Charge-carrier collective motion in germanium detectors for ββ-decay searches
mmanuscript No. (will be inserted by the editor)
Modeling the collective motion of charge carriers ingermanium semiconductor detectors
Tommaso Comellato , Matteo Agostini , Stefan Sch¨onert Physik Department E15, Technische Universit¨at M¨unchen, James-Franck-Straße 1, 85748, Garching, Germany Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UKReceived: date / Accepted: date
Abstract
The time analysis of the signal induced bythe drift of charged carriers in high purity germaniumdetectors provides information on the event topology.Millions of charge carriers are produced in a typicalevent. Their initial distribution, stochastic diffusion andCoulomb self-repulsion affect the time structure of thesignal. We present the first comprehensive study of theseeffects and evaluate their impact on the event discrim-ination capabilities of interest for neutrinoless double-beta decay experiments.
Keywords1 Introduction
Since the invention of transistors in 1948 [1], germaniumhas been used in a broad variety of applications, rang-ing from gamma-ray detection [2] to fiber optics [3, 4] tosearch for dark matter [5–7]. The state-of-the-art tech-nology allows the production of detector blanks withlengths and diameters of 8-9 cm using the Czochral-ski method. With a level of impurities of the order of10 atoms/cm , such crystals can be converted intoHigh Purity Germanium (HPGe) detectors. A HPGedetector is a semiconductor device. Two electrodes onthe crystal surface are used to apply a bias voltage andextend the semiconductor junction throughout the fulldetector volume. When a gamma-ray or charged parti-cle interacts within the detector it creates a large num-ber of charge carriers, i.e. electrons and holes. Chargecarriers of the same sign drift together towards the elec-trodes as cluster, following the electric field lines. Their a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] motion induces a signal on the electrodes that is typ-ically read-out by a charge sensitive amplifier. Similarto a time projection chamber, the analysis of the timestructure of the read-out signal contains informationon the topology of the event, i.e. on the number andlocation of the energy depositions.An important field of applications for germaniumdetectors is the search for neutrinoless double-beta (0 νββ )decay, a nuclear transition predicted by many exten-sions of the Standard Model of particle physics in whichtwo neutrons decay simultaneously into two protonsand two electrons. For this search, detectors are fab-ricated from germanium material isotopically enrichedto ∼
90% in the candidate double-beta decaying isotope Ge. Thus, the decay occurs inside the detector andthe electrons are absorbed within O ( mm ), producing apoint-like energy deposition. For double-beta decay ex-periments it is hence of primary interest to discriminatesingle site energy depositions (typical of the sought-after signal) from multiple-site energy depositions (typ-ical of background events induced by multi-Comptonscattering), as well as surface events (which, for geo-metrical reasons, are more likely to be external α or β particles).The time development of the signal depends on thegeometry of the detector, its electrode scheme, and itsimpurity concentration. Thus, an accurate modeling ofthe signal formation and evolution is an essential ingre-dient to design the detector and enhance the accuracyof the topology reconstruction and event discrimina-tion. As an example, simulations have been extensivelyused in gamma-spectroscopy, such as modeling the seg-mented detectors of AGATA and GRETA [8, 9], whilein double-beta decay experiments they led to BroadEnergy Germanium (BEGe) and P-type Point Contact(PPC) detectors [10, 11]. In the effort to increase the de- tector mass, new geometries such as the Inverted Coax-ial (IC) [12] have recently drawn increasing attention.In this new type of detectors, the time needed to collectelectrons and holes is much longer than in the aforemen-tioned geometries.In this article we investigate the collective effectsin a cluster of charge carriers and their impact on thesignal formation in the detector geometries of interestfor double-beta decay searches. These collective effectsinclude the self-repulsion between charge carriers andtheir thermal diffusion, whose impact depends on theinitial spatial distribution within the cluster. We per-formed comprehensive simulations of germanium de-tectors and validated them against the data acquiredwith a custom designed IC detector produced in col-laboration with Baltic Scientific Instruments (BSI) andHelmholtz Research Center (Rossendorf). Its geometryis the one used as reference for this paper. Our workbuilds on the results of [13], which reports the first ob-servation of such effects in PPC detectors and discusseshow to accurately model them. Our simulations havebeen carried out with the Mage [14] software frame-work based on
Geant-4 [15], and a modified versionof the SigGen software package [16] which already in-cluded the modeling of the collective effects and wasused in [13]. More details on simulations are given inAppendix A.
