Nuclear Instruments and Methods in Physics Re%carch A311 (1992) 105-112North-Holland

Timing response of silicon position-sensitive detectorsS.J . Bennett, J.B.A. England, M. Freer, B.R. Fulton and J.T . MurgatroydSchool of Physics and Space Research, Oriversii, o/' Birmingham, P.O. Box 36.3, Binningham B15 27T. UK

Received 15 July 1991

A simple RC-delay line model of a silicon position-sensitive detector has been extended to cover the case when one end of theresistive layer has been terminated . The timing predictions of the model have been compared with heavy-ion experimental data andgood agreement is found. The insight provided by the model has been used to improve the measured timing resolution obtainedusing thin ( - 35 Rm) passing detectors by typically a factor of five .

1. Introduction

Since their invention in the early 1960s [1-21, theapplication of one-dimensional position sensitive de-tectors (PSD) to experimental nuclear physics has beenextremely wide spread . However, little theoretical workhas been done to understand the physical processeswhich lead to the development of position x energyand energy signals from such detectors. Kalbitzer and

0

Rt

X

X

Fig. 1 . Equivalent circuit for a one-dimensional position-sensitive surface barrier detector terminated at one contact (PO. A particle

incident at X = x is represented as an instantaneous current source discharging a corresponding local capacitor . The energy and

position signals are obtained via charge sensitive preamplifiers on the contacts E and P i respectively . The other symbols aredescribed in the text .

0168-9002/92/$05 .0() (D 1992 - Elsevier Science Publishers B.V . All rights reserved

Melzer [31 have applied a simple RC-delay line modelto simulate a one-dimensional PSD which has beenearthed at both ends of the resistive contact. In prac-tice, PSDs must be correctly terminated at at least oneend of the resistive layer to prevent reflections occur-ing along the delay line which can lead to a consider-able deterioration of the position [41, energy and timeresolutions . For thin detectors ( - 30 lLm), of the typeoften used for AE passing detectors in heavy-ion telc-

Energy Signal

Ci

VL

Dnsir_f n Çi~nai

N R. 111 11 - 'sawlIN pMYRES~EARCM

Se-ctow, A

106

scopes, satisfactory impedance matching between thetermination resistor and the delay-line means that thisresistor is often comparable in size to the total resis-tance of the position contact of the detector itself. Thiscan introduce a marked asymmetry in the timing rc-sponse of a PSD with the position of the incidentparticle, and it becomes necessary to include theseeffects if realistic comparisons are to be made withpractical detectors .

It is the purpose of this article to extend the RC-de-lay line model originally developed by Kalbitzer andMelzer to include the effects of a termination resistorand to compare the results of this simple model withthose obtained in a typical heavy-ion experiment. Theunderstanding obtained from this work has allowedcorrection of position and energy dependent timewalkof a detector's timing signals, thus improving the meas-ured time resolution of the PSD.

2. Theory of operation

The PSD will be discussed in the following calcula-tions on the basis of the simplified equivalent circuitshown in fig. 1 . In this circuit the PSD is described as ahomogeneous RC-delay line which has a resistive con-tact of resistance R, and a continuously distributedjunction capacitance C. Whilst this simple model ne-glects such complications as non-ohmic contacts inpartially depleted detectors, it should nevertheless pro-vide a good description of well depleted devices, suchas A E passing detectors, which have a thin, highlydoped front layer formed by evaporation, diffusion orion-implantation techniques. To further simplify thecalculations, it is assumed that the PSD can be repre-sented as a one-dimensional RC-line . In reality, detec-tors often have appreciable width as well as length, butprovided they are operated in a one-dimensional mode,it is only the total current flow in the longitudinaldirection which is important . Under these conditions,Kalbitzer and Stumpfi [5] have shown that the two-di-mensional solution for a PSD grounded at both endsreduces to the one-dimensional solution obtained ear-lier [3].

