About the precision of time and amplitude measurements in drift detectors

NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH

Nuclear Instruments and Methods in Physics Research A326 (1993) 279-283 North-Holland

Section A

About the precision of time and amplitude measurements in drift detectors E. Gatti and G. Fusillo

Poluecnico di Milano, Dipartimento di Elettromca, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

The correct statistics of the shape fluctuations in charge clouds diffusing in drift detectors is developed under the hypothesis of fixed number of signal electrons. Its consequences on the precision of time and amplitude measurements are examined and compared with the classical Poisson treatment. It is shown that the time resolution remains the same, while the amplitude resolution can be more or less affected, depending on the relative values of the widths of signal pulse and filter weighting function . The well-known global fluctuations of released charges due to a single spectral line as well as electronic noises are easily composed with the calculated contribution due to the shape fluctuations .

1. Introduction The theory of time resolution for output pulses of a semiconductor drift detector has been developed since a few years [1]. Also the theory of amplitude resolution is well known in the case of other detectors supplying an approximately Dirac QS(t) current pulse to the front-end electronics, or when the pulse is supplied in a limited time interval At with unknown wave shape. In this latter case, to avoid ballistic deficit, the optimum weighting function is calculated with the constraint of a flat top region of duration At [2]. In the case of semiconductor drift detectors excited by minimum ionizing particles incident on the wafer, or by X-rays causing an almost punctual release of primary ionization, electron pulses arrive at the anode, because of diffusion during the drift time, as a current signal well approximated by a Gaussian function". In the presence of parallel noise, it would be penalizing to ignore this information and use simply a flat top filter in order to avoid "ballistic deficit" errors .

* Research supported by the Italian INFN, MURST and CNR. 01 Indeed, the expected Gaussian pulse width is known only when the drift time in the detector for the analyzed event has been measured . However, it must be observed that this measurement can be relatively inaccurate for the problem we are dealing with, since the Gaussian pulse width related to diffusion is a slowly varying function of carriers drift time . So, we do not consider here shape fluctuations due to an imperfect knowledge of the drift time .

However, in order to exploit this mean shape information, we must also evaluate the statistics of the Gaussian current pulse. In previous treatments, the statistics of input electrons was supposed to be a Poisson distribution, that is, a time-dependent white noise associated to the signal . This involved an associated variance in the total number of collected electrons equal to N, if N was the number of released electrons. Clearly, this inevitable consequence of Poisson statistics is unacceptable if we suppose that there is no recombination process along the drift path ; in fact, in the hypothesis of a capture free semiconductor medium, if a given number of electrons is released by an ionizing event, exactly the same number is collected at the anode . In this paper we develop a statistical treatment of the shape fluctuations in the signal coming from a drift detector that takes into account the rigid constraint of a fixed number of electrons collected at the anode . The purpose is the evaluation of the effect of this statistical approach, more in line with the physical reality than the classical Poisson model, on the resolution of time and amplitude measurements . 2. Statistics of the detector signal The first step is the statistical description of the detector signal by means of the calculation of its autocorrelation function . We imagine to divide the time axis into equal elementary intervals At, and we associate to the ith interval the random variable x, representing the number of electrons of the signal pulse portion included in the interval At, .

0168-9002/93/$06 .00 © 1993 - Elsevier Science Publishers B.V . All rights reserved

VII. IONIZATION-BASED

280

E. Gatti, G Fustllo / Time and amplitude measurements m drift detectors

At first, we examine the case of a signal formed by a single electron . In this case, x, = 1 if this electron drops into the interval At,, otherwise x, = 0. Let be p, the probability that the electron drops into the interval At, ; the probabilities p, are different in general, since they are proportional to the mean amplitude of the signal at time t, . It is fairly easy to write the expression of the generating function Fl associated to the joint probability P(x,, x,_ ., xM ) for the set of integral valued random variables x, [3]. We obtain F,(S  52, . . ., SO 1

xI

1

. . .