When gamma-rays or charged particles interact withinthe germanium detector they release energy. About 10 electron-hole pairs are created for each MeV releasedin the active detector volume. Once produced, the twokinds of carriers drift as two clusters in opposite direc-tions following the electric field lines until they reachthe electrodes. The signal induced by the motion ofthese charges can be to a first approximation modeledby the Shockley–Ramo theorem [17, 18]. The theoremstates that the instantaneous current I ( t ) induced at agiven electrode by a drifting cluster of charge q is givenby I ( t ) = q v ( r ( t )) · E ω ( r ( t )) (1)where v ( r ( t )) is the instantaneous drift velocity and E ω ( r ( t )) is the weighting field at position r ( t ). Theweighting field is defined as the electric field createdby the considered electrode set at 1 V, all other elec-trodes grounded and all charges inside the device re-moved. Thus, the signal induced at the electrode is the product of the instantaneous drift velocity and the pro-jection of the weighting field in the direction of motion,weighted by the deposited charge.Often events induced by gamma-rays result in mul-tiple energy depositions well separated compared to thedimension of the charge clusters. In this case, each clus-ter drifts independently of the others and the resultingsignal is the superposition of the individual signals, eachof them weighted by the charge in each cluster.Three illustrative HPGe detector geometries are an-alyzed in this article. These are the geometries used bythe current and future double-beta decay experiments: Gerda [19],
Majorana Demonstrator ( MJD ) [20],
Legend [21]. All of them are p-type detectors, with aLithium-diffused n + electrode and a B-implanted p + electrode. The three detector types are shown in Fig. 1along with the resulting weighting field and illustrativetrajectories.The PPC detectors have a cylindrical shape andhave masses up to 1 kg. Their geometry is characterizedby a small ( ∼ + electrode on one of theflat surfaces, while the rest of that flat surface is passi-vated. The remaining surface of the detector is coveredby the n + electrode. Electrons are collected on the n + electrode that is kept at a few kV operational voltage,while holes on the p + electrode, that is grounded andused to read-out the signal. This geometry creates aweighting field that increases rapidly in the immediatevicinity of the p + electrode. This results in a character-istic peak-like structure in the current signal when thehole clusters approach the p + electrode.Compared to PPC detectors, the BEGe detectorsare shorter but have a larger radius. The major dif-ference between the two geometries is the structure ofthe electrodes: the p + electrode is larger for BEGe (upto ∼
15 mm diameter) and surrounded by a passivatedgroove with typical depths of ∼ + electrode extends down to the groove, wrap-ping around the crystal on all surfaces. This structurehas a strong impact on the trajectories of the carriers,as it creates a funnel effect [22]: holes are pushed to-wards the center of the detector and then move to thep + electrode along a fixed path that is independent bytheir starting point (see central plot of Fig. 1). Sincethat is the volume in which the weighting field is high-est, according to Eq. 1, the majority of the inducedsignals in a BEGe detector share the same maximumvalue of the current I ( t ).The inverted coaxial detector has the same electrodestructure as a BEGe, though it is about twice as long.In order to keep a high electric field throughout thewhole volume, a hole is drilled on the opposite side ofthe p + electrode and constitutes part of the n + contact. Fig. 1: Weighting field E ω for a cross section of the three geometries used in current and future double-beta decayexperiments: (from left) PPC, BEGe and inverted coaxial. The thick black and gray lines are the p + and n + electrode, respectively. The yellow points are locations of an energy deposition, the white trajectories connectingthem to the p + electrode are the drift paths of holes and those connecting them to the n + electrode are the driftpaths of electrons.It normally extends down to within 25-35 mm from thep + electrode. With the wrap-around n + electrode, thefunneling is preserved and the trajectories converge inthe region of high weighting field (see Fig. 1). The modeling of the signal formation presented in theprevious section does not account for the cluster spatialextension that is O ( mm ) for a MeV energy deposition.It can be extended to account for the non-null dimen-sions of the cluster. The instantaneous signal inducedat the electrode will be the integral of equation 1 overthe spatial charge distribution g ( x ) of the cluster: (cid:101) I ( t ) = (cid:90) dx g ( x ( t )) I ( t ) . (2)If the electric field varies on scales similar to the clus-ter size, charges at the opposite side of the cluster willexperience different forces (accelerations), leading to adeformation of the cluster during its drift towards theelectrodes. Moreover, the stochastic diffusion and self-interaction of the charge carriers will progressively in-crease the size of the cluster during its motion. Thediffusion consists of a random thermal motion of thecarriers while the self-interaction is the result of theCoulomb force. In this work, such processes are treatedas collective effects [16]. That allows an analytical treat-ment and keeps the computational requirements to anaffordable level. We compared this approximated col-lective description with a full multi-body simulation We simulated the individual motion of 10000 charges in thefield generated by the detector and the instantaneous config-uration of the other charges. and found that it does not introduce noticeable inaccu-racies.In our collective treatment, the dynamics of driftingcharges in the presence of mutual repulsion and diffu-sion is described by the continuity equation [23]: ∂ Q∂r − r ∂Q∂r − D ∂Q∂t − Q ∂Q∂r V T π(cid:15)r = 0 (3)where Q ( r, t ) is the charge contained in a sphere of ra-dius r at time t , D is the diffusion coefficient, (cid:15) thepermittivity in germanium and V T the thermal voltage V T = k B T /q with q being the elementary charge. Thegeneral solution of Eq. 3 when the Coulomb repulsionterm is neglected describes a gaussian profile for thecharge cluster, whose width is σ D = √ Dt. (4)When charges drift in an electric field, the diffusioncoefficient D has a longitudinal and transverse compo-nent. Both are calculated in SigGen [16] in the respec-tive direction, but only the longitudinal is the responsi-ble for the deformation of the signal. As reported in [24],this component is lower as the electric field strength in-creases. This implies that, with a sufficiently high impu-rity concentration, the effect of diffusion can be stronglylimited (as stated also in [13]).Neglecting the diffusion term, the width of the chargecluster increases as σ R = (cid:114) µq π(cid:15) N t (5)where N is the number of charges in the distribution.