Lastly, certain assumptions are required about thenature of the charge collection process itself. Althoughthe size of the plasma column formed by the initialionizing event of, say, a heavy-ion of several tens ofMeV is small, the subsequent movement of the carriersin their own, and eventually the externally appliedelectric field will cause a certain spread in their spatialdistribution . However, it is reasonable to assume thateven at the end of this process, the region of junctionso affected will represent only a small fraction of thetotal detector's volume . The affected region will bedischarged by the carriers produced by the ionizing

S.J. Bennett et al. / Timing response ofsilicon detectors

event, and in this treatment specific details of thetemporal process by which the local capacitance be-comes discharged will be neglected, and idealized by a8-function . This assumption considerably simplifies themathematical treatment although it will not be applica-ble for all cases . Nevertheless, this idealization doesnot seem to affect the final results significantly asindicated by the comparison with experimental mea-surements shown in section 3.

The state described above will be taken as theinitial condition for the mathematical description ofthe equalization process during which all the chargeconcentrated at the position of incidence x will bedelivered to the terminals of the RC-line . The integra-tion capacitors, C;, are taken to be very much largerthan the junction capacitance C, as is typically thecase . This process is described by the diffusion equa-tion :

au

L2 a2U

at

RC aX2 'where U is the voltage on the line, L the length of thedetector, t the time after the initial ionizing event andX the position co-ordinate of the line.

The boundary condition at P l(X = L) is U = 0, forall t due to the virtual earth of the pre-amplifier input.The situation is more complicated at P2(X = 0) due tothe inclusion of the termination resistor. This boundarycondition is mathematically identical to that of the"radiation" boundary condition (or "Newton Cooling")in thermal conduction problems [6] and is described bythe differential equation :

au R-+ U=0,aX

LR t

for all t, where R, is the value of the terminationresistor. The initial conditions are : U=- for X = xand U = 0 elsewhere, with x * 0 or L. The solution toeq. (1) under these conditions is calculated in theappendix to be :

Q� = fi � dt is the total charge released by the ionizingevent . r, = R,/R and a � are the infinite number of

aU(X,

2Q � _x, t) = ca 2r~ RC)C F, e a � cos a,tn=1

XL

1 X an(x)+ -sin

rtan-L ßn

(2)

withx 1 x

an(x) an cos( an-)

+ -sinL

(an-),L

(3)r,

and1 1

a + 2 +-* (4)rt rt

10.0

9.0

8.0

7.0

Û6.0

CF 5.0

D 4.0

3.0

2.0

1 .0

0.00.0 0.2 0.4 0.6 0.8 1 .0

X/LFig. 2. The voltage distribution along the RC-delay line aris-ing from a particle incident at a position of x = 03L, withrt = 0.41 . At t = 0, the distribution is given by a 8-function atx = 03L . At latter times, the distribution remains practicallysymmetric about x = 0.3L up to t = RC/200 before develop-ing marked asymmetries . The effect of the termination resistor is clearly visible in terms of the non-zero voltage sup-

ported at the X = 0 end of the line .

positive solutions to the cyclic transcendental equationtan an = -anr,

The voltage distribution U(X, x, t ) arising from aparticle incident on a line which has r t = 0.41 (chosenfor comparison with the experimental measurementsdiscussed in section 3), at a position of x = 0.3L hasbeen calculated for a number of different times usingan IBM 3090-200S computer and is shown in fig . 2 . Fortime zero, the voltage is given by a 8-function atx = 0.3L . With increasing time the voltage distributionbroadens, but remains practically symmetric with re-spect to x = 0.3L up to a time of about RC/200 . Fortimes after this, the distribution becomes increasinglyasymmetric about x = 0.3L and the effect of the termi-nation resistor can be seen by the non-zero voltage atX=0.

