Majorana Collaboration

# Charge Trapping Correction and Energy Performance of the Majorana
Demonstrator

I.J. Arnquist  Pacific Northwest National Laboratory, Richland, WA 99354, USA
F.T. Avignone III  Department of Physics and Astronomy, University of South
Carolina, Columbia, SC 29208, USA  Oak Ridge National Laboratory, Oak Ridge,
TN 37830, USA  A.S. Barabash  National Research Center “Kurchatov Institute”
Institute for Theoretical and Experimental Physics, Moscow, 117218 Russia
C.J. Barton  Department of Physics, University of South Dakota, Vermillion, SD
57069, USA  K.H. Bhimani  Department of Physics and Astronomy, University of
North Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  E. Blalock  Department of Physics, North
Carolina State University, Raleigh, NC 27695, USA  Triangle Universities
Nuclear Laboratory, Durham, NC 27708, USA  B. Bos  Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  M. Busch  Department
of Physics, Duke University, Durham, NC 27708, USA  Triangle Universities
Nuclear Laboratory, Durham, NC 27708, USA  M. Buuck  Present address: SLAC
National Accelerator Laboratory, Menlo Park, CA 94025, USA  Center for
Experimental Nuclear Physics and Astrophysics, and Department of Physics,
University of Washington, Seattle, WA 98195, USA  T.S. Caldwell  Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514,
USA  Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA  Y-D.
Chan  Nuclear Science Division, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  C.D. Christofferson  South Dakota Mines, Rapid City,
SD 57701, USA  P.-H. Chu  Los Alamos National Laboratory, Los Alamos, NM
87545, USA  M.L. Clark  Department of Physics and Astronomy, University of
North Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  C. Cuesta  Centro de Investigaciones
Energéticas, Medioambientales y Tecnológicas, CIEMAT 28040, Madrid, Spain
J.A. Detwiler  Center for Experimental Nuclear Physics and Astrophysics, and
Department of Physics, University of Washington, Seattle, WA 98195, USA  Yu.
Efremenko  Department of Physics and Astronomy, University of Tennessee,
Knoxville, TN 37916, USA  Oak Ridge National Laboratory, Oak Ridge, TN 37830,
USA  H. Ejiri  Research Center for Nuclear Physics, Osaka University, Ibaraki,
Osaka 567-0047, Japan  S.R. Elliott  Los Alamos National Laboratory, Los
Alamos, NM 87545, USA  G.K. Giovanetti  Physics Department, Williams College,
Williamstown, MA 01267, USA  M.P. Green  Department of Physics, North Carolina
State University, Raleigh, NC 27695, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  Oak Ridge National Laboratory, Oak Ridge,
TN 37830, USA  J. Gruszko  Department of Physics and Astronomy, University of
North Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  I.S. Guinn  Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  V.E. Guiseppe  Oak
Ridge National Laboratory, Oak Ridge, TN 37830, USA  C.R. Haufe  Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514,
USA  Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA  R.
Henning  Department of Physics and Astronomy, University of North Carolina,
Chapel Hill, NC 27514, USA  Triangle Universities Nuclear Laboratory, Durham,
NC 27708, USA  D. Hervas Aguilar  Department of Physics and Astronomy,
University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  E.W. Hoppe  Pacific
Northwest National Laboratory, Richland, WA 99354, USA  A. Hostiuc  Center for
Experimental Nuclear Physics and Astrophysics, and Department of Physics,
University of Washington, Seattle, WA 98195, USA  M.F. Kidd  Tennessee Tech
University, Cookeville, TN 38505, USA  I. Kim  Los Alamos National Laboratory,
Los Alamos, NM 87545, USA  R.T. Kouzes  Pacific Northwest National Laboratory,
Richland, WA 99354, USA  T.E. Lannen V  Department of Physics and Astronomy,
University of South Carolina, Columbia, SC 29208, USA  A. Li  Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514,
USA  Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA  J.M.
López-Castaño  Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA  E.L.
Martin  Present address: Duke University, Durham, NC 27708  Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514,
USA  Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA  R.D.
Martin  Department of Physics, Engineering Physics and Astronomy, Queen’s
University, Kingston, ON K7L 3N6, Canada  R. Massarczyk  Los Alamos National
Laboratory, Los Alamos, NM 87545, USA  S.J. Meijer  Los Alamos National
Laboratory, Los Alamos, NM 87545, USA  S. Mertens  Max-Planck-Institut für
Physik, München, 80805, Germany  Physik Department and Excellence Cluster
Universe, Technische Universität, München, 85748 Germany  T.K. Oli  Department
of Physics, University of South Dakota, Vermillion, SD 57069, USA  G. Othman
Present address: Universität Hamburg, Institut für Experimentalphysik,
Hamburg, Germany  Department of Physics and Astronomy, University of North
Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  L.S. Paudel  Department of Physics,
University of South Dakota, Vermillion, SD 57069, USA  W. Pettus  Department
of Physics, Indiana University, Bloomington, IN 47405, USA  IU Center for
Exploration of Energy and Matter, Bloomington, IN 47408, USA  A.W.P. Poon
Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA
94720, USA  D.C. Radford  Oak Ridge National Laboratory, Oak Ridge, TN 37830,
USA  A.L. Reine  Department of Physics and Astronomy, University of North
Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  K. Rielage  Los Alamos National Laboratory,
Los Alamos, NM 87545, USA  N.W. Ruof  Center for Experimental Nuclear Physics
and Astrophysics, and Department of Physics, University of Washington,
Seattle, WA 98195, USA  D.C. Schaper  Los Alamos National Laboratory, Los
Alamos, NM 87545, USA  D. Tedeschi  Department of Physics and Astronomy,
University of South Carolina, Columbia, SC 29208, USA  R.L. Varner  Oak Ridge
National Laboratory, Oak Ridge, TN 37830, USA  S. Vasilyev  Joint Institute
for Nuclear Research, Dubna, 141980 Russia  J.F. Wilkerson  Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514,
USA  Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA  Oak
Ridge National Laboratory, Oak Ridge, TN 37830, USA  C. Wiseman  Center for
Experimental Nuclear Physics and Astrophysics, and Department of Physics,
University of Washington, Seattle, WA 98195, USA  W. Xu  Department of
Physics, University of South Dakota, Vermillion, SD 57069, USA  C.-H. Yu  Oak
Ridge National Laboratory, Oak Ridge, TN 37830, USA  B.X. Zhu  Present
address: Jet Propulsion Laboratory, California Institute of Technology,
Pasadena, CA 91109, USA  Los Alamos National Laboratory, Los Alamos, NM 87545,
USA

