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Optimization of silicon detector layout andassociated front-end electronics for timing

performance of a silicon PET through simulationAndrej Studen, Neal H. Clinthorne, Harris Kagan, Marko Mikuz

Abstract—There are indications that PET imaging may profitby incorporating position sensitive silicon detectors with intrinsicexcellent spatial resolution into the scanner setup [1]. However,poor timing performance of detectors can compromise mentionedbenefits.

Our group is developing a PET probe with silicon detectors.Our detectors come in two varieties, both with square pads anda thickness of 1 mm, but the pads are either 1 mm or 1.4 mm insize. The signals are amplified through VATAGP brand ASICs.For the timing signal, the ASIC provides a charge-sensitivepre-amplifier with a fast (75-150 ns shaping time) CR-RC2 shaper,and a fixed-level discriminator to generate a trigger signal.

There are three contributions to timing resolution: the time-walk, the jitter and the broadening related to depth of interaction.The paper concentrates on the last contribution, assuming aviable time-walk compensation [2] and negligible jitter for signalslarge compared to the discriminator threshold. The interactionof photons causes local depositions of ionization. The shapeof the signals vary with impact position. Through simulation,we estimated the impact of variation on timing resolution. Weevaluated the present systems and estimated improvement foralternative detector and electronics designs.

The simulation consists of four major steps. First, a particletracking tool (GEANT4 [3]) is employed to track the interactionsof a 511 keV photon in a silicon detector. A TCAD suite is usedto determine the electric and the weighting field in a detectorfor a given electrode and doped implant arrangement. Next, acharge propagation model [4] using TCAD calculated fields isemployed to determine the paths of secondary ionization andtheresulting charge induced on electrodes, and finally, a modelofthe first stage of the signal amplification - the pre-amplifier, theshaper and the discriminator - is used to generate the timingsignal.

Index Terms—silicon pad detectors, timing resolution, medicalimaging.

I. I NTRODUCTION

SILICON is a promising detecting material for a PETapplication. It offers great energy and spatial resolution,

insensitivity to magnetic fields, roubstness and compactnessneeded for dense probes. However, the stopping power is only2% of 511 keV photons in a mm of silicon, with most of thoseundergoing Compton scattering rather than photoabsorbiton.The obvious remedy is an increase in sensor thickness. Theionization caused by (mostly) Compton electrons will be localin nature – the range of recoil electron with a kinetic energy

A. Studen and M. Mikuz are with Jozef Stefan Institute, Ljubljana, Slovenia.M. Mikuz is also with University of Ljubljana, Ljubljana, Slovenia.

N. H. Clinthorne is with the University of Michigan, Ann Arbor, MI, USA.H. Kagan is with the Ohio State University, Columbus, OH, USAA. Studen can be reached atandrej.studen@ijs.si.

of 300 keV is only about 300µm – so the signals inducedon (surface) electrodes in thick(1 mm) detectors will varydepending on the depth of interaction1. This variation willbe reflected in an additional depth-related broadening of thetiming resolution.

This paper reports the results of the simulation of thesignals in a 1 mm thick p+nn+ detector in a pad geometry.The following section describes the steps taken for realisticcalculation of signal shapes in a silicon detector - first thesimulation of the recoil electron track, then the simulation ofthe signal current pulse at each point along the track (and inthe detector) and finally, the virtual electronics applied to theraw current pulse. Next section shows the results: agreement ofsimulation with the data, simulated variation with bias voltage,shaping time, pad size, shaper order, secondary threshold andreadout strategy. We also introduce an alternative detectordesign with segmented readout of the backplane and compareresults to conventional detectors.

II. T HE SIMULATION

The simulation was realized in three very distinct stepswhich are explained in the following subsections:

A. Recoil electron track

Geant 4 [3] with low-energy scattering extension [8] wasused to generate tracks of recoil electrons for interactions of511 keV photons from a point source placed 5 cm in frontof the 1 mm thick detector. Secondary particles generationthreshold of Geant 4 was set to its minimum at 1 keV. TheFigure 1 shows a typical track in a sensor. Each sphererepresents a (measureable) charge deposit by the track of therecoil electron in a 1 mm3 cube. This illustrates the locallityof the photon interaction and the resulting diversity of inucedsignal currents.