The currents flowing from the position and energycontacts are derived from eqs. (2)-(4) by means of therelation

ip,(x, t) = - 2Qo Y, e_(° �'--C)( Un

cos(an)RC n=1

rt

-a2 sin(an) an(x))

,F' n

S.J. Bennett et al. / Timing response ofsilicon detectors

x2Qo

C -la;,t/l2C)

an ) an(x)`

RC n =1

rt

ßnx

'L-(XIt)- 2Qo F e_can~~KC) ancos(an)

RC

� _ 1

( rt

an an(x)-an sin(an)

rt F'n

107

Expressions for the charge flowing out of the positionand energy contacts can be obtained by integrating thecurrent equations with respect to time under the as-sumption that the integrating capacitors Ci are verylarge compared with the junction capacitance .

a

QP,(x , t) = -2Qo F [1 - e-(anIIRC)~n=1

X

1

cos(an) - sin(an)

an(x))

(

, (5)an rt

ßn

(

1

) an(x)QPZ(x , t) = 2Qo F, [1 -

e_lanr~RC)1

,n=1

anrt 6n

QE(x, t) = 2Qo

[1 - e-la"`~RC)~

a1rcos(an)

n=1

n t

-sin(a n ) -

1

Ian(x)

.

(7)anrt F' n

Fig . 3 shows the results of numerical computations forthe amount of charge, QE, delivered to the energy

(6

t/RCFig. 3 . The charge collected at the energy contact as afunction of time for various positions of incidence . The valueof the termination resistor has been chosen to be R, = 0.41 R .The distribution is seen to be asymmetric with x, withthe longest risetimes occurring for positions of incidence of

x - 0.3L .

108

contact as a function of time and for a number ofpositions of incidence . The time taken for the chargepulses to rise is seen to increase with distance of theincident particle from the terminals at x = 0, L. In theabsence of a termination resistor, this increase is sym-metrical about the centre of the detector [3]. The effectof the termination resistor is to break this symmetry,with the longest delays before the signals rise nowoccurring for a position of incidence of x - 0.3L .

Fig . 4 shows calculations of the amount of chargedelivered to the position contact, P1 , at X = L as afunction of time, and for several different positions ofincidence of the initial ionizing particle . The risetimeof the signal is slowest for small values of x, becomingfaster as the position of incidence of the ionizing eventmoves closer to the position contact from which thesignal is derived . In agreement with the simplificationsof the model, which neglects any additional seriesresistances and assumes immediate discharging of thelocalized infinitesimal capacitor, the charge is instanta-neously delivered to the position contact four incidenceat x = L, resulting in the step-function shown in fig. 4.The termination resistor is seen to have the expectedresult of producing a non-zero position signal for inci-dence at x = 0, a feature commonly used to increasethe size of the position signal above the detector noise.

The total charge collected at each contact can beobtained from eqs. (5)-(7) after infinite time :

QP,(x, t = °°)00 (

1

n(x)=-2Qo

cos(an) - sin(an)a

n=1 anr,

Onx

rt +-Q°

Lrt +1 '

00

QPZ(x, r =°°) = 2Qon-1

(

a

1

nrt

) an(x)

F'n

QE(x1, t = 00)

=2Q00n=1

= _ Qo

S.J. Bennett et al. / liming response ofsilicon detectors

1

1

) an(x)COs(an) - Sin(an) -an`,

unrt f 13n(10)

The summations can be verified by expanding the righthand sides of eqs . (8)-(10) in terms of the series givenin eqs. (A.4) and (A.5) and recalling that tan an =-anrt . These equations confirm that a terminatedPSD can be thought of as a linear charge divider. Thecharge collected at the position contacts is a strictly

1 .21 .11 .0

1 .00.9

0.9

0.8

0.7

Q-0.7

( ",

___ 0.5Y 0.6

0.5+ / /

0.3

0.4

0.3

0.2

0.1

0.00.0 0.2 0.4 0.6 0.8 1 .0

0.10.0

t/RCFig . 4 . The amount of charge collected at the position contact,Pt, at X = Las a function of the time after the initial ionizingevent for a line with Rt = 0.41R . The time for the positionsignal to rise is peen to decrease with increasing x and

ultimately a step function is obtained for x /L =1 .

linear function of the position x where the particleentered the detector . The charge collected at the en-ergy contact is independent of position and is oppositein polarity to that of the position signals.

3. Experimental measurements on PSD timing

In order to compare the predictions of this simplemodel with heavy-ion data taken using real PSDs, it isnecessary to extend the model somewhat, to includethe effects introduced by the preamplifier and timingcircuitry used in an experiment. The experimentalmeasurements discussed here were made at the Nu-clear Structure Facility at Daresbury, UK, using a 200MeV 28Si beam incident upon a 400 p,g/cm 2 naturalcarbon foil .

