Download Statistical modeling of pulse height spectrum of gamma

APPLIED PHYSICS LETTERS
VOLUME 75, NUMBER 2
12 JULY 1999
Statistical modeling of pulse height spectrum of gamma-ray
spectrometers limited by incomplete charge collection
Y. Nemirovskya)
Kidron Microelectronics Research Center, Department of Electrical Engineering, Technion—Israel Institute
of Technology, Haifa 32000, Israel
~Received 11 February 1999; acceptd for publication 17 May 1999!
This letter presents an analytic approach to the calculation of the pulse height spectrum, using a
statistical model, which simultaneously considers a random point of photon absorption ~i.e.,
nonuniform absorption! and a random drift time for each carrier. The pulse height spectrum of
gamma-ray spectrometers is calculated as a function of photon energy, electron and hole
mobility-lifetime products and applied electric field. For spectrometers with a uniform electric field
the pulse height spectrum is obtained by a single numerical integration of an analytical expression.
© 1999 American Institute of Physics. @S0003-6951~99!02728-X#
This letter presents an analytic approach to the calculation of the pulse height spectrum, using a statistical model,
which simultaneously considers a random point of photon
absorption ~i.e., nonuniform absorption! and a random drift
time for each carrier. The pulse height spectrum of gammaray spectrometers is calculated as a function of photon energy, electron and hole mobility-lifetime products, and applied electric field. For spectrometers with a uniform electric
field the pulse height spectrum is obtained by a single numerical integration of an analytical expression.
The charge transport of wide band gap semiconductors
for room temperature nuclear detection applications is
strongly affected by trapping of carriers, in particular holes,
at impurities and defects.1–6 Charge collection efficiency and
the variance of the collected charge of semiconductor
gamma-ray spectrometers has been recently analytically
modeled for detectors with a uniform electric field.7 This
letter presents the statistical model of the pulse height spectrum, using some of the results of Ref. 7.
The physical model under consideration is a planar semiconductor detector with two biased electrodes separated by a
distance L, with a uniform electric field E5V/L where V is
the applied bias voltage. A single gamma-ray photon, which
is incident on the negative electrode, is absorbed at a random
distance x from the negative electrode. Upon absorption, a
random number N of electron-hole pairs is generated where
the expectation of N is determined by the photon energy.
Instantly each of the N electrons and each of the holes drift
with constant velocity, which is given by m n E, m p E for
electrons and holes, respectively, where m n , m p are the respective electron and hole mobilities. Each drifting electron
and hole induces an instantaneous current into the electric
circuit connected to the terminals and the integral of the current over the carrier random drift time is defined as the induced charge.
The statistical model is defined below: The electrons and
the holes that are generated at a distance x are statistically
independent. We use the following notation: for any two
random quantities x,Y the conditional expectation of Y for a
a!
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given value of x, is denoted by E (x) $ Y % . The conditional
expectation of the charge induced by the ith electron generated at a distance x from the negative electrode, is
m n ~ x ! 5E (x) $ q n,i % 5
S
D
L2x
ql n
12e 2 l n ,
L
~1!
where l n 5 m n t n E is the electron drift length, m n t n is the
electron mobility-lifetime product and q is the charge of a
single electron. Similarly, the conditional expectation of the
charge induced by the jth hole generated at a distance x from
the negative electrode is
S
D
x
ql p
12e 2 l p ,
L
m p ~ x ! 5E (x) $ q p, j % 5
~2!
where l p 5 m p t p E is the hole drift length, m p t p is the hole
mobility-lifetime product and e is the charge of a single electron. The conditional expectation of the mean square of the
induced charge of a single electron generated at a distance x
from the negative electrode is
E (x) $ q 2n,i % 52
S DS
ql n
L
2
12e 2
L2x
ln 2
D
L2x 2 L2x
e ln .
ln
~3!
Similarly, the conditional expectation of the mean square of
the induced charge of a single hole generated at a distance x
from the negative electrode is
E (x) $ q 2p, j % 52
S DS
ql p
L
2
x
12e 2 l p 2
D
x 2 x
e lp .
lp
~4!
