Analytical model for event reconstruction in coplanar grid CdZnTe detectors
Matthew Fritts, Jürgen Durst, Thomas Göpfert, Thomas Wester, Kai Zuber
aa r X i v : . [ phy s i c s . i n s - d e t ] J a n Analytical model for event reconstruction in coplanar grid CdZnTe detectors
Matthew Fritts a, ∗ , Jürgen Durst b , Thomas Göpfert a , Thomas Wester a , Kai Zuber a a Institut für Kern- und Teilchen-Physik (IKTP), Technische Universität Dresden, D-01062 Dresden, Germany b Erlangen Centre for Astroparticle Physics (ECAP), Friedrich-Alexander Universität Erlangen-Nürnberg, Erwin-Rommel-Str. 1,D-91058 Erlangen, Germany
Abstract
Coplanar-grid (CPG) particle detectors were designed for materials such as CdZnTe (CZT) in which chargecarriers of only one sign have acceptable transport properties. The presence of two independent anode sig-nals allows for a reconstruction of deposited energy based on the difference between the two signals, and areconstruction of the interaction depth based on the ratio of the amplitudes of the sum and difference of the sig-nals. Energy resolution is greatly improved by modifying the difference signal with an empirically determinedweighting factor to correct for the effects of electron trapping. In this paper is introduced a modified interactiondepth reconstruction formula which corrects for electron trapping utilizing the same weighting factor used forenergy reconstruction. The improvement of this depth reconstruction over simpler formulas is demonstrated.Further corrections due to the contribution of hole transport to the signals are discussed.
Keywords:
Coplanar grid, CdZnTe, Depth-sensing, γ -ray spectroscopy, Semiconductor detector
1. Introduction
CZT as a γ -ray detector material has a numberof attractive properties such as a large band-gapand good performance at room temperature. TheCOBRA experiment [1] uses CZT detectors in asearch for neutrinoless double-beta decay ( νββ )due to the presence of several νββ candidate iso-topes in CdZnTe and its low natural background ra-dioactivity. Currently COBRA utilizes a × × array of 1 cm CPG detectors operating under lowbackground conditions at the Gran Sasso NationalLaboratory (LNGS), with plans to upgrade to 64 de-tectors in early 2013.Good energy resolution is an important consid-eration in gamma spectroscopy. It is also key forCOBRA in order to enhance the sensitivity to a νββ line (produced by the summed energy of the ∗ Corresponding author. Tel 49 351 46334568; fax 49 35146337292.
Email addresses: [email protected] (Matthew Fritts), [email protected] (JürgenDurst), [email protected] (ThomasGöpfert),
[email protected] (Thomas Wester), [email protected] (KaiZuber) two betas emitted) and to distinguish it from otherbackgrounds, including the continuous spectrumof the competing process of neutrino-accompanieddouble-beta decay ( νββ ). The depth informationprovided by the CPG design is also very important,for two reasons. First, an important category ofbackground is α - or β -radiation at the cathode andanode surfaces. The short penetration of such back-ground allows for its efficient discrimination, if theinteraction depth is well known. Second, for certainregions in the detector distortion of the energy recon-struction is an unavoidable result of the CPG design.Depth information can be used to identify events inthese regions and remove them from considerationas νββ candidates. For both of these phenomenaanalysis cuts are necessary, and an accurate assess-ment of the selection efficiency of such cuts requiresthe most accurate depth information achievable.Such considerations give strong motivation for anaccurate depth reconstruction for CPG detectors fortheir use in the COBRA experiment, and for all ap-plications in which accurate knowledge of the ac-tive volume is required. In addition, due to the largenumber of detectors used in COBRA, an automaticmethod for generating depth formulas unique to eachdetector is highly desirable. Material properties andoptimal operational parameters differ significantly Preprint submitted to Elsevier July 23, 2018
CPGPRINCIPLES 2from detector to detector, necessitating unique for-mulas for both energy and depth reconstruction.