Heavy-ion detector telescopes, consisting of 10 mmx 10 mm 35 Wm thick 0E detectors backed by 10in x 10 mri 650 p,m stopping detectors were placed

at 13 ° on either side of the beam axis. Timing infor-mation was recorded whilst the detectors were illumi-nated by a wide variety of reaction products rangingfrom 4He up to 28Si. The position and energy signalsfrom the AE detectors were amplified by CooknellEC572 charge-sensitive pre-amplifiers, together withOrtec 572 shaping amplifiers before being digitized forevent-by-event storage. Timing information for eachevent was obtained from the Count Rate Meter (CRM)output of the Ortec 572. This signal is produced by asimple leading edge discriminator which fires when the

derivative of the pre-amplifier output signal crosses apreset threshold . Timing information for an event wasstored in the form of a time-to-amplitude convertersignal measuring the time difference between the two0 E energy signals.

The effect of a simple charge-sensitive pre-amplifiercircuit upon the detector energy signals can be mod-elled by the expression :

V(Qo, x, t) _QE(x, t)

f

where V is the output signal of the charge-sensitivepre-amplifier and Cf the feedback capacitor . Thistreatment ignores the discharging effect of the largeresistor usually included in the feedback loop, but thetime constant for this is usually - 50 ps and so is on atimescale around three orders of magnitude largerthan the risetime effects discussed here. The conditiongoverning the firing of the leading edge discriminator

(Qo , ,aV

x t) =

2Qo

Y e-canrt,,/RC ) a" cos(a )at

CfRC "=1

rt

n

-an sin(an) -an ) an(x)-rt 16n

= vth,

S.J. Bennett et al. / Timing response ofsilicon detectors

where Vth is the preset voltage level at which theoutput pulse is generated, and tth is the time betweenthe ionizing event occurring and the discriminator fir-ing .

In order to investigate the energy and positiondependence of the discriminator signals, this equationhas been solved numerically to locate the .root, tth . Thisroot has been found for a wide range of position andenergy parameters: 0.05 < x/L < 0.95 and 0.4q < Qo <6q where q is an arbitrary unit of the charge releasedby an ionizing event. The threshold voltage was chosento be -0.9q/(CfRC), which was as low as could beachieved whilst maintaining good numerical stability inlocating the root .

The dependence of tth with the total charge re-leased by the ionizing particle (i .e . the energy of theincident particle) is shown in fig. 5 for a variety ofvalues of x . The curves show that at extreme values ofx the timing signal is very prompt gird alti.OS*L indepen -dent of the energy deposited in the detector . However,for events closer to the centre of the detector thetiming signal becomes increasingly retarded overall andthe leading edge nature of the discriminator leads to astrong dependence with particle energy becoming ap-parent. The effect of the termination resistor is againvisible as a shift in the timing properties away from thecentre of the detector, with the largest retardation

0.010-

0.005-

0.040

Fig. 5 . Calculated energy dependence of the time taken foraV/at to reach the threshold voltage - 0.9q/(CfRC) forvarious values of particle position x/L. The RC line is takento have Rt = 0.41 R . For extreme values of position the timingsignal is almost independent of energy, whilst for positionsclose to the centre, a marked energy dependence develops.

occurring for signals arising from particles incident atx - 0AL.

This behaviour also appears in the complementary

0.0 2.0 4.0 6.0 8.0

---------------- ----------------------------------------- ----

Qo/q

109

x/LFig. 6 . Calculated position dependence of the time taken for8V/at to reach the threshold voltage -0.9q/(CfRC) forvarious values of particle position, x /L. The RC line is takento have Rt=0.41R. Clearly visible is the asymmetry arisingfrom this termination resistor and the way in which this

asymmetry increases for decreasing particle energy .