=0x 2 =0

x, X s1x, s2_

. . .

1 Y-

x,yt =0 x,y

SM

=

P(xl x2 , . . , -M) M

~=1

P's

in which M is the number of time intervals; we have also used the relation EM I p, = 1. Then, taking into account the theorem on the composition of independent events described by means of generating functions [41, it is immediate to calculate the generating function Fn, when we consider the real case of a signal formed on the basis of a fixed number N of electrons released at a point-instant; in fact, it is sufficient to raise F, to the Nth power, as the random walk of each electron is independent from the others . As is well known, the knowledge of the generating function makes it possible to calculate the variance of x, [4] as Var(x,) =~(x, - (x,») z> _

a 2FN asI

VA, +-as,

aFN ( as, )

At last, going from discrete to continuum, At, = 0tk _ . . . = At, - 0, and noting that for At, - 0 the second term in Var(x,) is infinitesimal of higher order than the first, we obtain the following form of the autocorrelation function of the signal at the filter input 1 Qtl, t2) =J( 1 1) S ( tl - t2) - NJ(tl)J(tz)

= CO(tl> t2) +C ,2( t l , t2) . We remark that the term C, l(t l , t2) is identical to the autocorrelation according to Poisson statistics ; the second term accounts for the constraint imposed of fixed electrons number . Such constraint introduces a correlation among signal values at different times. This correlation, as intuitive, is negative, since in a process realization, if the signal amplitude in some instant is greater than its mean value, in some other instant the amplitude must become smaller, in order to comply with the constraint of a constant pulse area . When the signal collected at the detector anode 1s sent to a filter with impulse response f(t), the output signal autocorrelation function, as is well known, is Cu( t l ,1 2)

-

J «J

Xf(tz-T2) dr, dr

Cu(tl, t 2)

+ J( T 1)f( tl - r l)f( t 2 -T I) d ,r l 1

as, as k - ( as, ) ( ask )

-Np,pk

If we name J(t) the mean electron current coming from the detector, we can establish the following relationships J(0'11 , - NA , J(tk)O t k = NPA . These equations enable us to rewrite the variance and covariance expressions referring directly to the mean electron current rather than to the probabilities p, . We find J t Ot, 2 Var(x,)=J(t,)Ot,- [ ( ,) ] , N J(t,)Ot,J(t k )Otk COV(x xk) = N

J(T1)f(tl - ?1)drl

X I +~J(r2)f(tz - rz) dr2.

If we name NO the output signal, that is, the convolution of J(t) and f(t), we can write

We remember that all the partial derivatives of FN appearing in these relations must be calculated for s1=s2= . . . .

=SM=1

+x

-NL

= Np, ( 1 - p) ,

COV(x xk) = (( xi -(x,))(Xk-(xk))) 2FN N N

C~( T 1 ,T 2)f( t l - 7 1)

and in our case

2

and the covariance of x, and x k as

~

C u( t l , t2)

+ J(rl)f(tI -7 1)f( t2 -r 1) drl 1 - NU(tl)U(tz)

= Cal( t l1 t2) + Cu2(t1, t2) . Here we can repeat considerations analogous to those made for C,(t l , t z ); in particular, C,,l(t,, tz ) corresponds to the output autocorrelation function according to Poisson statistics, while C2(t  t 2) is the negative correlation term, arising directly from C, 2(t,, tz ). 3. Consequences on the resolution of time and amplitude measurements From now on, for simplicity of calculation and without loss of generality, we consider a noncausal filter

