Charge trapping correction and energy performance of the Majorana Demonstrator

$P$-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 ${}^{76}\mathrm{Ge}$, 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 ${}^{76}\mathrm{Ge} Q$ value. An alternative approach achieving similar resolution enhancement is also presented.

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 ultrapure 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.

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. [26].

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.

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 $\mu \mathrm{s}$. 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.

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 ${\tau }_{\mathrm{ct}}=100\mu \mathrm{s}$ using Eq. (1).

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. [26].

Figure 7

The 2615-keV peak from ${}^{208}\mathrm{Tl}$ 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.

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 $\tau =800\mu \mathrm{s}$ are not visible due to overlaps. Horizontal grey lines indicate the inferred values of ${\tau }_{\mathrm{ct}}$. 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.

Figure 11

The timing bias calculated using Eq. (6) between coincidence events measured during calibration of the Majorana Demonstrator using a ${}^{228}\mathrm{Th}$ 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. 3, and on the bottom we see the bias after applying the correction described in Sec. 6.

Figure 12

Top: the mean energy shift in natural isotopic abundance BEGe detectors measured during a ${}^{228}\mathrm{Th}$ calibration run between the energy estimator using an uncorrected $t_0$ ($E_{t_0\text{uncorr}}$) and the estimator that does perform this correction ($E_{t_0\text{corr}}$) 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 ${}^{55}\mathrm{Fe}$ and ${}^{68}\mathrm{Ge}$, with energies at 6.49 keV and 10.37 keV, respectively. Comparing the $E_{t_0\text{corr}}$ spectrum (solid blue) to the $E_{t_0\text{uncorr}}$ spectrum (red line), we see the 0.16 keV shift produced by this correction so that the corrected peaks accurately measure the expected energies.

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-$\mu \mathrm{s}$ 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.