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Determination of space charge distributions in highly segmented large volume HPGe detectors from capacitance–voltage measurements
Nuclear Instruments and Methods in Physics Research A 640 (2011) 176–184
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Determination of space charge distributions in highly segmented large volume HPGe detectors from capacitance–voltage measurements B. Birkenbach, B. Bruyneel , G. Pascovici, J. Eberth, H. Hess, D. Lersch, P. Reiter, A. Wiens Institut f¨ ur Kernphysik, Universit¨ at zu K¨ oln, Z¨ ulpicher Straße 77, 50937 K¨ oln, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 November 2010 Received in revised form 21 February 2011 Accepted 21 February 2011 Available online 23 March 2011
The space charge distribution of a large volume highly segmented HPGe detector was determined by a non-destructive capacitance–voltage measurement. The capacitances between the 36 segments and the core were measured simultaneously with a precision pulser which was implemented in the core preamplifier. The pulser measurement was compared to and validated by direct capacitance measurements. The three-dimensional doping profile was reconstructed using analytical and numerical methods. Consistent values for the impurity concentration in the range of 0.5 and 1.5 1010 cm 3 were obtained. & 2011 Elsevier B.V. All rights reserved.
Keywords: Impurity concentration Segmented HPGe-detectors Capacitance–voltage Current–voltage characteristic
1. Introduction The final configuration of the 4p spectrometer Advanced GAmma Tracking Array (AGATA) [1] will consist of 180 highly segmented high purity germanium (HPGe) detectors and will provide best detection efficiency and performance yet employing the novel g-ray tracking [2] principle for position dependent g-ray detection. The main ingredient of g-ray tracking is the individual position of each energy deposition point after interaction of g-rays inside the HPGe crystal. To achieve the required position resolution of a few millimeters inside the detector volume the high segmentation of the detectors alone is insufficient and pulse shape analysis methods [3] are applied. Also other experiments like the neutrinoless beta decay experiment GERDA [4] employ PSA on comparable HPGe detectors for background recognition. A crucial step of the PSA procedure is a comparison of all the detected real pulses with a well-known data set of pulses with precise position information as a reference. These data sets are created either by computer simulations [5,6] or can be determined experimentally by position dependent scanning of the individual detectors [7]. In order to obtain very reliable computer simulations the characteristics of the individual HPGe detectors must be known very precisely and the simulations must be verified with experimental data. Recently major advances were achieved to understand the relevant parameters of the highly segmented HPGe detectors. For
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E-mail address: [email protected] (B. Bruyneel). 0168-9002/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2011.02.109
example, improved models were developed to describe the mobility of the charge carriers [8–10]. The electronic properties of the novel detectors were investigated in detail and crosstalk contribution, caused by the capacitive coupling between the segments and the core [11,12], were described successfully. Another important quantity which characterizes the HPGe detectors is the remaining impurity concentration of charge carriers throughout the Ge crystal volume. The impurity concentration affects directly the necessary operation voltage of the detector and the resulting electric field strength. Despite the high impact of the impurity concentration on the detector performance a direct measurement after crystal processing is prevented by the manufacturing techniques. Therefore, non-destructive capacitance–voltage measurements were investigated in Ref. [13] to determine the depletion regions inside the Ge crystal as a function of applied bias voltage. The depleted regions can be reconstructed from the capacitance values measured between the common core electrode and all the individual segments. Finally the impurity concentration is deduced from the variation of the depletion zone as a function of the high voltage. In this article we report on the first-time reconstruction of the doping profile inside a large volume, highly segmented HPGe detector. The new approach is based on simultaneous capacitance measurements with a 36-fold segmented AGATA detector employing a precision pulser of the AGATA core preamplifiers [14]. Combined with results of the computer simulation and reconstruction methods described in Ref. [13] the three-dimensional impurity concentration of an AGATA detector was determined. The paper is organized in the following way: In the first and second parts the relevant electronic properties of the highly
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segmented detector and the new method to measure the capacitance between core contact and segment electrodes with a pulser are introduced. In the central part the results of the capacitance– voltage measurements are presented. From these findings the doping profile is reconstructed finally using three different approaches with increasing refinement and complexity.
2. Electrical model of an AGATA detector An AC equivalent electrical detector model for the 36-fold segmented AGATA HPGe detector consists of 36 capacitances with 36 parallel and serial resistors (see Fig. 1). All capacitance and resistor values are not constant but change with the applied bias voltage. The capacitors Ci are strongly dependent on the depleted volume between the segment electrodes and the core electrode. The small leakage current through the segments is due to the not infinite but large resistors Rpi in parallel with the capacitors. The resistivity of the undepleted region between segments and core is modeled with the series resistances Rsi. The total capacitance for a whole AGATA detector is typically 4571 pF at full depletion and is given by the manufacturer [15]. Since the capacitance is a monotonically decreasing function with bias voltage, this value represents a lower limit. For the 36 individual segments, the minimum capacitance is therefore of the order of 1 pF. A lowest voltage of 10 V was applied for the measurements. At this bias voltage, the capacitance should be on the order of 3 nF, depending on the impurity concentration. The total series resistivity is largest when no bias voltage is applied and disappears at full depletion. Its value is strongly dependent on temperature and impurity concentration. Typical values for the bulk series resistivity (at 10 V bias and LN2 temperature) are of the order of 1 kO [15]. The parallel resistor values are extremely high for reverse biased diodes and therefore challenging to measure. At full depletion, typical values are in the T O region. To ensure that these values were also reached at the lowest bias voltages, a dedicated measurement was performed. The leakage current of a symmetric AGATA detector was measured as a function of very low bias voltage. The AGATA detector was mounted in the symmetric AGATA triple cryostat [16]. For this occasion, the standard preamplifier electronics were completely removed allowing direct access to core and segment electrodes. The setup is shown schematically in Fig. 2. All the segment electrodes were short connected and grounded via a Keithley 486 picoammeter. The core electrode was directly connected to a high voltage module omitting the usual 1 GO protection resistor. The leakage current was measured for different bias voltages ranging from 0.1 to 6 V. A self-made module was used which was based on the 12 V of the NIM power supply, a voltage divider and
Fig. 2. Setup for the measurement of the leakage current of a 36-fold segmented, symmetric AGATA detector.
Fig. 3. I–V characteristics of the symmetric AGATA detector.
filter. Results are shown in Fig. 3. These values were corrected for the current offset reading at 0 V. Errors are mainly induced by the variation of this offset while powering up the detector. Therefore, after each measurement, the detector was ramped down to perform a new offset measurement and to minimize the changes in offset during the measurement. These values are in line with measurements performed at higher bias voltages using commercially available HV modules. However, these modules produced considerably larger fluctuations in the current offset, implying larger errors at very low voltages. From the values shown in Fig. 3 the total parallel detector resistance was calculated. The differential resistivity is given as @U=@IjVbias . Above 10 V this resistance is already in the region of TO, and therefore its contribution will be completely negligible at all bias voltages of interest.
3. Principle of the pulser method
Fig. 1. Electrical model of a 36-fold segmented detector.
