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Measurements of Fano factors in silicon and germanium in the low-energy X-ray region
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Nuclear
EUEWIER
Instruments
and Methods
in Physics
Research
A 399 (1997) 3544364
SectIon A
Measurements of Fano factors in silicon and germanium in the low-energy X-ray region B.G. Lowe OxfOrd Instruments pk.
Ana&tical &-terns DitGion. -70Nufield Wav, Abingdon OX14 I TX, UK
Received 23 January
1997; received
in revised form 10 April 1997
Abstract
Estimates of Fano factors have been made in the low-energy X-ray region for both silicon and germanium (109eV to The measurements are based on observations on a large number of X-ray detectors taking into account the effects of incomplete charge collection. The values at an X-ray energy of 5.9 keV are 0.1161 + 0.0001 and 0.1057 f 0.0002, respectively, for silicon and germanium at cryogenic temperatures. The values are found to converge and increase to about 0.15 at 183 eV. Comparisons are made with theroretical predictions and the difficulties of making measurements in the 100 eV region are discussed. 5.9keV for silicon and 183eV to 5.9keV for germanium).
1. Introduction Estimates of Fano factors have been made in the low-energy X-ray region for both silicon and germanium (109 eV to 5.9 keV for silicon and 183 eV to 5.9 keV for germanium). Such measurements are scarce in silicon [l-3] and values for such low energies in germanium have not been reported previously. In reporting Fano factors (F) from spot measurements using the full volume of one, or a few, detectors, it has previously been pointed out [4,5] that there is a danger that systematic errors may be introduced due to variations inherent in the detector, such as variations in the field and material or material processing. What is actually reported is often an “effective Fano factor” (F’). For example Palms et al. [4], Bilger [6] and others have made measurements for a range of detector biases and attempted to extrapolate to infinite field to elimi0168-9002/97/$17.00 c 1997 Elsevier Science B.V. All rights reserved PII SO168-9002(97)00965-O
nate incomplete charge collection (ICC) effects. One problem is that it is not clear where the limiting value of F’ (F) is reached (see e.g. Ref. [6]). Sher et al. [S] attempted to overcome the problem of weak fields by collimating the radiation to a fine beam so that the best regions of the detector were selected. ICC is likely to influence F’ for very low-energy X-rays where the attenuation length is very small (of the order of 100 nm for C K, X-rays in silicon and germanium) and carrier diffusion takes place against the field to the front contact [7]. For such low-energy X-rays fine-beam collimation is not appropriate. Owens et al. [2] made measurements on CCD devices and did not consider ICC to be significant in these. The approach taken here is to look at a large number of detectors manufactured over a period of many years with their variations in characteristics as well as the many manufacturing improvements made to reduce ICC. In fact, the ICC is now so
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small that the physical interactions within the detector are becoming more transparent [S]. Improvements have also been reported by other manufacturers [9]. The hope is that definite trends can be discerned and measurements in the Fano limit can be made for the ionising media at these energies.
2. Procedure The detectors studied had either lithium-drifted silicon (Si(Li)) or high-purity germanium (HPGe) crystal transducers. Rather than studying a few detectors in great detail, the experimental data has been obtained from a large number (> 500 Si(Li) and > 170 HPGe) of detectors collected as part of the technical specification and statistical process control in detector production at Oxford Instruments. A similar approach has been taken to set up a model of mercuric iodide X-ray detectors by studying the characteristics of a large number of commercially manufactured devices [lo]. All the detectors selected for the present study were 10 mm2 active area and 3 mm deep, for the sake of consistency. All were collimated in the same way and were biased at - 500 V. The temperature of the silicon was 120 f 10 K and of the germanium was 95 + 1OK. All the detectors reported on here did not necessarily pass the test specifications at the time of testing. Bench tests, using a radio-active source (Fe-5S), were made on detectors on the basis of the IEEE standards for resolution and peak-to-background on the Mn K, peak [l 1) as well as in-house specifications on ICC and stability. The calibration was determined by an internal strobe oscillator [12] which gives true zero and the Mn K, peak at 5895.1 eV. The zero strobe is crucial for low-energy measurements in ensuring stability with no off-set. It also allows an accurate measure of the true system noise peak full-width half-maximum (FWHM) to be made during the data runs. All detectors were monitored over many runs to check for instabilities in charge collection, as indicated by increasing low-energy tailing on the Mn K, peak or increase in the peak-to-background. In the case of completely windowless detectors, a special interface
355
with a Be window was attached for these bench tests. The resolution and noise were calculated for each run as well as the average dispersion for each series of runs. These terms are explained and discussed in more detail below. The standard deviations on the FWHM for the peaks in each individual run was typically < 1% and the total cumulative counts in the Mn K, peak on which the data reported is based were usually of the order of 9 x lo6 for Si(Li) detectors and 3 x lo7 for HPGe detectors. Additional tests on an SEM were carried out to check low-energy performance and overall system performance. Besides Mn, the elements fluoresced included F, C, B and Be in the case of Si(Li) detectors and Al, F, C and B in the case of HPGe detectors. The choice of Al in the case of HPGe detectors is dictated by the fact that the Al K, energy (1486.6 eV) is just above the L-absorption edges of germanium (1217-1420 eV) and, therefore, the peak is susceptible to the effects of ICC. The Be K, X-ray (108.8 eV) is not normally detectable with a HPGe detector because of the attenuation in the cryogenic IR screen necessary to limit the crystal leakage current [13]. The peaks were accumulated at 10 eV per channel over 100 s live time periods and the net counts in the peaks were typically in the range 50000-150000 for Al, F, C, and B but usually only a few thousands in the case of Be. All measurements were made on the same pulse processing unit and pulse height analyser. The disadvantage of this approach is that the statistics on the SEM data are not ideal for the individual runs and there may be small calibration and statistical software errors added to random ADC effects. However, neither of these should show any systematic bias and the errors should show up over the many runs as a scatter about valid means. The 0 of the scatter on peak median for the SEM runs varied with energy from 1.3 eV on the B K, peak to 3.6 eV on the Al K, peak.
