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Nuclear Instruments and Methods in Physics Research A335 (1993) 44-58North-Holland

Spatial resolution of silicon microstrip detectors

R. Turchetta

LEPSI, ULP/CNRS, Strasbourg, France

Received 19 March 1993 and m revised form 14 June 1993

The spatial resolution of silicon microstrip detectors is studied as a function of the main detector parameters and of the trackangle. Several algorithms for finding the position of particle hits are presented and analysed . Analytic expressions of the spatialresolution are derived for the main algorithms. Using a Monte Carlo simulation, the spatial resolution is calculated for eachalgorithm and, for each detector design and track geometry, the algorithm that gives the best resolution is determined .

1 . Introduction

Silicon microstrip detectors are widely used as pre-cise vertex detectors in both fixed target and colliderhigh energy physics experiments [1-4]. For tracks im-pinging the detector along a line perpendicular to thedetector surface (0° tracks, see fig . 2), spatial resolutionof less than 1 .8 win [5] has been measured, using a 25win pitch detector . In recent years, a growing interesthas been expressed in the measurement of the spatialresolution for non-perpendicular (inclined) tracks [6-9] .This interest is related to the development of collidervertex detectors based on double-sided microstrip de-tectors, where particles traverse the detector at largeangle i9 . For example in DELPHI, tracks with an angleof up to 45° are detected in the Microvertex upgradeproject with double-sided detectors, and in the For-ward Barrel Microvertex project tracks could traversethe detector at angles larger than 70° [10] . A similarsituation is likely to be found in future collider vertexdetectors [11,12] .

In this paper, we discuss the limits on the spatialresolution for different detector configurations and fora wide angular range, from 0 to 75°. Different algo-rithms are considered and compared using a MonteCarlo simulation (section 2) . In sections 3 and 4, thebest algorithms for perpendicular and inclined tracksrespectively are presented, and their spatial resolutionis thoroughly analysed . In section 5, the behaviour ofspatial resolution with respect to the different detectorparameters and to the geometry of the tracks is dis-cussed, and our results are compared to the availableexperimental data . Our analysis shows that for eachgiven set of track angle and detector parameters, aparticular algorithm must be chosen in order to fullyexploit the resolution of silicon microstrip detectors .

2. Monte Carlo simulation

0168-9002/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

2.1 . The construction of the charge signal

NUCLEARINSTRUMENTS& METHODSIN PHYSICSRESEARCH

Section A

In the Monte Carlo simulation, only the total chargemeasured by the charge preamplifiers is calculated .The analytic expression of the electric field in a planarabrupt p-n junction [13] is used . In the construction ofthe charge signal at the output of the preamplifier, wecan distinguish four phases .

2.1 .1 . Charge generationThe minimum ionizing particle (MIP) traversing the

silicon loses energy that goes into the generation ofelectron-hole pairs at the mean ratio of 3.7 eV perpair . In the simulation, energy loss is calculated inde-pendently in each slice of 10 win of path m siliconaccording to the corresponding Landau distribution(most probable energy loss equal to 2.1 keV), convo-luted with a Gaussian of standard deviation

where SZ= 1000 keV 2/cm, and t,,,,, = 10 Rm. Thevalue of SZ has been chosen m order to have thecorrect value of the most probable energy loss for athickness of 300 win. No delta ray production has beenconsidered .

2.1 .2. Charge drift and diffusionA drift time t, is calculated for each 10 win slice;

this time depends on the slice position as well as on thetype of carriers we are considering: junction side (p-side) strips collect holes, while ohmic side (n-side)strips collect electrons. For each slice, diffusion isconsidered to be exactly Gaussian, i.e . the charge cloudsarriving at the strips is a Gaussian with a standard

deviation given by

0-ddf -

2Dtd ,

where D is the diffusion coefficient .

2.1.3. Capacitive couplingAfter steps 2.1 .1 and 2 .1 .2, a total charge signal

arriving at the strips can be calculated by summing upthe individual contributions of each 10 Vm slice. Foreach strip, the fraction of the signal that is detected bythe corresponding amplifier depends on the equivalentelectrical network of the detector. Strip-to-backplanecapacitance can be neglected, since its value is about10 times smaller than interstrip capacitance CIS inmost of the geometry [14,15] . This means losses ofsignal towards the backplane are not considered . Thecapacitance of a strip towards next-to-the-closestneighbours can be neglected since it is usually quitesmaller than the capacitance of a strip towards closestneighbours [14] . In AC-coupled detectors [16], a decou-pling capacitance CDEC Is integrated between the stripand the readout electronics . The equivalent electricalnetwork is shown in fig . 1 . A simple calculation showsthat a fraction K of the charge arriving at one stripappears at the output of the neighbouring amplifiers .This fraction is given by

CNEI

CDEC + 2CNEI

where CNEI is given by

C DECCIS

CDEC + CIS .

K=

CNEI -

Since we can usually say that CDEC >> CIS, we haveK- CIS/CDEC

In DC-coupled detectors, the strips are directlyconnected to the electronics ; the capacitance of a striptowards the input of the amplifier is then given by theMiller capacitance of the preamplifier, that is generallyquite higher than the interstrip capacitance . In thiscase, we can consider K - 0.

In order to reduce the number of readout channelswithout spoiling the spatial resolution, intermediate

Q DIFF S

CDEC

£----IiC is 7

CDEC

Fig . 1 . The detector capacitive network considered in thesimulation .