A built-in programmable pulser is installed on the core preamplifier board [14]. Different from standard pulser inputs, the pulser signal is injected at the source pin of the core contact FET which is connected to ground level via a 1:8 O resistor. A schematic drawing of the detector electronics is shown in Fig. 4. In order to extract capacitance values with the pulser signal the exact relationship between pulser line positions and capacitance is mandatory [17]. The preamplifier’s transfer function for pulser input signals is derived based on the circuitry in Fig. 4. The measured pulser positions in the spectra are compared to values from a g-ray calibration source. The response to currents Ig induced by gamma radiation is also derived. In the calculations, the two 1 GO resistors of Fig. 4 can be neglected. In the feedback
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It should be noted that this approximation is not per se valid. Especially at low bias voltages, t1 can be in the range of 1 ms, which is dangerously close to the typical integration times used for standard analog main amplifiers. Visual inspection of the long rise times with a scope can be helpful. Measurements should therefore be performed with the largest possible energy filter settings. If the total series resistance is negligible for the measurement of the total capacitance using Eq. (6), then also the series resistance values in Fig. 1 are negligible. The core-to-segments 1 capacitances are roughly 36 of the total capacitance, while the individual series resistances will be about 36 times larger than the total series resistance Rdet. Then the product CjRsj from Fig. 1 will be comparable to t1 . The current measured in each of the segments j is proportional to the core-to-segment capacitance Cj. According to the current divider rule, we obtain Fig. 4. Circuit diagram of the detector from the viewpoint of the core preamplifier (see text). The detector model shown in Fig. 1 is substituted with the total detector capacitance Cdet and series resistivity Rdet as indicated with the symbol. Stray capacitances Cstray produce an offset between the core pulser line position and the sum of the segment pulser line positions.
loop, the time constant tfb ¼ Cfb Rfb is 1 ms. Therefore, Rfb is only relevant for effects on a timescale larger than 1 ms. For similar reasons, the 1 GO protection resistor can be neglected. 3.1. The pulser transfer function In the case where a rectangular pulse is injected at the noninverting input of the preamplifier, this potential is ‘copied’ into the inverting input of the preamplifier: Vin ¼ Vpulser :
ð1Þ 5
This behavior is caused by the large open loop gain A C10 : if the output is not saturated – implying Vout ¼ AðV þ V Þ o1 V – the potential difference at the input nodes is smaller than the thermal noise. If no pulser signal is applied, the preamplifier input is acting as a virtual ground. This concept is already indicated in Fig. 4. Here the segments are considered as virtually grounded. Evaluating the current flow at the node Vin, the pulser transfer function is obtained to be Zfb Vin Vout Vin Vout ¼ ) ¼ 1þ : Zdet Zfb Vpulser Zdet
ð2Þ
Cj Ij ¼ P36
i¼1
Ci
Iin :
Eq. (7) summarizes the basic principle for capacitance measurements employing a pulser. In reality one usually observes that the total current in the segments is slightly lower than the current in the core. Or equivalently, the sum of the gain matched pulser positions in the segments is a bit lower than the core pulser line position. This observed difference is independent of the applied bias voltage. Such effects are created by stray capacitances Cstray (indicated in brackets for example in Fig. 4) and add to the total load capacitance seen by the core preamplifier, but remain invisible to the segment preamplifiers. A measurement was setup to investigate the sensitivity to the series and parallel resistance. A test circuit with adjustable resistors and capacitance was build into an AGATA test cryostat. For fixed pulser amplitudes, the amplitude of the core output was measured using a digital oscilloscope. Capacitances Cx from 10 pF to 1 nF were inserted. For each of these capacitances, a serial resistor was inserted varying from 0 to 1 kO and a parallel resistor of 1 GO was used. Fig. 5 shows the measured amplitudes of the core pulser line as a function of the total capacitive load of the preamplifier, being Cx in series with the coupling capacitor Cac. The parallel resistor showed no influence. Within the tolerances of the capacitances, the measurement shows a linear relationship and demonstrates validity of Eq. (6). The slightly different results for fixed load capacitances correspond to systematic errors induced by the different series resistances. These errors were
The second term Zfb =Zdet can be rewritten as the Laplace transform of an exponential decaying function: t=t1,2 a1,2 e ¼ a1,2 L1 ð3Þ 1 þst1,2 t1,2 with a1 ¼
Cdet Cac Cfb ðCdet þ Cac Þ
ð4Þ
Cdet Cac : Cdet þ Cac
ð5Þ
t1 ¼ Rdet
t2 and a2 will be used later in the Eqs. (10) and (11). The time constant t1 represents the additional rise time added to the pulser signal at the core preamp output. If t1 is sufficiently small with respect to the typical integration time, the exponential function can be considered as a delta function in the time domain. For this high frequency limit, the transfer function becomes Vout Cdet Cac : ¼ 1þ Vpulser Cfb ðCdet þCac Þ
ð7Þ
ð6Þ Fig. 5. Total load capacitance versus amplitude of the core pulser signal.
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maximally 5% for the 1 kO resistor which is within acceptable limits.
preamplifier rise times equivalently to the cold resistor model [18–21]. Using the current divider rule, one obtains:
3.2. Signal transfer function
Iin 1=sC det : ¼ Ig 1=sC det þ ½Rdet þ 1=sC ac þ Zeq
Opposite to the pulser signal, a real g-ray event induces currents Ig injected between the detector capacitance and the serial resistance in Fig. 4. This difference in injection point will result in a dissimilar bias voltage dependence. This difference will be shortly discussed here with respect to the core preamplifier. The transfer function, given by Iin =Ig , corresponds to the fraction of registered current in the core preamplifier relative to the total g-induced charge in the depleted germanium crystal. The equivalent input impedance Zeq of the preamplifier in Fig. 4 is given by Zeq ¼ A Zfb þ Rrise :
ð8Þ
The first term represents the Miller equivalent input impedance. The second term – the resistor Rrise [12] – can be used to model realistic
This can be rewritten in the form of Eq. (3) with coefficients: 1 C C a2 ¼ 1 þ det þ det Cac AC fb
t2 ¼ Cdet ½Rrise þ Rdet :
ð9Þ
ð10Þ ð11Þ
The rise time t2 is mainly the intrinsic rise time Cdet Rrise of the preamplifier. The preamplifier reaches a rise time at full depletion which is increased by the depletion voltage dependent fraction Cdet Rdet (see Eq. (11)). The latter information, unfortunately, can in reality not be used to measure Rdet as the currents induced from g-rays themselves are not ideally peaked, but have a finite position dependent width. The calibration peak positions are expected to shift as a function of bias voltage. The experimentally observed shift in peak position of core and segment for the 60 keV Am line is shown in Fig. 6. The core energy position varies strongly with bias voltage according to Eq. (4), while segments are nearly unaffected. Remark that this behavior is different than for pulser induced signals: the amplitude for pulser induced signals grows linearly with load capacitance as shown in Fig. 5, while the amplitude for gamma-induced currents is decreasing with load capacitance. At low bias voltages, the line width grows drastically due to the increased detector capacitance. The reduced energy resolution is indicated by the FWHM value of the peak in Fig. 6 (size of error bars).
4. Capacitance–voltage measurements
Fig. 6. Variation of the Am peak position with detector bias voltage. The error bars indicate the FWHM of the energy peak. They do not represent an uncertainty.
For the pulser based capacitance measurement the symmetric AGATA detector S002 was mounted in a standard single test cryostat. A space charge distribution of 0.5 (back) to 1.8 (front) 1010 cm 3 is
Fig. 7. C–V characteristics of detector S002 on linear and logarithmic scale. The capacitance measurements using the pulser method is shown with diamond symbols. Other data points belong to direct measurements using various power supplies.
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Capacitance [pF]
Capacitance [pF]
Capacitance [pF]
180
100 50 20 10 5 2 1 100 50 20 10 5 2 1 100 50 20 10 5
A1 B1 C1 D1 E1 F1
A2 B2 C2 D2 E2 F2
A3 B3 C3 D3 E3 F3
A4 B4 C4 D4 E4 F4
A5 B5 C5 D5 E5 F5
A6 B6 C6 D6 E6 F6
2 1 10 20
50 100 200 500 1K 2K Bias [V]
5K 10 20
50 100 200 500 1K 2K
5K
Bias [V]
Fig. 8. C–V characteristics of the segments of detector S002.