0.040

0.035 i x/Lii 0.1'

0.030-vi 0.3

0.025-U r 0.5

0.020- i 0.7r eIt

0.015- - I v1 w

0.9

graph shown in fig . 6, which instead shows the depen-dence of tth on particle position for a variety of in-jected charges. Clearly evident for events towards thedetector centre is both the overall retardation and themuch wider range of possible delays due to the in-creased energy dependence in the timing signal . Alsovisible is the asymmetric nature of the detector's timingresponse arising from the termination resistor at X = 0.It can be seen that the asymmetry becomes moremarked the lower the energy of the incident particle .The full interdependence of the energy and position ofan event on the timing of the discriminator logic signalis shown in the time response surface plot of tth(QO, x)in fig . 7.

The actual time difference between events in a pairof telescopes measured by a TAC would be altered bythe time response surfaces of the start and stop dis-criminator signals. The delay introduced by the startsurface acts to reduce the TAC conversion time, whilstthe delay from the stop surface acts to extend it . Thiswill result in the TAC signal becoming dependent onthe energies and positions of the events and increasesthe measured time resolution quite markedly. Fig. 8shows the TAC spectrum recorded in the experimentalmeasurements discussed earlier. The measured full-width at half-maximum is around 130 ns, this is consid-erably greater than the intrinsic time resolution of thedetector due to time walks in the start and stop dis-criminators. These timewalks are clearly evident in fig .9, which shows two-dimensional spectra of the TACsignal vs the energy signal in the detector which stopsthe TAC; for narrow position slices across this "stop"detector after timewalk effects due to the start detec-tor have been removed . The spectra show the coinci-

SJ. Bennett et al. / miming response ofsilicon detectors

Fig . 7 . The time dependence of tth upon particle position,x/L, and particle energy, Q(,/q. The asymmetry in responsedue to the termination resistor (R t = 0.41 R) is clearly visible

at low energies .

200 300 400 500

Uncorrected TAC signal (ns)Fig. 8. TAC spectrum without correction for the position andenergy dependent timewalk of the start and stop discriminatorsignals. These time walks lead to broadening of the time

resolution to around 130 ns FWHM.

dent events as sharp ridges upon a random back-ground . The spectrum in fig . 9 (top) represents eventswhich were 9.4 mm from the detector's terminatededge (x = 0.94L) and shows very little dependencewith energy, whilst the spectra in fig . 9 (middle) (5.6mm from the edge) and fig. 9 (bottom) (1 .9 mm fromthe edge) show increasing energy dependence as thediscriminator signals for low energy events are re-tarded in time. Also apparent is an overall motion ofthe ridges towards later times as events move awayfrom the detectors' two edges . All of these effects arein good agreement with the predicted behaviour shownby the curves in fig . 5 which were calculated with thevalue of r t appropriate for the 35 l,m detectors used tomake these measurements . The characteristic timeconstant, RC, for these detectors is around 1 lLs, andso the delays predicted by this simple model rangefrom 0 up to 35 ns . This agrees reasonably well withthe experimental data shown in fig . 9, which showsdelays of up to about 50 ns .

The position and energy dependence of the discrim-irtator timing signals from the stop detector are sum-marised in the time response surface plot shown in fig.10 . This figure represents the evaluation of a bi-cubicspline fit to position, energy and TAC signal coordi-nates extracted from the data shown in fig. 9, togetherwith that taken from other position slices . This surfaceshould be compared with the theoretically calculatedresponse shown in fig . 7. The two surfaces show verygood qualitative agreement, though other dynamic con-tributions to the time delay may be present which have

Wd

TAC (ns)

S.J. Bennett et al. / Timing response ofsilicon detectors

200 300 400 500 600

Fig. 9. TAC vs energy spectra for various position slices acrossa heavy-ion 0E position-sensitive detector terminated with aresistor of R t = 0.41 R : The spectra of top, middle and bottomparts represent different position slices taken at x = 0.94L .x = 0.56L, and x = 0.19L respectively. Clearly visible is theincreasing energy dependence of the TAC signal and the

overall retardation of event signals .

Fig. 10 . Experimental time dependence of the discriminatorsignal used to stop the TAC with the position, x/L, and

energy E, of an event.

not been included in this model, and these are cur-rently being investigated.

