Determination of the Fano factor in germanium at 77°K

NUCLEAR

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NORTH-HOLLAND

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D E T E R M I N A T I O N OF THE FANO FACTOR IN G E R M A N I U M AT 77 OK A. H. S H E R * a n d B. D. P A T E

Department o f Chemistry, Simon Fraser University, Burnaby 2, B. C., Canada Received 23 D e c e m b e r 1968 T h e F a n o factor in g e r m a n i u m at liquid nitrogen t e m p e r a t u r e has been determined t h r o u g h the observed line widths in pulseheight spectra f r o m Ge(Li) detectors u p o n which g a m m a rays were incident with energies o f 122, 356, 1173 a n d 1333 keV. I n s t r u m e n t a l contributions to the line width were m e a s u r e d with

a pulse generator a n d corrected for; the variation o f charge collection with applied bias was d e t e r m i n e d a n d F a n o factor m e a s u r e m e n t s were m a d e in a region where 100% collection efficiency was evidently obtained. A value for t h e F a n o factor o f 0.132=t=0.008 has been obtained for the energy range studied.

1. Introduction

electrodes and the charge converted to a voltage signal in the preamplifier, whence it is processed by an amplifier and pulse-height analyzer. The statistical fluctuations in both the charge production and charge collection/signal processing stages adversely affect the system resolution as manifested by the width of a peak in the pulse-height spectrum. Since the Fano factor is associated only with the statistics of charge production, the experimenter must be able to account for and correct for fluctuations in all the subsequent stages which increase detector line-width, if erroneous values for the Fano factor are not to be obtained. Specifically, the following factors contribute to the line-widthS):

The measurement of the Fano factor in germanium at 77°K + remains of interest from both experimental and theoretical points of view as evidenced by the recent papers of Klein 3) and Bilger*). The latter work, an extensive experimental investigation, represents the most recent study in the series begun by Mann and collaboratorsS7). Recent studies8,9), however, have used F values ranging from 0 . 1 3 - 0.15 in applications relating to detector resolution. The current study, therefore, is presented in an effort to corroborate the value of F determined by Bilger 4) and to determine a statistically significant value of F on the basis of the two studies. The measurements for the present study were carried out independently of Bilger 4) and represent a somewhat different approach to the measurement problem xo). One can consider the process by which radiationinduced charge in a semiconductor detector is collected and analyzed by the appropriate electronic equipment to yield a pulse-height spectrum as occurring in two general stages. Firstly, gamma-rays interact with the semiconductor to produce electrons and holes. The mechanism of energy l o s s 3) is s u c h , that statistical fluctuations are produced in N, the average yield of ion pairs, with a 2, the variance in the yield given by 1112): a 2 = FN,

(1)

where F is by definition the Fano factor. In the second stage, charge carriers are collected at the detector * Present address: N a t i o n a l B u r e a u o f Standards, W a s h i n g t o n , D. C. 20234. t T h e research for this paper was s u p p o r t e d (in part) by the Defence Research Board o f C a n a d a , G r a n t N o . 1680-40. + It is c o m m o n practice to say that Ge(Li) detectors are operated at 77 °K even t h o u g h the detectors are not in direct contact with

liquid nitrogen and direct temperature measurements are not made. It has been shown 1'2) however that the detector resolution is fairly constant over the range 60-160 °K ; this is not to say that other features of the ionization process remain constant. 251

a. Electronic noise of the detector/preamplifier system; b. Gain instabilities in the preamplifier, amplifier, and analyzer, which may be partly counting-rate dependent; c. Statistics of carrier recombination, trapping and signal risetime variations. Factor (b) can be eliminated by proper choice of the electronic equipment used, by not using data obtained in a run in which any electronic instability was noted, and by adjusting the counting rate to a suitably low value. If peak shapes in a pulse-height spectrum are Gaussian, factor (a) is accounted for by subtracting in quadrature the system noise A E p (measured from a peak produced by a pulse generator applied to the preamplifier input) from the total system line-width represented by the gamma-peak resolution, A E r , to yield the detector resolution, AED: 2 2 2 A E D = A E T - AE(, .