B. Induced signal

The energy deposition of the recoil electron is converted topairs of electrons and holes. Each pad is essentially a diode,and the electric field caused by the applied reverse bias splitsthe electrons and holes. Holes drift towards the top (p-type)pad electrode which is connected to the electronics and heldat ground potential. The electrons drift to the backplane, held

1This fact is often used to deconvolute the signal depth basedon signalshape [5]–[7]

2

Fig. 1. Illustration of the recoil electron track in a 1 mm3 cube of silicon fora 511 keV incident photon. This track had a total deposit of 150 keV. Eachsphere represents a measureable deposit of energy, with radius of the sphereproportional the amount of deposition.

at positive potential. Both types of carriers induce signalonthe top electrode. The simulation is split in two parts - driftand induction.

• DRIFT. A series of steps move the charge from pointof creation (Geant4 energy deposit) to the top or bottomelectrode. Each step is a time shift∆t, during which thecarrier moves for a∆x equal to the sum of electric fieldE(x) related drift and diffusion:

∆x = µE(x)∆t + ∆xD, (1)

where ∆xD is drawn from a 3D gaussian distributionwith a mean of 0 and a variance ofD∆t in all threedimensions.D is the diffusion coefficient andµ is thecarrier mobility, and both are related through Einsteinequation. The mobility is parametrized as in [9] toaccount for velocity saturation at high electric fields.For a pad detector, the electric fieldE depends ondepth only and is linear for voltages aboveVFD. Fordetector with backplane divided into strips, a 2D fieldwas calculated on a finite grid (2.5µm step) with directmatrix inversion.

• INDUCTION. Given a series of steps, the carrier velocitiesv(x) = ∆x/∆t are obtained for each step from (1).The pulse shape is determined by the Shockley-Ramotheorem:

I = qv(x) · W(x), (2)

assuming a known weighting fieldW(x). The calculationof the later requires a 3D calculation in a pad sensor,which was realized through TCAD [10] simulation tool.

For a strip shaped electrode, a finite grid method wasused and a 7 strip detector was modelled.

• SIGNAL MATRIX . There is a randomness in calculationof carrier track (1) due to diffusion. Since there areapproximately 300 carriers generated per 1 keV of theenergy deposit, we tracked 100 drifts per packet for arealistic signal. As that would consume a lot of CPU time,a signal matrix of pre-calculated signals was prepared. A3D grid was mapped onto the sensor and the signals werecalculated at grid interstitions, with grid spacing between50 and 70µm (depending on the pad size). Interpolationwas used for intermediate points.

C. Electronics

Due to expected high count of channels in a silicon detectorfor a PET application, we concentrated on a simple electroniccircuits, such as the one present in VATAGP series by IDEAS[11]. Each electronic channel was equipped with a charge-sensitive preamplifier and a CR-RCn shaper. The shape wasdetermined with a convolution, using Simpson method forintegrating the product:

S(T ) =

∫ T

0

s(t)K(T − t)dt, (3)

with s(t) electronics input, K(t) the kernel of the electronicsand S(t) the output signal. A leading edge discriminatorwas used as a trigger signal generator, the threshold wasset to 15 keV. There was only a weak dependence of thetiming performance when threshold was varied between 10and 20 keV.

D. CPU usage

The grid infrastructure [12] was used for signal computa-tion. The heavy part, the matrix caclulation took about 8000CPU minutes for a single voltage/detector pair, using∆t of200 ps, 100 drifts per interstition and 80k interstitions.

III. R ESULTS AND COMPARISONS

A. The weighting field

Figure 2 shows profiles of the weighting field in a paddetector for different pad sizes. The red curves correspondtothe 1 mm pads and the black to pad size of 1.4 mm. A constantweighting field (equal to that of the backplane electrode) isshown for comparison.

B. Signal shape

Figure 3 shows pulse shapes obtained in a pad detector.Extreme casses are shown at a low voltage – 200 V comparedto 150 V full depletion voltage. For events near the backplane(depth=1 mm) the holes have to drift through the wholedetector before they reach the high weighting field (Figure2 hence their collection time is long. Increasing the reversebias does not remove the effect completely (blue curve on theFigure 3 for a 500 V reverse bias).