R . Turchetta / Spatial resolution of silicon nucrostrip detectors 4 5

Fig . 2. The geometry considered in the simulation . The impactpoint xIp is defined as the point where the particle crossesthe mid-plane of the detector . The sagitta is the projection ofthe track along the coordinate measured by the strips . x, givesthe position of the ith strip, while p, is the length of that part

of the track that traverses the volume of the ith strip .

strips can be introduced between two readout strips[17] . In this case charge collected on one intermediatestrip is seen by the two neighbouring amplifiers . Theratio of charge seen by the two preamplifiers is linear[13] and given by the ratio of capacitances .

2.1 .4. NoiseA random noise sample is added to the charge

signal, according to a Gaussian distribution, whosestandard deviation is given by the equivalent noisecharge (ENO of the detector-amplifier system .

2.2. Clusterfinding algorithms

Hit search is done using cluster-finding algorithms(CFA), tailored according to the geometry of the tracks[7] .

For tracks at 0° or slightly inclined, most of thecharge is collected by no more than two strips . Forinclined tracks, the charge is spread over several strips .The average number of strips in a cluster is given bygeometry (fig . 2) and equal to

sagitta t~Ncluster)'

P

=-tan 0,

(1)

where t is the detector thickness and P the readoutpitch. The sagitta is defined as the projection on thedetector plane of the track in the direction perpendicu-lar to the strips . Apart from the two strips on the edges(what we call the head and the tail), the most probablesignal on all the other strips in the cluster is the sameand is given by

S'PSs

t sin D '(2)

46

where S, is the most probable signal released by aparticle at 0° .

2.2 .1 . Algorithm for perpendicular or slightly inclinedtracks

It is used if the sagitta is smaller than 3P/2 . First asearch is done for strips with (S/N)str,p > 3, where(S/N)str,p is the signal-to-noise ratio of one strip . Thenall the neighbouring strips with positive signals areincluded in the cluster . This is accepted if: (S/N)elustar> Sa/(2 cos 0 - ENC), (S/N)cluster being the sum of(S1N),u,p over all the strips included in the cluster .

2.2 .2. Algorithm for inclined tracksIn this case, all the strips with (S/N) str,p > 3 are

included in the cluster . The search starts from the stripwith the highest (S/N)strlp and then moves to the leftand to the right, stopping whenever (S/N)strlp fallsdown the threshold. At this point, the two strips closestto the selected strips are included in the cluster if theirsignal is positive . The cluster is accepted if (S/N)clu,ter> Stl /(2 cos ,9 - ENC), and the total number of stripsper cluster is higher than half the sagitta (measured inpitch units) .

All the events where an accepted cluster has beenfound are considered as good events and are furtheranalysed . The number of good events divided by thetotal number of generated events gives the efficiency ofthe detector.

2.3. One dimensional position-finding algorithm

Several one dimensional position-finding algorithms(PEA), chosen because of their common use and sim-plicity, have been analysed.

2.3.1 . Digital PFAThe position of the cluster is given by the position

of the strip with the highest signal . For non-inclinedtracks, a simple calculation shows that the standarddeviation of the residue distribution generated withthis algorithm is

O'DIGP

12 .

2.3 .2. Center-of-gravity (COG) PFAThe position is given by

Xco" = Y_SIX,

S, ,cluster cluster

where x, is the position of the ith strip included in thecluster and S, the signal on that strip; the sums areover all the strips included in the cluster . Under given

R. Turchetta / Spatial resolution of silicon nucrostrip detectors

assumptions on noise, it can be shown [18] that thespatial resolution is slightly dependent on the positionof the impact point with respect to the collectingelectrodes, and, for a given position, it is linearlydependent on the ENC/S ratio . We can consider theimpact point position for which the spatial resolution isworst and write

O'COG ENCP =a

where a is a coefficient that depends on the hypothesison noise, as well as on the number of strips included inthe cluster . It is equal to 2.12 for 3-strips clusters [18] .

2.3 .3 . Linear PFAIt is the COG PFA applied to two strips . We prefer

to analyse it separately from the COG PFA becausefor small angles ,9, the signal spreads over at most twostrips . The spatial resolution as a function of the im-pact point xlp is given by

1-11

ENC

X tp

// XtP \I

P

S

Yl-2 P +2 \ P /2

ENC= S a(xlplP) .

(4)

The dependence of the resolution on the impactpoint is quite smooth, since the function a(xlp/P) has,is minimum for x,P/P = 1/2 where a = V/2 and itsmaximum at the edge of the definition interval : a(0) _a(1) = 1 . Eq. (3) still holds with a = l.

2.3 .4. Non-linear on rl PFAIn the linear PEA, the position is assumed to be

given by

P P

where xL is the positionand the variable 71 is defined by [19]

P =f( 77) +

of the left strip m the cluster

where SR(L) is the signal on the right (left) strip in thecluster.

In the 77 PEA, the position is assumed to be givenby

where f(rl) is an arbitrary monotonic growing functionof Y7, with f(0) = 0 and f(1) = 1 . In section 3, thisalgorithm will be thoroughly analysed because of itsimportance for tracks at small angles .

x DHT

B

Fig . 3 . Definition of regions A and B. The dashed line inregion B, represents the border between the two strips. Theelectric field forces charges on each side of the border to drift

towards the closest strip.