###### Abstract

$P$-type point contact (PPC) high-purity germanium detectors are an important
technology in astroparticle and nuclear physics due to their superb energy
resolution, low noise, and pulse shape discrimination capabilities. Analysis
of data from the Majorana Demonstrator, a neutrinoless double-$beta$ decay
experiment deploying PPC detectors enriched in 76Ge, has led to several novel
improvements in the analysis of PPC signals. In this work we discuss charge
trapping in PPC detectors and its effect on energy resolution. Small
dislocations or impurities in the crystal lattice result in trapping of charge
carriers from an ionization event of interest, attenuating the signal and
degrading the measured energy. We present a modified digital pole-zero
correction to the signal energy estimation that counters the effects of charge
trapping and improves the energy resolution of the Majorana Demonstrator by
approximately 30$\%$ to around 2.4 keV full width at half-maximum at 2039 keV,
the 76Ge $Q$ value. An alternative approach achieving similar resolution
enhancement is also presented.

germanium detectors, enriched 76Ge, neutrinoless double beta decay, signal
processing

###### pacs:

23.40-s, 23.40.Bw, 14.60.Pq, 27.50.+j

##  I Introduction

Germanium semiconductor detectors’ excellent energy resolution has made them
the technology of choice for radiation detection and spectroscopy for decades.
High resolution is particularly advantageous in searches for neutrinoless
double-beta decay ($0\nu\beta\beta$) Agostini  _et al._ (2022); Avignone and
Elliott (2019); Dolinski  _et al._ (2019); Barabash (2018); Vergados  _et al._
(2016); Dell’Oro  _et al._ (2016), a postulated process beyond the standard
model in which two neutrons in a nucleus transform into two protons and emit
two electrons but no antineutrinos, violating lepton number conservation.
Experimental observation of $0\nu\beta\beta$ decay would provide essential
insights into the observed matter-antimatter asymmetry of the universe. The
existence of this process would imply that neutrinos are Majorana particles,
i.e., their own antiparticles Schechter and Valle (1982). Current-generation
$0\nu\beta\beta$ decay experiments have recently placed lower limits on the
$0\nu\beta\beta$ half-life in various isotopes that in some cases extend
beyond $10^{26}$ years Azzolini  _et al._ (2019); Agostini  _et al._ (2020);
Arnquist  _et al._ (2022); Armengaud  _et al._ (2021); Adams  _et al._ (2022);
Abe  _et al._ (2022); Anton  _et al._ (2019), while some next-generation
experiments aim to reach sensitivities of $10^{27-28}$ years to test the
inverted neutrino mass ordering Agostini  _et al._ (2021); Abgrall  _et al._
(2021); Adhikari  _et al._ (2022); Armstrong  _et al._ (2019). To maximize the
discovery potential, these experiments require extremely low levels of
radioactive background, a large active detector mass, and sufficient energy
resolution. The latter is required both to distinguish the $0\nu\beta\beta$
peak from the irreducible $2\nu\beta\beta$ background, as well as to minimize
the impact of continuum backgrounds.

The Majorana Demonstrator Abgrall  _et al._ (2014) is aimed at searching for
$0\nu\beta\beta$ decay using high-purity germanium (HPGe) detectors enriched
to 88$\%$ 76Ge Abgrall  _et al._ (2018). The experiment is sited at the
4850-foot level of the Sanford Underground Research Facility (SURF) Heise
(2015) in Lead, SD. Initially 29.7 kg of enriched Ge was deployed along with
14.4 kg of natural-abundance Ge detectors, with 58 individual detector units
deployed in two vacuum cryostats. All enriched Ge detectors are $P$-type point
contact (PPC) HPGe detectors manufactured by AMETEK-ORTEC Inc. ort (2009)
while all natural Ge detectors are broad energy germanium (BEGe) detectors
manufactured by Canberra Industries can (2009). Each cryostat contains seven
detector strings, illustrated in Fig. 1. In 2020 the apparatus was upgraded,
with 5.41 kg of enriched PPC HPGe detectors replaced with 6.67 kg of inverted-
coax point-contact (ICPC) detectors manufactured by AMETEK-ORTEC Inc. Cooper
_et al._ (2011) The ICPC technology is being pursued as part of the research
and development program for the LEGEND experiment. This paper will focus on
data from the Majorana Demonstrator’s PPC detectors.

Figure 1: Rendering of one of the two Majorana Demonstrator vacuum cryostats
containing a close-packed array of AMETEK-ORTEC PPC (cyan) as well as Canberra
BEGe (grey blue) HPGe detectors. The primary structural material is ultra-pure
copper electroformed underground at SURF. A thermosyphon provides cooling to
liquid nitrogen temperatures via the cross-arm extending to the left, which is
also the path for signal and HV cabling to penetrate the copper, lead, and
plastic shielding that surrounds the cryostats. The height of the cryostat is
about 46 cm.

The Majorana Demonstrator achieved the best energy resolution of all large-
scale $0\nu\beta\beta$ decay experiments to date Aalseth  _et al._ (2018);
Alvis  _et al._ (2019); Arnquist  _et al._ (2022). A critical step in this
achievement was the implementation of a correction for energy degradation due
to charge trapping. Energy is measured in HPGe detectors by collecting and
measuring the charge carriers liberated in ionization events. As the charge
carriers drift in the detectors’ internal fields, they produce signals
sensitive to charge trapping effects. Charge carriers can be trapped if they
encounter impurities or dislocations in the crystal lattice. If they are not
released on a timescale that is short compared to the signal collection time,
the signal amplitude, and hence the energy, will be degraded. When the energy
degradation is on the order of typical fluctuations in charge carrier
generation, as it is in the Majorana Demonstrator, its net effect in a
population of signals with varying collection times is to widen the detector
response function. Correcting for this trapping can thus result in improved
energy reconstruction, as has been demonstrated for example in planar
germanium detectors Hull  _et al._ (2014).

In this paper, we discuss novel digital signal processing algorithms used to
correct for charge trapping during reconstruction of event energy in PPC
detectors. We then evaluate the algorithms’ performance using data collected
with detectors operating in the full Majorana Demonstrator array and in the
“string test cryostats” used for detector characterization and acceptance
tests. In the string test cryostats, detector temperatures are estimated to be
$\approx$95 K, roughly 15 K above the operating temperature of the full array.
The strong temperature dependence in release times resulted in different
charge trapping effects in these two systems.