3. Detector crystal dispersion The resolution R (FWHM) and the noise strobe N (FWHM) were measured by fitting Gaussian peaks to the respective peaks for each run. The
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centroids of the fitted Gaussians were also recorded. The statistical broadening due to the crystal alone was then expressed as the dispersion D, where D= =R= -N=.
(1)
By recording the noise FWHM and the photopeak positions, the dispersion D for each photo-peak of these elements could be measured, as well as the effective linearity of the crystal transducer. The intrinsic linearity of the system used for these measurements was checked using a pulse generator and precision attenuators. The deviation in linearity was -6.7 eV maximum in the energy range studied and the appropriate corrections were made on all peaks. The count rates were restricted during the data collection reported here to avoid any possible high-rate effects.
4. The Fano factor The Fano factor cannot be measured directly from the dispersion D as given by Eq. (1) as it is coupled to E (the average energy to produce one electron-hole pair of charge carriers in the material) by the equation D= =2.35=. &FE,,
(2)
where E0 is the energy of the incident X-ray. In this analysis I have, like most other workers [14], assumed the values of E for photons in silicon and germanium at 90 K provided by Pehl [ 151 although those measurements were not strictly made using X-rays. These measurements were 3.81 and 2.96 eV, respectively, with an error of _+ 0.02 eV on each. It is assumed in the first instance that there is no non-linearity due to a variation in E with energy. A recent absolute measurement of I-: in silicon at room temperature [16] gave a constant value in the energy range 50-1500 eV of 3.64 f 0.03 eV. Using the temperature dependence reported by Pehl [ 151 this corresponds to a value of 3.75 at 120 K. If this value is adopted then all the F values quoted here for silicon should be increased by 1.6%. Another recent measurement [2] shows a value of E increasing towards low energies by about 2.7% in the
range 1000-300 eV. Evidence of variations of F with energy are discussed in more detail below. In the case of the Mn K, peak, corrections were made to take account the unresolved 11 eV separation of the K,i and Km2 lines. In the case of Si(Li) detectors, this correction amounts to a 0.7% downward adjustment to F. Due to the better resolution of the HPGe detectors the adjustment for them is about twice this amount. No correction has been made for the expected non-Gaussian shape of any of the peaks [Z].
5. Fano factors for silicon It has long been appreciated [ 17,18,9] that one of the effects of ICC for low-energy X-rays manifests itself as an effective non-linearity in the photo peak position. As this is due to the loss of charge carriers it is naturally accompanied by a corresponding increase in the dispersion D and the effective Fano factor F’. Such a correlation of peak shift with broadening of the C K, peak was reported recently in silicon [19]. Thus, the negative displacement of the C K, peak position (AE) from the accepted energy of the X-ray can be used as a measure of ICC. Fig. 1 shows F’ as measured from the C K, peak as a function of AE (increasing ICC) for the C K, peak. All the Si(Li) detectors have been sorted by 1 eV intervals of AE and each data point represents the mean value of F’ for each set of detectors and its standard deviation. This shows two regimes. The regime for AE from -5 eV to about 2 eV shows no correlation (correlation coefficient =O.Ol) and the regime for AE >2 eV shows a linear correlation (correlation coefficient =0.6). The first regime can be interpreted as the limiting value F with no ICC. Indeed, Fig. 1 is reminiscent of inverse field plots [6]. The flat regime width is consistent with the statistical scatter in peak positions as mentioned above and gives rise to negative values of AE. The extreme values of AE in this and ensuing plots represent one or two detectors where the statistics on the peak position is poor. The limiting value of F from Fig. 1 is 0.157 f0.008. Fig. 2 shows the variation of F’ as measured from the B K, peak as a function of AE for the
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0.3
f
t
3
0.2
f
f
f
0.1
I 10
5
0
-5
15
A E (eV) Fig. I. The effective Fano
factor
for silicon for the carbon
K, peak plotted
as a function
of peak shift AE.
1 --
0.8 --
I
0.6 --
I
,5
0
5
I
I
IO
15
20
25
30
A E (eV) Fig. 2. The effective Fano factor
for silicon for the boron
K, peak plotted
as a function
of peak shift AE.
35
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Instr. and Meth. in Phys. Res. A 399 (1997) 354-364
B K, peak (the true position taken as being 183.4 eV). This again shows two regimes and a value of F consistent with the above value at 277.4 eV within the experimental errors. F’ as measured from the Mn K, peak (5985.1 eV) showed no correlation with AE for the C K, peak. F’ measured from the fluorine K, peak showed very little correlation and there was no peak shift from the expected value (676.8 eV). It was therefore assumed that ICC had negligible influence on F for these energies and the values for Mn and fluorine K, peaks were found to be 0.1161 + 0.0001 and 0.134 f 0.003, respectively.
6. The problems of lower energies In interpreting data from the Be K, peaks (108.8 eV) there are a number of problems which greatly reduce the confidence level in the estimates. Firstly, we have a fall-off to low energies in detec-
tion efficiency and corresponding counts in the peak due to absorption of the X-rays in the window and contact materials used. The detectors studied here had all been recently “conditioned” [20], so that, ice or other condensates were not expected to be a problem. Secondly, there is a more rapid fall-off in the electronic efficiency due to the noise in the pulse processor recognition channel [21]. Thirdly, the beryllium specimen used in the SEM is prone to oxidation and therefore self-absorption and requires frequent cleaning. All these effects reduce the counts in the photopeak as mentioned earlier. Whereas all the other peaks studied are on a low flat background (see e.g. spectra in Ref. [20]), this is not always the case for the Be K, peak, which may sometimes lie partly in the noise tail. This effect will broaden the peak and displace it to lower energies. A more serious distortion of the Be K, peak is caused by the fall-off in electronic detection efficiency mentioned above. Fig. 3 illustrates this effect which opposes the effects of noise and ICC,
Undistorted Peak
80 --
Electronic Efficiency
s 2 5 ._ .o z .s
60 --
40--
g
8t
w
Distorted Peak 20 --
-I
20
40
60
80
100
120
140
Energy (eV)
Fig. 3. Distortion
of the beryllium
K, peak due to electronic
efficiency.