2.3.5. Digital head-tail (DHT) PFAFor inclined tracks, a first approach in the calcula-

tion of the position consists of considering just thepositions, x h and xt , of the head and the tail ofcluster, respectively (fig . 2)

2.3.6. Analog head-tail (AHT) PFASince energy loss is roughly proportional to the

particle path in silicon, an AHT PFA can be defined inthe following way (fig. 2) :

XAHT,O

x DHT + Sh - St .P P 2S,y

2

In order to reduce the influence of Landau fluctua-tions, we can further refine the definition of the AHTby writing

XAHT - XDHT + min(S,, S,y) - min(St , S,9) .P P

2S,

Experimental results [7] as well as our Monte Carlosimulations show that the AHT defined with eq . (7)gives slightly better results than the one defined by eq .(6). In the following, unless otherwise specified, we willsuppose the AHT defined with eq. (7).

2.3 .7. One-sided analog head-tail (1SAHT) PFAThe algorithms presented in 2.3 .5 and 2.3.6 demand

the knowledge of both the head and the tail of thecluster . However, if we know the angle of the track, theposition and pulse-height of one of the two edges issufficient to find the position of the cluster . For exam-ple, using the head (left edge) of the cluster we can saythat the position of the impact point is given by

AHT,h

_xh

1_

_Sh

t tan 0P - P + 2 S,, + 2P

A similar definition applies for the tail strip .

R. Turchetta / Spatial resolution of silicon nucrostrip detectors

3. The n algorithm

3.1 . The rl uariable and its distribution

47

Before considering the effect of the vi PFA on thespatial resolution, we will first consider the distributionof the variable 71 and its physical origin .

Many experimental results are currently availableabout this fundamental distribution in microstrip de-tectors [6,7,19,20]. A common feature of all the experi-mental results is the presence of peaks in the 17 distri-bution . This means that charge division is far frombeing linear, because, if this was the case, the q

distribution should be flat . This observed non-linearitycan be fully explained by the width of the diffusioncloud.

For perpendicular tracks, charge spread is domi-nated by diffusion. The width of the charge clouddepends on detector parameters as the thickness, theresistivity and the applied voltage. For most of thedetectors used in high-energy physics, the thickness isaround 300 wm and the resistivity is a few kfZ cm; thisgives a width of the diffusion cloud of about 5-10 Wmfor a fully depleted detector. Thus, the charge cloud isfar smaller than the pitch in most of the microstripdetectors in use.

How can we see this effect in the experimentaldata? Let us consider a detector where every strip isread out. If a particle hits the detector in the middlebetween two strips (region B in fig . 3), charge divisionwill be effective (and roughly linear) . On the contrary,if the particle crosses the detector about 10 ltm awayfrom the strip frontier (region A in fig . 3), most of thecharge will flow to one strip. In this region, 77 willassume a value close to 0 or 1. The width of region B,where charge division is effective, depends only ondiffusion, thus the width of region A is larger for largerpitches. For example, in a 50 wm pitch detector, regionA and B have roughly the same width. Since in regionB, 77 spans over a large set of values while in A it isconfined in a small region around the edges, the 77

distribution will have peaks close to the value 77 = 0

and q = 1 .

If an intermediate strip is introduced, the peaks atrl 0 and 71 = 1 will still be there, but a new peak atrl = 1/2 will appear, corresponding to the intermediatestrip (see fig . 6) . The three peaks will also be lesspronounced than in the previous case, since the strippitch gets closer to the width of the diffusion cloud.

Diffusion alone can thus explain the main feature ofthe q distribution, i.e . its non-linearity, but cannotexplain all its characteristics. Let us again consider a 50l.Lm pitch detector, where every strip is read out. If thecapacitive coupling and the noise are switched off, twovery narrow peaks at 71 = 0 and q = 1 appear (fig . 4). Ifwe now introduce the capacitive coupling (fig. 4), the

48

two peaks move to the center by an equal quantityK/(1 - K) = K, because some fraction of the chargealways flows to neighbouring strips.

Now noise is switched on . The two peaks spreadout, and their width is determined by the noise magni-tude . Since 77 can be considered as the signal on onestrip normalized to the total signal in the cluster, thewidth of the two peaks is roughly given by

ENC lapeak V

( 9)S SIN

This equation is confirmed by experimental results (fig .5) as well as by the simulation .

3.2. The 17 algorithm

Since hits are uniformly distributed over the detec-tor, the position of the cluster with respect to the leftstrip can be calculated with :

~to dN d

Iâ~t~

xn-Po

1 dN

=Pf(17o),

fo

d 17~1d17

where dN/d17 gives the differential 77 distribution . Eq .(10) defines a non-linear algorithm (see section 2.3 .4)with f(v7) given by the integral of the 77 distribution,normalized to the total number of events in the distri-bution .

1 1600

ô~, 1400

Ez 1700

1000

800

600

400

200

K = 0 .00

I 1 02 03 04 OS 06 07 08 09

R. Turchetta / Spatial resolution of silicon nucrostrtp detectors

(10)

Fig. 4 . The distribution of the variable 77 obtained with theMonte Carlo simulation in the absence of noise and for twodifferent values of the capacitive coupling K (P = 50 win and

no intermediate strip) .

ENTREES

ENTREES

120

80

40

d zo-X =Pz

f( drl ) ~~ .

120

80

40

00 0,2 0,4 0,6 0,8 1 0 0,2 0,4 0,6 0,8 1

11

1 1

Fig . 5 The experimental distributions of the variable 17 with aGaussian fit on the peaks for two different values of the SINin a 50 win pitch detector without any intermediate strip (a)and (b) SIN= 16 ; (c) and (d) ; SIN= 10 . The inverse of thestandard deviation of the Gaussian fits are 15 .2 and 16 .5 in (a)and (b) respectively, and 9.3 and 11 .2 in (c) and (d) respec-

tively .