The position and energy dependence of the TAC,caused by the dependence of the corresponding dis-criminators, can be removed by generating a correctedTAC value :TACcorr = TAC -fi(àEl , Pl ) -f2(AE2, P2 ) + offset,

where fI and f2 are the corresponding surface fits,and AE and P the appropriate energy and positionsignais . Fig . 11 shows the results of correcting the TACdata shown previously in fig . 8. The full-width half-

C:mV

1000

900

800

700

600

500Q .

c 400

Û 300

200

100

0200 300 400 500

Corrected TAC signal (ns)Fig. 11 . TAC spectrum after correction for the timewalk ofthe start and stop discriminator signals . This figure should becompared with the uncorrected spectrum shown in fig . 8 andit is evident that the time resolution has been improved

considerably to 25 ns at F`dVHM .

maximum of the coincidence peak has been reduced to25 ns, compared to the uncorrected width of 130 ns,decreasing the contribution from random events by acorresponding factor .

Appendix A: Solution of the diffusion equation for aPSD terminated at one end

The voltage distribution arising from a particle inci-dent on a PSD is given by the solution of the diffusionequation :au a2U

with K = L2JRC being the diffusion constant. Theboundary conditions for the PSD are : at P,(X = L)U = 0 for all t, and at P2(X = 0)

8U R~X + LRt U = 0

for all t . The boundary condition at X = 0 arising fromthe presence of the termination resistor is identical tothe "radiation" boundary condition (or Newton Cool-ing) found in thermal conduction problems [6).

By inspection:

Un = e-(Kan /L2)r

An cos( anL) + Bn sin anX

,(

Lwhere An , Bn , and an are constants yet to be deter-mined, is a solution of eq. (,%.M . Un meets the bound-ary conditions at P,(X = L) provided

Combining the two conditions (A.2) and (A.3) yields:tan a,= -an r tIt is possible to show that the roots of this equationmust be real and an represents the infinite number ofpositive solutions to this transcendental equation .

The initial condition :

U(X, t=0)= QOL 8(X-x),C

can be met by a series expansion in terms of thesolutions Un ,

00

U( X, t = 0) = 1: An'Yn ,

(A.4)n=1

SJ Bennett et al. / 71ming response ofsilicon detectors

where

X 1 X-y,= cos an-

)+

sin an L( L anrt ( )

and eq. (A.3) has been used to eliminate Bn . Thecoefficients A n of each term in the series can beevaluated using the orthogonality condition

fLQ tmyn d X =

which can be derived in an analogous way to that usedby Carslaw and Jaeger [6] . Thus the coefficients, An,are in general :

2An

In

L fLYnU(X, t = 0) dX

(A.5)0

i) L+-rt rt

and for U(X, t = 0) = Q0LB(X-x)/C,

An

U(X, x,

2QOLC

References

a2+I )

n

2 L+ Lrt rttan

a 2n1 L

an + 2 L+-rt rt

00

-(Ka2 2 )!t) = Y Anyn e

~ ILn=1

8mn

x 1 xX

(cos(anL) + anrt

stn((InL )

.

The final solution to eq. (A.1) is thus :

[1] K.H . Lautedung, 1 . Pokar, B . Schimmer and R . Stäudner,Nucl. Instr . and Meth . 22 (1963) 117.

[2] R. Bock, H.H . Duhm, W. Melzer, F. Pilhlhofer and B.Stadler, Nucl . Instr. and Meth. 41 (1966) 190 .

[3] S. Kalbitzer and W. Melzer, Nucl . Instr . and Meth. 56(1967) 301 .

[4] J.B.A . England, Proc. Eighteenth Scottish UniversitiesSummer School in Physics, eds. S.J . Hall and J.M . Irvine(SUSSP, 1978) p . 225 .

[5] S . Kalbitzer and W. Stumpfi, Nucl. Instr. and Meth . 77(1970) 3(nß.

[6] H.S . Carslaw and J.C. Jaeger, Conduction of heat in solids(Clarendon Press, Oxford, 1959).

Antan an = - -nB,

(A .2)

and it meets those at P2(X = 0) providedA nB

= ant'n

(A.3)