3

m]µdepth [0 100 200 300 400 500 600 700 800 900 1000

]-1

wei

gh

tin

g f

ield

[cm

0

10

20

30

40

50

60

70

Weighting field in a pad detector -- profile along depth

1 mm pad, pixel center

1.4 mm pad, pixel center

1 mm pad, pixel corner

1.4 mm pad, pixel corner

backplane

Fig. 2. The graph of the weighting field as a function of depth.The blackcurves correspond to a pad size of 1.4 mm, the red curves to a pad size of1 mm. The dashed lines give the field in the corner of the pad/pixel, the solidlines at the pad center. The dotted line shows the constant weighting field ofthe large back-plane electrode.

time [ns]0 20 40 60 80 100 120 140

curr

ent

[pA

]

0

2

4

6

8

10

12

14

16

Pulse ShapesAt 200 V, per carrier pair

1 mm pad, top interaction

1 mm pad, bottom interaction

1.4 mm pad, top interaction

1.4 mm pad, bottom interaction

1.4 mm pad, bottom inter. 500 V

Fig. 3. Current pulses obtained on pad electrodes for a pad detector (thicknessis 1 mm) for a drift of a single electron-hole pair. Red curvescorrespondto 1 mm pad size, black to 1.4 mm; dashed curves to interactions at thedetector backplane (depth=0.9825 mm), solid curves for interactions closeto the readout pads (depth=0.0125 mm). The simulated detector voltage wasU=200 V, except for the blue curve, calculated for a 1.4 mm paddetectorat 500 V bias and interaction at the backplane. Extreme casesshown forillustrative purposes.

C. Comparison to measurement

The simulated results were compared to the data [13]. Thedata were taken with a 1.4 by 1.4 mm2 silicon detector, 1 mmthick, with a VATAGP3 ASIC used at the detector’s front end.In VATAGP3, the trigger is performed as a leading edge ofa CR-RC2 shaped output of the charge-sensitive preamplifier,the shaping time is approximately 200 ns. The threshold inmeasurement and simulation was set to 30 keV. A positronsource (giving back-to-back 511 keV photons) was used anda fast LYSO/PMT assembly gave a timing reference signal.Figure 4 shows (appropriately delayed) trigger time versusmeasured energy of the recoil electron. On the same plot, thesame graph is given for a comparable number of simulatedevents; the same detector geometry, voltage and virtual elec-tronics were used as for the measured events. Figure 5 shows

trigger time [ns]0 50 100 150 200 250

ener

gy

[keV

]

0

50

100

150

200

250

300

350

400

450

500

Energy vs. trigger time scatter plot

Measurement

=200ns, U=200V, thr=30keVτ,2Simulation(CR-RC

Fig. 4. A scatter plot of the 511 keV photon interactions in 1.4 by 1.4 mm2

pad and 1 mm thick silicon detector with respect to the trigger time and theenergy of the recoil electron. In red, the graph is shown for the simulatedevents with the bias voltage of 200 V (VF D=150 V), the leading edgethreshold set to 30 keV, a CR-RC2 shaper and a shaping time of 200 ns,approximately matching the conditions set to the measured data (black) at200 V and a single channel of the VATAGP3. An appropriate constant delaywas set to the measured data to match them to the simulation.

trigger time [ns]0 50 100 150 200 250

even

ts [

1 n

s b

ins]

0

20

40

60

80

100

120

140

160

180

200

Projection for secondary threshold=100 keV

Measurement

=200 ns)τ, 2Simulation (U=200V, CR-RC

(); scaledδSimulation, elec. only -

Fig. 5. The distribution of the trigger time for 511 keV photon interactionsin 1.4 by 1.4 mm2 pad and 1 mm thick silicon detector for events withrecoil electron energy above the secondary threshold of 100keV. The blackhistogram shows the measured data and the red the simulation; the blue thedistribution of trigger times for an ideal interaction (aδ(0) current pulse).The settings are the same as in Figure 4. Observe the good matching betweendata and the simulation.

the variation of the trigger time for a subset of events whereenergy of the recoil electron exceeded the secondary thresholdof 100 keV; the secondary threshold was set to compensatefor the time-walk of the trigger. The agreement between themeasured data (red) and the measurement (black) is good. Thediscrepancy can be attributed to the limited accuracy of someof the parameters of the simulation – most notably the fulldepletion voltage and the shaping time of the shaper. Alsoshown on the Figure 5 is the distribution for an ideal detectorsignal – a current pulse in the form of aδ(t = 0) function.This illustrates the constribution of time-walk only, withoutdepth or jitter related broadening.