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E . Gatti, G. Fusillo / Time and amplitude measurements in drift detectors

with response f(t) having a centre of symmetry in that is, the square of the noise-to-signal ratio, at the t = 0; we also suppose that the time origin is the measurement time excitation time of the filter itself . Introducing the 1+a N )2 1 ( 2 - ) weighting function w(t) = f(-t) and focusing on the (4) 1 instant t = 0, which is the measurement time according to our conventions, on the basis of eq . (1) the output We remind the reader that only the contribution assonoise power in that instant is ciated to the statistical shape fluctuations in the signal +. is included in the noise. cn(0, 0) j(,r)w2(-r) d7 [U(0)] 2 As one would have expected on the basis of the constraint of a constant signal area, when the input = C1 (0, 0) + C .2(0, 0) . (2) pulse is very short with respect to the shaping time (that is when a -> 0), the noise-to-signal ratio tends to 3.1 . Time measurement 0. In this case no error is made in the amplitude measurement because the charge in the input pulse is Eq . (2) has an important consequence in the meafully integrated, irrespective to its shape. surement of the pulse arrival time . In fact, since in Adding and subtracting 1/N to the right member of time measurements the shape of w(t) is such as to give eq. (4), we can better point out the physical origin of a zero-crossing point in the time origin, U(0) is 0 and the contributions to --A' So, we can write C2(0, 0) vanishes #2 . Therefore, in this case the term C 1(0, 0) alone contributes to the output noise power at 1+a2 Ns,2q=1+ 1 -1 . the measurement time, exactly as it would happen if 6-+ 2a 2 - ) ~ the assumed statistics was simply the Poissonian one. As a consequence, the results previously obtained by The first term of the right member of eq . (5) is related the authors about time measurement resolution with to the well-known Poisson relative amplitude covarian antisymmetric optimum filter are rigorous . ance, inversely proportional to the number of electrons in the signal ; the term within brackets accounts for the 3.2 . Amplitude measurement sensitivity of the measurement to the statistical shape fluctuations when each part of the pulse is not equally In order to analyse the effects of the new statistical weighted (this happens when o-, n /o f is not sufficiently description of the signal on the pulse amplitude resolusmall) ; the third term is the correction found as a tion, let us consider a Gaussian weighting function consequence of the constraint of a fixed electrons w(t) of standard deviation Qf , normalized in such a number . The last term cancels the first one and, in way as to give an output signal peak equal to N when conclusion, the resolution is entirely due to the menthe input signal J(t) is a Gaussian function of standard deviation o,,, and given area N,

N

=f

w(t) =

2

Pin 1 + ~2 e

r

1 zi2rrf

.

Using eq. (2), we can calculate the output noise power at the measurement time t = 0, obtaining the following expression C(0, 0)

1+a2

N

(

1 + 2a 2

1 , )

NE; FANG FACTOR

in which a = o', n /o'f . From this result we calculate, dividing by N 2, the relative amplitude covariance e A, 2

#2

Really, as one can see from eq . (1), the contribution of the negative correlation term Cn2(tm, t in ) to the output noise power at a measurement instant tin , is 0 every time that t in is a zero-crossing point for U(t), irrespective to the shape both of w(t) and At).

Fig. 1 . Relative amplitude covariance due to the statistical shape fluctuations multiplied by N, versus the ratio o,n /a", of signal pulse and weighting function widths . The value of N multiplied by the relative amplitude covariance due to the statistical fluctuations of signal area, that is the Fano factor (0 .12 in silicon), is also reported for reference. The contributions of shape fluctuations and area fluctuations are equal for Om /Qf -0 .9 .