160
180
200
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reducing the peak FWHM and shifting it to higher energies. The broader the peak the greater is this effect. In sorting detectors for magnitude of ICC (Figs 4-6), the value of AE (in 5 eV intervals) for the C K, peak was chosen as this is the highest energy peak exhibiting appreciable ICC and provides the best statistics of all the very low-energy peaks, Fig. 4 shows AE/Eo as a % shift for the peaks studied as a function of E. without any correction made to the Be K, peak. The lines are drawn to guide the eye and illustrate the effects of increasing ICC. The Be K, peak is just above the Si L,, and Si Liir absorption edges (98899 eV) and ICC is, therefore, expected to have the greatest influence here and result in the greatest % energy shift. Clearly, the uncorrected data of Fig. 4 is anomalous in this respect. If no correction is made, a value of F’ =0.19 f 0.04 at 108.8 eV is obtained which must represent a lower limit.
7. Correction for distortion of Be peak The electronic efficiency was measured using the precision pulse generator and an attempt was made to correct the measured peak positions and widths. The correction has the effect of approximately doubling F for the Be K, peak but the value has a large uncertainty. Fig. 5 is the corrected version of Fig. 4 and the effective Fano factors as a function of energy are shown in Fig. 6. The values of F for silicon are summarised in Table 1.
8. Fano factors for germanium
The Fano factor for germanium based on the Mn K, peak measured from the bench runs was about 9% lower than the corresponding value for silicon. Fig. 7 shows the effective Fano factor taken from
20
increasing ICC t
400
300 Energy (eV) Fig. 4. Peak shifts for silicon expressed
as a percentage
of the expected
position
and plotted
as a function
of energy.
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Instr. and Meth. in Phys. Res. A 399 (1997) 354-364
35 -Increasing
ICC
30 --
25 -s f %
20 --
% s 15 --
0
100
200
300
400
500
600
of the expected
position
700
Energy (eV) Fig. 5. Peak a function
shift for silicon
corrected
for electronic
efficiency
expressed
as a percentage
and plotted
as
of energy
the C K, peaks for HPGe detectors plotted as a function of AE for the C K, peak. It is clear from Fig. 7 that there is only one domain and no correlation with AE (correlation factor = -0.06). Fig. 8 shows that even for the B K, peak there is no correlation. This lack of correlation is repeated for all the peaks analysed and so the Fano factors can be deduced using all of the data. The physical reason for this is not clear but may be connected with the shorter range of the photo- and Augerelectrons in germanium compared to silicon. The Fano factors for germanium are summarised in Table 2. There was no dependency on the source of the HPGe material detected in this study.
9. Discussion Fraser et al. [14] also summarised the experimental measurements of E and F for silicon. For
Si(Li) crystals cooled by liquid nitrogen the authors cited use a value of E quoted from previous measurements, usually also from Pehl [15]. The values of F range 0.072-0.132 for the energies below 10 keV illustrating the difficulties in making such measurements. A recent measurement [22] also assuming e =3.81 eV and using monochromatic synchrotron X-rays gives F =0.114 for the range 700 eV-7 keV. Although the Fano factors for silicon agree quite well in the region 200-300 eV with the calculations of Fraser et al. (Table l), the values at higher energies do not. The calculation of F by Alig [23] of 0.113 f 005 for silicon agrees better with the value at 5895 eV found here. Although F is found to increase at lower energies, it does so only by 35% around 300eV and not by the factor of about 3 recently reported for a silicon diode [3]. Table 3 summarises the results for silicon in terms of the product EF. The experimental values
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361
2
T
1.8 --
A
1.6 --
Increasing
ICC
0.8 --
0.6
100
0
400
300
200
500
600
700
Energy (eV) Fig. 6. Effective Fano
factors
for silicon corrected
for electronic
Table 1 Data for Si(Li) detectors Peak
Mn F C B Be
Energy
Fano factor
@V)
(F)
5895.1 676.8 217.4 183.4 108.8
0.1161 0.134 0.157 0.15 0.4
f f k + *
0.0001 0.003 0.008 0.01 0.2
Fraser et al.
Alig
[14] theory 0.16 0.15 0.14 0.15 0.16
[23] theory 0.113 f 0.005
agree reasonably well, including Owens et al. [2] of 0.54 eV near the fluorine K, line and an “asymptotic” value of about 0.48 eV at higher energies. These values were measured at 170 K and would correct to about 0.52 and 0.46 eV, respectively, at 120 K by their calculations. However, their theoretical value of 0.58 eV [ 141 at 5895 eV is still too high.
efficiency
and plotted
as a function
of energy.
As stated previously, no experimental values of E; have been published to date in the energy ranges discussed here for HPGe. However, Knoll [24] summarises values from Ge(Li) detectors for higher energies as between 0.057 and 0.129 again illustrating the difficulty in making such measurements. Pehl et al. [25] made measurements with Cu K, X-rays (8041 eV) in several small Ge(Li) detectors obtaining a value of 0.083. Alig [23] calculated a value of 0.126 f 0.005, higher than that for silicon, contrary to the present finding for Mn K, X-rays. Fig. 9 shows F as a function of energy (logarithmic scale) for both silicon and germanium. The value for the Be K, peak has not been included due to the high uncertainty. It is remarkable how similar the Fano factors of germanium are to those of silicon at the same energies, despite the considerable differences in dispersion measured. There is a similar upward trend in F as lower energies are approached. Fraser et al. [2,14] predict and
362
B.G. LowelNucl.
Instr. and Meth. in Phys. Res. A 399 (1997) 354-364
nc
FIl.51
AE (eV) Fig. 7. The Fano factor for germanium
for the carbon
K, peak plotted
as a function
of peak shift AE.
nc
0.5 F
1
0.3 --
0.2 --
I
”
”
f 0
i
f
f !
0.1 --
c -5
n 10
5
0
15
AE(eV)
Fig. 8. The Fano factor
for germanium
for the boron
K, peak plotted
as a function
of peak shift AE.
B.G. Lowe/N&.
