Simulations and Test Results of Large Area Continuous Position

Diamond & Related Materials 65 (2016) 115–124

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Simulations and Test Results of Large Area Continuous Position Sensitive Diamond Detectors M. Ciobanu a,⁎, M. Pomorski b, E. Berdermann c, C. Bunescu a,d, H. Comişel a, V. Constantinescu a, M. Kiš c, O. Marghitu a, M. Träger c, K.-O. Voss c, P. Wieczorek c a

Institute of Space Sciences, Bucharest, Romania CEA-Saclay, France GSI Helmholtzzentrum, Darmstadt, Germany d Jacobs University Bremen, Bremen, Germany b c

a r t i c l e

i n f o

Article history: Received 27 November 2015 Received in revised form 9 February 2016 Accepted 21 February 2016 Available online 2 March 2016 Keywords: diamond detector polycrystalline diamond large area continuous position

a b s t r a c t Continuous Position Sensitive Diamond Detector (CPSDD) development started by using the single crystal (sc) diamond material. The intrinsic high detection efficiency of sc diamond, providing a high Signal to Noise (S/N) ratio, allowed the full testing of CPSDD with α-particles. However, due to the size limitations of sc diamond, the development of Large Area CPSDD (LACPSDD) naturally evolved towards the use of polycrystalline (pc) diamond material, produced by chemical vapor deposition (CVD). The charge generated by the particle or radiation impact is collected through diamond like carbon (DLC) layers and associated metallic electrodes deposited on the sides of the pc diamond plate. The incident particle position can be obtained via charge division measurement by using charge sensitive amplifiers (CSA) connected to each electrode. In this paper we report the improvement in LACPSDD design by showing results obtained for two pc diamond detector (pcDD) structures. The first pcDD has a DLC layer with four electrodes at the corners of the front side, whereas the back side is fully metallized. The second pcDD has DLC layers on both sides of the detector plate, each equipped with two parallel electrode strips, along the x and y axis, respectively. Experimental results on the first pcDD showed an ion rate limitation, caused by the increase in the detector time constant (because of the larger detection area), and a low S/N ratio, due to the specific reduced signal associated with low Charge Collection Efficiency (CCE) of pc diamond. Subsequently, by using an optimized electronics and a better pc diamond (higher CCE), the second pcDD shows a higher S/N ratio, as well as a lower time constant. This paper presents simulation results on the time constant and an analytical evaluation of the S/N ratio, which serve to optimize the pc LACPSDDs. We also show experimental test results with α-particles, as well as 54Ni (1.7 AGeV) and 12C (11.4 AMeV) ion beams. Prime Novelty Statement: This paper presents the first large area continuous position sensitive diamond detectors implemented on polycrystalline CVD diamond material for single ionizing particle detection. © 2016 Elsevier B.V. All rights reserved.

1. Introduction By chemical vapor deposition, one can obtain diamond materials with reduced atomic impurities and structural defects, appropriate for particle detectors with parallel plate geometry, and comparable to commercial silicon-diode detectors in terms of energy resolution. The main advantage of the scDD and pcDD with respect to the Si-diode detectors is related to the superior timing and radiation hardness properties [1,2]. The first CPSDD was made on a sc CVD diamond bulk material ⁎ Corresponding author. E-mail addresses: [email protected] (M. Ciobanu), [email protected] (M. Pomorski), [email protected] (E. Berdermann), [email protected] (C. Bunescu), [email protected] (H. Comişel), [email protected] (V. Constantinescu), [email protected] (M. Kiš), [email protected] (O. Marghitu), [email protected] (M. Träger), [email protected] (K.-O. Voss), [email protected] (P. Wieczorek).

http://dx.doi.org/10.1016/j.diamond.2016.02.016 0925-9635/© 2016 Elsevier B.V. All rights reserved.

(4 × 4 × 0.2 mm3), with a diamond like carbon (DLC) thin film used for the fabrication of the resistive sensing electrodes [3]. The scDD high efficiency provides a high enough S/N ratio which allows to test the CPSDD with α-particles. For the 5.5 MeV α-particle, the energy loss is reported to have a linearity of about 99% and the estimated position resolution is ~30–40 μm, depending on the interaction point (measured in ambient air conditions). The disadvantage of the scDD is its limited size, of only a few mm2. Therefore, the LACPSDD developed towards the use of pc diamond, which can be produced in significantly larger plates by using the CVD technique. The few available papers to date presenting pc position sensitive detectors concentrate on far UV and X rays beam profile studies, where the pcDD has a high responsivity. These 1D or 2D CPSD’s are made of pc CVD thin films and metallic resistivesensing electrodes, in the current-integrated readout mode [4–7]. This paper presents the development of two different LACPSDDs, based on the pc diamond material, for position detection applications