40 20 0 20 40 r [mm] z [ mm ] . . . . . . . . . . . . . . . . . . D r i f t v e l o c i t y [ mm / n s ] t [ns] D r i f t v e l o c i t y [ mm / n s ] Hole drift velocity I n s t a n t c h a r g e s i z e [ mm ] Self repulsion onlyDiffusion onlyAcceleration onlyAll effects t [ns] [ n s ] Self repulsion onlyDiffusion onlyAcceleration onlyAll effects
Fig. 2: Breakdown of the collective effects on a charge cluster. The top-left plot shows the drift velocity field of anIC detector with superimposed in brown the drift path of the holes’ cluster for an interaction location marked bythe star. The cluster’s drift velocity along the path is shown in the bottom-left plot. The evolution of the cluster’ssize and σ τ is displayed in the top-right and bottom-right plot, respectively. The initial size of the cluster is 0.5mm, the average for energy depositions of 1.6 MeV.Fig. 2 displays the contribution of the mentionedprocesses to the charge cluster deformation . The top-left plot shows the drift velocity field on an IC detectorcross section, where superimposed in brown is the tra-jectory of holes for an energy deposition on the positionmarked with the star. As holes travel through the de-tector, they experience accelerations (decelerations) ac-cording to the electric field, stretching (shrinking) thecluster size in the direction of motion as shown in thetop-right panel (light blue curve). In the same plot, thebroadening effect due to the described Coulomb and dif-fusion processes are shown with the yellow and greencurves, respectively: as described by equations 4 and 5,their effect is a monotonic enlargement of the clustersize. Finally, the dark blue curve shows the evolutionof the cluster dimensions, when all effects act simulta-neously. It’s worth mentioning that the total size is notjust the simple sum of the three contributions, as theyare not independent: an enlargement of the cluster size, The initial cluster size is given here in Full Width Half Max-imum, and it has been determined as a function of energythrough Monte Carlo simulation. See details in Appendix A. for instance due to Coulomb or diffusion effects, empha-sizes the difference in the drift velocity field of chargesat the edge of the distribution, thus amplifying the ef-fect of acceleration. These non-linear effects have alsobeen verified by our full multi-body simulation men-tioned at the beginning of this section.The impact of the different collective effects on thesignal formation can be characterized through the timespread of the cluster, which we define in the followingas σ τ ( t ). The evolution in time of such parameter isdisplayed in the bottom right plot of Fig. 2. The lightblue curve shows that σ τ is constant if only accelerationeffects are considered. As other effects are switched on,their interplay gives a total time spread which can beup to a factor 5 larger than the initial value.The enlargement of the cluster size through the pa-rameter σ τ as a function of the interaction position isshown in Fig. 3 (top), separately for the three consid-ered geometries. For PPC detectors, the maximum en-largement is for interactions in the corners, where σ τ reaches about 15 ns. The corners are the part of thedetector from which the hole drift path is the longest. For BEGe detectors the maximum is slightly larger, upto 20 ns for radii larger than 30 mm. For inverted coax-ial detectors the effect is much stronger, up to a factor2 and it affects more than half of the detector volume.The impact on the signal shape is shown in the bottomrow of Fig. 3, where signals are shown with (light blue)and without (dark blue) the deformation caused by col-lective effects. The difference between the two cases isless than 0.5% of the signal amplitude in BEGe andPPC detectors (see green curve), but it is larger for in-verted coaxials, where the maximum of the current sig-nal is lowered by ∼ Double-beta decay experiments using HPGe detectorsrely heavily on the analysis of the time structure of thesignal in order to reconstruct the topology of the energydeposition and thus discriminate between neutrinolessdouble-beta decay events and other backgrounds. Thiskind of analysis is commonly