2. CPG principles
The CPG detector design was introduced byP. Luke in 1994 [2]. A proper understanding of thedesign principles begins with a treatment based onthe Shockley-Ramo theorem. For a more detaileddescription of the application of the Shockley-Ramotheorem to CPG detectors see [3].In a CPG detector such as those used in COBRA,the cathode is patterned on one side as a uniformrectangle, while on the opposite side two isolatedanode grids are formed in the shape of interlockingcombs. Figure 1 is a schematic of a COBRA CPGdetector, manufactured by EI Detection & ImagingSystems [4]. In operation a large (approximately1 kV) bias (referred to as HV) is applied betweenthe cathode and one anode, while a smaller (typi-cally 50-100 V) bias (GB) is applied between theanode grids. The anode held at higher potential isreferred to as the collecting anode (CA); the otheranode is known as the non-collecting anode (NCA).Electrons excited into the conduction band by a par-ticle interaction somewhere in the bulk of the de-tector drift straight toward the anode plane (in thenegative- z direction) until they get close to the an-ode grid rails, at which time they are diverted by thenear-anode field to be collected in the CA.A first step in the application of the Shockley-Ramo theorem is to calculate the weighting poten-tials [5, p. 813-8]. Figure 2 shows the weighting po-tentials of the CA and NCA along a plane throughthe center of the detector, calculated using the elec-trode geometry of COBRA detectors. At most lo-cations in the detector, the two weighting potentialsare nearly equal, rising with a slope of 1/2 from thecathode plane to the anode plane. At the position ofa CA rail (normally the final location of the mobileelectrons created by ionization), the CA and NCAweighting potentials are 1 and 0, respectively, bydefinition.These properties of the weighting potential allowfor a difference signal based on the raw CA andNCA signals that is proportional to the charge fromionization and independent of the interaction depth.From the Shockley-Ramo theorem one can calculate In current operation the guard ring electrode (black in Fig-ure 1) is left unconnected. This information is incorporated intothe weighting potential calculations.
Figure 1:
Schematic electrode layout for CPG de-tectors used by COBRA. Two interleaved anodesand an outer guard ring (uninstrumented) are pat-terned on one side (top); the other side (bottom) isa uniform cathode. Shown here is the conventionused in this paper for the z axis (interaction depth).The dashed lines indicate the section over which thecalculated weighting potential is shown in Figure 2. ELECTRONTRAPPING 3 x z C A w e i gh t i ng po t en t i a l x z NC A w e i gh t i ng po t en t i a l Figure 2:
Weighting potentials of CA and NCA ascalculated for the COBRA CPG detector geometryacross a plane through the center of the detector.The coordinates are normalized to the detector di-mensions, thus ranging from 0 to 1. the amplitudes of the two anode signals produced byelectrons: ∆ q CA = 12 Q ( z + 1) (1) ∆ q NCA = 12 Q ( z − (2)Here ∆ q represents the change in induced charge,and thus the amplitude of the signal, for the corre-sponding anode. Q is the magnitude of the mobilecharge produced by the interaction. z is the distancebetween the interaction location and the anode planeexpressed as a dimensionless fraction of the detectorlength. z is referred to as the “interaction depth” (orsimply “depth”).The depth dependence in the two raw anode sig-nals is removed by subtraction: ∆ q CA − ∆ q NCA = Q (3)Note that ∆ q NCA is always negative, so the equationrepresents a sum of the absolute amplitudes. By cal-ibrating this signal using sources of known spectraone achieves a well-resolved reconstruction of theenergy deposited by an interaction.Similarly one can remove the charge magnitudedependence to reconstruct the interaction depth [6]: ∆ q CA + ∆ q NCA ∆ q CA − ∆ q NCA = z (4)The picture of CPG operational principles pre-sented in this section can be referred to as zeroth-order behavior. First-order effects, which complicatethis simple picture, are the subject of the followingsections.