4

trigger time [ns]0 20 40 60 80 100

even

ts/0

.5 n

s b

in

0

200

400

600

800

1000

1200

1400

Timing distribution with quantiles

Q0 Q

1Q

2 Q3

Q4

Q5

Q6

Q7

Q8

At 200V, CR-RC

=150 ns, 1 mm padsτ

Fig. 6. The distribution of the trigger time for simulated 511 keV photoninteractions in a 1 mm and 1 mm thick pad detector at a bias voltage of200 V, and a CR-RC shaping with a shaping time of 150 ns and a secondarythreshold of 100 keV. The lines delimit the one-eighth quantiles.

Q4-Q0 Q5-Q1 Q6-Q2 Q7-Q3 Q8-Q4

tim

e d

iffe

ren

ce [

ns]

0

10

20

30

40

50

60

70

80

Optimum timing window

Fig. 7. The timing windows required to collect half of the simulatedinteractions for distribution depicted in Figure 6, showing the minimumwindow for the first half of the events.

D. Time window studies

As a figure of merrit we chose the time window requiredto collect half of the interactions in silicon. This correlatesnicely with the requirements of a PET detector, since the timewindow limits the activity of the source the detector can beexposted to.

We assumed the pads of the detector to be connencted to anelectronics with a leading edge trigger on the output of a CR-RCn shaper. A secondary threshold was set to compensate forthe time-walk. The resulting timing distribution for a certainsetting of the shaper order, the bias voltage, the shaping timeand the pad geometry is shown in Figure 6. Superimposed onthat histogram are the one-eighth quantiles (Q0 correspondsone thousandth and Q8 to all but the last thousandth of events).To collect a half of the events, an interval stretching over 4of the quantiles determined above is required. Figure 7 showsthat such a window is minimized by simply selecting the firsthalf of the events.

The width of the window was then compared for differentpad geometries, shaping times, readout strategies, bias voltages

voltage [V]200 250 300 350 400 450 500

[n

s]0

-Q 4Q

0

2

4

6

8

10

12

14

16

18

20

Voltage, shaping time and pad size study

=150 nsτ1.4 mm pad size,

=75 nsτ1.4 mm pad size,

=150 nsτ1 mm pad size,

=75 nsτ1 mm pad size,

Single pad readoutCR-RC2nd thr=100 keV

=75 nsτtime-walk only, =150 nsτtime-walk only,

Fig. 8. The optimal time-window (Q4-Q0) dependency on voltage, shapingtime and pad size. For all calculations, CR-RC shaper was used and thesecondary threshold was set to 100 keV. The time-walk contribution isobtained by looking at an ideal detector response (δ()).

and shaper orders. The following figures compare the optimalwindow width for variation in some of the above mentionedparameters.

TABLE ITHE RELATIVE CHANGE ACHIEVED BY VARIATION IN LISTED PARAMETER.

THE REMAINING PARAMETERS WERE SET AS INFIGURE 8. THE RANGE

SHOWN IN WINDOW(A)/WINDOW(B) COLUMN IS COVERED BY DATA

POINTS SHOWN IN THE GRAPHS INFIGURE 8.

parameter value A value B window(A)/window(B)

bias voltage 500 V 200 V 0.59-0.62

shaping time 75 ns 150 ns 0.69-0.72

pad size 1.4 mm 1 mm 0.79-0.84

Figure 8 shows the dependence of the timing window whenvoltage, shaping time and pad size are varied. The voltageincrease from 200 to 500 V reduces the required windowfor about 40 %. Halving the shaping time reduces the timingwindow for 30 % only for selected portion of events. Andfinally, the pad size reduction of 30 % increases the timewindow, but only for about 20 %. The above numbers aregiven in Table I and vary very little among the eight possiblepairs in the figure. So a 1 mm pad detector with a 75 ns shapingtime will perform approximately 10 % better than a 1.4 mmpad detector with 150 ns shaping time for all voltages, a factthat can be seen in the figure. Time-walk, obtained by lookingat an ideal detector response –δ() is comperably small, givingwindow width of 1.2 ns for 75 ns shaping and 2.3 ns for 150 nsshaping.