80
10 V 25 V
50 V 75 V
100 V 200 V
500 V 1000 V
1500 V 4000 V
70
Capacitance [pF]
60 50 40 30 20 10 0 A1
A2
A3 A4 Segments of S002
A5
A6
Fig. 9. Capacitances of sector A at different voltages. The first segment A1 is located at the hexagonal front side of the detector shape. Segment A6 is at the end of the detector, where the detector has a cylindrical shape.
given by the manufacturer [15], for more details about the AGATA detectors see Ref. [16]. The 36 segments and core preamplifier signals were digitized using 10 XIA DGF-4C modules. The gain and the offset of the preamplifiers were matched at full voltage of 5000 V using the 1.3 MeV line from a 60Co source. The pulser amplitude in all segments and the core were measured at different high voltages. The pulser signal of the core amplitude was normalized to the total detector bulk capacitance of 46.5 pF, as provided by the manufacturer. The capacitance values were corrected for the coupling capacitor of Cac ¼875 pF and the stray capacitance of Cstray ¼8 pF. The results for the total capacitance as measured from the core pulser signals is shown in Fig. 7. The individual core-to-segment capacitances Ci are shown in Fig. 8. In the graphs the geometrical identical segments are grouped together ringwise. To illustrate the difference between the capacitance values for the segmentation in depth along the detector axis, the capacitances for the six segments in sector A are shown in Fig. 9 as a function of the measured bias voltages. The measured
capacitances reflect the detector geometry. The lowest capacitance value of segment A2 is caused by the smallest surface area of the second ring of segments (see Table 1 for numbers) with respect to all other segments. The peculiar crosstalk behavior of the second ring segments [11] is also caused by the smallest geometrical surface and resulting capacitance. In order to cross-check and validate the pulser measurements the capacitance of the whole detector was also measured independently in a direct way with a standard capacitance meter. Such measurement needs direct access to the electrodes while the measuring device needs protection for the applied high voltages. The detector was mounted in the same way as for the current measurement described in Section 2. An electrical protection circuitry was used to shield the capacitance meter from high voltage. The coupling capacitor was increased to 100 nF to allow direct measurements in the nF-range. To compensate the resulting increase in charging time constant t ¼ RC, the 1 GO protection resistor was replaced by a 22 MO resistor. For the measurement all segment electrodes were shorted to ground. The parallel capacitance of the protection circuitry and the cryostat were measured. This constant offset was subtracted from the measured result of the detector. Different high voltage modules were used to avoid systematic errors. Finally the results of the direct and the pulser measurement compare very well (see Fig. 7).
5. Reconstruction of the doping profile From the capacitance–voltage data, shown in Figs. 7 and 8, the impurity concentration or doping profile of the HPGe crystal is extracted applying the techniques developed in Ref. [13]. Three approaches with different levels of complexity were tested: The planar approximation, the cylindrical approximation and the numerical analysis through chi square minimization. The results are discussed and compared below. 5.1. Planar approximation The simplest way to determine the impurity concentration N near the outer surface of the detector is given by a planar
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Table 1 Surface areas (cm2) of segments for all four types of AGATA detectors: irregular shaped A-, B-, C- and symmetric detectors (for details of the detector geometry see Ref. [16]). Seg.
Red (A)
A1 A2 A3 A4 A5 A6 B1 B2 B3 B4 B5 B6 C1 C2 C3 C4 C5 C6 D1 D2 D3 D4 D5 D6 E1 E2 E3 E4 E5 E6 F1 F2 F3 F4 F5 F6 Sum
Green (B)
Blue (C)
Sym. (S)
8.08 4.99 6.13 7.70 7.86 7.91 8.21 5.05 6.19 7.78 7.95 7.99 7.42 4.53 5.55 6.92 7.03 7.06 7.23 4.25 5.18 6.52 6.62 6.62 7.99 4.88 5.98 7.48 7.64 7.68 8.22 5.07 6.22 7.79 7.96 8.00
7.77 4.71 5.70 7.17 7.27 7.29 8.23 4.98 6.06 7.47 7.57 7.60 8.64 5.24 6.39 7.94 8.09 8.13 7.23 4.55 5.53 6.90 6.98 6.99 7.58 4.69 5.74 7.09 7.19 7.23 8.36 5.04 6.16 7.81 8.00 8.02
7.77 4.74 5.82 7.27 7.40 7.41 8.02 4.81 5.88 7.42 7.55 7.55 7.83 4.85 5.94 7.36 7.49 7.53 7.98 4.79 5.85 7.37 7.52 7.56 8.03 4.88 5.97 7.51 7.67 7.71 7.89 4.82 5.91 7.35 7.47 7.50
6.59 4.98 5.54 7.22 7.50 7.55 6.59 4.98 5.54 7.22 7.50 7.55 6.59 4.98 5.54 7.22 7.50 7.55 6.59 4.98 5.54 7.22 7.50 7.55 6.59 4.98 5.54 7.22 7.50 7.55 6.59 4.98 5.54 7.22 7.50 7.55
245.65
247.91
246.38
237.50
approximation for every segment. It is based on the equations: CðdÞ ¼
eA
ð12Þ
d
NðdÞ ¼
C3
eeA2
dC dV
1 ð13Þ
with d the depletion depth below the surface. This approximation is just valid for a small depletion area at low voltages, such that the area of the depletion boundary A still corresponds to the area of the segments [13]. These surface areas were calculated for the different segments. The resulting values are listed in Table 1 for the four types of AGATA detector geometries (three irregular shaped detectors and the symmetric detector [16]). The results of the impurity concentration obtained with the planar approximation are in the range of 0.5 and 1.0 1010 per cm3 (see Fig. 10). These values compare reasonable well with the manufacturer’s values in the range between 0.5 and 1.8 1010 per cm3.
5.2. Cylindrical approximation The AGATA crystal has nearly a cylindrical shape at the back side and the last four rings of segments in the detector are close to a coaxial detector geometry. The assumption of a cylindrical symmetry was applied for these four rings. The reconstruction of the impurity profile is given as a function of the depletion
Fig. 10. Results of the three reconstruction methods: planar, cylindrical and numerical, for the space charge reconstruction along each of the six rings of segments. The positions of the data points along the crystal are given for the center of the segments. The dashed line corresponds to the cubic spline interpolation used in the minimization procedure.
radius and is performed with the Eq. [13]: Cðrd Þ ¼
2peH ln rr2d
ð14Þ
Nðrd Þ ¼
1 Cðrd Þ3 dC 4p2 eerd2 H2 dV b
ð15Þ
with rd the depletion radius and r2, H the radius and height of the cylindrical segment electrode. Compared to the planar approximation, this method is not limited to the lowest bias voltages or small depletion depth. A full reconstruction of the doping profile as a function of the radius was performed. The result of the reconstruction is shown in Fig. 11. The impurity concentration obtained with the cylindrical approximation is slightly higher than with the planar approximation. The values are within the range of 0.5 and 1.1 1010 per cm3 (see Fig. 10). A gradient is observed from back to front, in agreement with the planar approximation. The planar approximation underestimates the real values as discussed in Ref. [13] by an amount which depends on the impurity concentration and the segment radius. The results in Fig. 11 show that the impurity concentration within one ring is except for the boundaries fairly constant for each of the six segments. The depletion profile is highly axially symmetric. This is probably due to the rotation of the crystal during the growth process. In order to evaluate the deviations from a homogeneous distribution correctly, effects breaking the coaxial geometry have to be estimated. For this purpose, a simulation was performed assuming a homogeneous impurity distribution of N¼1.0 1010/cm3. The result of this simulation is shown in Fig. 12. The space charge profile is reconstructed very nicely. Exceptions arise only close to the core and at the largest radii. The large errors at the largest radii are caused by the numerical errors of the derivative of the C–V curve at the end point. The deviations at small radii are induced by the fact that the core electrode is depleted first at the front end and depletion moves then as a function of the applied voltage towards the back end of the detector. While the core electrode might already be depleted near the front electrodes, the undepleted areas near the back of the detector still contribute to the capacitance in the front electrodes. Remark that the full geometry of core and segment defines core-tosegment capacitance, and not just the limitation of this geometry to the ring interval in depth to which the segment is confined. Therefore,
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Fig. 11. Reconstruction of the doping profile using the coaxial approximation for the last four rings of electrodes in the AGATA detector. The four graphs correspond to the four rings of segments. The graphs are ordered according to the physical order in the rings, starting with the back ring with largest diameter on top. In each of the four plots, six graphs are shown corresponding to the six segments in the corresponding ring.