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of single ionizing particles. The first pcDD, hereafter called as fourcorner (or tetra-lateral) structure, was developed in 2012 on a 30 × 30 × 0.35 mm3 pc diamond plate. The second, duo-lateral pcDD structure, was developed in 2014 on a diamond plate of 20 × 20 × 0.18 mm3. For a comparison between different material properties, in Fig. 1 we show the energy spectra of α-particles measured with scDD (≠ 480 μm), pcDD (≠ 210 μm), and DoI DD (≠ 300 μm). For α-particle source we used 241 Am which deposits for each event the same energy, ~ 4.95 MeV, and generates in the detector bulk a charge of ~ 62 fC (the particles are stopped in the detector material). The scDD spectrum is a narrow Gaussian line, located near the charge equivalent of the deposited energy (CCE ~ 100 %), and limited in resolution by the CSA noise. The DoI DD spectrum is slightly broader (CCE ~ 88 – 95 %), while the pcDD spectrum, corresponding to a low CCE (~ 12 – 37 %), has an asymmetric pffiffiffiffiffiffi specific shape (with mean value λP and standard deviation λP ), with the most probable value lower than the mean value. Due to the presence of grain boundaries, causing an in-homogeneous charge trap distribution in the pc diamond, the CCE varies significantly with the interaction position and the position detection is accordingly very challenging. Further details of the three pcDDs referred above are presented in Section 2. Section 3 addresses the position reconstruction for the fourcorner structure and presents experimental results obtained for the 54 Ni (1.7 AGeV) ion beam. Section 4 shows simulation results for a simplified detection structure, based on the charge diffusion in the DLC layers, which points out the possible miss-match between the detector time constant and the integration time of the electronics. An analytical evaluation of the S/N ratio, together with the implications on the position reconstruction error, are presented in Section 5. Test results of the duo-lateral structure - which benefited from upgraded electronics, following the simulations on the detector time constant, as well as from higher S/N ratio, due to the enhanced CCE of the pc diamond - are presented in Section 6. The conclusions and prospects are summarized in Section 7.

2. Continuous position sensitive diamond detectors The diamond material intrinsic high damage threshold made it a proper choice for particle detector sensors. The specific high breakdown field, allowing the polarization of the detector with strong electric fields, drives the secondary particles (electrons and holes produced by ionization) towards the collecting electrodes. The induced charge on the electrodes, obtained by the integration of the induced currents, can be further used to infer the particle impact position on the diamond

Fig. 1. Energy spectra of α-particles measured with a scDD, pcDD, and CVD Diamond-onIridium (DoI) DD; the last peak represents an artifact due to the control signal from a pulse generator.

plate. Basically, the reconstructed position, along the x and y axis, is obtained by charge division between the left/right and up/down electrodes (in duo-lateral or four-corner configuration). The charge fractions are normalized to the total charge and scaled by the length of the detector along the respective axis. While a detailed description of the reconstruction procedure for the two pcDD is included in Sections 3 and 4, here we briefly discuss known hardware means to reduce the reconstruction errors on the four-corner structure. Previously, the four-corner configuration was implemented on Si base materials and preferred because of using a single resistive layer [8–13]. Such studies reported a systematic pin-cushion distortion of the reconstructed position [9,11,13]. A minimization of this distortion is obtained by using a resistive contour strip, deposited along the entire edge of the detector plate [9]. By considering a main DLC layer of surface S resistivity RM S and a superposed supplementary DLC contour layer of RL resistance per unit length, the Doke condition [9] for minimum distortion errors is given by: RDoke ≡

RM S L  RSL

N10;

ð1Þ

where the superscripts M and S denote the main and respectively the supplementary DLC layer, while the subscripts S and L indicate the surface and linear resistivity. By considering the general case, in which the M and S resistive DLC layers have different resistivities, ρM and respectively ρS, Eq. (1) can be written as: RDoke ≡

ρM T S W S ; ρS T M LS

ð2Þ

where ρ, T, W, and L denote the resitivity, thickness, width and length of the respective region, indicated by the corresponding superscript (M and S). According to Eq. (2), in a fixed geometry and with approximately equal thicknesses of the two DLC regions, the remaining parameters to control the Doke condition are provided by ρM and ρS. Thus, the contour strips should have a lower resitivity (metal deposition) as compared to the main DLC layer (in particular for square detectors, where W = L). In Section 3 we show that the reconstructed position for the four-corner configuration has a similar pin-cushion distortion, partially caused by the failure to fulfill the Doke criterion. The four-corner LACPSDD structure, hereafter labeled as DD1, can be easily implemented on the substrate side, which is usually polished. Our first LACPSDD, shown in Fig. 2a, was made on a 30 × 30 × 0.35 mm3 pc diamond plate obtained by CVD [14,15]. The resistive layer is made on the substrate side and the growth side is fully metalized. After cleaning the diamond surfaces by hot acid, the DLC layers and metal electrodes (Ohmic contact behavior) were deposited by RF-sputtering in Argon plasma, by means of a laser cut stainless steel shadow mask. Reference [3] demonstrates that the DLC material is very robust, the metallization of DLC is reliable and can be used to cover the interval RS =3–30 kΩ/ ⃞ . An industrial quality graphitic target was the source of C atoms for the DLC deposition, performed under the following conditions: base vacuum 10–5 mbar, RF power 150 W, Argon flow ~ 42 sccm, deposition pressure ~6×10–3 mbar. Depending on the deposition time, hence layer thickness, the sheet resistivity of the DLC layer could be tuned from 1 kΩ/ ⃞ to 10 MΩ/ ⃞ . The DLC layer was made in two steps: first a ~ 200 nm layer (10 kΩ/ ⃞ surface resistivity), then the four low resistivity strips around the detector (~600 nm thickness, ~ 3 kΩ/ ⃞ surface resistivity). Using these resistivities and considering the DD1 characteristic geometry, Eq. (2) provides a value of the Doke parameter RDoke ≈0:1, well below the threshold, therefore we expect a significant distortion in the reconstruction (Section 3). This simple estimate indicates the necessity of using extremely thin metallic contour strips for future pcDDs of four-corner configuration. The duo-lateral LACPSDD structures, hereafter labeled as DD2 was implemented on a pc diamond with higher CCE, electronic grade,