Figure 9 shows the time-window dependency on the sec-ondary threshold, shaper order and readout strategy. For thisstudy, the pad size of 1 mm was chosen, the bias voltageof 500 V and the shaping time of 75 ns2. A word on thereadout strategy – it is advantageous to make the Ramo field asconstant as possible to remove depth of interaction dependency[2]. This is achieved by summing the central and all 8 adjacent

2Using the results from Table I the approximate values at different param-eter settings can be estimated.

5

secondary thresholde [keV]50 60 70 80 90 100

[n

s]0

-Q 4Q

0

2

4

6

8

10

12

14

16

Secondary threshold, shaper order and readout strategy

CR-RC, Single pad readout

, Single pad readout2CR-RC

CR-RC, 9-pad readout

, 9-pad readout2CR-RC

Fig. 9. The optimal time-window (Q4-Q0) dependency on the secondarythreshold, order of the shaper and readout strategy. The other settings of thecalculation were: U=500 V,τ=75 ns, 1 mm pad size.

pads, making the Ramo field (only approximately in a realdetector) constant.

TABLE IITHE RELATIVE CHANGE ACHIEVED BY VARIATION IN THE LISTED

PARAMETER. THE REMAINING PARAMETERS WERE SET AS INFIGURE 9.THE RANGE SHOWN IN WINDOW(A)/WINDOW(B) COLUMN IS COVERED

BY DATA POINTS SHOWN IN THE GRAPHS INFIGURE 9.

parameter value A value B window(A)/window(B)

shaper CR-RC CR-RC2 0.72-0.77

secondary threshold 100 keV 50 keV 0.81-0.83

readout strategy 9-pad single pad 0.55-0.58

The most dramatic improvement is achieved by summingthe pads, reducing the window size to approximately half itssize (exact values in Table II). However, one must consider thejitter trade-off; assuming that the input capacitance dominates,the jitter would increase three-fold for such an arrangement,possibly dominating the timing resolution3. Where jitter isconsidered, the higher order shaping is more efficient in noisefiltering (reduction rate of 174/190 for CR-RC2 versus CR-RC [14]). However, Figure 9 and Table II show that this gainis overwhelmed by the depth related broadening, with CR-RC2

shaper requiring a 30 % larger time window. The time-walkcontribution is not as pronounced - only 20 % is gained whenthe secondary threshold is increased from 50 to 100 keV.

The backplane offers a very good timing signal, because ofthe truly constant weighting field. However, it also has a verylarge capacitance and hence, plenty of jitter. We looked at anintermediate solution where the backplane is fragmented intosmaller electrodes with bearable capacitance. We simulateda 1 mm thick silicon detector with the top side fragmentedinto 1 mm pads (p+ implants on ann-material), and thebottom side segemented into 0.4 mm wide strips at a pitchof 0.5 mm (n+ implants, p-stops implicitly assumed), asschematically shown in Figure 10. The bottom electrodes were

3Assuming a 75 ns shaping time, a noise of 1000 electrons and thesecondary threshold of 100 keV the jitter for 511 keV photon interactionspectrum has an RMS of approximately 1.6 ns per channel

Fig. 10. Schematic drawing of the replacement of the solid backplane (left)with a strip-shaped electrodes (right) for a bottom timing readout.