referred to as Pulse ShapeAnalysis (PSA). Double-beta decay events are charac-terized by a single energy deposition while backgroundcan be generated by gamma-rays scattering multipletimes within the detector, or α and β particles deposit-ing energy next to the detector surface . PSA tech-niques are based on the recognition of a few specificfeatures of the signal time evolution which allows fora discrimination between signal- and background-likeevents. The effects discussed in the previous sectionhave the net result of blurring these features and, con-sequently, of worsening the performance of any PSAtechnique. In this section we evaluate their impact ona particular PSA technique that is the standard in thefield: the so called A/E method [10].The
A/E technique is based on a single parame-ter that is the maximum value of the current signal( A ), normalized by the total deposited energy ( E ) (or q in Eq. 1). In case of a single energy deposition, thesignal has a single peak structure with amplitude A ,which corresponds to the moment when the holes’ clus-ter passes through the region of maximum weightingfield.If the energy is deposited in multiple locations, mul-tiple clusters are simultaneously created and the totalsignal is the superposition of the signal induced by themotion of each of them. Different clusters will reach theregion of maximum weighting field at different times, These surface events generate peculiar pulse-shapes, therecognition of which is beyond the scope of this work. creating a multiple peak structure. Since the amplitudeof each peak is proportional to the total charge in thecluster generating it, events with multiple energy depo-sitions E i ∝ q i will have a lower A/E value comparedto single site events in which all energy is concentratedin a single cluster E ∝ (cid:80) i q i . When normalized to thetotal charge q , the signal from a multiple energy de-position gives lower A/E values compared to a singleenergy deposition. More details are given in AppendixB. The
A/E parameter is independent of the interac-tion position and its discrimination efficiency is con-stant throughout the whole detector volume. This isdue to the fact that the holes approach the region ofmaximum weighting field along the same trajectory ,independent of the original location where the clusterwas created. Without considering the collective effects,the A/E parameter is expected to have the same valuefor clusters with a given energy generated in most ofthe detector volume. The only exception is for interac-tions nearby the read-out electrode, for which the
A/E parameter is larger than usual because of extra con-tribution of the electrons’ cluster that now moves in aregion of strong electric and weighting field and its con-tribution on the signal shape is not negligible as in therest of the detector. The uniformity of the
A/E param-eter in the detector volume has been studied in detailin [22]. Collective effects depend on the interaction po-sition – as shown by the σ τ parameter in Fig. 3 – andthis creates an A/E dependence from the interactionposition.Fig. 4 shows the value of the
A/E parameter formono-energetic energy depositions simulated through-out the whole detector volume considering the collec-tive effects described in Sec. 3. The
A/E value variesby a few percent between the corners and the centerof the detector in the BEGe and PPC geometry. Asalready mentioned, the value is significantly amplifiedonly in about 3% of the detector volume around thep+ electrode. For inverted coaxial detectors, while thebottom half of the volume exhibits features similar tothe BEGe geometry, the upper part shows a consistent0.3% reduction of the
A/E value. This reduction of the
A/E has been experimentally confirmed by studyingthe response of our prototype inverted coaxial detectoragainst low-energetic gamma-rays used to create well-localized energy depositions on different parts of thedetector surface. This is true for BEGe and IC detectors. The funneling effectis not present in the PPCs, because for that geometry theweighting field at the p + electrode is spherical, hence thesignal does not depend on the angle from which the holesarrive.