A comparison between the plot of the true impactpoint as a function of vl [6) and the function f(q)shows this algorithm correctly reconstructs the impactpoint position . On the contrary, a linear algorithmgenerates systematic errors .We remark that the 17 algorithm uses only intrinsic

properties of the detector in order to correct for itsnon-linearities.

3.3. Spatial resolution to the case of non-linear chargedivision

Eq . (4) gives the spatial resolution that can beexpected when using the linear PFA, under the hy-pothesis that this algorithm correctly reconstructs theposition in the absence of noise. But, as we saw in theprevious section, this is not the case in microstripdetectors for tracks at 0° (or slightly inclined, so thatdiffusion is still the dominant factor in the chargedivision). In this case, the q PFA gives the correctposition . Using eq . (10) and error propagation, we canwrite

The error o,n on the determination of 77easily obtained from eq. (4), observing thatlinear algorithm x/P = 77 . Thus,

ENC=

S

y, '1 - 2,7 + 2772

and, from eq . (10),

dN

df _

d77d7l

I dNd

'

fd7

that is dfldq is given by the number of entries in the77 distribution normalized to the total number of en-tries Noc . The spatial resolution obtained with the Y7algorithm is then given by

dNENC~ =

~1d

PS -271+2r2 ,dN

(~

d~lfo d71

ENC

1 dN- a(rl) --,S

Not d77

can bein the

ENC-P S

R(x) .

(12)

3.4. How to get a linear charge division

R. Turchetta / Spatial resolution of silicon microstrip detectors

where 77 =f-'(x/P) .As we described in section 2.3 .3, the function a(71)

is quite flat, ranging from V /2 up to 1. On thecontrary, the distribution of rI is far from being flat .The term dN/d77 in eq . (11) gives then rise to dra-matic non-uniformities in the spatial resolution . Wecan define a resolution function R(x) such that

Microstrip detectors have thus a highly non-uniformspatial resolution, as already observed in ref. [6], andresidue distribution is far from being Gaussian . TheFWHM of the residue distribution reflects the re-sponse of region B, where spatial resolution has itsoptimum, while a Gaussian fit represents, in somesense, an average of the detector spatial resolution .We can ask ourselves the question : what is the

function f that gives the best resolution? It is easy toshow (see appendix A) that at equal signal-to-noiseratio the best resolution is obtained for the functionf(77)=-l, that is in the case of linear charge division .

It is thus important to get a charge division as linearas possible . In section 3.1, we showed non-linearitiesarise because of the shape of the diffusion cloud. Ifo-d,ff is the standard deviation of this cloud, non-lineari-ties are large if the ratio P/Q,,ff is large. We need thento reduce this ratio .

côâDE

360

320

280

240

200

160

120 1-

80

40

Fig . 6. The simulated distribution of the variable 77 for adetector with one intermediate strip, P = 50 wm and SIN=

20.

The first way is to increase od,ff . This can be doneby thickening the detector and applying the minimumbiasing voltage capable of fully depleting the detector .This is not always possible because other considera-tions, such as a minimum amount of matter, in order toreduce multiple scattering, set a limit to the thicknessof the detector . Moreover, a thicker detector suffersfrom a higher probability of 6-rays production . 8-rays

400

E3 300z

250

200

150

100

50

LJ

02 03 04 0b 06 07 08 09

1 7

49

0 01 02 03 04 05 06 07 08 09 1

Fig. 7 . The simulated distribution of the variable 17 for adetector with no intermediate strip, P= 50 wm, SIN= 20 atan angle 0 =10° . The asymmetry in the p distribution is due

to the geometry .

50

can displace the centroide of the charge cloud andlimit the spatial resolution [21].We are thus left with the second possibility, i .e .

reducing P . This can be done without increasing the

20

12

10

8

6

2

6

2

0 I

I

I

I20 30 40 50 60 70 80

13 (In degrees)

0

I

I20 22 24 26 28 30 32 34 36 38 40

R Turchetta / Spatial resolution of silicon microstnp detectors

,~ (In degrees)

number of electronic readout channels by introducingan intermediate strip between two readout strips forcharge interpolation . The resulting -q distribution (fig .6) is less peaked and charge division is almost linear .

20

c

cO

OrnN

6

2

20

18

16

14

12

0

I

I

I

I20 24 28 32 36 40 44 48 52 55

10

8

6

2

0 L`20

13 (In degrees)

I

I

I

I

I

I

I

I

I

I21 22 23 24 25 26 27 28 29 30

'3 (In degrees)

Fig . 8 . Comparison of Monte Carlo calculation of the spatial resolution (dashed line) and of the values given by eq . (18) . Thefunction 7r(p) has been calculated using the analytic expressions for the most probable value and the FWHM of the energy lossdistribution given on page 648 in ref. [22] . For each angle, eq . (18) has been calculated for the different positions of the impactpoint and then an average value has been calculated . In the figure the comparison is done for a 50 Win pitch detector, assumingC =1 . Similar agreement is obtained for different pitches . The curves are given only in the angular range between ,9- and a_

where eq . (18) applies . (a) S/N=25 ; (b) S/N=20 ; (c) S/N=15 ; (d) SIN= 10 .