3. Electron Trapping
Electron trapping in CZT detectors is a strongenough effect that, for COBRA detectors, the signalamplitude of an interaction occurring at large depthis reduced by about 10% relative to that of an inter-action occurring at small depth. This depth depen-dence would significantly degrade energy resolutionwere it not corrected. Such a correction is possibledue to the depth information intrinsic in the raw sig-nals.P. Luke and E. Eissler introduced a simple weight-ing factor (referred to here as w ) to modify the dif-ference signal in order to correct for electron trap-ping [7]: ∆ q CA − w ∆ q NCA (5) ELECTRONTRAPPING 4
NCA signal height-600 -500 -400 -300 -200 -100 0 100 C A s i gna l he i gh t Figure 3:
CA versus NCA signal heights for
Thcalibration data for one COBRA detector. γ -linesfrom the source appear as upward-sloping lines inthis plot. The red line indicates 2.615 MeV eventsfrom Tl decay. The slope of this line, 0.80, is theempirically-determined weighting factor w . In theabsence of trapping this slope would be 1.The weighting factor is empirically determined andis always less than one. The correction for trappingcan be understood qualitatively: as interaction depthincreases, − ∆ q NCA becomes smaller while the sig-nal reduction due to trapping becomes larger. Under-weighting the NCA amplitude thus artificially low-ers the amplitudes of low-depth interactions to bet-ter match the trapping-degraded amplitudes of high-depth interactions.The weighting factor w can be determined by anoptimization procedure seeking the best energy res-olution. Alternatively if information from both sig-nals is recorded separately the weighting factor canbe determined from a single calibration run by anal-ysis of the CA and NCA signal amplitudes for a cal-ibration line. Figure 3 illustrates this procedure asperformed in COBRA.In the following paragraphs an analytical treat-ment of the electron trapping effect in CPG detec-tors is presented, using a few simplifying idealiza-tions. This treatment is similar to the derivation in the Appendix of [8], although here the anode signalsare separately derived. First assume a mean trappinglength λ that is valid for the entire electron drift path.This is only true to the extent that both the crystalproperties and the electric field are uniform through-out the detector volume. In actuality the electric fieldis nearly uniform for a large majority of the electronpath for most interaction locations, and thus for mostof the drift distance in which trapping occurs. Thusthe magnitude of the charge as a function of driftdistance d is taken to be Q = Q e − d/λ (6)As with z , λ and d are expressed in terms of detec-tor length and are thus dimensionless. λ is consid-ered a free parameter in this treatment. Physically itdepends upon the electron mobility-lifetime productfor the crystal and the voltage applied to the crystal,both of which vary by detector. Thus λ is detectordependent.A second simplification is to assume that theweighting potentials of both anodes take their bulkform along the entire electron drift path up until theanode plane is reached, at which point the CA andNCA weighting potentials abruptly change to 1 and0, respectively. The idealized weighting potentialsas functions of depth z are illustrated in Figure 4.In this model the electron drift is reduced to one di-mension, with the drift entirely in the negative z di-rection. For most interaction locations this is a rea-sonable approximation in that, again, most of thetrapping occurs before deviations in the weightingpotential become significant and before the electronpath is diverted by the near-grid field. The expectedsize of the error introduced by this approximation isdiscussed in a later section.Given these assumptions the expected anode sig-nal amplitudes can be calculated. We divide the cal-culation into two parts. The first part correspondsto the electrons drifting through the region of uni-form weighting potentials, where dV w dz = − . Sincethe charge magnitude changes along the drift pathwe calculate this with an integral form of the Ramoequation, yielding ∆ q = Z z Q e − ( z − z ) /λ dz = 12 Q λ (cid:16) − e − z /λ (cid:17) (7) This is true considering the electrode geometry if one as-sumes that space charge has a negligible effect. The shape ofcharge pulses in COBRA detectors show no evidence of signifi-cant field distortions due to space charge.