(2)

Since the effects of factor (c) will be systematically dependent on the applied detector fieldS), one can, by obtaining pulse-height spectra with varying detector fields, choose data in the range where these effects are

252

A. H. S H E R A N D B. D. P A T E

independent of applied voltage. Variations in the appliedvoltage will also affect factor (a) to some extent through its effect on the capacitance of the detector and hence noise in the preamplifier. Thus one must use a detector with as small a variation of capacitance with voltage as possible and discard those data obtained in regions where capacitance varies with applied bias. No attempt was made to correct for effects of risetime variation and ballistic defects on A E D. Only those data points were used in the determination of F where such effects were supposed absent, as in the study by Bilger4).

tailing on the low energy side of the gamma-peak). Below approximately 20 V/ram (dependent on gammaray energy) tailing on the low energy side of the peak was observed; thus over a comparatively small range of applied fields, we obtained detector characteristics ranging from excellent to poor. It was then possible to select the range of operation in which field independent characteristics obtained. Leakage current at the maximum field (40 V/mm) was approximately 500 pA. Between approximately 16 V/mm (80 V applied) and 40 V/mm, the capacitance varied less than 2%. 3. Treatment of data

2. Experimental apparatus The major portion of the data to be reported here was obtained using a 0.85cm 3 Ge(Li) detector, D-74, coated with 150 A of CaF213). An Ortec model 118 F E T preamplifier operated at room temperature and a Tennelec TC-200 main amplifier operated in the single-RC dipping mode with 1.6 #s filter time constants were used in conjunction with a Nuclear Data ND-160 multichannel analyzer. Test pulses were obtained from a Berkeley Nucleonics Corporation RP-2 pulse generator, pulses being fed directly into the preamplifier input circuitry. Pulser stability was better than +0.01% in the course of the measurements. Source to detector distances were adjusted to yield approximately 10 counts/sec in the full-energy peak of interest with the pulse generator usually matched to this rate, unsynchronized with the line frequency. Counting times used were between 20 and 40 minutes, depending upon gamma-ray energy, previous experiments 14) having shown that gain instabilities were not detectable in the system for much longer counting periods. Spectra were recorded in 1024 channels without the use of gain stabilization devices. In addition, some data were obtained using an 8 cm 3 Ge(Li) detector HD-1 fabricated in this laboratory, an Ortec model l18A FET preamplifier operated at room temperature, and a Victoreen SCIPP 104TP analyzer with spectra stored in 3200 channels. The characteristics of the 0.85 cm 3 Ge(Li) detector (0.5 cm thick) and its suitability for use in such measurements as described herein bear discussion. This detector was of the type reported earlier 15) in which the reverse leakage current increased almost monotonically with applied bias; thus, bias greater than 200 V could not be applied without introducing excessive noise into the system. Yet, this field, 40 V/mm, was sufficient to collect the charge deposited in the detector without the observation of trapping (through

Pulse-height spectra were plotted on semilog paper to emphasize any tailing on the low energy side of the peak that might be present. Since the method of calculation described earlier required that the peaks be Gaussian-shaped, some means was required to test the peak shape. Fig. 1 illustrates the technique adopted applied to some typical data. Shown plotted are the pulser peak and a 122 keV gamma-ray peak from measurements at 150 V detector bias using a 57Co source. The full widths of each peak at 1/5 and 1/10 maximum height were obtained, and the ratio of these widths (fwxm) to the full width at half maximum (fwhm) were calculated and compared to the ratios for a Gaussian peak in the manner of Camp 11). A difference of less than 10% was taken to indicate Gaussian shape. This comparison for the spectrum of fig. 1 is shown in the table 1.