3.5 . Slightly inclined tracks

In this paragraph we consider angles i9 small enoughso that most of the charge is collected by two strips .The rl distribution becomes less and less peaked forincreasing angle (fig . 7), that is charge division becomesmore and more linear . We thus expect to find a mini-mum for the spatial resolution for the angle i9 suchthat

sagitta = 1

or

PvOopt = tan -1 ( t ) .

(13)

At i9 - i0npt the charge division is almost linear andthus spatial resolution can be roughly calculated witheq . (3), with a = 1 .

4 . The analog head-tail algorithms

4.1 . Analog head-tail. Spatial resolution

In this paragraph we present an approximate calcu-lation of the spatial resolution obtained with the AHTPFA. In order to simplify the calculation, we considerexpression (6) instead of (7) .We define Sh(S t ) as the most probable energy loss

on the head (tail) strip corresponding to the path ofthe particle in the detector . We can write for the trueimpact point (see fig . 2)

xti>

xh +x t

St - Sh t sin 10

P 2 2So P(14)

If we consider to be able to correctly reconstructthe cluster, the measured position will be given by

XMEAS _ xh+xt

St+N-Sh-Nn t sin *+P 2

2So P(15)

where Nh and Nt express the deviation of the mea-sured pulse-height from the most probable value . Theerror on the measurement of the position is then givenby:

Ox Nt -Nh tsinP

P 2So P

and the spatial resolution is given by the standarddeviation of this quantity. The deviations Nh and Ntare generated by two noise sources : i) Landau fluctua-tions ; ii) electronic noise . We can suppose these twosources are uncorrelated, and write, for an arbitrarystrip

z )z

zN

= ( NLandau ) + ~NElectromcs )'

R. Turchetta / Spatial resolution of silicon mierostrip detectors

The second term is easily evaluated and given by

NÉlectrorncs> = ENC2 .

t1 x z

Pz = {2 ENCz + [7r(Ph)PhSO1POC]z

+[77. (Pt)PtS,v/PoC]z}l(2S,9)2

1

ENC t sin tiz- ( 2

)2~2(

So

P

)

51

(16)

The first one cannot be calculated in an analyticfashion, so we need some approximate expression . Thedispersion of values around the most probable one Spis usually described by the FWHM wp of the distribu-tion . The ratio-rr(p) = wP/SP is not constant, but tendsto increase for decreasing path length p, and it rangesfrom about 1 for a pathlength of a few microns toabout 0.4 for p = 300 ~tm [22] . We can then write

1 z(NLandau)

_

7r(p)SPC1

(17)

where C is a coefficient that gives the ratio betweenthe full width at half-maximum and the standard devia-tion of the distribution, and it is equal to 2.36 in thecase of a Gaussian distribution . For the energy lossdistribution, no simple analytical expression of its stan-dard deviation exists so that we can just rely on approx-imations . The results obtained in this paragraph are ingood agreement with those of the simulation for C = 1(fig . 8).

Using eqs . (16) and (17), and assuming Nt and Nhare uncorrelated, the standard deviation of the erroron the position measurement is thus given by:

+ [1(Ph)Ph/POI2 + [17"(Pt)Pt/Po]

z

(18)C

where Sh =phSa/po and po =P/sin i9 .In this formula we recognize two terms . The first

one is due to the electronic noise : it is inversely pro-portional to the signal-to-noise ratio (measured at 0°),and depends on a geometric factor t sin(t9)/P = So/S,9that takes into account the reduction of the strip signalwith the angle .

The second term in eq . (18) is due to Landaufluctuations . We observe that this term is dependenton the impact point position in a complicated way,through the function 7r(p) and Ph and pt . We will notanalyse this dependence since it does not give rise toany large non-uniformity in the spatial resolution as wedescribed in section 3 . Since, on the average, thelength of the path in the volume of one strip decreaseswith the angle, and that 7r(p) is a decreasing functionof the path, the total average effect is that the secondcontribution to eq . (18) also increases with the angle .

52

20FYc 18 r

ô

c0 , 14r

12

8

6

4

2

0 130 40 50 60 70 80

15 (in degrees)

Fig . 9. The relative contributions of electronic noise (dottedline) and Landau fluctuations (dashed line) are compared tothe spatial resolution for a 50 win pitch detector with nointermediate strip and SIN= 25 . The curves are given onlym the angular range between 4~mm and ,9m_ where eq (18)applies Similar curves are obtained for different detector

designs .

Before discussing the range of application of eq .(18), we will compare the relative importance of thetwo terms appearing there . We suppose P = 100 p,mand So/ENC=15 . We also suppose that 7r(Ph) =7r(p t) = 0.6 and p h/po =pt/p, = 0.5 ; taking C= 1, thesecond term is equal to 0.180, while the first one isgiven by 0.045 (sin 0) z, that is the error on the deter-mination of the position is dominated by Landau fluc-tuations (fig . 9) .

Eq . (18) cannot be applied on the whole range of ,9 .First of all, we supposed that eq . (14) gives the correctimpact point. This is true when the sagitta is largerthan 2P . Using eq . (1), the minimum angle for whichwe can apply eq . (18) is then given by

2P~mm ~ tan

-l l

) .

(19)t

This angle depends only on the ratio P/t, beinglarger for larger pitch. For P = 50 Win, Pm,n = 18°, andfor P = 200 ltm, Amm = 55°.