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Instr. and Meth. in Phys. Res. A 399 (1997) 354-364
measure a general increase in F for silicon below about 2000 eV with a transitional maximum over the Si K absorption edge at 1838 eV. The Si(Li) detector peak positions without ICC effects are estimated to be 278.0 + 0.5 eV for C K, and 184.0 + 0.5 eV for B K,. These are consistent with the accepted values and therefore do not support any change in E between 183.4 and 5895 eV as predicted and measured by Fraser et al. [2,14] and supported by Lechner et al. [3]. As pointed out by
Table 2 Data for HPGe detectors Peak
Energy (eV)
&.F(eV)
Fano factor
Mn Al F C B
5895.1 1486.6 676.8 277.4 183.4
0.317 0.35 0.387 0.467 0.48
(corrected) 0.1057 + 0.0002 0.121 f 0.001 0.133 _+0.003 0.155 + 0.005 0.15 + 0.01
Table 3 Experimental data for Si(Li) detectors Peak
Energy (eV)
E.F (eV)
Owens et al. Ref. [2]
Mn F
5895.1 676.8
0.4423 k 0.0004 0.51 * 0.01 0.60 + 0.03 0.58 + 0.04 1.5 _+0.8
“0.48 0.54
277.4 183.4 108.8
C B Be
McCarthy
Lepy
Musket
Ref. [9]
Ref. [22]
Ref. [18]
0.423
0.434 0.4 * 0.2
0.75 0.55
a Asympotic value
0.2 T 0.19 -F 0.18 -0.17 -0.16 --
.
m I 1000 Energy (eV)
Fig. 9. Fano factors of silicon and germanium plotted as a function of energy.
10000
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B.G. LowelNucl.
Table 4 Peak positions Peak
Mn Al F C B
for HPGe
Energy
(eV)
Instr. and Meth. in Phys. Res. A 399 (1997) 354-364
detectors Position
i: (Ge)
(eV)
(eV)
(eV)
5895.1 1486.6 676.8 217.4 183.4
(calibration) 1488.4 + 0.6 681.2 + 0.5 280.3 k 0.5 187.9 + 0.5
2.96 2.96 2.94 2.93 2.89
(assumed) f 0.001 f 0.002 f 0.005 + 0.008
Scholze [16] this may be due to the effects of ICC not being taken fully into account. The respective peak position values for germanium are summarised in Table 4. There is a small positive peak shift, increasing with decreasing energy. This is opposite to the expected effect of ICC and hence might be interpreted as a decrease in E from the value at 5895 eV. The consequence of this interpretation on E is also shown in Table 4 and the corrections to F for this effect in Table 2.
Acknowledgements
I would like to express my gratitude to the detector production team at Oxford Instruments Microanalysis Group in making the measurements and recording the data so diligently.
References 111 R.G. Musket, Nucl. Instr. and Meth. 117 (1974) 385. 121A. Owens, G.W. Fraser, A.F. Abbey, A. Holland, K. McCarthy, A. Keay, A. Wells, Paper B (Measurements), Nucl. lnstr. and Meth. A 382 (1996) 503. H. Soltau, L. Struder, Nucl. c31 P. Lechner, R. Hartmann, lnstr. and Meth. A 377 (1996) 206. Rao. R.E. Wood, Nucl. Instr. r41 J.M. Palms, P. Venugopala and Meth. 76 (1969) 59.
c51 A.H. Sher, W.J. Keery, IEEE Trans. Nucl. Sci. NS-17 (1970) 39. C61H.R. Bilger, Phys. Rev. 163 (1967) 238. c71 J. Llacer, E.E. Hailer, R.C. Cordi, IEEE Trans. Nucl. Sci. NS-24(1977) 53. PI J.L. Campbell, J.A. Maxwell, T. Papp, G. White, 1996 EDXRS 96 Conf. Lisbon June 1996, X-Ray Spectrometry 26 (1997) 223. in: Williams, Goldstein, Newbury (Eds.), c91 J.J. McCarthy, X-Ray Spectrometry in Electron Beam Instruments, Plenum Press, New York, 1995 p. 67. J. Henderson, Proc. of the 9th Int. Cl01 X.J. Bao, M. Natarajan. Workshop on Room Temperature Semiconductor X-Ray and Gamma-Ray Detectors, Associated Electronics and Applications, Grenoble, September 1995. Standard Test Procedures for Semi1111IEEE Std.759-1984 conductor X-Ray Energy Spectrometers. Cl21 K. Kandiah, IEEE Trans. Nucl. Sci. NS-22 (1975) 2058. Cl31 C.E. Cox, B.G. Lowe, R.A. Sareen, IEEE Trans. Nucl. Sci. 35 (1988) 28. A. Cl41 G.W. Fraser, A.F. Abbey, A. Holland, K. McCarthy, Owens, A. Wells, Paper A (Theory), Nucl. Instr. and Meth. A 350 (1994) 368. Gives a summary of measurements of F with references. D.A. Landis, M. Lenzlinger, Cl51 R.H. Pehl, F.S. Goulding, Nucl. Instr. and Meth. 59 (1968) 45. [I61 F. Scholze, H. Rabus. G. Ulm, 1996, submitted to Appl. Phys. Lett. also SPIE 1996 (Denver) paper 2808-50. 1171J. Jacklevic, IEEE Trans. Nucl. Sci. NS-18 (1971) 187. X-Ray Spectrometry, L181 R.G. Musket, Energy Dispersive National Bureau of Standards Special Publication, Vol. 604, 1981, 97. D. Hauff, P. Lechner, R. Richter, L. Struder, Cl91 R. Hartmann, J. Kemmer, S. Krisch, F. Scholze, G. Ulm, Nucl. Instr. and Meth. A 377 (1996) 191. Goldstein, Newbury (Eds.). cw B.G. Lowe. in: Williams, X-Ray Spectrometry in Electron Beam Instruments, Plenum Press, New York, 1995. p. 7. Energy Dispersive X-Ray Spectrometry, c-211P.J. Statham, National Bureau of Standards Special Publication, Vol. 604. 1981. 127. P. Stemmler, G. Ban, L. Beck, L-w M.-C. Lepy, J. Plagnard, P. Dhez, EDXRS 96 Conf. Lisbon, June 1996, X-Ray Spectrometry 26 (1997) 195. ~231 R.C. Alig, Phys. Rev. B 27 (1983) 968. Radiation Detection and Measurement. ~241 G.F. Knoll, Wiley, New York. 1989. 1251 R.H. Pehl, F.S. Goulding, Nucl. Instr. and Meth. 81 (1970) 329.