##  II Charge Trapping in PPC Detectors and Effective Pole-Zero Correction

HPGe detectors have a “p-i-n” structure, where a large crystal of nearly
intrinsic p- or n-type material is modified by the addition of p+ and n+
contacts on its surface. In the Majorana Demonstrator PPC detectors, the
p-type impurity concentration in the bulk material is typically on the order
of $10^{10}$ atoms/cm3. The p+ contact is created by boron implantation in a
small “point contact” on one face of the crystal. The n+ contact is created by
lithium diffusion and covers the opposite face and cylindrical side of the
crystal. A passivated surface insulates the boundary between contacts, and a
1–5 kV reverse bias voltage between the contacts depletes the intrinsic region
of any free charge carriers. Any ionizing particle or process within the
depletion region will create charge clouds of electron-hole pairs that are
respectively transported to the n and p sides of the junction by the combined
electric field of the bias voltage and the charge impurity gradient. The
electrical signal produced at the n+ and p+ contacts can be understood in
terms of the Shockley-Ramo theorem Shockley (1938); Ramo (1939), which states
that the amplitude of the charge signal induced by a moving carrier depends on
the change in the weighting potential between its initial and final positions.
The weighting potential depends only on the geometry of the electrode
configuration and is obtained by setting the contact of interest to unit
voltage and grounding all other contacts.

PPC detectors have a sharply peaked weighting potential near the point
contact, as shown in Fig. 2. Most electron-hole pairs are generated at a
distance from the point contact. Since only the holes traverse the region with
high weighting potential, PPC signals tend to be hole-dominated. The shape and
timing of the induced signal depends on charge carrier drift paths, so a great
deal of information can be extracted from each pulse Alvis  _et al._ (2019).
The entire signal formation process can be simulated to high numerical
accuracy, allowing detailed comparisons of simulation and data to explore
subtle effects of detector properties on pulse shape. In this work, we use a
version of the HPGe pulse shape simulation package siggen specialized for
Majorana Demonstrator detectors Radford (2014).

Figure 2: Simulated weighting potential in a PPC detector, showing its
localization near the point contact region at the bottom center. The lithium
diffused surface covers the top and the side region of the detector. The
passivated surface surrounds the point contact at the bottom region. The white
lines are isochrones of equal drift time for holes to reach the point contact
spaced by 200 ns. Figure from Ref. Alvis  _et al._ (2019).

Charge carriers in events undergoing longer drift have an increased
probability to encounter traps from impurities or dislocations in the crystal
lattice. The trapped carriers are subsequently released at a rate dependent on
the effective depth of the trap and the detector temperature. The release time
increases with trap depth. As the temperature of the detector is lowered, the
release time can become long compared to the signal collection time, leading
to pulses with degraded amplitude. In this regime, under the assumption that
the trapping probability is independent of location within the crystal, a
constant Poisson probability for charge trapping per unit time during the
charge drift leads to exponential charge loss, with time constant $\tau_{\rm
ct}$. As charges are trapped, their drift ceases to contribute to the signal
amplitude. This amplitude degradation is visualized in the blue simulated
single-site pulses in Fig. 3, which are generated with unit amplitude at
detector positions with short, intermediate, and long drift times. Although we
did not rigorously validate our assumption of trapping probability uniformity,
this exponential decay model yielded excellent results here.

In the Majorana Demonstrator, detector signals are read out using charge
sensitive preamplifiers with a resistive feedback loop Abgrall  _et al._
(2022). Pulse traces are recorded with a 14-bit 100 MHz digitizer, capturing
20–50 $\mu$s of data. AC coupling in the second-stage amplification gives the
recorded signals an exponential decay constant of $\tau_{RC}\approx
72~{}\mu$s, as simulated in the red traces in Fig. 3. A standard pole-zero
correction Leo (1994), applied offline as a digital filter, can correct for
this decay in the digitized signals prior to trapezoidal filtering to estimate
the pulse amplitude Jordanov and Knoll (1994). However, such fully pole-zero
corrected traces simply recover the situation shown in the blue traces in Fig.
3, exhibiting significant amplitude dependence on drift time. As a result,
detector energy resolution can be heavily degraded.

If drift times can be extracted directly, energies can be manually corrected
to reduce this detector resolution degradation Kephart  _et al._ (2009);
Martin  _et al._ (2012). In practice, drift time can be difficult to extract,
and algorithms can be sensitive to various kinds of noise or other anomalies
or exhibit energy- as well as drift time-dependent biases that are hard to
avoid. In addition, because the drifting charges contribute to the signal the
entire time they are drifting, the degradation is dependent not just on the
drift time, but also on the trajectories of the drifting charges.

In this work we implemented a different energy reconstruction strategy that
corrects for the signal degradation more directly. Instead of fully pole-zero
correcting the pulses, an “effective” pole-zero correction is performed using
a time constant ($\tau$) given by the difference between the charge-trapping
and resistive-capacitive (RC) decay constants:

$\frac{1}{\tau}\equiv\frac{1}{\tau_{RC}}-\frac{1}{\tau_{\rm ct}}.$ (1)

As described in detail in Appendix A, this correction effectively deconvolves
the electronics response from the waveform, and then convolves it with an
exponential response governed by $\tau_{\rm ct}$. The latter convolution has
the effect of adding back in, sample by sample in the waveform, the charge
lost due to trapping during each digitization time step.

Post-filtering, the pulses exhibit exponentially decaying tails with time
constants equivalent to $\tau_{\rm ct}$, as shown in the green traces in Fig.
3. The alignment of the tails of these traces demonstrates that for any fixed
time interval following the onset of the pulse, all pulses lose the same
fractional amplitude. We thus first extract the pulse onset time ($t_{0}$),
and then evaluate the energy of the pulse from the amplitude of the
trapezoidal-filtered effective-pole-zero-corrected trace at a fixed time
interval following $t_{0}$.

Figure 3: Simulated PPC waveforms from siggen with exaggerated drift times and
preamplifier decay. Red dashed: no correction (raw signal), blue solid:
standard $\tau_{RC}$ correction, green dotted: effective $\tau$ correction.
The pulse with a large step in the rising edge is due to a “multisite” event
in which simultaneous energy depositions occur at multiple positions in the
detector. The inset zooms in on the upper left region of the figure.

##  III Energy and $t_{0}$ Evaluation

The energy estimation of Majorana Demonstrator events is done in two stages.
An uncalibrated energy is determined by applying a recursive trapezoidal
filter Jordanov and Knoll (1994). The uncalibrated energy is then converted to
keV based on fits to peaks of known energy in calibration spectra. A
trapezoidal filter includes a ramp, a flat-top, and a falling region with
durations $t_{\text{ramp}}$, $t_{\text{flat}}$, and $t_{\text{fall}}$,
respectively. The ramp and fall times are selected to give optimal estimates
of the voltage levels before and after the incident pulse, and the flat time
is chosen to be slightly longer than the longest drift times in the detector.
For symmetric trapezoidal filters, $t_{\text{fall}}=t_{\text{ramp}}$. We will
denote the rise, flat-top, and fall times of trapezoidal filters using the
notation [rise, flat, fall ], where each time is given in units of $\mu$s.