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DETERMINATION OF THE FANO FACTOR

considered explicitly; fitting o f curves to the data points inevitably involves s o m e sort o f approximation.

TABLE 1

Fwxm/fwhm ratios. x 1/5 1/10

pulser * 1.52 1.85

122 keV *

4. Experimental results

Gaussian 12)

1.58 1.92

1.524 1.823

* An uncertainty of 2~5% is attached to these values, representing upper and lower limits of the measurement. As the width of the peak can be ascertained to within

0.5 channels, the data in fig. 1, where the energy spread is 1 4 0 e V / c h a n n e l ,

253

are estimated to be accurate to

within _+5% (as calculated at the halfwidth). Using eq. (2), F can be calculated as follows:

The data plotted as d E 2 vs reciprocal field strength a) I/e, are shown in figs. 2, 3 and 4 for g a m m a - r a y energies of 122, 356, 1173 and 1333 keV, respectively. Each point represents the m e a n value of the results f r o m 4 to 6 measurements at each field value; error bars are the rms deviations. Values of F were calculated for each field strength where the criteria for Gaussian shape of both pulser and gamma-ray peaks as outlined above were met. A horizontal line representing the average value o f F is drawn for each g a m m a - r a y energy; table 2 summarizes the average values o f F obtained f r o m each gamma-ray energy. 5. Discussion of results

F = (AED/AEpo~ . . . . )2,

(3)

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The data in table 2 m a y be averaged to obtain a m e a n value of the Fano factor in germanium at (r.7) ~

in which E v is the gamma-ray energy in keV, and ~ is the radiation ionization energy taken to be 2.984-0.01 eV/ion-pairl6). The method used for

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Fig. 3. Square of detector peak width vs reciprocal field strength for the 356 keV gamma-ray of taaBa.

254

A. H. S H E R A N D B. D. P A T E

77 °K of F = 0.132+0.008, in good agreement with the values of 0.129__+0.003 determined by Bilger 4) and 0.13 calculated by Klein3). TABLE 2 Fano factor F.

AED (keV

G a m m a - r a y energy (keV)

122 * 356 1173 1333

F 1"

fwhm)

0.49 0.88 1.61 1.72

0.120±0.040 0.132d_p.004 0.138 ~0.005 0.137 -L0.004

* Includes data from 8 cm 3 detector. t Uncertainties are the estimated standard deviations.

From data similar to those shown in figs. 2, 3 and 4, Mann et al. 5) inferred that in order to obtain a measure (2 5) z I

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of the "intrinsic" Fano factor, it was necessary to extrapolate to infinite electric field. Bilger 4) did not use such an extrapolation; however, no specific comments on the method previously described were made. Such an extrapolation does not appear to the present authors to be necessary; experimental observations suggest that a region of detector bias values can be chosen in which charge collection from the detectors used in the present study was essentially 100%. The data on this are three-fold. The detector resolution, AEo, was observed to improve (become smaller) with increasing applied bias up to some limiting value. Thereafter AED was independent of further increases in the applied bias value (until increased leakage current caused deterioration of overall resolution). A parallel behaviour was observed in the relative pulse-height of full energy events in a pulse-height spectrum with increasing detector bias. This behaviour is shown in fig. 5 where the variation of relative pulseheight with bias is plotted for three of the incident g a m m a radiations investigated. Here relative pulseheight is defined as:

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relative pulse-height = 1 - {(Eo - E)/Eo} , where E 0 is the ultimate value of pulse-height obtained and E is the value observed at a particular applied bias. It is seen that above a limiting applied bias value the relative pulse-height values are static at 100%, i.e. no increase in relative pulse-height is observed. A third observation which bears on charge collection efficiency is the appearance of "tailing" on the low energy side of the full energy peaks under investigation.

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Fig. 5. Relative pulse height (%) vs detector field strength for gamma-ray energies of 122, 356 and 1173 keV.