EUEWIER
Instruments
and Methods
in Physics
Research
A 399 (1997) 3544364
SectIon A
Measurements of Fano factors in silicon and germanium in the low-energy X-ray region B.G. Lowe OxfOrd Instruments pk.
Ana&tical &-terns DitGion. -70Nufield Wav, Abingdon OX14 I TX, UK
Received 23 January
1997; received
in revised form 10 April 1997
Abstract
Estimates of Fano factors have been made in the low-energy X-ray region for both silicon and germanium (109eV to The measurements are based on observations on a large number of X-ray detectors taking into account the effects of incomplete charge collection. The values at an X-ray energy of 5.9 keV are 0.1161 + 0.0001 and 0.1057 f 0.0002, respectively, for silicon and germanium at cryogenic temperatures. The values are found to converge and increase to about 0.15 at 183 eV. Comparisons are made with theroretical predictions and the difficulties of making measurements in the 100 eV region are discussed. 5.9keV for silicon and 183eV to 5.9keV for germanium).
1. Introduction Estimates of Fano factors have been made in the low-energy X-ray region for both silicon and germanium (109 eV to 5.9 keV for silicon and 183 eV to 5.9 keV for germanium). Such measurements are scarce in silicon [l-3] and values for such low energies in germanium have not been reported previously. In reporting Fano factors (F) from spot measurements using the full volume of one, or a few, detectors, it has previously been pointed out [4,5] that there is a danger that systematic errors may be introduced due to variations inherent in the detector, such as variations in the field and material or material processing. What is actually reported is often an “effective Fano factor” (F’). For example Palms et al. [4], Bilger [6] and others have made measurements for a range of detector biases and attempted to extrapolate to infinite field to elimi0168-9002/97/$17.00 c 1997 Elsevier Science B.V. All rights reserved PII SO168-9002(97)00965-O
nate incomplete charge collection (ICC) effects. One problem is that it is not clear where the limiting value of F’ (F) is reached (see e.g. Ref. [6]). Sher et al. [S] attempted to overcome the problem of weak fields by collimating the radiation to a fine beam so that the best regions of the detector were selected. ICC is likely to influence F’ for very low-energy X-rays where the attenuation length is very small (of the order of 100 nm for C K, X-rays in silicon and germanium) and carrier diffusion takes place against the field to the front contact [7]. For such low-energy X-rays fine-beam collimation is not appropriate. Owens et al. [2] made measurements on CCD devices and did not consider ICC to be significant in these. The approach taken here is to look at a large number of detectors manufactured over a period of many years with their variations in characteristics as well as the many manufacturing improvements made to reduce ICC. In fact, the ICC is now so
B.G. LowelNucl.
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small that the physical interactions within the detector are becoming more transparent [S]. Improvements have also been reported by other manufacturers [9]. The hope is that definite trends can be discerned and measurements in the Fano limit can be made for the ionising media at these energies.
2. Procedure The detectors studied had either lithium-drifted silicon (Si(Li)) or high-purity germanium (HPGe) crystal transducers. Rather than studying a few detectors in great detail, the experimental data has been obtained from a large number (> 500 Si(Li) and > 170 HPGe) of detectors collected as part of the technical specification and statistical process control in detector production at Oxford Instruments. A similar approach has been taken to set up a model of mercuric iodide X-ray detectors by studying the characteristics of a large number of commercially manufactured devices [lo]. All the detectors selected for the present study were 10 mm2 active area and 3 mm deep, for the sake of consistency. All were collimated in the same way and were biased at - 500 V. The temperature of the silicon was 120 f 10 K and of the germanium was 95 + 1OK. All the detectors reported on here did not necessarily pass the test specifications at the time of testing. Bench tests, using a radio-active source (Fe-5S), were made on detectors on the basis of the IEEE standards for resolution and peak-to-background on the Mn K, peak [l 1) as well as in-house specifications on ICC and stability. The calibration was determined by an internal strobe oscillator [12] which gives true zero and the Mn K, peak at 5895.1 eV. The zero strobe is crucial for low-energy measurements in ensuring stability with no off-set. It also allows an accurate measure of the true system noise peak full-width half-maximum (FWHM) to be made during the data runs. All detectors were monitored over many runs to check for instabilities in charge collection, as indicated by increasing low-energy tailing on the Mn K, peak or increase in the peak-to-background. In the case of completely windowless detectors, a special interface
355
with a Be window was attached for these bench tests. The resolution and noise were calculated for each run as well as the average dispersion for each series of runs. These terms are explained and discussed in more detail below. The standard deviations on the FWHM for the peaks in each individual run was typically < 1% and the total cumulative counts in the Mn K, peak on which the data reported is based were usually of the order of 9 x lo6 for Si(Li) detectors and 3 x lo7 for HPGe detectors. Additional tests on an SEM were carried out to check low-energy performance and overall system performance. Besides Mn, the elements fluoresced included F, C, B and Be in the case of Si(Li) detectors and Al, F, C and B in the case of HPGe detectors. The choice of Al in the case of HPGe detectors is dictated by the fact that the Al K, energy (1486.6 eV) is just above the L-absorption edges of germanium (1217-1420 eV) and, therefore, the peak is susceptible to the effects of ICC. The Be K, X-ray (108.8 eV) is not normally detectable with a HPGe detector because of the attenuation in the cryogenic IR screen necessary to limit the crystal leakage current [13]. The peaks were accumulated at 10 eV per channel over 100 s live time periods and the net counts in the peaks were typically in the range 50000-150000 for Al, F, C, and B but usually only a few thousands in the case of Be. All measurements were made on the same pulse processing unit and pulse height analyser. The disadvantage of this approach is that the statistics on the SEM data are not ideal for the individual runs and there may be small calibration and statistical software errors added to random ADC effects. However, neither of these should show any systematic bias and the errors should show up over the many runs as a scatter about valid means. The 0 of the scatter on peak median for the SEM runs varied with energy from 1.3 eV on the B K, peak to 3.6 eV on the Al K, peak.