The variance of the collected charge of an electron is given
by
s 2n ~ x ! 5E (x) $ q 2n,i % 2 ~ E $ q n,i % ! 2 ,
~5!
where s n (x) is the standard deviation of the collected charge
of a single electron generated at a distance x. Similarly, the
variance of the collected charge of a hole is given by
s 2p ~ x ! 5E (x) $ q 2p, j % 2 ~ E $ q p, j % ! 2 ,
~6!
where s p (x) is the standard deviation of the collected charge
of a single hole generated at a distance x.
The following additional assumptions are made.8
0003-6951/99/75(2)/298/3/$15.00
298
© 1999 American Institute of Physics
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Appl. Phys. Lett., Vol. 75, No. 2, 12 July 1999
Y. Nemirovsky
299
The expectation of a random process compromised of
statistically independent random processes is the sum of the
expectations. Accordingly, the expectation of the collected
charge of each electron-hole pair is
m ~ x ! 5m n ~ x ! 1m p ~ x ! ,
~7!
and the total collected charge of N electron-hole pairs is
Nm(x). The variance of a random process compromised of
statistically independent random processes is the sum of the
variances. Accordingly, the variance of the collected charge
of each electron-hole pair is
s 2 ~ x ! 5 s 2n ~ x ! 1 s 2p ~ x ! ,
~8!
and the total variance of N electron-hole pairs is N s 2 (x).
The probability density distribution function of the normalized sum of a random variable approaches that of a Gaussian
distribution as the number of components increases ~the
‘‘central limit theorem’’!. Accordingly, the probability density distribution of N electron-hole pairs absorbed at a distance x from the negative electrode is given by
1
A2 p N s ~ x !
U
exp
U
$ @ Q2Nm ~ x !# 2 %
2N s ~ x !
2
~9!
,
where Q is the pulse height and 0,Q<qN.
In the experimentally measured pulse height spectra the
abscissa is a linear scale of the total collected charge Q while
the ordinate is the differential number of pulses per differential pulse height. Each pulse results from a photon interacting
within the detector. Therefore, it is the probability density
distribution function of all the events that occur at any distance x. The latter is obtained by averaging the Gaussian
distribution with the probability density for a photon to be
absorbed at a distance x from the negative electrode, which is
given by
p x~ x ! 5
S
a
12e 2 a L
D
e 2ax.
Accordingly, the pulse height spectrum is given by
S~ Q !5
1
Es
L
A2 p N
H
0
3exp 2
1
~x!
$ @ Q2Nm ~ x !# 2 %
2N s 2 ~ x !
J
•p x ~ x ! dx.
~10!
Thus, the spectrum can be calculated as a function of pulse
height Q where 0,Q<qN. It is worthwhile noting that
* qN
0 S(Q)dQ51 and this can be used to check the calculation of the spectrum.
Figure 1 exhibits the validity of the model. The energy
pulse height spectrum is calculated with the expression for
S(Q) and compared to a computer simulation. The simulation routine assumes that the photon is absorbed at a random
location x determined by the probability function
p x~ x ! 5
a e 2ax
12e 2 a d
,
where a is the absorption coefficient and d is the thickness of
the spectrometer. Each interacting photon generates N
FIG. 1. Calculated ~solid line! and simulated ~dots! spectra for 60 keV
photons emitted by Am251 ( a 540 cm21 ). The spectrometer thickness is 0.2
cm, the applied bias voltage is 200 V and the assumed mobility-lifetime
products of electrons and holes are, respectively m n t n 5131023 cm2 V21 ,
m p t p 5131026 cm2 V21 .
electron-hole pairs. Instantly, each of the N electrons ~holes!
drifts with constant velocity towards the positive ~negative!