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Fig. 2. The LACPSDD structures are realized on pc diamond base material. Only the substrate side is polished. The left and right plots show the substrate and respectively the growth side of the pc diamond plate. (a) The four-corner DD1 with the dimensions 30 × 30 ×0.35 mm3 The DLC layer, deposited only on the substrate side, consists of a central domain and 4 lateral strips on the edges. The lateral strips have a conductivity ~ 3 times higher than the central region. The square shaped Al collecting electrodes are located at the corners. The growth side is fully metallized (Al); (b) The duo-lateral DD2, 20 × 20 × 0.18 mm3, has DLC layers on both sides. All pc diamond plates are mounted and bonded on PCB supports.

grown at Element Six Ltd. and supplied by the former Diamond Detectors Ltd. We used the previously described technology for the DLC layers and for the collecting metallic electrodes (Ohmic contact behavior) deposition, whereas the masks were obtained by lithography. Fig. 2b shows DD2, realized on a 20 × 20 × 0.18 mm3 plate. DD2 has a symmetrical U-I characteristic, with a dynamic resistance of 210 GΩ and breakdown limits outside the measurements range (±500 V). Its surface resistivity is 30 kΩ/ ⃞ . For a comparison between the three pcDD structures discussed here, in Table 1 we show a brief overview of their main parameters. Regarding the detector time constant, τD = RSCD, the most suitable structure appears to be DD2 (low capacity). In the following we detail the measurements made by DD1 and DD2. Details on the measurement of the detector capacity, CD, can be found in [16]. 3. The four-corner LACPSDD structure The current signals are obtained from the four corner electrodes on the front side and from the back side electrode (Fig. 2a). These currents are read out by CSAs [17] and the shapers used are of CR-RC type (minimum ballistic deficit). The peaking time of the shaper is set to 0.18 μs. The corner electrodes are marked as 1, 2, 3, 4 starting from Table 1 Characteristics of the pcDD structures.

DD1 DD2

growth

substrate

CD

breakdown

RS (kΩ/ ⃞ )

RS (kΩ/ ⃞ )

(pF)

(V)

metallic 29.5

14.1 34.7

133 101

-400, 350 b -500, N500

the upper-left corner, in c.w. direction. Q BCK represents the charge collected by the back side electrode, and Q SUM the total charge collected by the four electrodes on the front side, Q SUM = Q1 + Q 2 + Q 3 + Q 4. Due to charge conservation, Q BCK should be equal to QSUM. The event position is calculated by using the following charge division equations: L Q2 þ Q3  Q1  Q4 ; 2 Q1 þ Q2 þ Q3 þ Q4 L Q1 þ Q2  Q3  Q4 y¼ : 2 Q1 þ Q2 þ Q3 þ Q4 x¼

ð3Þ

If the DC positive biasing high voltage is applied on the back electrode, it will collect the generated electrons, while the four corner electrodes will collect the generated holes. The CSA connected to the back electrode includes one additional inverting stage (gain − 1) in order to obtain the same polarity for all signals. The data acquisition system consisted of NIM and CAMAC modules and CAMDA software. This pcDD has a very low CCE (b0.1), therefore due to the low S/N ratio we were not able to separate the response produced by α-particles. Next, the detector was placed in a high energy beam, 54Ni (1.7 AGeV), at the end of the beam line, after the Ni target and a few other detectors. Thus, it was subjected to both 54Ni ions and reaction fragments. As expected, these ion beam tests showed a bad correlation between QBCK and QSUM charges, as shown in Fig. 3a. Among the possible reasons of this weak correlation we mention: (i) the very high beam rate (1.7 ⋅ 109 events/s), saturating the CAMAC ADC (~ 104 events/s); ii) the high detector time constant, due to the larger detection area, as compared to the scDD [3]. Fig. 3b shows the impact position reconstruction obtained by using Eq. (3). We recognize the classical “cushion” error. The lack of symmetry