VII. IONIZATION-BASED

282

E Gatti, G . Fusillo / Time and amplitude measurement m drift detectors

tioned shape fluctuations, whose effects are present only for a > 0. Fig. 1 shows a plot of eq . (5). We note that, adopting Poisson statistics, one would find the same curve, simply shifted of 1 upwards; in fact, there would be no more the third term in eq . (5). 4. Effect of the Fano factor on the amplitude measurement resolution Now we want to take into account also the variance of the global number of electrons released in pulses belonging to a class of events spending the same energy in the detector . If we call n the actual number of electrons forming the signal pulse in a given realization of the measurement process, we can assume for the random variable n a Gaussian probability density, centred on the mean value N and with variance that, according to Poisson statistics, would be still equal to N, but that, according to the Fano theory, is reduced to FN, where F is the Fano factor (F is about 0.12 for silicon) . This Gaussian function represents the weight according to which filter output statistics, calculated for a given n, contributes to determine the noise power at the measurement time . Such output statistics asserts that the signal peak amplitude has still a Gaussian probability density, centred on the mean value n and with variance equal to the autocorrelation C(0, 0) calculated with Eq . (3) replacing N with n . Now it is straightforward to modify the result obtained for the output noise power under the hypothesis of fixed N in such a way as to take into account the statistical nature of n; in fact, it is sufficient to convolve the two Gaussian functions we are dealing with . We find that the effects of signal shape fluctuations and signal area variations add together quadratically, giving the following expression of the relative amplitude covariance at the measurement time

ä

_

_

-~S12-

N~

- F+

1 + a2

1 +2a 2

1

From this relation, we can see that, taking o-,/o-i close to 0, the contribution due to the statistical shape fluctuations vanishes and the only term due to the global area fluctuations remains; on the other hand, increasing am /a,, the term related to the statistical shape fluctuations becomes the dominant one. For example, at a-mlo-f = 1, this term is equal about 0.15/N, and so it is already greater than the other. Incidentally, we recall that the choice o-, = o-f is suggested by the theory when the white parallel noise due to the detector leakage current is dominant with respect to the other sources of noise.

5. Conclusions Classical statistics of electron clouds diffusing in drift detectors has been modified in such a way as to overcome its incompatibility with the hypothesis of no recombination process in the detector . The correct statistics, applied to time filtering, has given the same resolution as the classical theory, essentially because the measurement instant is coincident with the zero-crossing point of the output signal . The contribution of the statistical shape fluctuations to the amplitude measurement resolution has been found 0 in the case of a perfect integration of the detector signal, as intuitive. For a large ratio am/at of input signal pulse and weighting function widths, EÂ becomes asymptotically equal for Poisson statistics and the one introduced in this paper. The Fano factor contribution to the amplitude resolution has been found dominant over that of the shape fluctuations only when u,/a-f is somewhat smaller than l . Finally, we note that it is easy to write the differential equation for the optimum amplitude weighting function wt, P ,(t), accounting for the calculated correction in the noise contribution due to the signal shape fluctuations . A procedure similar to the one used in ref. [1] (sec eqs. (9) and (10)) can be used . We find for wopt(t) the following equation : -

C2 e`

g21V2

2 2v Wopt(t) + NS(t)Wopt(t) + N2 WOPt(t) 1+EN mm

-2\

N~s(t)I

' s T WOP,(T) dT,

in which C is the total input capacitance, e,2, the power spectrum density of the series noise of the preamplifier, q the electron charge, N the mean number of signal electrons, s(t) the probability density of signal electrons, v the mean current of leakage electrons, EÂ,mm the minimum value of EA, obtained when w(t) _ W" P t(t ) .

The term I IN within brackets in the right member of eq . (6) is the one accounting for the negative correlation found in the statistics of the electron cloud. A solution for w pt (t) can be found in a closed form when the resolution is limited by the series noise of the preamplifier only, or when this noise contribution is absent . The general solution requires numerical methods of analysis . Acknowledgements We wish to thank G. Bertuccio, S. Cova, D. De Falco, A. Lacaita, P . Rehak and M. Sampietro for the interesting discussions and comments on the argument.

E . Gatti, G. Fusillo / Time and amplitude measurements in drift detectors References [1] E. Gatti, P . Rehak and J .T. Walton, Nucl . Instr . and Meth . 226 (1984) 129 . [2] E. Gatti and P .F. Manfredi, Riv . Nuovo Cimento 9 (1) (1986) p . 89 .

283

[3] W . Feller, An Introduction to Probability Theory and Its Applications, vol . I, 2nd ed . (Wiley, 1957) pp. 248-249, p . 261 . [4] W . Feller, op . cit ., p . 250 .

VII . IONIZATION-BASED