Fig. 12. Simulated doping profile equivalent of Fig. 11 for reference. The capacitances at different bias voltages were calculated assuming a fixed doping profile of N ¼1.0 1010/cm3. The impurity profile was reconstructed using the coaxial approximation.
the capacitance in the first ring will still change with bias voltage until the whole detector is depleted. This effect is disregarded in the cylindrical approximation and causes the overestimation of the space charge at smaller radii in Fig. 11 and 12. The comparison between simulation and experiment is consistent with a homogeneous radial impurity concentration.
A small deviation is visible as a small bump at a radius rd ¼2 cm in the last ring. This might be caused by a disturbance of the field outside the detector geometry, e.g. by HV insulation material. The average value for each ring is also included in Fig. 10 for comparison. The outermost section of the reconstruction profile was excluded to avoid residual geometry effects.
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Considering the expected underestimation of the planar approximation [13], the results of the planar and cylindrical solution compare well.
Fig. 13. The chi square minimization scheme.
5.3. Numerical method Finally the space charge was determined by a chi square optimization procedure using the numerical computation program described in Ref. [13]. Since the space charge was proven to be axial symmetric and radius independent using the cylindrical approximation, only the variation in depth was left to fit. The numerical code is known to be less accurate at lower bias voltages, due to the finite grid resolution. Therefore, only the capacitances above 100 V bias were included in the comparison to guarantee capacitance simulation errors below 10%. Above 2000 V this detector shows constant capacitance values or full depletion and the values above 2000 V bias voltage were also excluded from the comparison. A scheme of the chi square minimization procedure is shown in Fig. 13. Since the detector has a sixfold segmentation in depth, the space charge was modeled with 6 parameters to prevent underdetermination. These six parameters were chosen to correspond to the space charge in the middle of each segment. The space charge, assumed axially symmetric and radially independent, was interpolated using a cubic spline interpolation [22] in intermediate positions. Starting from the doping profile determined by the cylindrical approximation the depletion boundaries were calculated for the 16 bias voltage values between 100 and 2000 V. For each of these solutions, the 37 voltage dependent weighting potentials belonging to the core and each of the segments were determined. From these weighting potentials, the core-to-segment capacitances were calculated which allow direct comparison with the experimental values. The optimal solution is also shown in Fig. 10 for comparison with the two other approaches. The dashed line is the spline interpolation which is used in the optimization routine. The result is nearly linear and varies between 1.5 and 0.5 1010 per cm3 from front to back of the crystal. The solution compares well for the four highest rings with both other methods. For the two lower rings, where the cylindrical approximation cannot be used, the discrepancy with the planar approximation is striking. Of the two methods, the chi square method is the most accurate. The planar approximation differs here for the reason discussed earlier. A direct comparison between the experimental capacitances and the simulated capacitances for the optimum solution obtained through chi square minimization is shown in Fig. 14.
Applied Voltage [V]
Fig. 14. Comparison of capacitances between the optimum simulated solution and the experimental values. The insert corresponds to the data range included in the optimization.
183
Fig. 15. Plot of the relative deviation for the optimal solution between simulation and experiment for all 36 segments as a function of applied high voltage.
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To acquire a value for the achieved accuracy, the simulation was recalculated twice, adding 710% to the optimum impurity concentrations shown in Fig. 10. Since the agreement is much better than these limits, the obtained space charge concentrations are determined with an accuracy better than 710%. The agreement of the individual core-to-segment capacitances for the optimal parameters, under assumption of a coaxial space charge distribution is shown in Fig. 15. Most segment capacitances are well below the 710% relative error. An exception is segment E1, which shows consistently lower capacitance values. The experimental data, shown in Fig. 14, indicate that the detector depletes earlier than expected from the physical geometry of the crystal. Note that also in Fig. 11 the 5 mm core radius is not reached at full depletion. The capacitance at full depletion is also higher than was expected from simulation. Comparable effects have been observed in all AGATA detector geometries. This difference is caused by the effective thickness of the lithium diffused layer near the core contact. A thickness of about 1.5 mm is estimated from the difference in capacitance.
were observed. A nearly linear gradient in depth was measured going from 1.5 1010 in the front to 0.5 1010 per cm3 in the back of the crystal, that is comparable to the linear gradient of 1.8 1010 in the front to 0.5 1010 per cm3 in the back of the crystal given by the manufacturer. We were able to determine this space charge profile with good precision and demonstrated that the sensitivity of the method is well below 10%. In future this new and non-destructive method will be applied to determine the impurity concentration and its gradient along the detector axis for operational HPGe detectors after processing the Ge crystals.
Acknowledgments This research was supported by the German BMBF under Grants 06K-167 and 06KY205I.
References 6. Discussion and outlook The full three-dimensional distribution of the impurity profile of a large volume, irregular shaped, highly segmented HPGe detector was determined for an operational detector via a capacitance–voltage measurement. The acquired capacitance values were obtained by employing a pulser and the capacitive coupling of the pulser signal through the bulk capacitance of the HPGe crystal. The main advantage of the new method is that it can be applied to the working detector and does not require modifications of the cryostat. The method was validated by comparison to a direct measurement of the capacitance. The capacitance values were not affected by the serial detector resistance. Three types of methods were applied for the reconstruction of the impurity concentration from the CV data. The planar approximation, which is only valid for small depletion depth [13], yielded results consistent with the two other approaches, but underestimated the impurity concentration in the front part of the detector. The cylindrical method and the numerical method compare very well with the impurity concentration and its gradient given by the manufacturer. The two methods are complementary: In the numerical method, a tradeoff has to be made between numerical precision, fit parameters and calculation time. The cylindrical approximation on the other hand is very fast and retrieves the maximum of information. However, deviations from a cylindrical detector shape limit the applicability of this approach. The final results for the symmetric AGATA crystal showed a highly cylindrical space charge distribution. No radial variations
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AGATA, Official website, /http://www-w2k.gsi.de/agataS. I.Y. Lee, Nucl. Instr. and Meth. Phys. Res. A 422 (1999) 195. D. Bazzacco, Nucl. Phys. A 746 (2004) 248. ¨ S. Schonert, et al., Nucl. Phys. B (Proc. Suppl.) 145 (2005) 242. B. Bruyneel, B. Birkenbach, Automated optimization of detector simulation, in: Sixth AGATA week, Legnaro 2007, 2007. /http://agata.lnl.infn.it/ AgataWeek2007S. B. Bruyneel, B. Birkenbach, AGATA detector simulation library (ADL). /http:// www.ikp.uni-koeln.de/research/agata/S. L. Nelson, M. Dimmock, A. Boston, H. Boston, Nucl. Instr. and Meth. Phys. Res. A 573 (2006) 153. ¨ L. Mihailescu, W. Gast, R. Lieder, H. Brands, H. Jager, Nucl. Instr. and Meth. Phys. Res. A 447 (2000) 350. B. Bruyneel, P. Reiter, G. Pascovici, Nucl. Instr. and Meth. Phys. Res. A 569 (2006) 764. B. Bruyneel, P. Reiter, G. Pascovici, Nucl. Instr. and Meth. Phys. Res. A 569 (2006) 764. B. Bruyneel, P. Reiter, A. Wiens, J. Eberth, H. Hess, G. Pascovici, N. Warr, D. Weisshaar, Nucl. Instr. and Meth. Phys. Res. A 599 (2009) 196. B. Bruyneel, Differential crosstalk in segmented detectors, in: Seventh AGATA week, Uppsala 2008, 2008. /http://www.agata.org/aw7/presentationsS. B. Bruyneel, B. Birkenbach, P. Reiter, Space charge reconstruction in highly segmented HPGe detectors through capacitance–voltage measurements, in press, doi:10.1016/j.nima.2011.02.110. G. Pascovici, A. Pullia, F. Zocca, B. Bruyneel, Low noise, dual gain preamplifier with built in spectroscopic pulser for highly segmented high-purity germanium detectors, in: M.P. Brown, K. Austin (Eds.), WSEAS Conference, Bucharest, Romania, 2008. M. Allali, Encapsulated germanium detector operating manual. A. Wiens, H. Hess, B. Birkenbach, B. Bruyneel, J. Eberth, D. Lersch, G. Pascovici, P. Reiter, H. Thomas, Nucl. Instr. and Meth. Phys. Res. A 618 (2010) 223. ¨ E.W. Kuhn, A.N.F. Schroeder, Nucl. Instr. and Meth. 79 (1970) 304. E. Gatti, P. Manfredi, Il Nuovo Cimento 9 (1986) 1. E. Gatti, P. Manfredi, IEEE Trans. Nucl. Sci. NS-25 (1) (1978) 66. V. Radeka, IEEE Trans. Nucl. Sci. NS-21 (1) (1974) 51. F. Zocca, A. Pullia, G. Pascovici, IEEE Trans. Nucl. Sci. NS-55 (2) (2008) 695. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, 1992.