D E T E R M I N A T I O N OF THE FANO FACTOR.

Tailing which was present at low bias values was observed to decrease with increasing applied bias and to disappear at bias values near those at which the constant AED and relative pulse-height values were obtained. Presumably the tailing arises from variation in charge collection efficiency from region to region within the detector. A quantitative estimate of the extent of tailing can be made by using measured values of the full width of the peak at 1/10 of its maximum height. Dependent somewhat on incident gamma-ray energy, this width was observed at bias values below the limiting value to be increased by a factor of from 1.5 to 5 over the width observed with bias values where the relative pulse-height was 100%. This is in agreement with the observations reported by Campl~). From the calculations of Chartrand and Malml~), it is possible to determine in a semi-quantitative manner whether the observed small decreases in relative pulse-height (a few tenths of a percent) and increases in peak width of the order observed are mutually consistent. For the case examined by them in which only one type of charge carrier was completely collected, the increase in peak full width at 1/10 maximum height for a decrease in pulse-height o f 0.1% is approximately a factor of 2; this calculation was performed for a gamma-ray energy of 662 keV, a Fano factor value of 0.13 and a trapping length for the collected carrier 100 times the active detector thickness. One would expect that increases in line width with variation in field strength would be less for lower energy gamma-rays and more for higher energy gamma-rays"). All of the foregoing corresponds quite well to the present observations reported above. Thus it seems reasonable to attribute the presently

255

observed variations in line width to variations in charge collection efficiency of ion pairs rather than to irregularities in the detector field due to uncompensated impurities in the crystal as did Bilger4). It would, perhaps, be instructive, in this connection, to make measurements of the Fano factor in germanium using a collimated beam of gamma-rays ~8) so that the separate contributions of hole collection and electron collection to the Fano factor determination could be measured independently. References 1) E. Sakai, H. L. Malta and 1. L. Fowler, AECL-2762 (May, 1967). 2) E. Sakai and H. L. Malm, Appl. Phys. Letters 10 (1967) 268. ~) C. A. Klein, IEEE Trans. Nucl. Sci. NS-15, no. 3 (1968) 214. 4) H. R. Bilger, Phys. Rev. 163 (1967) 238. 5) H. M. Mann, H. R. Bilger and I. S. Sherman, IEEE Trans. Nucl. Sci. NS-13, no. 3 (1966) 252. 6) H. M. Mann, Bull. Am. Phys. Soc. 11 (1966) 127. 7) H. M. Mann, IEEE Trans. Nucl. Sci. NS-14, no. 6 (1967) 10. 8) j. E. Cline, IEEE Trans. Nucl. Sci. NS-15, no. 3 (1968) 198. 9) L. L. Makorsky, N. B. Strokan and N. I. Tisnek, IEEETrans. Nucl. Sci. NS-15, no. 3 (1968) 304. 10) A. H. Sher, Ph. D. dissertation (Simon Fraser University, 1967). 11) D. C. Camp, UCRL-50156 (March, 1967). 12) G. Dearnaley and D. C. Northrop, Semiconductor counters for nuclear radiations, 2nd ed. (Wiley, New York) p. 25. 13) A. H. Sher, B. D. Pate, J. F. O'Hanlon and R. R. Haering, Nucl. Instr. and Meth. 53 (1967) 341. 14) A. H. Sher and B. D. Pate, Nuclear Physics A l l 2 (1968) 85. 15) A. H. Sher and B. D. Pate, Nucl. Instr. and Meth. 53(1967) 339. 10) S.O. Antman, D . A . Landis and R. H. Pehl, Nucl. Instr. and Meth. 40 (1966) 272. 17) M. G. Chartrand and H. L. Maim, AECL-2764 (June 1967). 18) p . p . Webb, H . L . Maim, M . G . Chartrand, R. M. Green, E. Sakai and I. L. Fowler, Nucl. Instr. and Meth. 63 (1968) 125.