In contrast with the common method of using the maximum value of a symmetric
trapezoidal filter as the uncalibrated energy estimate, the Majorana
Demonstrator uses a “fixed-time pickoff” technique in which we first determine
the start time of the pulse in the filtered signal, $t_{0}$, and then the
uncalibrated energy is taken to be the value of the filtered signal at the
fixed time $t_{0}+\delta t$, as shown in Fig. 4. The advantage of the fixed-
time pickoff is two-fold. First, the noise in this energy estimate is governed
by the statistics of the single picked-off sample, rather than the maximum of
a series of samples, and thus avoids the bias to higher energy of the maximum
value estimator. This bias is relevant for low-energy signals (below
$\approx$10 keV). Second, careful selection of the short-rise-time trapezoidal
filter’s parameters can provide an accurate $t_{0}$ time that naturally
encodes the drift time of the signal. By combining this fixed-time pickoff
with the effective pole-zero correction described in Sec. II, an optimized
energy estimator is obtained with a free parameter $\tau$ that can be tuned
for individual detectors.

Figure 4: The fixed-time-pickoff technique used to estimate the uncalibrated
energy. Top: A normalized raw waveform from data. Middle: Symmetric leading-
edge (cyan) and energy (orange) trapezoidal-filtered signals. Bottom:
Asymmetric leading-edge (blue) and energy (orange) trapezoidal-filtered
signals. The filters are labeled using the convention $[rise,flat,fall]$ in
$\mu$s. The dashed lines show $t_{0}$, and the dotted lines indicate the
energy pickoff time. Note that the rising edge of the asymmetric filter more
closely resembles the rising edge of the waveform, producing a better $t_{0}$
estimation.

The start time $t_{0}$ of each pulse is evaluated using two different methods,
each using a leading-edge algorithm applied to a filtered signal. Our initial
$t_{0}$ algorithm, used in Refs. Aalseth  _et al._ (2018) and Alvis  _et al._
(2019), used a trapezoidal filter with a symmetric shape of $[1,1.5,1]$. The
second algorithm, used in Ref. Arnquist  _et al._ (2022), uses an asymmetric
trapezoidal filter with a shape of $[0.04,0.1,2]$. For both filters, the pulse
start time $t_{0}$ is identified by stepping backwards from the maximum of the
filtered signal until the first crossing of a very low (2 ADC $\approx 1$ keV)
threshold, and then interpolating between the samples before and after the
threshold crossing. Since they rely on a threshold crossing, both algorithms
result in a bias toward later times (discussed in Sec. VI). However the
asymmetric $[0.04,0.1,2]$ trapezoidal filter results in a more accurate and
precise $t_{0}$ determination, since it is more optimized to preserve timing
information in the rising edge, while still minimizing the impact of both low
and high frequency noise. Figure 4 shows the symmetric and asymmetric filters,
as well as the $t_{0}$ values determined by these algorithms.

After obtaining $t_{0}$, we then use a $[4,2.5,4]$ trapezoidal filter to
evaluate the uncalibrated energy with a fixed-time pickoff at $t_{0}+\delta
t$, where

$\displaystyle\begin{split}\delta t=\
&(t_{\mathrm{ramp}}+t_{\mathrm{flat}})-0.5~{}\mu\text{s}.\end{split}$ (2)

In this equation, the time constants refer to those of the $[4,2.5,4]$ energy
filter. The value 0.5 $\mu$s was determined empirically to assure that the
pickoff time is in the flat top of the filtered signal before the falling edge
as shown in Fig. 4.

##  IV Energy Resolution Optimization

The Majorana Demonstrator data are divided into data sets (DS) separated by
hardware or software-based changes to the experimental configuration and are
described in Ref. Alvis  _et al._ (2019). The analysis in this paper primarily
uses data taken during weekly 228Th source calibrations Abgrall  _et al._
(2017a). Waveform data for each physics event from the data acquisition system
is processed and stored for offline analysis.

To obtain the energy of an event, the Majorana Demonstrator uses the
$[4,2.5,4]$ trapezoidal filter with the effective pole-zero correction and
fixed-time pickoff energy estimator described earlier. Calibration constants
determined using 228Th data are then applied to convert the uncalibrated
energy to keV. The full width at half-maximum (FWHM) of the 208Tl peak at 2615
keV is measured by fitting that peak with a peak shape function consisting of
a Gaussian, a step function (an erf), a low-energy tail modeled by an
exponentially modified Gaussian, and a quadratic background Alvis  _et al._
(2019).

In the Majorana Demonstrator algorithm, the effective time constant ($\tau$)
that optimizes the energy resolution of the 2615 keV peak is determined
detector by detector for each data set. This is done by varying $\tau_{\rm
ct}$ in 10 $\mu$s steps from 60 to 250 $\mu$s, determining the FWHM of the
2615 keV peak at each value of $\tau$, and then fitting FWHM/$\mu$ versus
$\tau^{-1}$ to a quadratic function, where $\mu$ is the fit peak position:

$\mathrm{FWHM}/\mu=p_{0}+p_{1}\ \tau^{-1}+p_{2}\ \tau^{-2}.$ (3)

This optimization process is illustrated in Fig. 5 for detector P42575A using
a 1-h 228Th calibration. For this detector, a clear minimum is found near
$\tau^{-1}\approx 0.004~{}\mu\mathrm{s}^{-1}$; we choose the numeric minimum
corresponding to $\tau_{\rm ct}=90~{}\mu$s according to Eq. 1. This is a
reasonable choice when the detector energy gain and operating bias voltage
remain roughly constant over time, which is the case for most calibration
periods in the Majorana Demonstrator, as shown below in Fig 9.

Figure 5: FWHM/$\mu$ versus $1/\tau$ fit to a quadratic function. For this
detector, a clear minimum is found at $1/\tau\approx 0.004$ which is close to
the point where $\tau_{\rm ct}=100\mu$s using Eq. 1.