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Fig. 3. 54Ni beam measurements on DD1. The top and bottom panels show the results for auto-triggered and externally triggered mode, respectively. The external trigger consisted in an additional 2 mm scDD. (a) QSUM as a function of QBCK scatter plot; (b) Reconstructed position; (c-d) Same quantities as (a-b) for the externally triggered mode.

of the reconstructed position is probably related to additional calibration errors and to the ballistic deficit of the used shaper. In order to decrease the errors caused by the high event rate, we replaced the auto-triggered mode by an external trigger. Thus, a 2 mm diameter scDD was mounted in front of the DD1 detector and connected to a 2 GHz bandwidth current amplifier. In Fig. 3c we can see a much better correlation between the total collected charges QBCK and QSUM. The impact position reconstruction (Fig. 3d) has a main, disc shaped, central population surrounded by an asymmetric hallow, presumably related to the already mentioned error sources. Considering the results of this first pcDD design as quite promising, we further concentrated on the correction of the error sources. In this context, the simulations performed on different detector geometries suggested the use of the duo-lateral pcDD configuration (Fig. 2b). In the following section we present simulation results for a simplified duo-lateral detection structure, based on charge diffusion. 4. Simulations of charge diffusion in the DLC layer We simulated the duo-lateral pcDD structure by using a generalpurpose software platform. The geometry consisted of a stack of three bricks. For the middle domain, of size 20 × 20 × 0.18 mm3, the material characteristics were chosen similar to sc diamond (relative permittivity εr = 5.5, conductivity σ = 10–17 S/m). The top and bottom adjacent domains have the same volume, 19.2 × 19.2 × 0.18 mm3, and material properties similar to DLC materials (εr = 1, σ = σDLC). In this simulation setup σDLC is considered as a variable parameter and in the following we show the results for different values of the surface resistivity. Each DLC block has two metallic terminals along the opposite edges. Thus, on the top layer we define the left terminal (LT) and the right terminal (RT) and on the bottom layer the up terminal (UT) and down terminal (DT), see also Section 2. We accepted a compromise in the DLC thicknesses chosen (0.18 mm), in order to reduce the complexity of this 3D structure (the actual mesh consists of 90129 elements and the total memory required is about 3.5 GBytes). The conductivity of the DLC

layer implies a skin depth of a couple of mm, much larger than the thickness of the DLC layer used in simulations. As the first step, the DLC conductivity was calibrated using 1 V steps at LT relative to 0V at RT and − 1 V steps at UT relative to 0 V at DT. Fig. 4a-b show the terminal current transient evolution in the calibration simulation for two σDLC values, corresponding to 3 kΩ/ ⃞ and 30 kΩ/ ⃞ surface resistivity. Due to the distributed resistance and capacitance of the DLC layer, the stationary current value is reached after a certain time interval. The detector time constant, τD = RSCD, is τD3k = 300 ns, respectively τD30k = 3000 ns in these two cases. While the average energy loss of charge carriers through the matter is given by the Bethe-Bloch formula, the distribution of the energy loss by ionization in thin layers is theoretically described by a Landau distribution function. We assume that this distribution of the energy loss translates to a similar shape of the temporal dependence of the current generator. Thus, by using a current generator (Eq. (4)) of a similar shape with the Landau distribution function, we simulate the ionization process produced by a charge carrier inside the diamond plate, and subsequently we only look for the diffusion of this charge inside the structure. After the calibration stage, the diffusion of charge, generated by an event located at any point (x, y) was simulated, by using two equal current generators, i(t) and –i(t), located at the same x and y coordinates, on the front and respectively the back DLC layer. The current generator used has a time dependence given by the following equation:   t  t iðt Þ ¼ i0 1  e τ0 e 3τ0 ;

ð4Þ

where τ0 and i0 indicate the scaling parameters of the current generator. For the results shown here we used τ0 =1 ns and i0 = 1 mA. The current shape for these parameters corresponds to an asymmetric dependence, with a sharp increase to the maximum, ~ 0.47 mA, in about 1.4 ns, followed by a long attenuation in about 20 ns. The associated total charge is about 2.33 pC.

M. Ciobanu et al. / Diamond & Related Materials 65 (2016) 115–124

Fig. 4. The transient response provided by the terminals of the simulated structure. The right and left plots show the response for RS = kΩ/ ⃞ and respectively RS = 30 kΩ/ bottom the plots show the transient response for: a-b) the calibration; c-d) charge injection at x=0, y=0; e-f) charge injection at x=9 mm, y=9 mm.