top readout bottom strip bottom 3-strip solid backplane ()δ

[n

s]0

-Q 4ti

me

win

do

w Q

0

1

2

3

4

5

6

7

8

Bottom side readout -- different options

6.56 ns

5.35 ns

3.34 ns2.95 ns

1.17 ns

=75 ns, 2nd thr=100 keV,τCR-RC, U=500 V,1 mm pad size, strip pitch 0.5 mm

Fig. 11. The optimal 50 % time window (Q4-Q0) for two detector geometries.Top readout and solid backplane correspond to a normal pad detector (left-hand side of Figure 10), while the bottom strip and bottom 3-strip readoutcorrespond to the strip readout of a modified detector with the backplanesegmented into 0.4 mm wide strip electrode on a 0.5 mm pitch. The bottomelectrodes were at +500 V potential relative to the top electrodes. Theelectronics used for all signals was a charge-sensitive preamplifier with aCR-RC shaping and a shaping time of 75 ns, the secondary threshold wasset to 100 keV. Theδ() corresponds to an ideal current pulse and showscontribution of time-walk only.

set to +500 V potential relative to the top electrodes and theelectronics used was the same as for the top readout – chargesensitive preamplifier with a CR-RC shaper and a shapingtime of 75 ns. The top electrodes would be used for energyand spatial information while the bottom electrodes providedtiming information.

Figure 11 shows that a single 0.5 mm strip gives only aslight improvement over the normal top readout. However asum of 3-strips – effectively equal to a strip pitch of 1.5 mm– already approximates the solid backplane (jitter neglected)performance. Theδ() function shows contribution of time-walk only.

IV. SUMMARY AND CONCLUSIONS

The paper describes the simulation of the timing propertiesof the interactions of 511 keV photons in thick p+nn+ padsilicon detectors. For the trigger signal, a leading edge triggeron a CR-RCn shaped and preamplified raw signal from thesensor was used, the threshold fixed at 15 keV. The calcula-tions were made for 1 mm thick sensors with a pad size of 1and 1.4 mm, the bias voltage of 200 to 500 V, shaping time

6

of 75 and 150 ns, shaper orders n=1 and n=2. As a time-walkcompensation, a secondary threshold on the recoil electronenergy was used and varied between 50 and 100 keV. As afigure of merrit, the time window neccesary to collect half ofthe events was choosen. Results were compared to collecteddata and reasonable agreement was found.

The results can be grouped by the parameter which wasvaried:

• VOLTAGE: Increasing the sensor voltage always helps,with reduction to half the original time window as voltageis increased from 200 V to 500 V, at a full depletionvoltage of 150 V.

• SHAPING TIME: The reduction in shaping time from150 ns to 75 ns still helps, but its effect is far less than onehalf. In most simulated cases, the charge collection time isalready appreciable, giving a reduction of approximately30 %.

• PAD SIZE: Smaller pad sizes compromise the timingresolution, since the weighting field is contained closerto a smaller pad and the depth related signal variation ismore pronunced. The effect adds about 20 % to the timingwindow when pad size is reduced from 1.4 to 1 mm.

• SHAPER ORDER: Using high order shapers is misleading.Although the electronic noise is reduced, the additionalbroadening due to a weaker slope of the shaper outputovercomes the apparent benefits.

• SECONDARY THRESHOLD: The effect is not pronunced.A 20 % reduction when going from 50 to 100 keV.

An additional study was made in adjusting detector geom-etry and readout strategy. However, in those cases one mustconsider additional circumstances, e.g. the effects of thejitteras the capacitance of the readout channel is changed. Thefollowing results are more of a trade-offs which have to beconsidered at specific total detector dimensions and involvedcomplexities:

• 9-PAD READOUT: To improve the timing performance,sum of central and 8 adjacent pads was compared toa single pad readout. The reduction of depth relatedbroadening is a solid 50 %, but increase in jitter is(naively) 3-fold.

• BOTTOM STRIPS: Producing a detector where strips onthe bottom side are used for the timing information onecan gain another factor of 2 on the timing window.However, the detector size will determine the amplifierinput capacitance and the resulting jitter, which maycompromise the benefits. Also, there is an additionallevel of complexity involved since: a) the bottom side ofthe detector has to be processed and b) an AC coupledelectronics over 500 V potential has to be used.

ACKNOWLEDGMENT

The research for the paper was performed within the EF-7sponsored Madeira project.

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