This effective charge trapping correction results in an average 30$\%$
improvement in the FWHM at 2615 keV for all operating detectors in DS6a, as
shown in Fig. 6. The energy resolution over the entire spectrum is
parametrized by

$\sigma(E)=\sqrt{\sigma_{n}^{2}+\sigma_{f}^{2}E+\sigma_{c}^{2}{E}^{2}}$ (4)

where $\sigma_{n}$ accounts for electronic noise, $\sigma_{f}$ accounts for
the Fano factor Fano (1947), and $\sigma_{c}$ is a term resulting in a linear
dependence at high energy that is dominated by residual charge trapping. The
inclusion of all data incorporates statistical variation between calibrations,
different detectors, and gain instabilities that can contribute to the
$\sigma$ parameters; here we report the results using the statistics of the
calibrations only, to highlight the impact of the charge trapping correction.
Figure 7 shows the peak shape at 2615 keV for all operating enriched detectors
combined. The charge trapping correction results in a markedly sharper peak,
with FWHM improved from 5.1 keV to 2.9 keV. Figure 8 shows the energy
resolution curve as a function of energy for the sum of all operating
detectors in DS0–DS6a (the dataset reported in Alvis  _et al._ (2019)) with
and without the charge trapping correction applied. Prior to application of
the charge trapping correction, resolution scales mostly linearly with energy
due to the $\sigma_{c}$ term; with the charge trapping correction, the Fano
term $\sigma_{f}$ dominates. Figure 9 shows the FWHM of the combined spectrum
of each calibration subset as a function of time from DS0–DS6a. The
instability in resolution seen during DS5 is due to noise introduced by
imperfect grounding during the final stages of construction of the outer poly
shield of the experiment. Finally, Table 1 lists the FWHM at the
$0\nu\beta\beta$ $Q$ value of 2039 keV determined by fits to Eq. 4, for each
data set published in Ref. Alvis  _et al._ (2019). There is no significant
difference in the energy resolution of the enriched and natural detectors,
which were manufactured by different companies with different designs, after
the charge trapping correction. Both have an average resolution of about
2.2–2.4 keV FWHM without considering the uncertainties due to nonlinearities
and gain stability, which increase the FWHM to 2.5 keV Alvis  _et al._ (2019).

Figure 6: Energy resolution (FWHM, statistics only) at 2615 keV for all operating detectors in DS6a before (blue points) and after (red points) the charge trapping correction. The $x$-axis is the detector ID. Detector IDs beginning with “B” and “P” are natural and enriched Ge detectors, respectively. The average resolution after the charge trapping correction (red line) represents a $\approx$30% improvement. The variation between detectors is mostly due to differing crystal impurities and maximum drift times. Figure from Ref. Alvis  _et al._ (2019).  Figure 7: The 2615-keV peak from 208Tl in calibration data with all operating enriched detectors in DS0–DS6a. The fits reproduce the measured peak shapes nearly perfectly. Figure 8: Resolution (FWHM, statistics only) vs energy for the combined energy spectrum of all operating detectors in DS0–DS6a with fits to Eq. 4. The energy resolution of this curve at the $Q$ value is about 2.4 keV with the charge trapping correction. Inclusion of systematic uncertainty increases the FWHM to 2.5 keV. Figure 9: Energy resolution (FWHM, statistics only) of all operating detectors at 2039 keV versus time over each weekly calibrations in DS0–DS6a. The instability in DS5 is due to construction activities that induced higher noise. Table 1: Optimized FWHM energy resolution (statistics only) in keV at 2039 keV for DS0–DS6a, showing enriched, natural, and all detectors combined, determined by fitting Eq. 4. Systematic contributions raise the final combined FWHM to 2.5 keV. |  enrGe |  natGe | All
---|---|---|---
DS0 | 2.472(5) | 2.214(4) | 2.403(5)
DS1 | 2.465(6) | 2.131(3) | 2.441(5)
DS2 | 2.468(6) | 2.118(4) | 2.444(6)
DS3 | 2.477(6) | 2.295(4) | 2.430(5)
DS4 | 2.475(4) | 2.405(4) | 2.456(4)
DS5a | 2.815(7) | 2.508(4) | 2.727(6)
DS5b | 2.434(5) | 2.256(4) | 2.381(5)
DS5c | 2.444(2) | 2.260(2) | 2.392(2)
DS6a | 2.381(3) | 2.225(2) | 2.334(3)
DS0-DS6a | 2.484(5) | 2.299(3) | 2.438(5)

##  V Temperature Dependence of Charge Trapping

Acceptance testing for Majorana Demonstrator PPC detectors was carried out
underground at SURF both in vendor-provided cryostats as well as, for some
detectors, in custom-built string test cryostats. The string test cryostats
were designed to operate strings of 3–5 detectors using the same detector
mounts, front ends, amplifiers, and readout electronics as the Majorana
Demonstrator. Charge trapping corrections were not applied for the acceptance
tests, which still showed uniformly good energy resolution usually below 3 keV
FWHM at 2615 keV and within $\approx$10% of the values measured in-situ in the
Majorana Demonstrator modules with optimized charge trapping corrections.

Thirteen detectors were operated in both string test cryostats and the
Majorana Demonstrator modules, including two BEGe detectors fabricated from
natural Ge. The string test cryostat data were reanalyzed including the charge
trapping correction described above. The optimal values of $\tau_{\rm ct}$ are
compared in Fig. 10. Twelve of the detectors showed negligible charge trapping
in the string test cryostats, while only one of the BEGe detectors showed
improvement with a minimal charge trapping correction. Both BEGe detectors and
two of the enriched detectors required only a minimal charge trapping
correction in the Majorana Demonstrator modules for all data sets, while the
remaining nine enriched detectors required significant charge trapping
corrections to achieve optimal energy resolution.

Figure 10: Optimal values of $\tau$ for data taken in string test cryostats
(STC) and Majorana Demonstrator modules for 13 detectors. The $x$-axis is the
detector ID; those starting with “B” are BEGe detectors, the remaining are
enriched PPC detectors. Some symbols (such as the STC value for B8470 at
$\tau=800~{}\mu$s are not visible due to overlaps. Horizontal grey lines
indicate the inferred values of $\tau_{\rm{ct}}$. The symbol $\infty$
indicates that charge trapping corrections do not improve the energy
resolution, implying that there is little to no charge trapping. We find
slight variations between data sets.

The primary difference in operating conditions between the string test
cryostats and the Majorana Demonstrator detector modules (Fig. 1) is the
detector temperature. The string test cryostats were cooled by a solid copper
cold finger dipped in a liquid nitrogen dewar. The modules of the Majorana
Demonstrator, on the other hand, are cooled by a more sophisticated
thermosyphon configuration Aguayo  _et al._ (2013) that held the coldplate and
detectors at temperatures near 80 K. The less efficient cooling mechanism of
the string test cryostats was estimated to cool the detectors only to
$\approx$95 K, roughly 15 K warmer than in the Majorana Demonstrator modules.
The vendor-provided cryostats used the same LN-cooled cold finger
configuration as the string test cryostats.

Drifting charges that get captured in charge traps are re-released on a time
scale that varies strongly with temperature. The characteristic re-release
time is distinct from the trapping time constant $\tau_{\rm ct}$ introduced
earlier, although both depend on the trap depth. At high temperatures, the re-
release time can be short relative to the drift time, so that trapped charges
are essentially immediately released and collected in time for the energy
measurement. At very low temperatures, the trapping duration can be long
compared to the interval between events in the volume of the charge cloud
trajectory, in which case the traps “freeze-out” and remain occupied, and are
thus unavailable for further trapping. It is only at intermediate temperatures
that the traps can hold on to the charge long enough to degrade energy, but
release the charge soon enough that the trap is active again for the next
charge cloud traversing its vicinity.