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Determination of space charge distributions in highly segmented large volume HPGe detectors from capacitance–voltage measurements B. Birkenbach, B. Bruyneel , G. Pascovici, J. Eberth, H. Hess, D. Lersch, P. Reiter, A. Wiens Institut f¨ ur Kernphysik, Universit¨ at zu K¨ oln, Z¨ ulpicher Straße 77, 50937 K¨ oln, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 November 2010 Received in revised form 21 February 2011 Accepted 21 February 2011 Available online 23 March 2011
The space charge distribution of a large volume highly segmented HPGe detector was determined by a non-destructive capacitance–voltage measurement. The capacitances between the 36 segments and the core were measured simultaneously with a precision pulser which was implemented in the core preamplifier. The pulser measurement was compared to and validated by direct capacitance measurements. The three-dimensional doping profile was reconstructed using analytical and numerical methods. Consistent values for the impurity concentration in the range of 0.5 and 1.5 1010 cm 3 were obtained. & 2011 Elsevier B.V. All rights reserved.
Keywords: Impurity concentration Segmented HPGe-detectors Capacitance–voltage Current–voltage characteristic
1. Introduction The final configuration of the 4p spectrometer Advanced GAmma Tracking Array (AGATA) [1] will consist of 180 highly segmented high purity germanium (HPGe) detectors and will provide best detection efficiency and performance yet employing the novel g-ray tracking [2] principle for position dependent g-ray detection. The main ingredient of g-ray tracking is the individual position of each energy deposition point after interaction of g-rays inside the HPGe crystal. To achieve the required position resolution of a few millimeters inside the detector volume the high segmentation of the detectors alone is insufficient and pulse shape analysis methods [3] are applied. Also other experiments like the neutrinoless beta decay experiment GERDA [4] employ PSA on comparable HPGe detectors for background recognition. A crucial step of the PSA procedure is a comparison of all the detected real pulses with a well-known data set of pulses with precise position information as a reference. These data sets are created either by computer simulations [5,6] or can be determined experimentally by position dependent scanning of the individual detectors [7]. In order to obtain very reliable computer simulations the characteristics of the individual HPGe detectors must be known very precisely and the simulations must be verified with experimental data. Recently major advances were achieved to understand the relevant parameters of the highly segmented HPGe detectors. For
Corresponding author. Tel.: þ 33 1 69 08 75 53.
E-mail address: [email protected] (B. Bruyneel). 0168-9002/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2011.02.109
example, improved models were developed to describe the mobility of the charge carriers [8–10]. The electronic properties of the novel detectors were investigated in detail and crosstalk contribution, caused by the capacitive coupling between the segments and the core [11,12], were described successfully. Another important quantity which characterizes the HPGe detectors is the remaining impurity concentration of charge carriers throughout the Ge crystal volume. The impurity concentration affects directly the necessary operation voltage of the detector and the resulting electric field strength. Despite the high impact of the impurity concentration on the detector performance a direct measurement after crystal processing is prevented by the manufacturing techniques. Therefore, non-destructive capacitance–voltage measurements were investigated in Ref. [13] to determine the depletion regions inside the Ge crystal as a function of applied bias voltage. The depleted regions can be reconstructed from the capacitance values measured between the common core electrode and all the individual segments. Finally the impurity concentration is deduced from the variation of the depletion zone as a function of the high voltage. In this article we report on the first-time reconstruction of the doping profile inside a large volume, highly segmented HPGe detector. The new approach is based on simultaneous capacitance measurements with a 36-fold segmented AGATA detector employing a precision pulser of the AGATA core preamplifiers [14]. Combined with results of the computer simulation and reconstruction methods described in Ref. [13] the three-dimensional impurity concentration of an AGATA detector was determined. The paper is organized in the following way: In the first and second parts the relevant electronic properties of the highly
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segmented detector and the new method to measure the capacitance between core contact and segment electrodes with a pulser are introduced. In the central part the results of the capacitance– voltage measurements are presented. From these findings the doping profile is reconstructed finally using three different approaches with increasing refinement and complexity.
2. Electrical model of an AGATA detector An AC equivalent electrical detector model for the 36-fold segmented AGATA HPGe detector consists of 36 capacitances with 36 parallel and serial resistors (see Fig. 1). All capacitance and resistor values are not constant but change with the applied bias voltage. The capacitors Ci are strongly dependent on the depleted volume between the segment electrodes and the core electrode. The small leakage current through the segments is due to the not infinite but large resistors Rpi in parallel with the capacitors. The resistivity of the undepleted region between segments and core is modeled with the series resistances Rsi. The total capacitance for a whole AGATA detector is typically 4571 pF at full depletion and is given by the manufacturer [15]. Since the capacitance is a monotonically decreasing function with bias voltage, this value represents a lower limit. For the 36 individual segments, the minimum capacitance is therefore of the order of 1 pF. A lowest voltage of 10 V was applied for the measurements. At this bias voltage, the capacitance should be on the order of 3 nF, depending on the impurity concentration. The total series resistivity is largest when no bias voltage is applied and disappears at full depletion. Its value is strongly dependent on temperature and impurity concentration. Typical values for the bulk series resistivity (at 10 V bias and LN2 temperature) are of the order of 1 kO [15]. The parallel resistor values are extremely high for reverse biased diodes and therefore challenging to measure. At full depletion, typical values are in the T O region. To ensure that these values were also reached at the lowest bias voltages, a dedicated measurement was performed. The leakage current of a symmetric AGATA detector was measured as a function of very low bias voltage. The AGATA detector was mounted in the symmetric AGATA triple cryostat [16]. For this occasion, the standard preamplifier electronics were completely removed allowing direct access to core and segment electrodes. The setup is shown schematically in Fig. 2. All the segment electrodes were short connected and grounded via a Keithley 486 picoammeter. The core electrode was directly connected to a high voltage module omitting the usual 1 GO protection resistor. The leakage current was measured for different bias voltages ranging from 0.1 to 6 V. A self-made module was used which was based on the 12 V of the NIM power supply, a voltage divider and
Fig. 2. Setup for the measurement of the leakage current of a 36-fold segmented, symmetric AGATA detector.