40 20 0 20 40 r [mm] z [ mm ]
40 20 0 20 40 r [mm]
40 20 0 20 40 r [mm] [ n s ]
200 400 600 800 1000 t [ns] N o r m a li z e d I ( t ) NO CEwith CE 400 600 800 1000 1200 t [ns]
NO CEwith CE 1200 1400 1600 1800 2000 2200 t [ns]
NO CEwith CE 210123 R e l a t i v e V a r i a t i o n [ % ] Fig. 3: Top: values of the σ τ parameter as a function of the interaction position, for the three geometries considered.Bottom: simulated signals for the interactions and drift paths indicated by the brown point and curve, with andwithout Collective Effects (CE). Higher values of σ τ , as in inverted coaxial detectors, imply lower values of thecurrent I ( t ).Maximizing the detector volume is of primary im-portance for neutrinoless double-beta decay experiments.However, the larger the collection path, the stronger theimpact of these collective effects will be. In the follow-ing we evaluate the event-reconstruction performanceof inverted coaxial detectors and discuss possible anal-ysis techniques to correct for these collective effects. Toquantify the performance we focus on the acceptanceof 0 νββ -like events and of typical backgrounds of theexperiments.The event discrimination based on the A/E param-eter is calibrated using the Double Escape Peak (DEP)events from
Tl as a proxy for 0 νββ decay events,as they both consist in a single energy deposition (formore details on the calibration of the analysis, we referto Appendix B). The
A/E distribution of DEP eventsis used to set a cut value which keeps 90% of their to-tal number. This value cannot be directly translated to0 νββ decay acceptance, for two reasons: the first is thatDEP and 0 νββ events have a slight different topology .The second, DEP events are concentrated on corners,0 νββ decays are homogeneously distributed. DEP events consist in an electron and positron sharing 1.6MeV, while 0 νββ events produce two 1 MeV electrons. Thischanges the initial cluster size, as well as the Bremsstrahlungprobability.
In order to estimate the 0 νββ decay acceptance, weperformed a Monte Carlo simulation of the energy de-posited in 300000 0 νββ and DEP events. The MonteCarlo simulation takes into account all the physical dif-ferences between the two classes of events and their spa-tial distribution within the detector. For each event, thetotal signal is computed using the modeling describedin Sec. 2 and 3 and analyzed to extract the
A/E pa-rameter. From the
A/E distribution of DEP events,we set the cut value and applied it to the 0 νββ pop-ulation. This resulted in a final 0 νββ acceptance of(86 . ± . Tl, we alsoextracted the
A/E distributions of events from
TlFull Energy Peak (FEP),
Tl Single Escape Peak (SEP)as well as from the Compton continuum (CC) from
Tl and
Bi, which constitute background at Q ββ .We applied the cut obtained from DEP events to thesedistributions and obtained the survival fraction of (5 . ± . . ± . . ± . . ± . ββ from Tland
Bi, respectively. The values, reported in Tab. 1, z [ mm ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40 30 20 10 0 10 20 30 40 r [mm] z [ mm ]
40 30 20 10 0 10 20 30 40 r [mm]
40 30 20 10 0 10 20 30 40 r [mm] N o r m a li z e d c u rr e n t a m p li t u d e R i s e T i m e [ n s ] Fig. 4:
A/E (top) and rise time (bottom) values for the three analyzed geometries. In PPC and BEGe detectorsrise times range up to 600-800 ns, while for inverted coaxials they can be twice as big, and saturate for highz-positions, where the threshold at 0.5% is no longer a good approximation of the beginning of charge collection.A correlation between
A/E and rise time is visible for the inverted coaxial detector.are in agreement with the typical theoretical values forBEGe detectors [22].As pointed out above, the impact of the collectiveeffects is correlated with the time needed to collect thehole cluster. Following the proposal of [25], we testeda correction on the
A/E parameter based on the re-constructed collection time of the signals, in order torestore the position independence. In this work we re-construct such a quantity by taking the time betweentwo arbitrary thresholds on the signal, i.e. what is calledthe rise time . Noise conditions can prevent accuratedetermination of the start time for thresholds below0.5% at the energies of interest for double-beta decaysearch. Hence, for this work we refer to rise time as thetime between 0.5% and 90% of signal development . Amap of the mean rise time as a function of the interac-tion position within the detector is shown in Fig. 4 forthe three geometries considered. These rise time and Normally, the thresholds are set on the signal which is ex-perimentally accessible, which means the output of the chargesensitive pre-amplifier. That is the charge signal V ( t ), whichis the integral of the current signal I ( t ). Other techniques, based on the convolution of the signalfunction with a well tuned impulse response function, couldlead to the identification of lower thresholds, such as 0.1% ofthe signal amplitude.