We conclude that in the Majorana Demonstrator modules, the detector
temperatures are in this intermediate range where charge trapping impacts are
significant. The temperature in the string test cryostats seems to have been
sufficiently high to lead to prompt re-release, consistent with the lack of a
need for significant charge trapping corrections in those data.

##  VI Correction for $t_{0}$ estimation bias

Because the $t_{0}$ estimation algorithms described in Sec. III rely on
identifying a threshold crossing, the $t_{0}$ values obtained are biased
towards later times than the true $t_{0}$. Since after the charge trapping
correction the energy estimation trapezoidal filter’s flat top decays with a
time constant of $\tau_{\rm ct}$, the bias $\Delta t_{0}$ is propagated into a
bias $\Delta E$ in energy measurements according to

$\Delta E=\frac{\Delta t_{0}}{\tau_{\rm ct}}E.$ (5)

Furthermore, as the ratio of the energy to the $t_{0}$ threshold decreases,
the size of the $\Delta t_{0}$ increases, resulting in energy scale non-
linearity. By estimating the size of $\Delta t_{0}$ as a function of energy,
one can correct for this bias and improve the linearity of the energy
response.

The size of this bias can be measured by comparing $t_{0}$ to the time of a
pulse from a coincidence event in a second detector. Ideally, the second
detector would have a much faster response than a PPC detector. However, for
Majorana Demonstrator data we can still measure this bias by comparing
coincidences between two PPC detectors. Because $\gamma$ rays emitted in a
cascade will be absorbed by separate detectors within 10 ns of each other, the
coincidence events they produce are effectively observed simultaneously, so
any difference in $t_{0}$ reflects measurement bias.

The timing bias between detectors $i$ and $j$ is determined using

$\Delta t_{ij}=t_{DAQ,i}+t_{0,i}-t_{DAQ,j}-t_{0,j}$ (6)

where $t_{DAQ}$ is the trigger time of the event according to the data
acquisition (DAQ) system, and $t_{0}$ is the measured start time of the
waveform relative to the first recorded sample. Note that $\Delta t_{ij}$ is
not a clean measurement of timing bias for a single detector, but reflects the
bias of all detectors involved. The 228Th sources used for detector
calibration will emit 583 keV $\gamma$ rays followed promptly by $2615$ keV
$\gamma$ rays at a high rate, which will often produce coincidence events. By
measuring $\Delta t_{ij}$ for coincident events in which one detector measures
583 keV, we can observe the energy dependence of $\Delta t_{0}$ for the other
detector across a wide range of energies, as shown in the top panel of Fig.
11. This allows us to extract a correction based on the statistics of the
recorded $\Delta t_{0}$ values.

Figure 11: The timing bias calculated using Eq. 6 between coincidence events
measured during calibration of the Majorana Demonstrator using a 228Th line
source. The coincidence events shown have one detector in the 583 keV peak,
while the energy of the other detector is shown on the x-axis. The red points
represent the mean bias measured for a given energy bin, relative to the 583
keV event, and the green lines show the standard deviation. On top, we see the
timing bias using the $t_{0}$ estimate described in Sec. III, and on the
bottom we see the bias after applying the correction described in Sec. VI.

For each detector $i$, we model its (unknown) mean $\Delta t_{0}$ as

$\langle\Delta
t_{i}(E)\rangle=\begin{cases}\frac{a_{0}}{\sqrt{E}}-a_{1}&E>E_{c}\\\ m\cdot
E+b&E\leq E_{c}\end{cases}$ (7)

where $a_{0}$, $a_{1}$, and $E_{c}$ are floating model parameters, and the
mean refers to the average over detector signal shapes. The
$\frac{a_{0}}{\sqrt{E}}$ term can be derived by assuming that the waveform
begins rising linearly from the baseline with a slope that is a constant
fraction of the total energy, with a kink at the true event start time. The
constant $a_{1}$ term represents a fixed offset produced by the DAQ channel.
This model breaks down at low energies, so we transition to a linear model
below energy $E_{c}$, with $m$ and $b$ chosen so that the function is
continuous and differentiable at this transition. This transition typically
occurs around 10 keV, below which the threshold crossing occurs after the
assumption of a linear rise no longer applies. In addition, we model the
variance of the timing bias vs energy, using

$\langle\Delta t_{i}(E)^{2}\rangle=\sigma_{0}^{2}+\frac{\sigma_{1}^{2}}{E}$
(8)

where $\sigma_{0}$ and $\sigma_{1}$ are additional floating model parameters.

Using this model, we can measure the timing bias as a function of energy for
each detector by performing a simultaneous fit of the measured timing biases
$\Delta t_{ij}$ of coincidence events in 228Th calibration data. In order to
maximize the number of available events and reduce statistical error, all
coincidence events are used as opposed to just those involving a 583 keV hit
shown in Fig. 11; while not necessarily optimal, this approach works well, as
seen below. We construct the following unbinned negative log-likelihood
function:

$-2\log\mathcal{L}=\sum_{i,j}\frac{\big{(}\Delta t_{ij}-\langle\Delta
t_{i}(E_{i})\rangle+\langle\Delta
t_{j}(E_{j})\big{\rangle})^{2}}{\langle\Delta
t_{i}(E_{i})^{2}\rangle+\langle\Delta t_{j}(E_{j})^{2}\rangle},$ (9)

where the sum runs over all pairs of hits $i,j$ that trigger within 4 $\mu$s
of each other. The MIGRAD algorithm from ROOT’s minuit package Brun and
Rademakers (1997) is used to minimize this likelihood function in order to
obtain best fit parameters. This fit is performed on a weekly basis, for each
calibration of the Majorana Demonstrator. After obtaining the best fit
parameters for each channel, we correct the $t_{0}$ parameters for detector
$i$ as follows:

$t_{0,corr}=t_{0}-\langle\Delta t_{i}(E)\rangle$ (10)

where the corrected start time $t_{0,corr}$ is then used in identifying the
pickoff time for the procedure described in Sec. III. The bottom panel of Fig.
11 shows the timing bias for coincidence events in 228Th data involving 583
keV hits after applying this correction.