Fig. 3. I–V characteristics of the symmetric AGATA detector.
filter. Results are shown in Fig. 3. These values were corrected for the current offset reading at 0 V. Errors are mainly induced by the variation of this offset while powering up the detector. Therefore, after each measurement, the detector was ramped down to perform a new offset measurement and to minimize the changes in offset during the measurement. These values are in line with measurements performed at higher bias voltages using commercially available HV modules. However, these modules produced considerably larger fluctuations in the current offset, implying larger errors at very low voltages. From the values shown in Fig. 3 the total parallel detector resistance was calculated. The differential resistivity is given as @U=@IjVbias . Above 10 V this resistance is already in the region of TO, and therefore its contribution will be completely negligible at all bias voltages of interest.
3. Principle of the pulser method
Fig. 1. Electrical model of a 36-fold segmented detector.
A built-in programmable pulser is installed on the core preamplifier board [14]. Different from standard pulser inputs, the pulser signal is injected at the source pin of the core contact FET which is connected to ground level via a 1:8 O resistor. A schematic drawing of the detector electronics is shown in Fig. 4. In order to extract capacitance values with the pulser signal the exact relationship between pulser line positions and capacitance is mandatory [17]. The preamplifier’s transfer function for pulser input signals is derived based on the circuitry in Fig. 4. The measured pulser positions in the spectra are compared to values from a g-ray calibration source. The response to currents Ig induced by gamma radiation is also derived. In the calculations, the two 1 GO resistors of Fig. 4 can be neglected. In the feedback
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It should be noted that this approximation is not per se valid. Especially at low bias voltages, t1 can be in the range of 1 ms, which is dangerously close to the typical integration times used for standard analog main amplifiers. Visual inspection of the long rise times with a scope can be helpful. Measurements should therefore be performed with the largest possible energy filter settings. If the total series resistance is negligible for the measurement of the total capacitance using Eq. (6), then also the series resistance values in Fig. 1 are negligible. The core-to-segments 1 capacitances are roughly 36 of the total capacitance, while the individual series resistances will be about 36 times larger than the total series resistance Rdet. Then the product CjRsj from Fig. 1 will be comparable to t1 . The current measured in each of the segments j is proportional to the core-to-segment capacitance Cj. According to the current divider rule, we obtain Fig. 4. Circuit diagram of the detector from the viewpoint of the core preamplifier (see text). The detector model shown in Fig. 1 is substituted with the total detector capacitance Cdet and series resistivity Rdet as indicated with the symbol. Stray capacitances Cstray produce an offset between the core pulser line position and the sum of the segment pulser line positions.
loop, the time constant tfb ¼ Cfb Rfb is 1 ms. Therefore, Rfb is only relevant for effects on a timescale larger than 1 ms. For similar reasons, the 1 GO protection resistor can be neglected. 3.1. The pulser transfer function In the case where a rectangular pulse is injected at the noninverting input of the preamplifier, this potential is ‘copied’ into the inverting input of the preamplifier: Vin ¼ Vpulser :
ð1Þ 5
This behavior is caused by the large open loop gain A C10 : if the output is not saturated – implying Vout ¼ AðV þ V Þ o1 V – the potential difference at the input nodes is smaller than the thermal noise. If no pulser signal is applied, the preamplifier input is acting as a virtual ground. This concept is already indicated in Fig. 4. Here the segments are considered as virtually grounded. Evaluating the current flow at the node Vin, the pulser transfer function is obtained to be Zfb Vin Vout Vin Vout ¼ ) ¼ 1þ : Zdet Zfb Vpulser Zdet
ð2Þ
Cj Ij ¼ P36
i¼1
Ci
Iin :
Eq. (7) summarizes the basic principle for capacitance measurements employing a pulser. In reality one usually observes that the total current in the segments is slightly lower than the current in the core. Or equivalently, the sum of the gain matched pulser positions in the segments is a bit lower than the core pulser line position. This observed difference is independent of the applied bias voltage. Such effects are created by stray capacitances Cstray (indicated in brackets for example in Fig. 4) and add to the total load capacitance seen by the core preamplifier, but remain invisible to the segment preamplifiers. A measurement was setup to investigate the sensitivity to the series and parallel resistance. A test circuit with adjustable resistors and capacitance was build into an AGATA test cryostat. For fixed pulser amplitudes, the amplitude of the core output was measured using a digital oscilloscope. Capacitances Cx from 10 pF to 1 nF were inserted. For each of these capacitances, a serial resistor was inserted varying from 0 to 1 kO and a parallel resistor of 1 GO was used. Fig. 5 shows the measured amplitudes of the core pulser line as a function of the total capacitive load of the preamplifier, being Cx in series with the coupling capacitor Cac. The parallel resistor showed no influence. Within the tolerances of the capacitances, the measurement shows a linear relationship and demonstrates validity of Eq. (6). The slightly different results for fixed load capacitances correspond to systematic errors induced by the different series resistances. These errors were
The second term Zfb =Zdet can be rewritten as the Laplace transform of an exponential decaying function: t=t1,2 a1,2 e ¼ a1,2 L1 ð3Þ 1 þst1,2 t1,2 with a1 ¼
Cdet Cac Cfb ðCdet þ Cac Þ
ð4Þ
Cdet Cac : Cdet þ Cac
ð5Þ
t1 ¼ Rdet
t2 and a2 will be used later in the Eqs. (10) and (11). The time constant t1 represents the additional rise time added to the pulser signal at the core preamp output. If t1 is sufficiently small with respect to the typical integration time, the exponential function can be considered as a delta function in the time domain. For this high frequency limit, the transfer function becomes Vout Cdet Cac : ¼ 1þ Vpulser Cfb ðCdet þCac Þ
ð7Þ
ð6Þ Fig. 5. Total load capacitance versus amplitude of the core pulser signal.
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maximally 5% for the 1 kO resistor which is within acceptable limits.
preamplifier rise times equivalently to the cold resistor model [18–21]. Using the current divider rule, one obtains:
3.2. Signal transfer function
Iin 1=sC det : ¼ Ig 1=sC det þ ½Rdet þ 1=sC ac þ Zeq
Opposite to the pulser signal, a real g-ray event induces currents Ig injected between the detector capacitance and the serial resistance in Fig. 4. This difference in injection point will result in a dissimilar bias voltage dependence. This difference will be shortly discussed here with respect to the core preamplifier. The transfer function, given by Iin =Ig , corresponds to the fraction of registered current in the core preamplifier relative to the total g-induced charge in the depleted germanium crystal. The equivalent input impedance Zeq of the preamplifier in Fig. 4 is given by Zeq ¼ A Zfb þ Rrise :
ð8Þ
The first term represents the Miller equivalent input impedance. The second term – the resistor Rrise [12] – can be used to model realistic
This can be rewritten in the form of Eq. (3) with coefficients: 1 C C a2 ¼ 1 þ det þ det Cac AC fb
t2 ¼ Cdet ½Rrise þ Rdet :
ð9Þ
ð10Þ ð11Þ
The rise time t2 is mainly the intrinsic rise time Cdet Rrise of the preamplifier. The preamplifier reaches a rise time at full depletion which is increased by the depletion voltage dependent fraction Cdet Rdet (see Eq. (11)). The latter information, unfortunately, can in reality not be used to measure Rdet as the currents induced from g-rays themselves are not ideally peaked, but have a finite position dependent width. The calibration peak positions are expected to shift as a function of bias voltage. The experimentally observed shift in peak position of core and segment for the 60 keV Am line is shown in Fig. 6. The core energy position varies strongly with bias voltage according to Eq. (4), while segments are nearly unaffected. Remark that this behavior is different than for pulser induced signals: the amplitude for pulser induced signals grows linearly with load capacitance as shown in Fig. 5, while the amplitude for gamma-induced currents is decreasing with load capacitance. At low bias voltages, the line width grows drastically due to the increased detector capacitance. The reduced energy resolution is indicated by the FWHM value of the peak in Fig. 6 (size of error bars).
