (t) :

(19)

Equation (19) is adjusted for the new coordinate system. We regard the situation t ! 1, where jstep has reached its final value:

lim !1 lim t!1 t

step (t) + j step (t) jright left step (t) 0 j step (t) jright left

= jDp 2Luc = jDp :

(20)

In the forthcoming step, we make use of the distributive property of the linear filters gn t , together with the following relations:

()

Cn (uc ) + Cn(0uc ) =2cos n uLc 1 cos 12 n Cn (uc ) 0 Cn(0uc ) = 0 2sin n uLc 1 sin 21 n:

(21)

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

lim !1

t

sine (t) + j sine (t) jright left

= jDp 41Lu gcos (uc; t) 3 sin !t

(22)

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

Fig. 3. Performance of the PTF (for a PSD with t = 0 (u(t) = u + 0:2 1 s(t)).

L

= 20 mm,

R

= 700 k and

and the difference

lim

t!1

205

C

= 28 pF): computation of ~

In (26),

= 0 jDp 41Lu gsin (uc ; t) 3 sin !t

sine (t) 0 jleft sine (t) jright

(23)

with:

u

for a sudden change in position at

gPSD (t) is the PSD Transfer Function: gPSD (uc; t) = 02gcos (uc; t) + 4 uLc gsin (uc ; t):

(27)

VI. RESULTS

gcos(uc; t) = gsin (uc ; t) =

N n=1 N n=1

cos n uLc 1 cos 21 n 1 e0 sin n uLc 1 sin 21 n 1 e0

"g n "g n

(t)

(t):

(24)

~ ()

If we apply this analysis to (19), we find for uPSD t (remember that / jright and ileft / 0jleft ):

iright

u~PSD (t) =

L 2Lu 0 41Lu gcos (uc ; t) 3 sin !t 2 1 0 41u gsin (uc ; t) 3 sin !t : L

1

(25)

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

O(1

)

u u~PSD (t) uc 0 2gcos (t) 3 1u sin !t + 4 c gsin (t) L 31u sin !t = gPSD (uc ; t) 3 up (t): (26)

In this section we compare the results of the PTF to Pstar simulations ([8]). In the Pstar simulations a fine grain RC transmission line is modeled. Light source movement is modeled by changing the position of a current source along the RC transmission line. Fig. 3 shows the results for u ~PSD (t) for both the PTF [computed using (26)] as well as the Pstar simulations at three different position on 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 in position of the light source can be described as a linear filter operating on the “real” movement of the light source [see (26)]. This filter is introduced as the PSD transfer function (PTF) [gPSD (u; t) in (27)], and depends on the u-position along the PSD. The assumptions made in the analysis are that the total light intensity remains constant, and that the change in position is small with respect to the PSD length L. PSD response computations for a sudden change in position show that the 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 photoeffect ,” Proc. IRE, pp. 474–483, 1957. [2] G. Lucovsky, “Photoeffects in nonuniformly irradiated p–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 reversebiased mode,” IEEE Trans. Electron Devices, vol. ED-19, pp. 591–596, 1971.

[5] C. A. Klein and R. W. Bierig, “Pulse-response characteristics of position-sensitive photodetectors,” IEEE Trans. Electron Devices, vol. ED-22, pp. 532–537, 1974. [6] C. Narayanan et al., “Position dependance of the transient response of a 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 Value Problems, 4th ed. New York: McGraw-Hill, 1987. [8] “Philips EDT/Analogue Simulation,” in Pstar User Guide for Pstar 3.0. Eindhoven, The Netherlands: Philips Electronics, 1994.

Fig. 4. f

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

–I

characteristics of the lateral NPN and PNP devices.

The base resistance of this device is higher as compared to the vertical bipolar transistor as well as the lateral BJT reported earlier [7]. The higher base resistance is because of the lateral base contact as well as no silicide used in the process. But the lateral BJT reported earlier has top base contact, and silicide has been used to minimize the base resistance [7]. The values of other basic parameters such as Re, Rc, and Va of the lateral NPN are 290 , 9.7 k , and 13 V, respectively. The values of the junction capacitances Cje , Cjc , and Cjs of the lateral NPN are 0.29 fF, 0.28 fF, and 2.9 fF, respectively. A noise figure of 2.3 dB at 2 GHz with VCE = 2 V was obtained for the lateral NPN. The high frequency performance of the lateral bipolar devices also depends on the collector width. The collector width of the lateral bipolar devices is approximately 1 m. The high frequency performance can be further improved by reducing the base and collector widths, and also by using silicide to further reduce the base resistance. The further improvement in device performance was predicted using 2-D device simulator MEDICI [9]. Simulation results show that by narrowing the base width to 0.08 m and collector width to 0.5 m, the lateral NPN can achieve a fmax of 47 GHz. CMOS transistors with a channel length of 0.5 m and a channel width of 5 m were characterized. The threshold voltages of the NMOS and PMOS devices are +0.8 V and 00:80.8 V, respectively. IV. CONCLUSION In summary, simple and high performance TFSOI lateral complementary BJT structures were implemented in a TFSOI CMOS process with minimized base and external base linkage regions (for minimized overall base resistance). The experimental results showed that this TFSOI C-BiCMOS technology is very promising for high-level integration of RF mixed-signal circuits for wireless communication applications.