3. Detector crystal dispersion The resolution R (FWHM) and the noise strobe N (FWHM) were measured by fitting Gaussian peaks to the respective peaks for each run. The
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centroids of the fitted Gaussians were also recorded. The statistical broadening due to the crystal alone was then expressed as the dispersion D, where D= =R= -N=.
(1)
By recording the noise FWHM and the photopeak positions, the dispersion D for each photo-peak of these elements could be measured, as well as the effective linearity of the crystal transducer. The intrinsic linearity of the system used for these measurements was checked using a pulse generator and precision attenuators. The deviation in linearity was -6.7 eV maximum in the energy range studied and the appropriate corrections were made on all peaks. The count rates were restricted during the data collection reported here to avoid any possible high-rate effects.
4. The Fano factor The Fano factor cannot be measured directly from the dispersion D as given by Eq. (1) as it is coupled to E (the average energy to produce one electron-hole pair of charge carriers in the material) by the equation D= =2.35=. &FE,,
(2)
where E0 is the energy of the incident X-ray. In this analysis I have, like most other workers [14], assumed the values of E for photons in silicon and germanium at 90 K provided by Pehl [ 151 although those measurements were not strictly made using X-rays. These measurements were 3.81 and 2.96 eV, respectively, with an error of _+ 0.02 eV on each. It is assumed in the first instance that there is no non-linearity due to a variation in E with energy. A recent absolute measurement of I-: in silicon at room temperature [16] gave a constant value in the energy range 50-1500 eV of 3.64 f 0.03 eV. Using the temperature dependence reported by Pehl [ 151 this corresponds to a value of 3.75 at 120 K. If this value is adopted then all the F values quoted here for silicon should be increased by 1.6%. Another recent measurement [2] shows a value of E increasing towards low energies by about 2.7% in the
range 1000-300 eV. Evidence of variations of F with energy are discussed in more detail below. In the case of the Mn K, peak, corrections were made to take account the unresolved 11 eV separation of the K,i and Km2 lines. In the case of Si(Li) detectors, this correction amounts to a 0.7% downward adjustment to F. Due to the better resolution of the HPGe detectors the adjustment for them is about twice this amount. No correction has been made for the expected non-Gaussian shape of any of the peaks [Z].
5. Fano factors for silicon It has long been appreciated [ 17,18,9] that one of the effects of ICC for low-energy X-rays manifests itself as an effective non-linearity in the photo peak position. As this is due to the loss of charge carriers it is naturally accompanied by a corresponding increase in the dispersion D and the effective Fano factor F’. Such a correlation of peak shift with broadening of the C K, peak was reported recently in silicon [19]. Thus, the negative displacement of the C K, peak position (AE) from the accepted energy of the X-ray can be used as a measure of ICC. Fig. 1 shows F’ as measured from the C K, peak as a function of AE (increasing ICC) for the C K, peak. All the Si(Li) detectors have been sorted by 1 eV intervals of AE and each data point represents the mean value of F’ for each set of detectors and its standard deviation. This shows two regimes. The regime for AE from -5 eV to about 2 eV shows no correlation (correlation coefficient =O.Ol) and the regime for AE >2 eV shows a linear correlation (correlation coefficient =0.6). The first regime can be interpreted as the limiting value F with no ICC. Indeed, Fig. 1 is reminiscent of inverse field plots [6]. The flat regime width is consistent with the statistical scatter in peak positions as mentioned above and gives rise to negative values of AE. The extreme values of AE in this and ensuing plots represent one or two detectors where the statistics on the peak position is poor. The limiting value of F from Fig. 1 is 0.157 f0.008. Fig. 2 shows the variation of F’ as measured from the B K, peak as a function of AE for the
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0.3
f
t
3
0.2
f
f
f
0.1
I 10
5
0
-5
15
A E (eV) Fig. I. The effective Fano
factor
for silicon for the carbon
K, peak plotted
as a function
of peak shift AE.
1 --
0.8 --
I
0.6 --
I
,5
0
5
I
I
IO
15
20
25
30
A E (eV) Fig. 2. The effective Fano factor
for silicon for the boron
K, peak plotted
as a function
of peak shift AE.
35
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B K, peak (the true position taken as being 183.4 eV). This again shows two regimes and a value of F consistent with the above value at 277.4 eV within the experimental errors. F’ as measured from the Mn K, peak (5985.1 eV) showed no correlation with AE for the C K, peak. F’ measured from the fluorine K, peak showed very little correlation and there was no peak shift from the expected value (676.8 eV). It was therefore assumed that ICC had negligible influence on F for these energies and the values for Mn and fluorine K, peaks were found to be 0.1161 + 0.0001 and 0.134 f 0.003, respectively.
6. The problems of lower energies In interpreting data from the Be K, peaks (108.8 eV) there are a number of problems which greatly reduce the confidence level in the estimates. Firstly, we have a fall-off to low energies in detec-
tion efficiency and corresponding counts in the peak due to absorption of the X-rays in the window and contact materials used. The detectors studied here had all been recently “conditioned” [20], so that, ice or other condensates were not expected to be a problem. Secondly, there is a more rapid fall-off in the electronic efficiency due to the noise in the pulse processor recognition channel [21]. Thirdly, the beryllium specimen used in the SEM is prone to oxidation and therefore self-absorption and requires frequent cleaning. All these effects reduce the counts in the photopeak as mentioned earlier. Whereas all the other peaks studied are on a low flat background (see e.g. spectra in Ref. [20]), this is not always the case for the Be K, peak, which may sometimes lie partly in the noise tail. This effect will broaden the peak and displace it to lower energies. A more serious distortion of the Be K, peak is caused by the fall-off in electronic detection efficiency mentioned above. Fig. 3 illustrates this effect which opposes the effects of noise and ICC,
Undistorted Peak
80 --
Electronic Efficiency
s 2 5 ._ .o z .s
60 --
40--
g
8t
w
Distorted Peak 20 --
-I
20
40
60
80
100
120
140
Energy (eV)
Fig. 3. Distortion
of the beryllium
K, peak due to electronic
efficiency.