EVALUATIONOFTHETRAPPINGMODEL 5 z0 0.2 0.4 0.6 0.8 1 W e i gh t i ng po t en t i a l Collecting Anode (CA)Non-collecting Anode (NCA)
Figure 4:
Idealized weighting potentials of CAand NCA corresponding to path of charges driftingthrough a CPG detector.This result is valid for both the CA and NCA sig-nals. It is closely related to the Hecht equation [5,p. 489], differing only by the factor one-half. In asignal formed by the sum of the CA and NCA sig-nals (sometimes referred to as the cathode signal) theexact Hecht equation is recovered. This correspondsto the weighting potential of a simple parallel-plateelectrode geometry.The second part of the calculation of the anodesignal amplitudes corresponds to the abrupt changesin weighting potential at z=0: +1/2 for CA and -1/2for NCA. Inserting the final charge into the Ramoequation one finds ∆ q ,CA = 12 Q e − z /λ (8) ∆ q ,NCA = − Q e − z /λ (9)The full expressions for the anode signals are ∆ q CA = 12 Q h λ (cid:16) − e − z /λ (cid:17) + e − z /λ i (10) ∆ q NCA = 12 Q h λ (cid:16) − e − z /λ (cid:17) − e − z /λ i (11)By inspection we see that, in the limit of large λ , the corresponding no-trapping forms (Equations 1and 2) are restored.As in the zeroth-order case the position depen-dence can be removed by a linear combination of the two signals to reconstruct the charge excited intothe conduction band by the interaction: ∆ q CA − λ − λ + 1 ∆ q NCA = λλ + 1 Q (12)It is noteworthy that the trapping effects, themselvesof exponential form, are canceled exactly with a lin-ear combination. For this it is necessary only thatthe z -dependent terms in equations 10 and 11 areidentical in form. On the left-hand side of equa-tion 12 is a weighted difference signal equivalent tothe one introduced by Luke. On the right-hand sideis the excited charge reduced by a constant factor, asexpected from the qualitative understanding of theweighted difference. The constant is of course ul-timately absorbed into the energy calibration. Therelationship between the parameter λ and the empir-ical weighting factor w is λ = 1 + w − w (13)An equivalent relationship is derived in [8].It is also possible to construct an analytical ex-pression for the interaction depth z from the rawsignal formulas by eliminating Q from equations10 and 11: λ ln (cid:18) λ ∆ q CA + ∆ q NCA ∆ q CA − ∆ q NCA (cid:19) = z (14)Note that as expected the zeroth-order depth for-mula is recovered in the limit of large λ . Thistrapping-corrected depth formula is easily imple-mented because the parameter λ can be calculatedfrom w which is already determined during the en-ergy correction procedure.
4. Evaluation of the trapping model
In the framework of the simple model for trap-ping presented here we can predict the performanceof the zeroth-order depth formula, now seen as anapproximation. It is clear that it is non-linear; more-over, the zeroth-order formula will overestimate theinteraction depth in a detector-dependent way. Fromthe empirical weighting factors w determined forCOBRA detectors the overestimation can be ex-pected to reach a maximum of around 3-10% for z = 1 .Figure 5 illustrates the advantages of the newdepth formula. Sources of alpha radiation at thecathode surfaces for the detectors currently in op-eration at LNGS produce an excess of events near ADDITIONALFIRST-ORDEREFFECTS 6 z0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 N u m be r o f e v en t s Zeroth-order formulaTrapping-corrected formula
Figure 5:
Interaction depth z of near-cathode eventsin the COBRA data collected at LNGS in 31 detec-tors with deposited energies greater than 2 MeV. Anexcess of interactions occurring very near the cath-ode is known to exist. Uncorrected depth reconstruc-tion results in depths overestimated by a few percentand a wider distribution due to detector differences. z = 1 . The true interaction depth for these events isexpected to be very close to 1. The figure comparesthe depth distribution using both the zeroth-orderand trapping-corrected formulas. The predicted ad-vantages are evident: the corrected depths are muchcloser to one, and they are much more tightly dis-tributed.The empirically determined parameter λ can beused to estimate the electron mobility-lifetime prod-uct ( µτ ) for each detector as follows: µτ = L λV bulk (15)where L is the detector length (1 cm) and V bulk is calculated to be the potential difference betweenthe cathode and the average of the anode potentials(HV - GB). Figure 6 shows the resulting µτ esti-mates for 30 COBRA detectors. The mean value of . × − cm / V is close to those quoted for CZTin several publications (see Table 7 in [9]). Howeverthe wide distribution of values suggests significantvariation across detectors, and possibly additionaleffects unaccounted for by this simplified model. One potential effect is an imbalance in the CA and NCAsignal amplification factors arising from the specific properties ofthe electronics. If such an imbalance exists it will contribute to theweighting factor w independently of electron transport properties. /V) (cm τµ N u m be r o f de t e c t o r s Figure 6:
Electron mobility-lifetime products of30 COBRA detectors, estimated by the method de-scribed in the text. The mean value is . × − cm / V.