The Majorana Demonstrator has performed two energy estimations which we will
compare. The first, referred to as $t_{0}$ uncorrected energy, includes the
charge trapping correction using a $[1,1.5,1]$ leading-edge trapezoidal filter
and no $t_{0}$ correction, and is calibrated to a linear function with an
offset at $E=0$ Aalseth  _et al._ (2018); Alvis  _et al._ (2019). The second,
referred to as $t_{0}$ corrected energy, uses the $[0.04,0.1,2]$ leading-edge
trapezoidal filter and includes the $t_{0}$ correction outlined here, and is
calibrated to a quadratic function with no offset at $E=0$ Arnquist  _et al._
(2022). Both estimators followed the resolution optimization procedure
described in Sec. IV. Figure 12 shows the energy shift between these
estimators. Prior to $t_{0}$ correction, the energy bias is greatest at low
energies, with an average value of $\approx 0.15$ keV, and reaching up to
$0.3$ keV in detectors with the largest effect, which is equivalent to
$\approx$1 FWHM of a low energy peak. By correcting this nonlinearity, we can
achieve accurate energy measurements at all energies, and we remove the need
for a calibration offset at $E=0$. This also removes the need for a separate
calibration at low energies as was performed in Ref. Abgrall  _et al._
(2017b), and aids in analyses that utilize events at all energies.

Figure 12: Top: the mean energy shift in natural isotopic abundance BEGe
detectors measured during a 228Th calibration run between the energy estimator
using an uncorrected $t_{0}$ ($E_{t_{0}\,\textrm{uncorr}}$) and the estimator
that does perform this correction ($E_{t_{0}\,\textrm{corr}}$) as a function
of energy. Bottom: a low energy background spectrum in natural Ge detectors,
with two peaks produced by electron capture in the bulk of the detectors by
cosmogenically produced isotopes 55Fe and 68Ge, with energies at 6.49 keV and
10.37 keV, respectively. Comparing the $E_{t_{0}\,\textrm{corr}}$ spectrum
(solid blue) to the $E_{t_{0}\,\textrm{uncorr}}$ spectrum (red line), we see
the $0.16$ keV shift produced by this correction so that the corrected peaks
accurately measure the expected energies.

##  VII An Alternative Correction

An alternative charge trapping algorithm was developed in parallel with the
$\tau$ optimization algorithm described above and implemented in a secondary,
redundant analysis. The alternative method was tested using Majorana
Demonstrator calibration data and gives results identical to the method
described in Sec. III within statistical uncertainty, although the alternative
method was found to sometimes out perform the $\tau$ optimization method.

In the alternative method, a standard pole-zero correction is applied using
the measured preamplifier time constant only, $\tau_{RC}$. This generates
outputs similar to the blue signals in Fig. 3. The start time, $t_{0}$, is
evaluated using the same method described in Sec. III. Then the pole-zero-
corrected waveform signal is integrated over two time intervals,
[$t_{0},t_{0}+2\mu s$] and [$t_{0}+2\mu s,t_{0}+4\mu s$], and the difference
$D$ between the two integrals is calculated. The 2 $\mu s$ integration time is
chosen because it is slightly longer than the maximum drift time for all of
the Majorana Demonstrator detectors. The integration windows and the
calculation of $D$ are illustrated in Fig. 13.

Figure 13: Visualization of the alternative charge trapping correction. The
value of $D$ is represented by the area shaded green, given by the difference
between the 2-$\mu$s integrated waveform regions represented by gray and red.
For reference, pulses with shorter and longer drift times are included; these
would have smaller and larger values of $D$, respectively.

As described in more detail in the Appendix, the signal degradation due to
charge trapping is nearly proportional to the integral of the uncollected
charge, which is approximately proportional to $D$. Thus the final step is to
add a small $D$-dependent correction to the uncalibrated energy calculated
using the standard trapezoidal filter. In the Majorana Demonstrator PPCs a
simple linear correction proved to be sufficient:

$E_{\mathrm{corrected}}=E_{\mathrm{uncorrected}}+\alpha D.$ (11)

The charge-trapping correction factor $\alpha$ is optimized by minimizing the
FWHM of the 2615-keV peak.

This alternative algorithm has several advantages. The primary advantage is
that optimization of $\alpha$ is much less computationally intensive than
optimization of $\tau$, which requires repeated digital signal processing.
This algorithm should also have some advantage at higher count rates because
it does not generate any undershoot at the end of the trapezoidal filter that
results from the incomplete pole-zero correction of the $\tau$ optimization
method. This algorithm is also superior to methods based on estimating the
drift time from the pulse rise time because the evaluation of $D$ is based on
an integral and requires the estimation of only one time point ($t_{0}$)
rather than two. Nonlinear dependence of the correction on $D$ can be easily
accommodated by extending Eq. 11 to quadratic or higher order. $D$ is also
useful in contexts outside of energy estimation, as is the pole-zero corrected
waveform based on $\tau_{RC}$ only.

This alternative method and the original $\tau$ optimization method assume
different time dependences for the untrapped charges. The $\tau$ optimization
method assumes exponential decay of untrapped charge during the charge drift,
while the alternative method assumes a linear decay. This linear approximation
is valid as long as the charge trapping rate is small, i.e., if the maximum
drift time is short compared to the mean free drift time. It would be possible
to extend the alternative method to accommodate exponential decay by modifying
Eq. 11. Both methods require an accurate determination of $t_{0}$ for each
signal, but the bias correction described in Sec. VI works equally well for
both. Finally, both algorithms assume that in a PPC detector it is only hole
trapping that significantly affects the extracted energy, and that the
interaction creating the electron-hole pairs is away from the point contact,
i.e. at a small weighting potential. Both of these assumptions are valid for a
majority of signals in PPC detectors.

##  VIII Conclusion

We reported two charge trapping corrections that result in a significant
improvement in the energy performance of the Majorana Demonstrator. One is
based on a modified pole-zero-corrected energy estimator, the other is based
on an integral of the uncollected charge. An analysis based on the former
gives an average FWHM after charge trapping corrections of 2.4 keV at the
$0\nu\beta\beta$ $Q$ value of 2039 keV and 2.9 keV at 2615 keV (statistical
uncertainties only). The presence or absence of significant charge trapping
appears to strongly correlate with detector temperature, requiring energy
estimation parameters to be tuned to cryostat-specific environmental
conditions. These correction algorithms are very flexible and suitable for
other arrays of HPGe PPC detectors, including the upcoming LEGEND experiment
Abgrall  _et al._ (2017c, 2021).