160
180
200
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reducing the peak FWHM and shifting it to higher energies. The broader the peak the greater is this effect. In sorting detectors for magnitude of ICC (Figs 4-6), the value of AE (in 5 eV intervals) for the C K, peak was chosen as this is the highest energy peak exhibiting appreciable ICC and provides the best statistics of all the very low-energy peaks, Fig. 4 shows AE/Eo as a % shift for the peaks studied as a function of E. without any correction made to the Be K, peak. The lines are drawn to guide the eye and illustrate the effects of increasing ICC. The Be K, peak is just above the Si L,, and Si Liir absorption edges (98899 eV) and ICC is, therefore, expected to have the greatest influence here and result in the greatest % energy shift. Clearly, the uncorrected data of Fig. 4 is anomalous in this respect. If no correction is made, a value of F’ =0.19 f 0.04 at 108.8 eV is obtained which must represent a lower limit.
7. Correction for distortion of Be peak The electronic efficiency was measured using the precision pulse generator and an attempt was made to correct the measured peak positions and widths. The correction has the effect of approximately doubling F for the Be K, peak but the value has a large uncertainty. Fig. 5 is the corrected version of Fig. 4 and the effective Fano factors as a function of energy are shown in Fig. 6. The values of F for silicon are summarised in Table 1.
8. Fano factors for germanium
The Fano factor for germanium based on the Mn K, peak measured from the bench runs was about 9% lower than the corresponding value for silicon. Fig. 7 shows the effective Fano factor taken from
20
increasing ICC t
400
300 Energy (eV) Fig. 4. Peak shifts for silicon expressed
as a percentage
of the expected
position
and plotted
as a function
of energy.
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35 -Increasing
ICC
30 --
25 -s f %
20 --
% s 15 --
0
100
200
300
400
500
600
of the expected
position
700
Energy (eV) Fig. 5. Peak a function
shift for silicon
corrected
for electronic
efficiency
expressed
as a percentage
and plotted
as
of energy
the C K, peaks for HPGe detectors plotted as a function of AE for the C K, peak. It is clear from Fig. 7 that there is only one domain and no correlation with AE (correlation factor = -0.06). Fig. 8 shows that even for the B K, peak there is no correlation. This lack of correlation is repeated for all the peaks analysed and so the Fano factors can be deduced using all of the data. The physical reason for this is not clear but may be connected with the shorter range of the photo- and Augerelectrons in germanium compared to silicon. The Fano factors for germanium are summarised in Table 2. There was no dependency on the source of the HPGe material detected in this study.
9. Discussion Fraser et al. [14] also summarised the experimental measurements of E and F for silicon. For
Si(Li) crystals cooled by liquid nitrogen the authors cited use a value of E quoted from previous measurements, usually also from Pehl [15]. The values of F range 0.072-0.132 for the energies below 10 keV illustrating the difficulties in making such measurements. A recent measurement [22] also assuming e =3.81 eV and using monochromatic synchrotron X-rays gives F =0.114 for the range 700 eV-7 keV. Although the Fano factors for silicon agree quite well in the region 200-300 eV with the calculations of Fraser et al. (Table l), the values at higher energies do not. The calculation of F by Alig [23] of 0.113 f 005 for silicon agrees better with the value at 5895 eV found here. Although F is found to increase at lower energies, it does so only by 35% around 300eV and not by the factor of about 3 recently reported for a silicon diode [3]. Table 3 summarises the results for silicon in terms of the product EF. The experimental values
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361
2
T
1.8 --
A
1.6 --
Increasing
ICC
0.8 --
0.6
100
0
400
300
200
500
600
700
Energy (eV) Fig. 6. Effective Fano
factors
for silicon corrected
for electronic
Table 1 Data for Si(Li) detectors Peak
Mn F C B Be
Energy
Fano factor
@V)
(F)
5895.1 676.8 217.4 183.4 108.8
0.1161 0.134 0.157 0.15 0.4
f f k + *
0.0001 0.003 0.008 0.01 0.2
Fraser et al.
Alig
[14] theory 0.16 0.15 0.14 0.15 0.16
[23] theory 0.113 f 0.005
agree reasonably well, including Owens et al. [2] of 0.54 eV near the fluorine K, line and an “asymptotic” value of about 0.48 eV at higher energies. These values were measured at 170 K and would correct to about 0.52 and 0.46 eV, respectively, at 120 K by their calculations. However, their theoretical value of 0.58 eV [ 141 at 5895 eV is still too high.
efficiency
and plotted
as a function
of energy.
As stated previously, no experimental values of E; have been published to date in the energy ranges discussed here for HPGe. However, Knoll [24] summarises values from Ge(Li) detectors for higher energies as between 0.057 and 0.129 again illustrating the difficulty in making such measurements. Pehl et al. [25] made measurements with Cu K, X-rays (8041 eV) in several small Ge(Li) detectors obtaining a value of 0.083. Alig [23] calculated a value of 0.126 f 0.005, higher than that for silicon, contrary to the present finding for Mn K, X-rays. Fig. 9 shows F as a function of energy (logarithmic scale) for both silicon and germanium. The value for the Be K, peak has not been included due to the high uncertainty. It is remarkable how similar the Fano factors of germanium are to those of silicon at the same energies, despite the considerable differences in dispersion measured. There is a similar upward trend in F as lower energies are approached. Fraser et al. [2,14] predict and
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nc
FIl.51
AE (eV) Fig. 7. The Fano factor for germanium
for the carbon
K, peak plotted
as a function
of peak shift AE.
nc
0.5 F
1
0.3 --
0.2 --
I
”
”
f 0
i
f
f !
0.1 --
c -5
n 10
5
0
15
AE(eV)
Fig. 8. The Fano factor
for germanium
for the boron
K, peak plotted
as a function
of peak shift AE.