[2] H. Ammo, H. Ejiri, S. Kanematsu, H. Kikuchi, M. Yano, and H. Miwa, “A complementary BiCMOS technology for low power wireless telecommunication applications,” in Proc. ESSDERC, 1999, pp. 444–447. [3] S. Parke, F. Assaderaghi, J. Chen, J. King, C. Hu, and P. K. Ko, “A versatile, SOI BiCMOS technology with complementary lateral BJT’s,” in IEEE IEDM Tech. Dig., 1992, pp. 453–456. [4] G. G. Shahidi et 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 for low power applications,” in IEEE IEDM Tech. Dig., 1993, pp. 449–452. [6] T. Shino, K. Inoh, T. Yamada, H. Nii, S. Kawanaka, T. Fuse, M. Yoshimi, lateral Y. Katsumata, S. Watanable, and J. matsunaga, “A 31 GHz f BJT on SOI self-aligned external base formation technology,” in IEEE IEDM Tech. Dig., 1998, pp. 953–956. [7] H. Nii, T. Yamada, K. Inoh, T. Shino, S. Kawanaka, M. Yoshimi, and on Y. Katsumata, “A novel lateral bipolar transistor with 67 GHz f thin-film SOI for RF analog applications,” IEEE Trans. Electron Devices, 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, and Ian T. Young Abstract—This article describes the dynamic behavior of position sensitive detectors. Earlier work has described the PSD response to changes in light intensity. Here, we investigate the PSD response to a change of light spot position. We call the linear filter describing this response the PSD transfer 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 to a nonuniform illumination over the PSD surface two lateral photocurrents are generated. The ratio of the photocurrents is a measure for the center of gravity of the intensity distribution of the illumination. This effect is known as the lateral photo effect, and J. T. Wallmark was one of the first researchers to recognize the possiblities [1]. G. Lucovsky described the theory behind the the lateral photo effect [2]. A model of a PSD is shown in Fig. 1. If the PSD layer is perfectly homogeneous, the position of light incidence is [3]:

up = L

iright : iright + ileft

(1)

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

ACKNOWLEDGMENT The authors would like to thank the fabrication staffs of the Microfabrication Facility at the HKUST for their help in the processing and constant support. REFERENCES [1] R. Reedy, J. Cable, and D. Kelly, “Single chip wireless systems using SOI,” in IEEE Int. SOI Conf., 1999, pp. 8–11.

Manuscript received May 25, 2001; revised August 21, 2001. This work was been supported by the Dutch Technology Foundation STW. The review of this brief was arranged bny Editor K. Najafi. M. de Bakker, P. W. Verbeek and I. T. Young are with the Pattern Recognition Group, Faculty of Applied Sciences, Delft University of Technology, NL-2628 CJ Delft, The Netherlands (e-mail [email protected]). G. K. Steenvoorden is with the TNO Institute of Applied Physics, 2600 AD Delft, 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

the intensity impulse response. Next, we show how we model a moving light 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

0 RC t = 0jpn L2 D

(2)

where represents the potential difference between the PSD layer and the substrate1 , R and C are the PSD resistance and capacitance, L the PSD length, and jpn the current density entering the PSD layer at w = 0. In the general situation and jpn depend on t as well as on u: (u; t) and jpn (u; t). III. INTENSITY IMPULSE RESPONSE In this section we compute the response of a PSD to an impulse in intensity. The starting point is the Lucovsky equation, (2), with boundary conditions ju=0;L = VR (an externally applied reverse bias voltage), and t > 0. If we write 0 = 0VR , and substitute jp 1 (u0up ) (t0t0 ) for jpn , we can write for (2):

0

uu

0 0 RC t = 0 jp 1 (t 0 t ) (u 0 up ) L D

(3)

0

2

with boundary conditions 0 ju=0;L = 0. The solution of (3) can be found by performing a finite Fourier sine transform (FFST) with respect to u and a Laplace Transform with respect to t. The theory on the FFST can be found in [7]. The result of this double transform we call 8~ (n; s):