4. Capacitance–voltage measurements
Fig. 6. Variation of the Am peak position with detector bias voltage. The error bars indicate the FWHM of the energy peak. They do not represent an uncertainty.
For the pulser based capacitance measurement the symmetric AGATA detector S002 was mounted in a standard single test cryostat. A space charge distribution of 0.5 (back) to 1.8 (front) 1010 cm 3 is
Fig. 7. C–V characteristics of detector S002 on linear and logarithmic scale. The capacitance measurements using the pulser method is shown with diamond symbols. Other data points belong to direct measurements using various power supplies.
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Capacitance [pF]
Capacitance [pF]
Capacitance [pF]
180
100 50 20 10 5 2 1 100 50 20 10 5 2 1 100 50 20 10 5
A1 B1 C1 D1 E1 F1
A2 B2 C2 D2 E2 F2
A3 B3 C3 D3 E3 F3
A4 B4 C4 D4 E4 F4
A5 B5 C5 D5 E5 F5
A6 B6 C6 D6 E6 F6
2 1 10 20
50 100 200 500 1K 2K Bias [V]
5K 10 20
50 100 200 500 1K 2K
5K
Bias [V]
Fig. 8. C–V characteristics of the segments of detector S002.
80
10 V 25 V
50 V 75 V
100 V 200 V
500 V 1000 V
1500 V 4000 V
70
Capacitance [pF]
60 50 40 30 20 10 0 A1
A2
A3 A4 Segments of S002
A5
A6
Fig. 9. Capacitances of sector A at different voltages. The first segment A1 is located at the hexagonal front side of the detector shape. Segment A6 is at the end of the detector, where the detector has a cylindrical shape.
given by the manufacturer [15], for more details about the AGATA detectors see Ref. [16]. The 36 segments and core preamplifier signals were digitized using 10 XIA DGF-4C modules. The gain and the offset of the preamplifiers were matched at full voltage of 5000 V using the 1.3 MeV line from a 60Co source. The pulser amplitude in all segments and the core were measured at different high voltages. The pulser signal of the core amplitude was normalized to the total detector bulk capacitance of 46.5 pF, as provided by the manufacturer. The capacitance values were corrected for the coupling capacitor of Cac ¼875 pF and the stray capacitance of Cstray ¼8 pF. The results for the total capacitance as measured from the core pulser signals is shown in Fig. 7. The individual core-to-segment capacitances Ci are shown in Fig. 8. In the graphs the geometrical identical segments are grouped together ringwise. To illustrate the difference between the capacitance values for the segmentation in depth along the detector axis, the capacitances for the six segments in sector A are shown in Fig. 9 as a function of the measured bias voltages. The measured
capacitances reflect the detector geometry. The lowest capacitance value of segment A2 is caused by the smallest surface area of the second ring of segments (see Table 1 for numbers) with respect to all other segments. The peculiar crosstalk behavior of the second ring segments [11] is also caused by the smallest geometrical surface and resulting capacitance. In order to cross-check and validate the pulser measurements the capacitance of the whole detector was also measured independently in a direct way with a standard capacitance meter. Such measurement needs direct access to the electrodes while the measuring device needs protection for the applied high voltages. The detector was mounted in the same way as for the current measurement described in Section 2. An electrical protection circuitry was used to shield the capacitance meter from high voltage. The coupling capacitor was increased to 100 nF to allow direct measurements in the nF-range. To compensate the resulting increase in charging time constant t ¼ RC, the 1 GO protection resistor was replaced by a 22 MO resistor. For the measurement all segment electrodes were shorted to ground. The parallel capacitance of the protection circuitry and the cryostat were measured. This constant offset was subtracted from the measured result of the detector. Different high voltage modules were used to avoid systematic errors. Finally the results of the direct and the pulser measurement compare very well (see Fig. 7).
5. Reconstruction of the doping profile From the capacitance–voltage data, shown in Figs. 7 and 8, the impurity concentration or doping profile of the HPGe crystal is extracted applying the techniques developed in Ref. [13]. Three approaches with different levels of complexity were tested: The planar approximation, the cylindrical approximation and the numerical analysis through chi square minimization. The results are discussed and compared below. 5.1. Planar approximation The simplest way to determine the impurity concentration N near the outer surface of the detector is given by a planar
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Table 1 Surface areas (cm2) of segments for all four types of AGATA detectors: irregular shaped A-, B-, C- and symmetric detectors (for details of the detector geometry see Ref. [16]). Seg.
Red (A)
A1 A2 A3 A4 A5 A6 B1 B2 B3 B4 B5 B6 C1 C2 C3 C4 C5 C6 D1 D2 D3 D4 D5 D6 E1 E2 E3 E4 E5 E6 F1 F2 F3 F4 F5 F6 Sum
Green (B)
Blue (C)
Sym. (S)
8.08 4.99 6.13 7.70 7.86 7.91 8.21 5.05 6.19 7.78 7.95 7.99 7.42 4.53 5.55 6.92 7.03 7.06 7.23 4.25 5.18 6.52 6.62 6.62 7.99 4.88 5.98 7.48 7.64 7.68 8.22 5.07 6.22 7.79 7.96 8.00
7.77 4.71 5.70 7.17 7.27 7.29 8.23 4.98 6.06 7.47 7.57 7.60 8.64 5.24 6.39 7.94 8.09 8.13 7.23 4.55 5.53 6.90 6.98 6.99 7.58 4.69 5.74 7.09 7.19 7.23 8.36 5.04 6.16 7.81 8.00 8.02
7.77 4.74 5.82 7.27 7.40 7.41 8.02 4.81 5.88 7.42 7.55 7.55 7.83 4.85 5.94 7.36 7.49 7.53 7.98 4.79 5.85 7.37 7.52 7.56 8.03 4.88 5.97 7.51 7.67 7.71 7.89 4.82 5.91 7.35 7.47 7.50
6.59 4.98 5.54 7.22 7.50 7.55 6.59 4.98 5.54 7.22 7.50 7.55 6.59 4.98 5.54 7.22 7.50 7.55 6.59 4.98 5.54 7.22 7.50 7.55 6.59 4.98 5.54 7.22 7.50 7.55 6.59 4.98 5.54 7.22 7.50 7.55
245.65
247.91
246.38
237.50
approximation for every segment. It is based on the equations: CðdÞ ¼
eA
ð12Þ
d
NðdÞ ¼
C3
eeA2
dC dV
1 ð13Þ
with d the depletion depth below the surface. This approximation is just valid for a small depletion area at low voltages, such that the area of the depletion boundary A still corresponds to the area of the segments [13]. These surface areas were calculated for the different segments. The resulting values are listed in Table 1 for the four types of AGATA detector geometries (three irregular shaped detectors and the symmetric detector [16]). The results of the impurity concentration obtained with the planar approximation are in the range of 0.5 and 1.0 1010 per cm3 (see Fig. 10). These values compare reasonable well with the manufacturer’s values in the range between 0.5 and 1.8 1010 per cm3.