In order to determine the maximum angle for whicheq . (18) can be used, we remark that we supposed thatthe cluster has been correctly defined, that is the headand the tail of the cluster are correctly identified andthat any error on the measured position comes from

R. Turchetta / Spatial resolution of silicon microstrip detectors

electronic noise and Landau fluctuations on thosestrips . As a matter of fact, another possible source oferror is an incorrect reconstruction of the cluster . Withthe CFA presented in section 2.2 .2, this happens when-ever the signal on one of the internal strips in thecluster (that is the strips that are neither the head northe tail) is lower than the threshold. If this happens thereconstructed cluster will contain less strips than ex-pected and the reconstructed position will be displacedby some amount . Since the most probable signal on theinternal strips is given by eq . (2), we can suppose thecluster will be correctly reconstructed if

S,

SriP

> constant * ENCy

t sin 0A reasonable choice of the constant is 4. This gives

for the maximum angle the following expression

I SO P19"'° `= stn

( 4 ENC t

and thus it depends on SIN as well as on the ratioP/t .

4.2. 1-sided analog head-tail. Spatial resolution

In the same way as we did in the previous para-graph, we can calculate the spatial resolution in thecase of the 1S-AHT PFA. The error on the position isnow given by

Ox

N t sin dP So P

and its variance is

(20)

X2 - ( ESC

)21

t sinn p ~2 + [,rr(pt)pt/po]7

21()

A comparison of formulae (18) and (21) shows thatthe spatial resolution of the 1S-AHT PFA is roughlythat of the AHT PFA times ~r2, that is usually the1 S-AHT algorithm performs worse than the ART PFA.However, at very large angles, that is for angles of theorder of T9m,x or larger, the simulation shows that the1S-AHT PFA gives better results than the AHT PFA.This effect can be explained as follows: in this angularrange the (S/N)str,p is poor, so that it is quite difficultto correctly reconstruct the clusters. A wrong recon-struction happens whenever the (S/N)str,p of one ofthe internal strips drops below the selected threshold.The loss of only one of the internal strips is sufficientto cause a wrong determination of one of the two edgestrips . In this case, the AHT PFA will always give awrong value of the cluster position, while each of thetwo 1S-AHT algorithms will correctly calculate thecluster position in half of the cases. Thus the 1S-AHTalgorithms give better results than the AHT PFA at

very large angles . However, we must be aware of thefact that in this angular range, the efficiency of thedetector is sensibly less than 100% (see fig . 13), so thatwe should avoid operating in this region .

5. Simulation results and comparison with experiments

5.1 . Simulation results

Before considering the simulation results, we willsummarize the analysis of sections 3 and 4.

At small angles, the charge division is non-linear,mainly due to diffusion. The non-linear algorithm basedon the differential distribution of the Y7 variable allowsa correction of this non-linearity . It should thus givethe best results among all the considered algorithms .Since it is based on the pulse-height of the two adja-cent strips that collect most of the charge, it loses itseffectiveness when the charge spreads over more thantwo strips . This constraint sets an upper limit to theangular range where the q algorithm should be ap-plied, and this upper limit can be determined by set-ting sagitta < 1, that is the upper limit is equal to ~0oP,(see section 3.5) . The spatial resolution obtained withthe 77 algorithm improves when increasing P from 0°to 1 ,,Pt, because the charge division becomes increas-ingly linear .

At large angles, the charge spreads over severalstrips but the information on the cluster position iscontained only in the position and the pulse-height ofthe two edge strips, the head and the tail of the cluster .The AHT algorithm exploits this property of the clus-ters and should thus give the best resolution at largeangles . As we discuss in section 4, the AHT is effectivein the angular range ~Om,n < 0 < Omax . Beyond this up-per limit the (S/N)s,T,P becomes so poor that it is verydifficult to correctly reconstruct the cluster, and thenthe two edge strips . For 19 > ~max, we expect the IS-APT to give better results, even though we should beaware that at these very large angles the efficiency ofthe detector is poor .

Our analysis shows what is the best algorithm to beused for the whole angular range, expect for the inter-

Table 1The algorithm that gives the best spatial resolution as afunction of the track angle ,9

R. Turchetta / Spatial resolution of silicon mecrostrtp detectors

EY

0

ô0

53

Fig . 10 . The spatial resolution as a function of 0 is shown fordifferent algorithms together with the best results . The simu-lated detector has P=50 win, SIN =15 and no intermediatestrip . A similar behaviour is obtained for different configura-tions . The lowest curve is the resolution measured taking thebest algorithm at each angle . The other four curves give theresolution measured with : (A) the 77 algorithm (dashed line) ;(B) the center-of-gravity algorithm (dotted line); (C) the ana-log head-tail algorithm (dash-dotted line) ; (D) the one-sidedanalog head-tail algorithm (upper full line). At 60°, the bestresults are obtained by using the other one-sided analoghead-tail algorithms (the two 1S-AHT give almost equal but

not identical values for the resolution) .

val ~.Pt < r9 < 1imm . Before considering the results ofthe simulation, we can try to find the best algorithm inthis range with some simple considerations . Fori9opt , the charge division is almost linear and most ofthe signal is collected by two strips so that the rlalgorithm computes the same cluster position as theCOG PFA. For P = Amts, most of the charge is spreadover three strips and we can find that in this case theCOG PFA gives roughly the same cluster position asthe AHT does (see appendix B) . Thus, the COG PFArepresents a sort of compromise between the 77 and theART algorithms and we can expect the COG PFA togive the best results in the angular region under con-sideration .