Fig. 4c-f show the terminal current transients during the time interval 0 – 1μs, for the two chosen values of the surface resistivity, RS = 3 kΩ/ ⃞ and respectively RS = 30 kΩ/ ⃞ . If the injection point is at the detector center (x=0, y=0), the terminal currents, shown in Fig. 4c-d, have the same shape; the main difference is due to the different detector time constants. If the injection point is near the detector corner (x=9 mm, y=9 mm), the currents collected by the four terminals, shown in Fig. 4ef, indicate a factor of ~400 between RT, UT and LT, DT peak amplitudes. As expected, the charge diffusion is slower to the more distant terminals, LT and DT, prompting also a delayed response as compared to RT and UT. Fig. 5a shows the case when the charge is injected at the (x =−9 mm, y= 0 mm) point, equally distanced (~9.2 mm) from the UT and DT terminals; the UT and DT currents have the same transient shape of the collected charge. The LT terminal is closest to the injection point (0.6 mm), whereas the RT terminal is the farthest (18.6 mm). At times

119

⃞ . From top to

t b50 ns, LT has the highest peak current, whereas RT has a bipolar current shape, observed as well for other duo-lateral structures [18]). In the 3D duo-lateral case, one layer will collect holes (biased with negative voltage) and the other layer, electrons (positive voltage). Due to charge conservation, Q L + Q R = Q U + Q D = Q. In Fig. 5a we see a loss of charge due to the direct coupling between layers. Similar to the four-corner configuration (Eq. (3)), the reconstruction of the charge impact position relies on charges collected by all four lateral terminals. Thus, the x and y coordinates are obtained using the following equations: L QR  QL ; 2 QR þ QL L QU  QD y¼ ; 2 QU þ QD x¼

ð5Þ

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constant and the integration time of the electronics. The relationship between the charge collection degree and the reconstruction error suggests a quantitative criterion to estimate the accuracy of the reconstruction. Fig. 6a shows the charge balance on both DLC layers, obtained by subtracting the total injected charge, Q =± 2.33 pC, from the sum of the charges collected by the respective two terminals. The charge balance is shown for the same values of the surface resistivity used in Fig. 5, RS = 3, 6, 15, 30 kΩ/ ⃞ . It is evident that the shapes are symmetrical to the horizontal median line: this suggest the presence of a coupling mechanism between layers (the charge injected at one point is partially balanced by the charge lost at the second point), with a systematic behaviour, which generates a systematic distorsion error on total layer charge and correspondingly to reconstruction error. Fig. 6b shows the total charge balance sheet which confirms that the simulation conserves the total charge within ± 0.06%. The conclusions of these simulations are: 1) the integration time must be longer than the drift time; 2) by monitoring the equality Q L + Q R = Q U + Q D we can get information on the quality of the detection; 3) New question raised: is it possible to use a 2D mathematical algorithm to correct the systematic errors of the reconstruction formula (1), to decrease the absolute error, with the target | εR | b 20μm or |εR%| b 0.1 % ? In the following section we provide an analytical description of the reconstruction error as a function of position, signal, and S/N ratio.

Fig. 5. (a) The transient responses at all terminals for charge injected at x=−9 mm, y=0, surface resistivity ~6 kΩ/ ⃞ ; (b) The injected point is swept by 1 mm pitch in the range x∈ (-9,9) mm, y=0. The absolute error is shown for four RS cases: 3 kΩ/ ⃞ , 6 kΩ/ ⃞ , 15 kΩ/ ⃞ and 30 kΩ/ ⃞ .

where the subscripts L, R, U, D denote, as before, the left, right, up, and down terminals. The absolute reconstruction error, εR, and the absolute error normalized to the side length, εR%, are given by the following equations (written only for x): ε R ¼ xI  xR ; ðxI  xR Þ εR% ¼  100; L

ð6Þ

with I and R indicating the true impact point and its reconstructed value. Fig. 5b shows the dependence of the reconstruction error on the injection point, along the horizontal central line of the detector, yI = 0 and xI from xI = − 9 mm to xI = 9 mm, with 1 mm pitch. The charge is estimated by integration over a 1 μs time interval. We used four surface resistivity cases: 3 kΩ/ ⃞ , 6 kΩ/ ⃞ , 15 kΩ/ ⃞ and 30 kΩ/ ⃞ . The results show a good correlation between the error level and the charge collection degree. Thus, the 1 μs integration time results in a complete charge collection for the cases with surface resistivity in the range ~ 6-15 kΩ/ ⃞ and the errors are accordingly small, | εR | b 130 μm and | εR% | b 0.7%. The output currents for the case RS = 30 kΩ/ ⃞ (Fig. 4d,f) show a much broader signal, extending well beyond the 1 μs integration time. As a consequence of the insufficient integration time, we observe higher errors, | εR | ≈ 1 mm and | εR% | ≈ 6% near the edges. The analysis of this simplified model illustrates the need of a proper match between the detector time

Fig. 6. (a) Charge balance is explored along the horizontal central line of each layer for four RS cases. The two sets of lines, corresponding to the two layers, are symmetrical (open versus closed symbols); (b) The total charge balance is performed for four RS cases.