A/E values are correlated in the inverted coaxial ge-ometry. This is shown explicitly in Fig. 5 for DEP (5a)and 0 νββ decay (5b) events. Both plots suggest that alinear correlation could be used to align the
A/E valuesin the bottom and top part of the detector volume.This double peak structure has been first reportedin [26, 27]. Its origin is connected by our work to thecollective effects and the spatial distribution of DEPevents within the detector. Indeed, the configurationof the inverted coaxial detector creates a region on thetop and one on the bottom part of the detector in whichrise time and
A/E saturate to a limit value, which getsmore represented than the others. This effect is evenmore pronounced for DEP events, which are more likelyto occur on the detector edges.Motivated by the correlation shown in Fig. 5, we ex-plored the impact of a first order linear correction of the
A/E value based on the rise time for each event. The
A/E maps before and after such correction are shownin Fig. 6. The linear correction reduces the differenceamong A/E values: the volume that exhibits an
A/E value of (1 . ± . A/E values get lowered by almost 0.5%.This is due to the interplay between collective effects,which combine in such a way that the cluster defor-
Simulations Data
IC BEGe[19, 22] IC (this work) BEGe [10, 22]Event class
Standard RT corr Standard Standard RT corr Standard
Tl DEP 90.00 (8) 90.08 (8) 90 (1) 90.1 (8) 90.1 (8) 90 (1)
Tl SEP 5.1 (3) 5.8 (3) 8 (1) 5.0 (3) 5.3 (3) 5.5 (6)
Tl FEP 7.4 (1) 8.1 (1) 12 (2) 7.64 (5) 7.92 (5) 7.3 (4)CC @Q ββ ( Tl) 45.1 (3) 46.7 (3) 42 (3) 32.3 (2) 33.1 (2) 34 (1)CC @Q ββ ( Bi) 20.3 (4) 21.8 (4) – – – 21 (3)0 νββ
Table 1: Percentage of events classified as single-site for different event samples and detectors, taken from simula-tions and experimental data. For inverted coaxial detectors, the results are given both before (
Standard ) and aftera correction based on the rise time (
RT corr ). R i s e T i m e . % - % [ n s ] A/E [a.u.] (a) DEP events from data (filled colored contour) and simu-lations (gray contour lines) R i s e T i m e . % - % [ n s ] A/E [a.u.] (b) 0 νββ events from simulations
Fig. 5: Distribution of the
A/E and rise time for (5a) DEP events and (5b) 0 νββ events. The distributions areshown for experimental data (color maps) and simulated data (contour lines).mation (hence
A/E ) is not univocally associated to thelength of the drift paths. In order to determine whetherit is convenient to apply the rise time correction or not,we tested it on the simulations of
Tl and 0 νββ . Theresults are reported in the second column of Tab. 1. Thesurvival fraction of 0 νββ events decreases after rise timecorrection from a value of 86.1% to 85.5%. In terms ofbackground, the rise time correction increases the sur-vival fraction of events at Q ββ by 1.5%. The correctiondoes not improve the overall efficiencies, but reducesthe volume dependence of the PSA performance, possi-bly reducing the systematic uncertainties of the exper- iment. It might become more and more relevant as thedetector volume keeps on increasing.The distribution of the A/E and rise time from ex-perimental data is shown in the coloured filled contourof Fig. 5a, in comparison with simulations, representedby the gray contour lines. The 0.3% displacement in
A/E between the two blobs is well reproduced by ourwork. This is not the case if collective effects are notincluded. The excess in data at low values of
A/E is ex-pected, as DEP events cluster on corners, where a frac-tion of events occurs in a transition layer where there isno electric field and the charged carriers move becauseof diffusion. This effect is not included in our simula-
40 30 20 10 0 10 20 30 40 r [mm] z [ mm ]
40 30 20 10 0 10 20 30 40 r [mm] N o r m a li z e d c u rr e n t a m p li t u d e Fig. 6:
A/E maps from Monte Carlo 0 νββ decay events. The left plot shows the values of
A/E normalized accordingto the energy correction (see Appendix B) and the right plot shows the values after rise time correction.tion. The rise time is systematically underestimated by ∼
30 ns in our simulation. This disagreement does notaffect the conclusions of our work and could in prin-ciple be improved by tuning the unknown parametersof the crystal, such as the impurity profile along thesymmetry axis, or the hole mobility.From
Th data, the measured survival fraction ofthe different classes of events has also been extracted,both before and after rise time correction. The num-bers, reported in Tab. 1, show an agreement < . ββ . This can been tracedback to inaccuracies in the positioning of the source.The distance between radioactive source and detectorchanges the fraction of multiple site events from cas-cade of gammas (this was also observed in [22]). Thisdoes not affect the populations of SEP and FEP events,since for them a statistical subtraction of the side-bandsis performed (details in Appendix B). The impact of therise time correction on data, even if not statistically sig-nificant, reflects what is found with simulations, namelythat it increases the acceptance of FEP and SEP events,as well as of background at Q ββ . In Summary, the mod-eling developed reproduces the A/E results within 0.2%and hence its systematic uncertainties are lower thanthe impact of the collective effects that we wanted tostudy.