Our expectation about the best algorithm are sum-marized in table 1. They are confirmed by the simula-tion (fig . 10).We have simulated 300 win thick detectors with

several values of the readout pitch and of SIN, and

Angular range Best algorithmSmall angles: 0 < ,9 < a%.,t ,7Intermediate angles : AoPt < 0 < 0_ COGLarge angles : 'amm < ,9 < t9max AHTVery large angles: 0 > am'. 1S-AHT

(but poorefficiency)

54

with zero or one intermediate strips . The simulationwas done for angles 49 ranging from 0 to 80° with a 5°step . Figs . 11 and 12 show the spatial resolution, ob-

c

O

OvoO

Ç40

3S

30

25

20

10

5

R. Turchetta / Spatial resolution of silicon mierostrip detectors

tained with a Gaussian fit to the residue distribution .At each angle, we considered the result of the algo-rithm that gives the best spatial resolution . In fig. 13

E3

cO

OrnN

O

N!Y

40

35

30

25

20

15

10

5

60

O 50

40

20

10

0

I

I0 10 20 30 40 50 60 70 80

~ (In degrees)

I

I

I

I

I

I

I

I

I0 10 20 30 40 50 60 70 80

iY (In degrees)

Fig. 11 . Spatial resolution as a function of 10 . No intermediate strip. The curve are drawn up to Om_ (a) Detector with a 50 Wmreadout pitch. The considered SIN are 25, 20, 15, 10, the lowest curve corresponding to the highest signal-to-noise ratio. (b)Detector with a 100 wm readout pitch. SIN= 20, 15 and 10. (e) Detector with a 150 [Lm readout pitch. SIN= 20, 15 and 10 . (d)

Detector with a 200 wm readout pitch. SIN= 20, 15 and 10 .

the efficiency is shown for detectors with no intermedi-ate strips and a pitch of 50 and 100 Rm .

Let us first consider the curves for detectors with nointermediate strips (fig . 11). Some common features

Ec

cO

O

NCC

E

O

ô

N

32

28

24

20

16

12

8

4

0

40

35

30

25

20

15

10

5

0

0 10 20 30 40 50 60 70 80

19 (in degrees)

0 10 20 30 40 50 60 70 80

R. Turchetta / Spatial resolution of silicon microstrip detectors

19 (in degrees)

Ec

cO

O

O

O

ON

40

35

30

25

20

15

10

5

0

35

4)Elf 30

25

20

15

10

5

0

0 10 20 30 40 50 60 70 80

19 (in degrees)

0 10 20 30 40 50 60 70 80

55

can be observed . The spatial resolution improves whengoing from 0° to 1%Pt , goes through a minimum at ,% Ptand then worsens steadily . The position of the mini-mum depends only on the pitch and not on the signal-

19 (in degrees)

Fig. 12 . Spatial resolution as a function of $. One intermediate strip . The curves are drawn up to an angle i9max . (a) Detector witha 50 wm readout pitch . The considered SINare 25, 20, 15, 10, the lowest curve corresponding to the highest signal-to-noise ratio .(b) Detector with a 100 [Lm readout pitch. SIN= 20, 15 and 10 . (c) Detector with a 150 [Lm readout pitch. SIN= 20, 15 and 10 .

(d) Detector with a 200 Wm readout pitch . SIN= 20, 15 and 10 .

56

to-noise ratio (see section 3.5) . At equal SIN, we getthe best spatial resolution with the smallest pitch, butthe differences are less large for large angle. Forexample, if we consider the curves for SIN= 20, at,g = 5° the spatial resolution for P = 50 win is slightlybetter than 4 [Lm while for P = 100 win it is about 20win. At a9= 15°, where the curve for P= 100 winreaches its minimum, the spatial resolutions are 6.2and a 7.9 win respectively, and for 0 = 60°, they areabout 18 and 20 win respectively . Moreover (see fig .13), at 0 = 60°, the efficiency of the 50 win pitchdetector has already dropped to 95% while that of the100 win pitch detector is still 100% .We can now consider the curves in fig . 12, obtained

for the same detector configurations as those of fig . 11,but with the addition of one intermediate strip . Atsmall angles, this strip improves the resolution of thedetector because it makes the charge division morelinear . For P= 50 win, the charge division is alreadyalmost linear at 0°, and the spatial resolution is at itsoptimum. For the other pitch values, the curves of fig .12 show minima, whose positions are shifted towardssmaller angle with respect to the minima in the curvesof fig . 11 . An analysis of these curves shows that theposition of the minimum can be still given by eq . (13),but where P is taken as the strip pitch, that is half thereadout pitch. At large angle, the charge division ismainly determined by Landau fluctuations and not by

R. Turchetta / Spatial resolution of silicon rnierostrtp detectors

i9 (in degrees)

diffusion, so that the intermediate strip does not playany role, and, for 0 > Omm, we obtain the same spatialresolution with or without any intermediate strip.

We must stress the fact that at equal strip pitch, butdifferent readout pitch, we get the best resolution withthe smallest readout pitch over the full angular range.This can be seen by comparing, for example, figs . I lband 12d, obtained for detectors with the same strippitch (100 win), but with different readout pitches (100and 200 win, respectively) .

5.2. Simulation versus experiments

Up to now only a few measurements of spatialresolution versus track angle are available . Moreover,our analysis has shown that different algorithms givedifferent values of the spatial resolution, so that adirect comparison is possible only for those measure-ments which employ the same position-finding algo-rithm.