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By replacing Eq. (12) in Eq. (11), the standard deviation of xn becomes:

5. The signal to noise limitation Eq. (5) are further processed by considering that GC is the total (CSA and ADC) conversion gain. By taking US and UN as the measured signal and noise, the charge collected by each terminal can be written as:

σN L σ xn ¼ 2  S 2

 U S ðt Þ þ U N ðt Þ Q p ðt Þ ¼  ¼ Sp þ N p ; GC p

Subsequently, the reconstructed position, xR, obtained by replacing Eq. (13) in Eq. (9), shows a dependence on ξ, σN, and S given by the following equation:

ð7Þ

where the subscript p = L, R, U, D denotes the charge collection terminal position for the duo-lateral detection structure. By Sp and Np we denote the corresponding charge signal and respectively the charge noise. In order to evaluate the influence of the S/N ratio on the reconstructed position, in the following we detail the calculation only for the x coordinate and later on adapt the results also for the y coordinate. We introduce a linear parametrization of the charge signals, SR and SL, over the x coordinate through the ξ variable as follows: SR ¼ Sð1  ξÞ; SL ¼ Sξ;

ð8Þ

where S = SL + SR is the total charge signal along the x axis on the respective layer and ξ∈ [0,1]. By replacing the collected charges, Qp, given by Eq. (7) in the x coordinate reconstruction equation (Eq. (5)) and considering the introduced linear parametrization (Eq. (8)), some simple algebra yields: xR ¼

  L 2ξN R þ 2ðξ  1ÞNL ð1  2ξÞ þ ≡ xξ þ xn ; S þ NR þ NL 2

xR ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ξ  2ξ þ 1:

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L σN 2 1  2ξ  2  2ξ  2ξ þ 1 : 2 S

ð13Þ

ð14Þ

Due to the symmetry, by considering a similar linear parametrization, SU = S(1 - η) and SD = Sη, with η ∈ [0, 1] the reconstructed yR position can be written as follows: yR ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L σN 2η2  2η þ 1 : 1  2η  2  2 S

ð15Þ

As expected Eqs. (14) and (15) show that the error in xR and yR scales directly with the noise amplitude, reflected by σn, and inversely with the signal; in other words, the larger the S/N ratio, the smaller the expected reconstruction error. The average signal, S ¼ S0 λP , depends on the material type (Section 1) through the λP parameter. The upper limit, λP = 1, is associated with the sc diamond, whereas for the pc diamond λP ≈ 0.1 - 0.4. The ratio of the two perturbation amplitudes,

ð9Þ

where by xξ and xn we denote the linear and respectively the noise term in xR. According to Eq. (9), the measurement accuracy of xR is embedded in the xn term. Therefore, we consider that the position xR is given by: xR ¼ xξ þ xn  σ xn ;

ð10Þ

where xn and σxn denote the mean and the standard deviation of the xn term in Eq. (9). In the following we assume NR and NL as uncorrelated random  L ¼ 0 ) and equal  R ¼ 0 and N variables, with zero mean values ( N standard deviations σN. While NR and NL can be taken as normally distributed white noise, the total charge, S, can be modeled as well by a random, uncorrelated variable. We assume that S follows a Landau distribution with the mean value, S ¼ S0 λP, and standard deviation,σ S ¼ S0 pffiffiffiffiffiffi λP . Under the mentioned assumptions on the noise and signal  R; N  L ; SÞ ¼ 0. Since NR, NL distributions, the mean value of xn is xn ¼ f ðN and S can be regarded as independent processes, the root mean square error of xn can be derived by error propagation:

σ xn

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  2  2 u ∂xn 2 ∂xn ∂xn ¼ t σ 2N þ σ 2N þ σ 2: ∂N R xn ∂NL xn ∂S xn S

ð11Þ

The expressions of the partial derivatives involved in Eq. (11) are given by the following equations: 

∂xn L 2ξ ¼ ; ∂NR x n 2  S  ∂xn L 2ðξ  1Þ ¼ ;  2 ∂N  S  L xn ∂xn ¼ 0: ∂S x n

ð12Þ Fig. 7. (a) Accuracy of position reconstruction (x coordinate) for σN/S0 =3/10 and σN/S0 = 3/100, λP =1 (scDD); (b) Reconstruction error (x coordinate) for σN/S0 =3/10,3/100, 3/ 1000, again with λP =1.