0

n L

2

+

RC s L2

1 8~ (n; s) = 0

jp D

1 sin

nup L

1 e0st

:

(4)

~ (n; s)=jp and rearrangement of terms leads to Solving (4) for 8 8~ (n; s) = jp

L2 RC

D

sin nu e0st L (n ) RC + s

1

=

1 2 n=1

sin(

Sn (up ) = sin n up L

nup nu ( )2 sin L sin L 1 gn (t 0 t0 )

L D n

pulse jright (up ; t) = +

(6)

+ 21

)

:

(9)

In the new coordinate system we can use symmetry properties of the PSD to compute the photocurrent to the right PSD contact, by replacing up by 0up . The impulse responses for the left and right PSD signals are in the new coordinate system:

(5)

1

jp 2 S (u )g (t 0 t0 ); D n=1 n n p n

1

jp 2 S (0up )gn (t 0 t0 ): D n=1 n n

(10)

Equation (10) also implies that the closer the light source originates to a contact, the faster the response is. IV. MOVING LIGHTSOURCE

with

( ) = 2n e0 t s(t);

2

gn t

n

) = (n RC

2

:

(7)

2

We introduce n rather than n in order to emphasize that the value is always positive, and proportional to n2 . In (7), s(t) is the unit step function. The (lateral) current density at the left contact (u = 0) is (note that u0 = u ): pulse

jleft

(up ; t) = 0 u (0 ; t) =0

jp D

1 2 sin nup gn (t 0 t0 ):

n=1 n

following notation is applied:

In this section we arrive at the goal of this article, i.e., the PSD response to a moving light source. We introduce up t , the position of a light spot, and we assume a constant intensity for t > . The current density to the left contact can then be described by the convolution of pulse jleft up t t with the unit step function

()

L

[email protected]

[email protected] and

= @[email protected]

0

( ()) pulse jleft (t) =jleft (up (t); t) 3 s(t) 1 t jp = 0 D n2 Sn (up (t0 )) gn (t 0 t0 )dt0 : n=1

0

(11) (8)

pulse The minus sign in jleft indicates that the current at the left contact flows in the negative u direction. In the remainder of this article we use

1The

a new coordinate system for the PSD, with the origin in the center of the PSD. This is achieved by replacing nup =L in (8) by

pulse jleft (up; t) = 0

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

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

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

up (t) = uc + 1u 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 with R

j =(D 1 RC ).

1

where uc is the central position of the oscillations, u is the amplitude and ! the angular frequency. Rewriting (11) [using (12)] leads to

j jleft (t) = 0 p D

1

t

1 2 S (u )cos n n c

0 n=1

+Cn (uc )sin in which

n

1u sin !t0 n gn (t 0 t0 )dt0 (13)

L

Cn(u) is (compare with (9)): Cn(u) = cos n uLp + 21 1 2 S (u ) n n c

j jleft (t) = 0 p D n=1

1

t 0

:

(14)

gn (t 0 t0 )dt0

2 Cn (uc ) 1 n 1u 0 jDp n L m=1

gn (t 0 t0 )sin !t0 dt0 : (15)

0

[0 ]

0

(

[

= 40

)sin

=

e0

=

step where jleft uc ; t is the intensity step response for u uc and it desine scribes the switching on of the light source at t . In (17), jleft uc ; t describes the PSD response to the sinusoidal movement of the light source [using (16) and taking the sum up to N ]:

=0

sine jleft (uc ; t) = 2 jDp 1Lu Cn (uc ) 1 e0 " gn (t) 3 sin !t: n=1

(

)

(18)

sine From (18) we learn that jleft can be described as a sum of linear filters, gn t , operating on the sinusoidal movement of the light source, u !t.

1 sin

()

=5

= = 04

( ) sin ( " g (t) 3 sin !t: n

In this section we investigate what the impact of the PSD response to a moving light source is to the position we determine using the PSD equation, (1). We define uPSD as the position we determine if we simply insert iright t and ileft t in the PSD equation (see the model of Fig. 1):

)

(16)

~ ()

()

u~PSD (t) =

]

)

04 [ 04 04 ]

)

(17)