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measure a general increase in F for silicon below about 2000 eV with a transitional maximum over the Si K absorption edge at 1838 eV. The Si(Li) detector peak positions without ICC effects are estimated to be 278.0 + 0.5 eV for C K, and 184.0 + 0.5 eV for B K,. These are consistent with the accepted values and therefore do not support any change in E between 183.4 and 5895 eV as predicted and measured by Fraser et al. [2,14] and supported by Lechner et al. [3]. As pointed out by
Table 2 Data for HPGe detectors Peak
Energy (eV)
&.F(eV)
Fano factor
Mn Al F C B
5895.1 1486.6 676.8 277.4 183.4
0.317 0.35 0.387 0.467 0.48
(corrected) 0.1057 + 0.0002 0.121 f 0.001 0.133 _+0.003 0.155 + 0.005 0.15 + 0.01
Table 3 Experimental data for Si(Li) detectors Peak
Energy (eV)
E.F (eV)
Owens et al. Ref. [2]
Mn F
5895.1 676.8
0.4423 k 0.0004 0.51 * 0.01 0.60 + 0.03 0.58 + 0.04 1.5 _+0.8
“0.48 0.54
277.4 183.4 108.8
C B Be
McCarthy
Lepy
Musket
Ref. [9]
Ref. [22]
Ref. [18]
0.423
0.434 0.4 * 0.2
0.75 0.55
a Asympotic value
0.2 T 0.19 -F 0.18 -0.17 -0.16 --
.
m I 1000 Energy (eV)
Fig. 9. Fano factors of silicon and germanium plotted as a function of energy.
10000
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Table 4 Peak positions Peak
Mn Al F C B
for HPGe
Energy
(eV)
Instr. and Meth. in Phys. Res. A 399 (1997) 354-364
detectors Position
i: (Ge)
(eV)
(eV)
(eV)
5895.1 1486.6 676.8 217.4 183.4
(calibration) 1488.4 + 0.6 681.2 + 0.5 280.3 k 0.5 187.9 + 0.5
2.96 2.96 2.94 2.93 2.89
(assumed) f 0.001 f 0.002 f 0.005 + 0.008
Scholze [16] this may be due to the effects of ICC not being taken fully into account. The respective peak position values for germanium are summarised in Table 4. There is a small positive peak shift, increasing with decreasing energy. This is opposite to the expected effect of ICC and hence might be interpreted as a decrease in E from the value at 5895 eV. The consequence of this interpretation on E is also shown in Table 4 and the corrections to F for this effect in Table 2.
Acknowledgements
I would like to express my gratitude to the detector production team at Oxford Instruments Microanalysis Group in making the measurements and recording the data so diligently.
References 111 R.G. Musket, Nucl. Instr. and Meth. 117 (1974) 385. 121A. Owens, G.W. Fraser, A.F. Abbey, A. Holland, K. McCarthy, A. Keay, A. Wells, Paper B (Measurements), Nucl. lnstr. and Meth. A 382 (1996) 503. H. Soltau, L. Struder, Nucl. c31 P. Lechner, R. Hartmann, lnstr. and Meth. A 377 (1996) 206. Rao. R.E. Wood, Nucl. Instr. r41 J.M. Palms, P. Venugopala and Meth. 76 (1969) 59.
c51 A.H. Sher, W.J. Keery, IEEE Trans. Nucl. Sci. NS-17 (1970) 39. C61H.R. Bilger, Phys. Rev. 163 (1967) 238. c71 J. Llacer, E.E. Hailer, R.C. Cordi, IEEE Trans. Nucl. Sci. NS-24(1977) 53. PI J.L. Campbell, J.A. Maxwell, T. Papp, G. White, 1996 EDXRS 96 Conf. Lisbon June 1996, X-Ray Spectrometry 26 (1997) 223. in: Williams, Goldstein, Newbury (Eds.), c91 J.J. McCarthy, X-Ray Spectrometry in Electron Beam Instruments, Plenum Press, New York, 1995 p. 67. J. Henderson, Proc. of the 9th Int. Cl01 X.J. Bao, M. Natarajan. Workshop on Room Temperature Semiconductor X-Ray and Gamma-Ray Detectors, Associated Electronics and Applications, Grenoble, September 1995. Standard Test Procedures for Semi1111IEEE Std.759-1984 conductor X-Ray Energy Spectrometers. Cl21 K. Kandiah, IEEE Trans. Nucl. Sci. NS-22 (1975) 2058. Cl31 C.E. Cox, B.G. Lowe, R.A. Sareen, IEEE Trans. Nucl. Sci. 35 (1988) 28. A. Cl41 G.W. Fraser, A.F. Abbey, A. Holland, K. McCarthy, Owens, A. Wells, Paper A (Theory), Nucl. Instr. and Meth. A 350 (1994) 368. Gives a summary of measurements of F with references. D.A. Landis, M. Lenzlinger, Cl51 R.H. Pehl, F.S. Goulding, Nucl. Instr. and Meth. 59 (1968) 45. [I61 F. Scholze, H. Rabus. G. Ulm, 1996, submitted to Appl. Phys. Lett. also SPIE 1996 (Denver) paper 2808-50. 1171J. Jacklevic, IEEE Trans. Nucl. Sci. NS-18 (1971) 187. X-Ray Spectrometry, L181 R.G. Musket, Energy Dispersive National Bureau of Standards Special Publication, Vol. 604, 1981, 97. D. Hauff, P. Lechner, R. Richter, L. Struder, Cl91 R. Hartmann, J. Kemmer, S. Krisch, F. Scholze, G. Ulm, Nucl. Instr. and Meth. A 377 (1996) 191. Goldstein, Newbury (Eds.). cw B.G. Lowe. in: Williams, X-Ray Spectrometry in Electron Beam Instruments, Plenum Press, New York, 1995. p. 7. Energy Dispersive X-Ray Spectrometry, c-211P.J. Statham, National Bureau of Standards Special Publication, Vol. 604. 1981. 127. P. Stemmler, G. Ban, L. Beck, L-w M.-C. Lepy, J. Plagnard, P. Dhez, EDXRS 96 Conf. Lisbon, June 1996, X-Ray Spectrometry 26 (1997) 195. ~231 R.C. Alig, Phys. Rev. B 27 (1983) 968. Radiation Detection and Measurement. ~241 G.F. Knoll, Wiley, New York. 1989. 1251 R.H. Pehl, F.S. Goulding, Nucl. Instr. and Meth. 81 (1970) 329.