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σxnpcDD/σxnscDD, depends on the λscDD /λpcDD ratio and can be ~ 2.5-10 P P times larger for pcDD. Fig. 7a-b show the reconstructed position and respectively the reconstruction error (x coordinate) for λP =1 and a couple of σN/S0 values. The side length is L=20 mm, similar to Section 4. As expected, small errors correspond to low σN/S0 values ( Fig. 7b). 6. Tests of the duo-lateral LACPSDD structure The duo-lateral structure DD2 (Fig. 2b) was first subject to α-particle tests. DD2 is bonded to a PCB and connected to 4 CSA amplifiers [17]. The shapers used are of CR-RC type (minimum ballistic deficit). The peaking time of the shaper is set to 1.5 μs to cover the charge drift time in DLC layers. One detecting layer has two additional inverting stages (gain − 1) in order to get all signals of the same polarity. The data acquisition system consists of NIM and CAMAC modules and CAMDA software (similar to the tetra-corner setup). The electrical calibration of the readout is made by applying pulse signals to all CSAs. The α-particle source used is 241Am, located parallel to the detector, at ~2.5 mm distance; each particle is stopped in ~15 μm detector material and the energy lost is Eα =5.486 MeV (Qα ~62 fC). A capton mask with thickness ~100 μm and a ~2 mm diameter hole is placed on the top DLC electrode. Fig. 8a shows the pulse height spectra of the charge collected by all four channels. Fig. 8b shows the total charge collected by each DLC layer: each spectrum has a mean value S0 ~11 fC with a standard deviation σN ~ 3 fC. In our case the ratio σN/S0 ~ 3/11, close to the first value shown in Fig. 7a, therefore one can expect some ±6 mm error in the image reconstruction. As estimated in [16], the charge generated in the pcDD is proportional to Z2 (with Z the atomic number). According to Fig. 7, for heavier ions, the S/N ratio is higher and the position reconstruction error becomes smaller. Fig. 8c shows the scatter plot of the collected charges on the top layer (y direction) as a function of the collected charge on the bottom layer (x direction). This plot confirms a good correlation between the total charge on the top and bottom layer (QX ≈ QY), as expected by charge conservation. However, the individual values have a big dispersion

and a low signal to noise ratio (S/N =3–30). Using Eq. (5) the reconstructed position of all events is shown in Fig. 8d together with the superposed black circle indicating the size and location of the α-particle source (the respective hole in the capton mask). The reconstructed distribution is significantly broader than the actual impact positions, in good agreement with the error estimate above. Next, the DD2 was exposed to 12C microbeam at 11.4 MeV/A beam energy (micro-probe ion beam facility at GSI) [15], which provides a ~25 times higher S/N ratio for the stopped particles. In order to explore the entire surface of the detector, the particle injection beam was moved, stepwise, into 62 positions, building an irradiation matrix of 8 × 8 points of equal 2.5 mm pitch in the x and y direction. Each point was irradiated by automatic micro-beam sweeps in small rectangles of areas (280 μm × 230 μm). During the initial stage of the measurement setup we performed the necessary electrical calibration of the detector ensemble. Thus, by using a pulse generator and a four ways distributor we were able to infer the different offsets and gains corresponding to the CSAs and ADCs of all channels. We consider that the electrical calibration done in the measurement environment provides reliable parameters for the subsequent data processing. However, for future experiments we also explore the possibility to derive the offsets and gains directly from the primary data, by numerical analysis, task beyond the scope of this paper. Following the on-site electrical calibration and the actual measurements, we reconstructed the position of each event by using Eq. (5). In Fig. 9a we show the reconstructed positions together with the associated beam injection foci, indicated by the little black squares. Note that the injection beam matrix has a small deviation from the regular grid at the beginning of the second injection line, counting from the bottom. This shift occurred during the operation and it is not caused by a data processing error. In addition to the 62 event distributions, nearby the beam grid points, we also obtain a central (x ≈ - 1.5 mm, y ≈ 0.5 mm) and lower density population of events (marked by the red circle). The analysis of the reconstructed positions associated with each of the 62 micro-beam spots shows that this population contains contributions

Fig. 8. The α-particle measurements: (a) Pulse height spectra for all channels; (b) Histograms of the collected charge on the top layer (x-axis), QX =QLT +QRT, and respectively the bottom layer (y-axis), Q Y =Q UT +Q DT; (c) The scatter plot of the collected charge on the x layer versus the collected charge on the y layer; (d) The reconstruction of all event positions. The superposed black circle indicates the location and size of the 2 mm diameter hole in the α-particle absorbing mask.

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shown in Section 4. By looking at the columns error dependence, starting from the top, we have an absolute error decreasing around the minimum error region followed by an increase towards the bottom edge, as previously shown in Fig. 5 for the S/N ratio around ~30 kΩ/ ⃞ . This suggests that we still have a mismatch between the signal shape and the ADC integration time. However, the columns error dependence is not similar for the matrix rows as one would expect. This could be related to the complex pc diamond structure in comparison to the simplified simulation setup, based only on diffusion, presented in Section 4. Preliminary results from a numerical technique to compensate the non-linear error distribution, are currently under evaluation and will be presented in an upcoming publication. In order to show the distribution of the absolute error of all events we first remove the outliers by taking into account only the events with a maximum error less than 2.5 mm, equal to the microbeam step. In Fig. 10 we show the histogram of the reconstructed error for the three regions indicated in Fig. 9b. The error histogram of all events (A) show a mean value of ~ 0.62 mm and a standard deviation of ~ 0.32 mm. The standard deviations of the B and C regions are ~0.22 mm and respectively ~0.17 mm. According to the Bethe equation, for other ion species, we expect to have a proportional dependence of the S/N ratio with the product of the ion mass and the square of the ion charge. This looks quite promising, in particular for the prospects of using the future DoI material, characterized by a CCE comparable to scDD. According to our estimates, the use of DoI material should lead to a noise error decrease by a factor of 2.5 to 10 times, as compared to pcDD. 7. Summary and conclusions