In ref. [7] the algorithms presented in this paperwere used in the analysis of the experimental data . Theexperimental curve obtained for a detector with 50 winreadout as well as strip pitch and a SIN= 27 is drawnin fig. 14 (black circles) together with the simulatedvalues (triangles) . In the region between 0 and 10°, theexperimental resolution is constant within the experi-mental error, even though a minimum seems to appear

08

06

04

(b( I

v (in degrees)

Fig . 13 . Efficiency as a function of 0 . No intermediate strip . (a) Detector with a 50 win readout pitch. The considered SIN are 25,20, 15, 10, the lowest curve corresponding to the lowest signal-to-noise ratio . (b) Detector with a 100 win readout pitch. SIN= 20,

15 and 10.

_22 5Ei

15

125

0

75

25

6. Conclusions

R. Turchetta / Spatial resolution of silicon microstrip detectors

10 20 30 40 50 60

19 (in degrees)

Fig . 14. Comparison between simulation (triangles) and exper-imental data (circles).

between 5 and 10°. This agrees with the simulatedresults where a slightly pronounced minimum is foundat 0 = 5°. For angles larger than 10°, both simulationand experiments show a constant decrease in resolu-tion . The simulation results are systematically betterthan the experimental ones, and this is probably due tothe fact that the simulated conditions are ideal (nonoisy or dead strip, ideal alignment, etc.).

Our simulation well reproduces the trend of experi-mental data in this case and we hope that in the nearfuture more experimental data will become availablefor comparison.

Our analysis shows the importance of the choice ofthe position-finding algorithm in the optimisation ofthe spatial resolution of a microstrip detector . Fromthe point of view of the choice of the best algorithms,four angular domains can be distinguished . At smallangle, where diffusion is important, the 77 algorithm,that takes into account non-linearities in charge divi-sion, gives the best performances . At large angle, asimple linear (the analog head-tail) algorithm that usesonly the signal of the two edge strips in the clustergives the best results . In the intermediate region be-tween the domains of small and large angles, thewell-known center-of-gravity algorithm represents a

compromise between the v1 and the AHT algorithmsand gives the best results. For very large angles, wherethe detector efficiency is substantially lower than 100%,a simplified version of the AHT PFA, the one-sidedanalog head-tail algorithm, has demonstrated bestperformances .

In terms of detector design, for small angle it isclear that at equal SIN the pitch should be kept assmall as possible in order to optimise the spatial reso-lution . At large angle this is still true, but a compro-mise must be found between good spatial resolutionand high efficiency . Eventually, the requirement ofhigh efficiency sets a limit on the pitch and, thus, onthe spatial resolution .

The algorithms presented here have been devel-oped and analysed in the case of microstrip detectors,but they can be applied to any segmented silicondetector . For example, the non-linear algorithm hasalready been successfully applied to silicon pixel detec-tors [23] . The algorithms for inclined tracks could alsobe used, thus giving a first approach towards the opti-mum algorithm in pixel detectors .

Appendix A: Determination of the function R(x)

We suppose that the spatial resolution o-DET isobtained by averaging over the spatial resolution as afunction of the impact point, i.e.

O'DET )z

~2

_P PIn the case we are considering it is

(O'D

T )z= (

ESC

z) f 1[R(x)]z dx

and we want to determine the function R(x) for whichthe integral is minimum.

We express the resolution function as the sum oftwo terms:R(x) = 1 +g'(x),

where Ox) is the derivative of an arbitrary functiong, with the condition that

g(0) = g(1) .This condition is not restrictive at all, because thefunction f is usually symmetric with respect to 1/2.

We thus have

f i [R( x)] 2 dx= fi [+g'(x)]2 dxo

=1 +2fotg'(x) dx+ fo i [g'(x)] 2 dx

= I + fol[g,(x)]z dx > 1,

57

58

where the equality holds only in the case where R(x)1 .In the case we are considering

1 dNl

0, d7lNR(x) = S a( 77)

Since a(rl) is flat and quite close to 1, we must havedN/d?7 = constant .

Appendix B. The center-of-gravity (COG) algorithm at

a = dmin

We consider the situation 0 = Dm�, . In this case (seeeq . (19)) the sagitta is equal to 2 (in pitch units) and wecan consider that the signal spreads over three strips,strip 0, 1 and 2. The cluster position calculated withthe COG PFA is

x~OG _ XOSo +x i Si +x252S,+2S2

P

SO +S, +S2

SO +S 1 +S2 '

while that measured by the AHT PFA is

xAHT __ XO +X2

S' - SOS'-SO

P

2

2S,

2S,9 '

where S, is the pulse-height measured on theWe can say that for the internal strip (strip no .

we have St = S,y, while for the total charge So , + SiS2 = 2S,9 . Thus

XCOG

S1 +2S~

So ~ + Si + S2 + S2 - So~

P SO +S i +SZ So +Si +S'

- 1 + S2-SO S2 - So - xAHT

S,,+ Si + Sz2S,y

P '

that is the COG and the AHT calculatesame cluster position .

Acknowledgements

R. Turchetta /Spatial resolution of silicon nucrostrcp detectors

tth strip.1)

roughly the

Many fruitful discussions with Peter Weilhammer,CERN, are at the origin of this work . Prof . M. Schaef-fer, LEPSI, has always warmly supported and encour-

aged this work . I also wish to acknowledge M. Tyndel,RAL, project leader of the DELPHI Microvertex Up-grade, for the kind permission of using the experimen-tal data taken in the DELPHI beam test . D. Husson,LEPSI, and R. Roy, LGME Strasbourg, carefully readand commented the draft of this paper.

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