5.2. Cylindrical approximation The AGATA crystal has nearly a cylindrical shape at the back side and the last four rings of segments in the detector are close to a coaxial detector geometry. The assumption of a cylindrical symmetry was applied for these four rings. The reconstruction of the impurity profile is given as a function of the depletion
Fig. 10. Results of the three reconstruction methods: planar, cylindrical and numerical, for the space charge reconstruction along each of the six rings of segments. The positions of the data points along the crystal are given for the center of the segments. The dashed line corresponds to the cubic spline interpolation used in the minimization procedure.
radius and is performed with the Eq. [13]: Cðrd Þ ¼
2peH ln rr2d
ð14Þ
Nðrd Þ ¼
1 Cðrd Þ3 dC 4p2 eerd2 H2 dV b
ð15Þ
with rd the depletion radius and r2, H the radius and height of the cylindrical segment electrode. Compared to the planar approximation, this method is not limited to the lowest bias voltages or small depletion depth. A full reconstruction of the doping profile as a function of the radius was performed. The result of the reconstruction is shown in Fig. 11. The impurity concentration obtained with the cylindrical approximation is slightly higher than with the planar approximation. The values are within the range of 0.5 and 1.1 1010 per cm3 (see Fig. 10). A gradient is observed from back to front, in agreement with the planar approximation. The planar approximation underestimates the real values as discussed in Ref. [13] by an amount which depends on the impurity concentration and the segment radius. The results in Fig. 11 show that the impurity concentration within one ring is except for the boundaries fairly constant for each of the six segments. The depletion profile is highly axially symmetric. This is probably due to the rotation of the crystal during the growth process. In order to evaluate the deviations from a homogeneous distribution correctly, effects breaking the coaxial geometry have to be estimated. For this purpose, a simulation was performed assuming a homogeneous impurity distribution of N¼1.0 1010/cm3. The result of this simulation is shown in Fig. 12. The space charge profile is reconstructed very nicely. Exceptions arise only close to the core and at the largest radii. The large errors at the largest radii are caused by the numerical errors of the derivative of the C–V curve at the end point. The deviations at small radii are induced by the fact that the core electrode is depleted first at the front end and depletion moves then as a function of the applied voltage towards the back end of the detector. While the core electrode might already be depleted near the front electrodes, the undepleted areas near the back of the detector still contribute to the capacitance in the front electrodes. Remark that the full geometry of core and segment defines core-tosegment capacitance, and not just the limitation of this geometry to the ring interval in depth to which the segment is confined. Therefore,
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Fig. 11. Reconstruction of the doping profile using the coaxial approximation for the last four rings of electrodes in the AGATA detector. The four graphs correspond to the four rings of segments. The graphs are ordered according to the physical order in the rings, starting with the back ring with largest diameter on top. In each of the four plots, six graphs are shown corresponding to the six segments in the corresponding ring.
Fig. 12. Simulated doping profile equivalent of Fig. 11 for reference. The capacitances at different bias voltages were calculated assuming a fixed doping profile of N ¼1.0 1010/cm3. The impurity profile was reconstructed using the coaxial approximation.
the capacitance in the first ring will still change with bias voltage until the whole detector is depleted. This effect is disregarded in the cylindrical approximation and causes the overestimation of the space charge at smaller radii in Fig. 11 and 12. The comparison between simulation and experiment is consistent with a homogeneous radial impurity concentration.
A small deviation is visible as a small bump at a radius rd ¼2 cm in the last ring. This might be caused by a disturbance of the field outside the detector geometry, e.g. by HV insulation material. The average value for each ring is also included in Fig. 10 for comparison. The outermost section of the reconstruction profile was excluded to avoid residual geometry effects.
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Considering the expected underestimation of the planar approximation [13], the results of the planar and cylindrical solution compare well.
Fig. 13. The chi square minimization scheme.
5.3. Numerical method Finally the space charge was determined by a chi square optimization procedure using the numerical computation program described in Ref. [13]. Since the space charge was proven to be axial symmetric and radius independent using the cylindrical approximation, only the variation in depth was left to fit. The numerical code is known to be less accurate at lower bias voltages, due to the finite grid resolution. Therefore, only the capacitances above 100 V bias were included in the comparison to guarantee capacitance simulation errors below 10%. Above 2000 V this detector shows constant capacitance values or full depletion and the values above 2000 V bias voltage were also excluded from the comparison. A scheme of the chi square minimization procedure is shown in Fig. 13. Since the detector has a sixfold segmentation in depth, the space charge was modeled with 6 parameters to prevent underdetermination. These six parameters were chosen to correspond to the space charge in the middle of each segment. The space charge, assumed axially symmetric and radially independent, was interpolated using a cubic spline interpolation [22] in intermediate positions. Starting from the doping profile determined by the cylindrical approximation the depletion boundaries were calculated for the 16 bias voltage values between 100 and 2000 V. For each of these solutions, the 37 voltage dependent weighting potentials belonging to the core and each of the segments were determined. From these weighting potentials, the core-to-segment capacitances were calculated which allow direct comparison with the experimental values. The optimal solution is also shown in Fig. 10 for comparison with the two other approaches. The dashed line is the spline interpolation which is used in the optimization routine. The result is nearly linear and varies between 1.5 and 0.5 1010 per cm3 from front to back of the crystal. The solution compares well for the four highest rings with both other methods. For the two lower rings, where the cylindrical approximation cannot be used, the discrepancy with the planar approximation is striking. Of the two methods, the chi square method is the most accurate. The planar approximation differs here for the reason discussed earlier. A direct comparison between the experimental capacitances and the simulated capacitances for the optimum solution obtained through chi square minimization is shown in Fig. 14.
Applied Voltage [V]
Fig. 14. Comparison of capacitances between the optimum simulated solution and the experimental values. The insert corresponds to the data range included in the optimization.
183
Fig. 15. Plot of the relative deviation for the optimal solution between simulation and experiment for all 36 segments as a function of applied high voltage.
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To acquire a value for the achieved accuracy, the simulation was recalculated twice, adding 710% to the optimum impurity concentrations shown in Fig. 10. Since the agreement is much better than these limits, the obtained space charge concentrations are determined with an accuracy better than 710%. The agreement of the individual core-to-segment capacitances for the optimal parameters, under assumption of a coaxial space charge distribution is shown in Fig. 15. Most segment capacitances are well below the 710% relative error. An exception is segment E1, which shows consistently lower capacitance values. The experimental data, shown in Fig. 14, indicate that the detector depletes earlier than expected from the physical geometry of the crystal. Note that also in Fig. 11 the 5 mm core radius is not reached at full depletion. The capacitance at full depletion is also higher than was expected from simulation. Comparable effects have been observed in all AGATA detector geometries. This difference is caused by the effective thickness of the lithium diffused layer near the core contact. A thickness of about 1.5 mm is estimated from the difference in capacitance.
were observed. A nearly linear gradient in depth was measured going from 1.5 1010 in the front to 0.5 1010 per cm3 in the back of the crystal, that is comparable to the linear gradient of 1.8 1010 in the front to 0.5 1010 per cm3 in the back of the crystal given by the manufacturer. We were able to determine this space charge profile with good precision and demonstrated that the sensitivity of the method is well below 10%. In future this new and non-destructive method will be applied to determine the impurity concentration and its gradient along the detector axis for operational HPGe detectors after processing the Ge crystals.
Acknowledgments This research was supported by the German BMBF under Grants 06K-167 and 06KY205I.
References 6. Discussion and outlook The full three-dimensional distribution of the impurity profile of a large volume, irregular shaped, highly segmented HPGe detector was determined for an operational detector via a capacitance–voltage measurement. The acquired capacitance values were obtained by employing a pulser and the capacitive coupling of the pulser signal through the bulk capacitance of the HPGe crystal. The main advantage of the new method is that it can be applied to the working detector and does not require modifications of the cryostat. The method was validated by comparison to a direct measurement of the capacitance. The capacitance values were not affected by the serial detector resistance. Three types of methods were applied for the reconstruction of the impurity concentration from the CV data. The planar approximation, which is only valid for small depletion depth [13], yielded results consistent with the two other approaches, but underestimated the impurity concentration in the front part of the detector. The cylindrical method and the numerical method compare very well with the impurity concentration and its gradient given by the manufacturer. The two methods are complementary: In the numerical method, a tradeoff has to be made between numerical precision, fit parameters and calculation time. The cylindrical approximation on the other hand is very fast and retrieves the maximum of information. However, deviations from a cylindrical detector shape limit the applicability of this approach. The final results for the symmetric AGATA crystal showed a highly cylindrical space charge distribution. No radial variations
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