Fig. 9. (a) The reconstruction of the events positions; (b) The error vectors are defined by the known microbeam focus and the fitted actual position of the center of the related reconstructed points.

from all 62 distributions, presumably because of the errors discussed in Sections 4 and 5. We note also the deviation of the distributions of reconstructed events from the expected circular shapes towards elliptical shapes, more pronounced near the edges of the pcDD. We presume that this deviation is partially caused by imprecision in offsets and gains determination, which supports the necessity of a future thorough analysis of calibration related issues. In order to better show the distribution of the reconstruction error over the entire surface of the detector, we computed the median center (xC, yC) for each of the 62 beam distributions. The (xC, yC) positions are not affected by the presence of events located in the central population (red circle). The accuracy of the derived (xC, yC) centers was also crosschecked by fitting the respective distributions with 2D elliptical gaussian functions, which provided similar results with respect to the center determination (analysis not included here). Fig. 9b shows the reconstruction error distribution on the surface through the vectors drawn from the microbeam central focus to the actual reconstructed center for each of the 62 distributions. The error distribution shows a nonlinear dependence over the surface, with a minimum value of ~7.3 μm around the center and a maximum of about 1.02 mm near the edges. The region of minimum errors is located mainly around the 3rd–5th rows of the injection matrix (from the bottom). The errors increase as one approaches the rows near the edges of the pcDD, where a sharp decrease can be noticed on the electrodes (red dashed strips). The error dependence is quite different when viewed separately on the rows and respectively on the columns. By looking only at the grid points inside the region bounded by the electrodes, we observe an error dependence along the columns in agreement with the results

The reconstruction of the charged particle impact position on the pc diamond sensor, assembled in four-corner (tetra-lateral) or duo-lateral configuration, was studied based on numerical simulations, experimental measurements, as well as an analytical exploration of the S/N ratio. Overall, we showed that a precise reconstruction of the impact position can only be obtained for a proper configuration of the charge collection elements, coupled with suitable front-end electronics (FEE), that matches the specific ensemble response of the detector, as well as good properties of the pc diamond. The technical design of the deposited DLC layers, correction contour strips, and charge collection electrodes play a major role in the observed systematic distortion of the position reconstruction. While the fourcorner structure is affected by a global pin-cushion distortion, the duo-lateral structure does not show a global distortion, but rather a small scale non-linear error distribution. Further work will address a numerical technique to parametrize and compensate the non-linear

Fig. 10. The absolute error histograms for the three regions indicated in Fig. 9b.

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error distribution. Future duo-lateral detection structures may benefit from a multi-electrode configuration to reduce the time difference in the response of the various electrodes. A FEE matched to the pcDD has to accommodate a large dynamic range of responses, depending on the surface resistivity and on the impact position, which is finally reflected by the charge collection degree and, accordingly, by the reconstruction accuracy. The simulation of charge diffusion produced by a current generator of similar profile to that observed experimentally, clarified the dependence of the structure response on the surface resistivity and on the impact position. Consequently, we emphasized the importance of a proper matching between the detector time constant and the FEE integration time. A good match between the two corresponds to a complete charge collection and implies a high accuracy of the reconstruction. A measure of the calibration quality (offsets and gains) of the detector is provided by the charge conservation on the structure, QX ≈ QY for the duo-lateral and QBCK ≈ QSUM for the four-corner structure. For heavier ions, the S/N ratio is higher and the reconstruction error smaller, as predicted by Fig. 7. The material properties are essential for the LACPSDD: higher CCE and lower detector time constant (low capacity) help in improving the S/N ratio and in preventing incomplete charge collection because of slow diffusion. The position reconstruction accuracy, as well as the response to lower energy particles, are expected to benefit from the use of the future DoI material. A DoI based duo-lateral structure, of ~ 60 × 60 mm2 area, ~ 1 mm thickness, and RS = 2 kΩ/ ⃞ of the DLC layers, will have a detector time constant τD2k ≈ 320 ns. A potential application of LACPSDD based on DoI is in cancer therapy [19], where C ions of ~ 50-430 MeV/A energy deposit in 1 mm thickness ~ 1.1-0.24 times the energy deposited by the 12C ion beam on the DD2 structure. The use of multiple parallel mounted LACPSDD detectors will facilitate the cross-check of the correct execution of an irradiation prescription.

Acknowledgments The work in Romania was supported by the DIADEMS project, STAR contract 61/2013 with the Romanian Space Agency. The support of the ADAMAS collaboration and the use of the infrastructure at GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, are acknowledged as well.

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