5. Additional first-order effects
In CZT holes have much poorer transport prop-erties than electrons; nevertheless they contributesignificantly to the raw signal amplitudes. Becauseholes drift much more slowly than electrons, it is firstnecessary to consider whether their contributions tosignal amplitudes are fully recorded in the data pro-cessing procedure. In standard COBRA pulse pro-cessing the effective time window within which theamplitude is measured is greater than 3 µ s. Thelifetime of holes in CZT is estimated at 1 µ s, sothat more than 95% of the hole charge is trappedwithin the processing time window—assuming thatthe holes do not reach an electrode within this time.Thus to a reasonable approximation we can assumeholes contribute fully to the processed signal ampli-tudes.In analogy to the parameter λ in the treatment ofelectron trapping, we take ρ to be the mean trappinglength for holes, normalized to the detector length.Because typical λ values are around 10, and becausethe mobility-lifetime product for holes in CZT is es-timated to be about 1% that of electrons, ρ is esti-mated to be around 0.1. Thus for many interactiondepths we do not expect holes to drift far enough to However for this study such imbalances were measured and the λ and µτ values corrected accordingly. Neither can temperature bea factor since all detectors are operated under essentially identicalconditions at room temperature. ADDITIONALFIRST-ORDEREFFECTS 7be collected at an electrode; rather they will be al-most completely trapped after drifting a fraction ofthe detector length.Qualitatively the hole effect can be pictured asfollows: holes will drift a short distance towardthe cathode, along a path in the bulk region of theweighting potential. They thus contribute equal pos-itive contributions to the CA and NCA signal am-plitudes. Since the energy reconstruction is based onthe difference of these signals, the hole contributionswill cancel each other. On the other hand the depthreconstruction is based on the sum of the anode sig-nals, and so the hole effect can lead to a significantoverestimation of interaction depth.Quantitatively we can consider the hole effect us-ing the same framework as was used for electrontrapping. Assuming that the interaction depth occursin the bulk region of the weighting potential, we find ∆ q holes = 12 Q ρ (cid:16) − e − (1 − z ) /ρ (cid:17) (16)This contribution applies to both anode signals.Considering the small value of ρ we see by in-spection that the hole contribution is approximately Q ρ for interaction depths far from the cathode(where − z is several times ρ ). In contrast theexpression falls to zero as z approaches 1, since fornear-cathode interactions the holes have no distanceto drift. Therefore the hole contribution is zero for z = 1 .The contribution of holes to the signals hasa second-order effect on the energy reconstruc-tion, producing a small reduction for near-cathodeevents [7, 8]. For interactions far from the cathodethe weighted difference signal (Equation 12) is en-hanced by approximately ρ/λ , on the order of 1%.This constant factor will be absorbed into the energycalibration. However the hole enhancement disap-pears for interactions very close to the cathode. Thusthe energy depositon of near-cathode events will beunderestimated by about 1%. This is comparable tothe energy resolution of COBRA detectors. The ef-fect is noticeable only in a layer whose thickness isa few percent of the detector length. Figure 7 showsthis small effect on a calibration line with roughlyuniform depth distribution.The hole effect on depth reconstruction is larger.It can be adequately described by considering the ad-dition of the hole contribution to the zeroth-orderequations (Equation 1 and 2) and the correspond-ing effect on the zeroth-order depth formula (Equa-tion 4). For interactions far from the cathode, the Energy (keV)500 550 600 650 700 750 800 z N u m be r o f e v en t s Figure 7:
Distribution of events in depth and energyin a COBRA detector irradiated with a
Cs source.The reconstructed energy of 662 keV γ events issomewhat underestimated at depths near 1 due to thehole effect leading to a deformation of the line nearthe top of the plot.depth will be overestimated by approximately ρ . Fornear-cathode interactions there is no overestimation,as is clear from Figure 5. Figure 8 illustrates therelationship between true and measured depth. Theclearest evidence of this effect from data is the smallnumber of events below a certain measured depth forpopulations of interactions known to have uniformdepth distributions, as exemplified in Figure 9.In principle of course one can also correct for thehole effect in the depth reconstruction. However itis more difficult to calculate the parameter ρ in thecalibration process. Another difficulty is that theanalytic expression for hole-corrected depth has nosolution if the zeroth-order depth is greater than 1.Since the finite resolution of measured depth some-times results in values greater than 1, this is problem-atic for an event-by-event hole correction algorithm. CPG event reconstruction depends on the nearlyequal CA and NCA weighting potentials at the inter-action point. However the two weighting potentialsdiffer significantly at locations near the anode plane,as illustrated in Figure 2.The corresponding effects are somewhat compli-cated, and depend not only on depth z but alsoon the x and y coordinates of the interaction point.The energy can be either underestimated or overes-timated depending on whether the interaction pointlies above a NCA or a CA rail. The thickness of the ADDITIONALFIRST-ORDEREFFECTS 8 z0 0.2 0.4 0.6 0.8 1 M ea s u r ed z Figure 8:
Relationship between measured depth andtrue depth (red) based on an analytical treatment ofthe hole effect, assuming ρ = 0 . . The measureddepth is corrected for electron trapping but not cor-rected for the hole effect.layer near the anode in which these effects are sig-nificant is greater near the lateral surface of the de-tector than it is near the center of the anode plane,where it is smaller than 0.5 mm. Where it is thinnest,holes can actually cancel the distorting effect, bydrifting toward the cathode into the bulk region ofthe weighting potential.Where the effective near-anode layer is thickest,holes will not drift far enough to cancel the distortingeffects. As with the reconstructed energy, the recon-structed depth is also distorted, but it will generallybe smaller than ρ , so that a judicious depth selectioncan remove such events from consideration in dataanalysis.Very near the anode the electric field in the de-tector is dominated by the grid bias, which pro-duces a different effect. The region of effect dependsstrongly on the applied bias, but is estimated to betypically around 0.1 mm. For interactions in this re-gion the holes will drift toward the NCA instead ofthe cathode. The corresponding hole contributionsto the signals will be opposite in sign, thus strongly The distortions in the weighting potential can be reduced ifa bias is applied to the guard ring [10]. However for simplicityCOBRA detectors are currently operated with the guard ring un-connected. z0 0.2 0.4 0.6 0.8 1 N u m be r o f e v en t s Figure 9:
Depth distribution of events from aCOBRA detector for data taken at LNGS in an en-ergy range of 180-300 keV. The events consist al-most entirely of intrinsic β -decay of Cd, andthus are known to be evenly distributed throughoutthe detector (although some events near the detec-tor surfaces will be lost due to escaping β particles).The low count of measured depths below about 0.1is evidence of the hole effect. The hole effect isalso partially responsible for the nonuniform distri-bution at greater depths. Distortions in energy- anddepth-reconstruction at lower depths due to the de-tails of the weighting potential also contribute to thisnonuniformity. SUMMARY 9enhancing the difference signal. Due to the shorthole drift distance, most holes are collected beforethey are trapped. The shape of the weighting poten-tials corresponding to the paths of both electrons andholes differ significantly from those correspondingto events in the bulk. The net effect is that the recon-structed energy will be approximately doubled [6].The reconstructed depth will be very close to zero.Signals from interactions far from the anode arealso affected by the detailed form of the weightingpotentials near the anode. If such effects are consid-ered Equations 10 and 11 must be corrected. Thiscorrection will vary with the x and y positions of theinteraction. However we can approximate the sizeof the correction by considering a specific case: aninteraction occurring on a line leading to the centerof a CA rail in the central region of the anode plane.In this case the weighting potential as a function of z fits well to an exponential form with characteristiclength α of approximately 0.015 (0.15 mm). We findthen that Equations 10 and 11 are corrected by termsof order α/λ . This is a third-order effect, much lessthan 1%, and thus wholly negligible considering theenergy and depth resolutions achievable with the de-tectors.
6. Summary
It has been shown that a simple model for elec-tron trapping with an idealized form for the weight-ing potential provides a mathematical framework toexplain the use of a weighted difference signal to re-construct event energies in a coplanar grid detector.Using the same model a new formula for the inter-action depth which corrects for electron trapping ef-fects has been introduced. This formula demonstra-bly improves the depth reconstruction, especially forevents near the cathode. Other effects that can dis-tort the energy and depth calculations have been dis-cussed in an effort to understand event reconstruc-tion in CPG detectors as accurately as possible.
Acknowledgments
The authors would like to thank Tobias Köttig foruseful discussions, the COBRA collaboration for useof data, and the Deutsche Forschungsgemeinschaftfor its support.EFERENCES 10