In this paper we discussed the collective effects in clus-ters of charge carriers in germanium detectors and the impact of such effects on signal formation, with partic-ular focus on the consequences for double-beta decayexperiments with Ge. We determined that the defor-mation of the signal due to collective effects is rele-vant for detectors with long drift paths. In particular,we observed in the inverted coaxial geometry a posi-tion dependence of the standard pulse shape discrim-ination parameter used in double-beta decay experi-ments (
A/E ). With the combined use of Monte Carloand pulse shape simulations of
Tl and 0 νββ decaysof Ge, we determined that such volume dependencedoes not impact the pulse shape discrimination perfor-mances significantly. This proved to be the case bothusing the standard
A/E analysis, and implementing acorrection based on the reconstruction of the drift path.As detector volumes keep on increasing, the impactof collective effects on
A/E might become stronger [27].Moreover, the background composition at Q ββ will change,too, for different detector geometries. With such con-ditions, it is meaningful to compare detector perfor-mances at the same 0 νββ acceptance. This could beused in the future to fix the A/E cut on DEP events.A visual representation of the 0 νββ acceptance as afunction of the acceptance of DEP events is displayedin Fig. 7, both before and after rise time correction. Noappreciable difference was observed when the true drifttime (extracted from the simulations) was used for thecorrection.As planned by
Legend , inverted coaxial detectorswill be deployed in environments which are more chal-lenging than a vacuum cryostat and exhibit differentelectronics noise conditions. In this work we exploredthe impact of a factor 5 higher noise level on PSD per- DEP acceptance (1-sided) a cc e p t a n c e before correctionafter RT correctionafter DT correction Fig. 7: Acceptance of 0 νββ events as a function ofDEP’s, in the case of no-correction on
A/E (bluecurve), or after rise time (green curve) and drift time(yellow curve) correction.formances. This yields (for a cut at 90% DEP accep-tance) an increase in the 0 νββ decay acceptance of 3%,but at the same time an increase of 5% in the back-ground events surviving the
A/E cut at Q ββ . This iscompatible with values of other BEGe detectors alreadyin use in Gerda [19]. We also explored the performancesof inverted coaxial detectors with lengths in the range8 − Acknowledgements
We are very grateful to David Radfordwho developed SigGen as an open source project. SigGen isthe software that we used to model the HPGe detector signaland included already the modeling of collective effect that weused to study the performance of our three detector geome-tries. We are also thankful to D. Radford for many suggestionsand enlightening discussions during the work as well as hishelp during the preparation of this manuscript. We are alsothankful to all the members of the GERDA and LEGENDcollaborations for their valuable feedback. This work has beensupported in part by the European Research Council (ERC)under the European Union’s Horizon 2020 research and in-novation programme (Grant agreement No. 786430 - GemX)and by the SFB1258 funded by the Deutsche Forschungsge-meinschaft (DFG).
Appendix A: Details on simulations
This section deals with the technical details of the sim-ulations carried out for this work. The physics modelfor 0 νββ and
Tl decays has been simulated withinthe
MaGe framework [14], while the generation of sig- nals in germanium detectors has been simulated usingthe SigGen software [16].Appendix A.1: Monte Carlo simulationsThe Monte Carlo simulations have been performed us-ing the
MaGe software, a
Geant
MaGe givesthe opportunity to select the track precision of the simu-lated particles, by choosing the realm . For this work, weused the
DarkMatter realm, in which the precision forgamma rays and e ± are 5 µ m and 0.5 µ m, respectively,which corresponds to ∼ νββ decays in the germa-nium detector volume, and Tl and
Bi sources placedat 20 cm far from the detector side. The hits in the ger-manium detector’s active volume have been stored andthen given as an input to the pulse shape simulationsoftware.Also, from the Monte Carlo simulation of
Tl, theenergy dependence of the starting size of the chargecarriers’ cluster has been extracted. This has been doneby using the R90 parameter, which is defined as theminimum radius of the sphere which contains 90% ofthe energy depositions. We selected 30 energy windowsin the range [1 . , .