## Acknowledgments

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Nuclear Physics under contract/award nos. DE-
AC02-05CH11231, DE-AC05-00OR22725, DE-AC05-76RL0130, DE-FG02-97ER41020, DE-
FG02-97ER41033, DE-FG02-97ER41041, DE-SC0012612, DE-SC0014445, DE-SC0018060,
DE-SC0022339, and LANLEM77/LANLEM78. We acknowledge support from the Particle
Astrophysics Program and Nuclear Physics Program of the National Science
Foundation through grant nos. MRI-0923142, PHY-1003399, PHY-1102292,
PHY-1206314, PHY-1614611, PHY-1812409, PHY-1812356, PHY-2111140, and
PHY-2209530. We gratefully acknowledge the support of the Laboratory Directed
Research & Development (LDRD) program at Lawrence Berkeley National Laboratory
for this work. We gratefully acknowledge the support of the U.S. Department of
Energy through the Los Alamos National Laboratory LDRD Program and through the
Pacific Northwest National Laboratory LDRD Program for this work. We
gratefully acknowledge the support of the South Dakota Board of Regents
Competitive Research Grant. We acknowledge the support of the Natural Sciences
and Engineering Research Council of Canada, funding reference number
SAPIN-2017-00023, and from the Canada Foundation for Innovation John R. Evans
Leaders Fund. This research used resources provided by the Oak Ridge
Leadership Computing Facility at Oak Ridge National Laboratory and by the
National Energy Research Scientific Computing Center, a U.S. Department of
Energy Office of Science User Facility. We thank our hosts and colleagues at
the Sanford Underground Research Facility for their support.

##  Appendix A Signal Induction with Charge Trapping

Signal induction in PPC HPGe detectors is governed by the Shockley-Ramo
theorem Shockley (1938); Ramo (1939): the induced current $I$ at a conductor
due to a charge $q$ moving with velocity $\vec{v}$ in its vicinity is given by

$I(t)=q\,\vec{v}\cdot\vec{\nabla}\phi,$ (12)

where $\phi$ is the weighting potential for the conductor, evaluated by giving
the conductor unit potential and grounding all other conductors in the
vicinity.

In PPC HPGe detectors, the signal at the point contact is induced by the
motion of both electron ($e$) and hole ($h$) charge clouds. The holes drift
toward the point contact while electrons drift away from it, with differing
drift speeds. In the presence of charge trapping, the charge $q$ in each cloud
decreases with time. Assuming a constant charge trapping probability that does
not vary with position, the charge in each cloud will decay exponentially,
with species dependent time constants. Ignoring phenomena like recombination,
diffusion, and space charge effects, the induced current at the point contact
can be written

$I(t)=q_{0}e^{-(t-t_{0})/\tau^{h}_{\rm
ct}}\frac{d\phi_{h}}{dt}-q_{0}e^{-(t-t_{0})/\tau^{e}_{\rm
ct}}\frac{d\phi_{e}}{dt},$ (13)

where $q_{0}$ is the initial charge in the hole cloud at time $t_{0}$,
$\tau^{k}_{\rm ct}$ is the charge trapping time constant for species $k$, and
$\frac{d\phi_{k}}{dt}$ is the rate of change of the weighting potential
traversed by species $k$. $\frac{d\phi_{k}}{dt}$ depends not only on the
charge species but also the trajectory, and is non-zero only for
$t_{0}<t<t_{0}+T_{k}$, where $T_{k}$ is the time it takes the charge cloud to
reach a detector contact.

This signal is measured by collecting the induced current on a capacitor and
measuring the charge $Q(t)$, given by the integral of Eq. 13. The electrons
and holes start at the same location with the same initial weighting potential
$\phi_{0}$. For negligible charge trapping ($\tau^{k}_{\rm ct}\gg T_{k}$), the
collected charge is given by

$Q_{\rm no~{}ct}(t)=q_{0}\Delta\phi(t)$ (14)

where $\Delta\phi(t)$ is the difference in weighting potential at the
positions of the hole and electron clouds at time $t$. This difference is zero
for $t<t_{0}$, and it is equal to 1 for $t>T_{e},T_{h}$.

###  A.1 Effective Pole-Zero Correction

The effective pole-zero correction described in Sec. II can be thought of as
first deconvolving the electronic response from the recorded signals, and then
convolving the signal with an exponential response with optimized time
constant $\tau_{\rm ct}$. The resulting signal has modified collected charge

$\tilde{Q}(t)=\int_{t_{0}}^{t}e^{-\frac{t-t^{\prime}}{\tau_{\rm
ct}}}\,I(t^{\prime})\,dt^{\prime}.$ (15)

The term $e^{t^{\prime}/\tau_{\rm ct}}$ can be combined with the
$e^{-t^{\prime}/\tau^{k}_{\rm ct}}$ terms in $I(t^{\prime})$ that represent
the charge loss during drift due to trapping. Thus the convolution has the
effect of “adding back” the lost charge. In PPC detectors the signal
generation for most events is dominated by the hole motion, so the optimal
value of $\tau_{\rm ct}$ will be close to $\tau^{h}_{\rm ct}$. Thus hole
trapping is almost fully compensated, while the electron trapping, which is
more rapid, remains only partially compensated but contributes less to signal
degradation. The resulting modified collected charge is thus approximately

$\tilde{Q}(t)\approx e^{-(t-t_{0})/\tau_{\rm ct}}\,q_{0}\,\Delta\phi(t),$ (16)

which differs from Eq. 14 by only a multiplicative exponential function of
time. As a result, the fixed-time pickoff energy estimation method described
in Sec. III will obtain a similar value for events with the same energy
deposition at different points in the detector, with minimal variation due to
charge trapping.

###  A.2 Uncollected Charge Integral Method

The alternative approach described in Sec. VII can be motivated by
approximating the exponential functions in Eq. 13 by their first-order Taylor
expansions and then directly integrating. The terms like $t\frac{d\phi}{dt}$
can be integrated by parts to give

$\int_{0}^{T}t\,\frac{df}{dt}\,dt=\int_{0}^{T}[f(T)-f(t)]\,dt.$ (17)

Using the fact that $\phi_{h}(t>t_{0}+T_{h})=1$ and
$\phi_{e}(t>t_{0}+T_{e})=0$, we find that at the end of the drift the
collected charge has approximately the constant value:

$Q\approx q_{0}-\frac{q_{0}}{\tau_{\rm
ct}^{h}}\int_{t_{0}}^{t}\left(1-\Delta\phi(t^{\prime})+\frac{\tau_{\rm
ct}^{h}-\tau_{\rm ct}^{e}}{\tau_{\rm
ct}^{e}}\phi_{e}(t^{\prime})\right)\,dt^{\prime},$ (18)

where $t$ is any time satisfying $t-t_{0}>T_{h},T_{e}$.

The quantity $q_{0}(1-\Delta\phi(t))$ represents the uncollected charge at
time $t$ (in the absence of charge trapping). To zeroth order in
$t/\tau^{h}_{\rm CT}$, its integral over the drift is proportional to the
quantity $D$ defined in Sec. VII. The third term in Eq. 18 gives a
contribution proportional to the integral of the signal due to the electron
drift only. The integrated electron signal is small for the vast majority of
events, and is comparable to the uncollected charge integral only for events
originating very close to the point contact. On average it contributes a shift
that is proportional to energy, and hence is accounted for in the energy scale
calibration. Thus, an energy correction proportional to $D$ is expected to
remove most of the variance due to charge trapping.

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