V. PSD TRANSFER FUNCTION

t

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 closer look at the intensity impulse response (10). Concerning the sum over n, we can divide the intensity impulse response in two domains (see Fig. 2(a) and (b): domain A: t 2 ; " and domain B: t 2 "; 1 , see also [5] and [6]). What we find is that on domain A, the summation over n has to be carried out up to n ! 1 to obtain the correct value. For domain B t > " , the summation converges for a low summation limit we call N . The summation limit N and the deadtime " depend on R and C as well as on the position uc . The closer the light source moves to the contact, the shorter " becomes, and the higher the required summation limit N is. However, the " and N that are required for u values close to a contact, are also sufficient for all points further away. In short, what we are going to do, is take the " and N corresponding to up 0 : L (i.e., close to the left contact) and use this for all positions up 2 0 : L; : L . For the PSD from Fig. 2, we find for up 0 : L (from numerical computations) N and " ns. Next, we compute the second integral of (15) by integrating up to t 0 ": t0" gn t 0 t0 !t0 dt0 e0 " gn t 0 " 3 ! t 0 "

(

step sine jleft (t) = jleft (uc ; t) + jleft (uc ; t)

N

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

1

In the last step we used the fact that " ! 01 . Next, we rewrite (15) to

(

L

1u sin !t0

= 700 k and C = 28 pF. The current densities in the graphs are normalized for

L iright (t) 0 ileft (t) 2 iright(t) + ileft (t) :

(19)

Equation (19) is adjusted for the new coordinate system. We regard the situation t ! 1, where jstep has reached its final value:

lim !1 lim t!1 t

step (t) + j step (t) jright left step (t) 0 j step (t) jright left

= jDp 2Luc = jDp :

(20)

In the forthcoming step, we make use of the distributive property of the linear filters gn t , together with the following relations:

()

Cn (uc ) + Cn(0uc ) =2cos n uLc 1 cos 12 n Cn (uc ) 0 Cn(0uc ) = 0 2sin n uLc 1 sin 21 n:

(21)

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

lim !1

t

sine (t) + j sine (t) jright left

= jDp 41Lu gcos (uc; t) 3 sin !t

(22)

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

Fig. 3. Performance of the PTF (for a PSD with t = 0 (u(t) = u + 0:2 1 s(t)).

L

= 20 mm,

R

= 700 k and

and the difference

lim

t!1

205

C

= 28 pF): computation of ~

In (26),

= 0 jDp 41Lu gsin (uc ; t) 3 sin !t

sine (t) 0 jleft sine (t) jright

(23)

with:

u

for a sudden change in position at

gPSD (t) is the PSD Transfer Function: gPSD (uc; t) = 02gcos (uc; t) + 4 uLc gsin (uc ; t):

(27)

VI. RESULTS

gcos(uc; t) = gsin (uc ; t) =

N n=1 N n=1

cos n uLc 1 cos 21 n 1 e0 sin n uLc 1 sin 21 n 1 e0

"g n "g n

(t)

(t):

(24)

~ ()

If we apply this analysis to (19), we find for uPSD t (remember that / jright and ileft / 0jleft ):

iright

u~PSD (t) =

L 2Lu 0 41Lu gcos (uc ; t) 3 sin !t 2 1 0 41u gsin (uc ; t) 3 sin !t : L

1

(25)

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

O(1

)

u u~PSD (t) uc 0 2gcos (t) 3 1u sin !t + 4 c gsin (t) L 31u sin !t = gPSD (uc ; t) 3 up (t): (26)

In this section we compare the results of the PTF to Pstar simulations ([8]). In the Pstar simulations a fine grain RC transmission line is modeled. Light source movement is modeled by changing the position of a current source along the RC transmission line. Fig. 3 shows the results for u ~PSD (t) for both the PTF [computed using (26)] as well as the Pstar simulations at three different position on 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 in position of the light source can be described as a linear filter operating on the “real” movement of the light source [see (26)]. This filter is introduced as the PSD transfer function (PTF) [gPSD (u; t) in (27)], and depends on the u-position along the PSD. The assumptions made in the analysis are that the total light intensity remains constant, and that the change in position is small with respect to the PSD length L. PSD response computations for a sudden change in position show that the 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 photoeffect ,” Proc. IRE, pp. 474–483, 1957. [2] G. Lucovsky, “Photoeffects in nonuniformly irradiated p–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 reversebiased mode,” IEEE Trans. Electron Devices, vol. ED-19, pp. 591–596, 1971.

[5] C. A. Klein and R. W. Bierig, “Pulse-response characteristics of position-sensitive photodetectors,” IEEE Trans. Electron Devices, vol. ED-22, pp. 532–537, 1974. [6] C. Narayanan et al., “Position dependance of the transient response of a 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 Value Problems, 4th ed. New York: McGraw-Hill, 1987. [8] “Philips EDT/Analogue Simulation,” in Pstar User Guide for Pstar 3.0. Eindhoven, The Netherlands: Philips Electronics, 1994.