2] MeV, extracted their relative R90value, and fitted the resulting energy dependence with alinear interpolation. The fitting function was then givenas an input to the pulse shape simulation software.Appendix A.2: Pulse Shape SimulationsThe SigGen software is a software tool to simulate elec-tric field and signals of a germanium detector. For thiswork we simulated the electric and weighting field on a0.1 mm grid, and signals on a time-step of 0.1 ns.For the physics of collective effects, signals havebeen simulated on a 0.5 mm grid on a detector cross sec-tion. For this analysis, a single carriers’ cluster has beensimulated, with a starting size given by Monte Carlosimulations, through the R90 parameter described above.To generate signals from Monte Carlo, we built theevent-waveform (e.g. a 0 νββ decay event) by summingup the hit-waveforms simulated for every energy de-position in germanium, each of them weighted by thedeposited energy. In order to take collective effects intoaccount, we used the position of the first energy de-position to calculate the σ τ parameter for the event.Before applying electronics response function, we thenperformed a convolution of the event-waveform with a gaussian function of width σ τ . Applying electronics re-sponse function (which has been tuned using the modeldeveloped by [28] on data from Am) to the convo-luted waveform gave the final event-waveform.On every event-waveform, electronics noise, takenfrom our experimental setup, has been added. The risetime has been extracted directly from the noisy wave-forms, while the A parameter has been calculated afterapplying 5 moving window averages of 100 ns width.Finally, the energy E , given by Monte Carlo, has beensmeared using a gaussian function whose width σ hadbeen inferred from the experimental resolution curve.For a comparison with data, the standard Gerda anal-ysis [29], as described in Appendix B, has been car-ried out for simulated data. Specifically, the
A/E en-ergy dependence has been extracted from simulations,giving values which are compatible with the experi-mental ones. The energy corrected
A/E has been usedto produce Fig. 5 and 6. The correction has provedto be effective, as the
A/E distributions of DEP and0 νββ decay events are centered around the same value.Finally, to calculate the survival fractions of the differ-ent event classes, the statistical subtraction described inAppendix B has been applied to the (energy corrected)
A/E distributions.
Appendix B: A/E cut calibration
This section describes in more detail the calibrationprocedure to set the
A/E cut and to calculate the sur-vival fractions of other events. This is entirely based onthe works in [10, 30].Appendix B.1: The
Th source
Th is the reference source in double-beta decay ex-periments for multiple reasons. First, its daughter
Tlhas a gamma at 2.6 MeV which can undergo pair pro-duction in the interaction with the detector. When thisis the case, the two 511 keV photons from the annihila-tion of the positron can either be absorbed in or escapethe detection volume. In case both are absorbed in thedetector, their energies sum up to that of the electron,thus falling into the Full Energy Peak (FEP) at 2.6MeV. When one of the two escapes detection, the detec-tor measures 2.6 - 0.511 MeV and the event is referredto as Single Escape Peak (SEP). If pair creation occurson corners, there is a significant probability that neitherof the 511 keV photons deposit any energy in the detec-tor. This case is known as Double Escape Peak (DEP)and is of particular importance for double-beta decayexperiments, as it consists of an electron and positron depositing 1.592 MeV in the detector, thus resemblingthe physics of the energy deposition from double-betadecay. For this reason, DEP events are used as a proxyfor signal-like events.On the other hand, SEP events, being composed ofan energy deposition of an electron-positron pair and agamma, are characterized by two energy depositions.Those events, together with those from the FEP of
Tl and
Bi, which are mainly composed of multipleCompton scattering, are used as reference to estimatethe event discrimination performance of a detector.Furthermore, what makes
Th also a valuable sourcefor 0 νββ decay search, is that at the energy of Q ββ =(2039 ±
35) keV, the spectrum is composed of eventswith different topologies, among which, a fraction canundergo single Compton scattering, and thus mimic thesignal of a 0 νββ decay. Knowing this fraction is thus animportant parameter when estimating the backgroundat Q ββ .Appendix B.2: A/E
AnalysisThe
A/E analysis is based on a single parameter thatis the maximum value of the current signal ( A ), nor-malized by the total deposited energy ( E ). In case ofa single energy deposition, A/E exhibits a value whichis higher than the case of a multiple energy deposition.This is due to the fact that a multiple energy depositiondistributes the total charge in several clusters, each gen-erating a current proportional to the charge containedin the cluster.As the starting size of the cluster increases with en-ergy, its time spread (our σ τ parameter) gets larger,generating lower values of A/E . This energy depen-dence is estimated by selecting an arbitrary number ofenergy regions in the Compton continuum in the range[1 . , .
3] MeV and extracting the
A/E values for eachregion. The dependence on energy is then fitted andcorrected with a linear interpolation.The standard analysis uses DEP events from
Thas a proxy of single energy depositions and fixes a lowcut value for
A/E by setting the acceptance of DEPevents to 90%. With this value, it computes the numberof events surviving the cut for different event classes:this is referred to as a 1-sided cut. In addition, in orderto reject surface events from regions which are close tothe p + electrode (which are potentially coming fromsurface contamination), it computes the mean µ andwidth σ of the distribution of A/E and sets the highcut to the value of µ + 4 σ : this procedure is referred toas a 2-sided cut.In order to extract the correct survival fractionsof DEP, SEP and FEP events, the standard analysis gets rid of the Compton scattering events which lie inthe same energy region of interest by a statistical sub-traction: for every peak, two sidebands are selected (atlower and higher energy), whose A/E distribution isthen subtracted from that of the peak of interest.
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