electrode. The drift terminates independently and randomly
by either trapping, recombination, or by sweepout to the respective electrode. The probability density for the electron
~hole! drift time is
P~ t !5
H
t 21 e 2t/ t
e
x
2l
S
x
d t2
mt E
t,
D
x
mE
x
t5
mE
J
,
where t is the electron ~hole! lifetime, m is the electron
~hole! mobility, and x is the random absorption depth. The
simulation calculates the induced charge by each carrier and
the total random induced charge collected at the output as a
result of each absorbed photon. The random variables are
obtained using a MATLAB function ~EXPRND!. The procedure is repeated for a large number of interacting photons
and the histogram shown in Fig. 1 is thus obtained. There is
a good agreement between the calculated and simulated
pulse height spectrum. It is worthwhile to note that the histogram has been obtained with a Pentium PC operating for
several hours while the calculated spectrum has been obtained using MATLAB for the single integration within a
few minutes.
Figure 2 exhibits several energy pulse height spectra calculated with the expression for S(Q). The spectrometer is
irradiated from the negative electrode by 60 keV photons
emitted by Am241. The detector thickness is 0.2 cm. The
applied bias voltage is V b 52200 V. Figure 2 exhibits the
effect of the electron and hole mobility-lifetime products on
the low-energy tailing, the resolution and the charge collection efficiency. The resolution is defined by the full width at
half maximum ~FWHM! divided by the collected charge at
the maximum and multiplied by 100 in order to present the
data in percent. The charge collection efficiency is defined
by the average collected charge divided by the total charge
produced by each interacting photon and multiplied by 100
in order to present the data in percent.
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300
Appl. Phys. Lett., Vol. 75, No. 2, 12 July 1999
Y. Nemirovsky
FIG. 3. Calculated spectra for 122 keV photons emitted by Co57 ( a 57
cm21 ) and detected by a CdZnTe spectrometer with 0.4 cm thickness, as a
function of the applied bias voltage. The assumed mobility-lifetime products
of electrons and holes are, respectively, m n t n 5531024 cm2 V21 , m p t p
5131026 cm2 V21 . The resolution is 11.9% for V52400 V, 9.9% for
V52600 V, 6.3% for V52800 V, 2.7% for V521000 V and 1.3% for
V521200 V.
FIG. 2. Calculated spectra for 60 keV photons emitted by Am241 ( a 540
cm21 ) and detected by a CdZnTe spectrometer with 0.2 cm thickness and
operated at a bias voltage of 2200 V; ~a! exhibiting the effect of hole
mobility-lifetime product on the low-energy tail. The assumed mobilitylifetime product of electrons is m n t n 5531024 cm2 V21 while several values of the mobility-lifetime product of holes are assumed: m p t p 5531026
cm2 V21 , m p t p 5131026 cm2 V21 , m p t p 5531027 cm2 V21 . The respective resolutions are 1%, 4.3% and 6.3%; ~b! exhibiting the effect of the
electron and hole mobility-lifetime products on the charge collection efficiency and the resolution. A ratio of 1000:1 is assumed between the electron
and hole mobility-lifetime products. For m n t n 5531023 cm2 V21 , m p t p
5531026 cm2 V21 the resolution is 0.5%. For m n t n 5131023 cm2 V21 ,
m p t p 5131026 cm2 V21 the resolution is 3.1%. For m n t n 5531024
cm2 V21 , m p t p 5531027 cm2 V21 the resolution is 6.3%.
In Fig. 2~a! the electron mobility-lifetime product is the
same for all the presented spectra ~the assumed value is 5
31024 cm2 V21 ) and the hole mobility-lifetime product
varies. The tailing increases and the resolution degrades as
the value of the hole mobility-lifetime product is reduced.
In Fig. 2~b! the ratio between the electron and hole
mobility-lifetime products is held constant ~the assumed factor is 1000! but the electron mobility-lifetime product varies.
The spectra exhibit the significant effect of the electron
mobility-lifetime product on the charge collection efficiency
as well as the resolution.
Figure 3 exhibits several energy pulse height spectra calculated with the expression for S(Q), assuming conditions
corresponding to an actual experiment, where a spectrometer
with a given transport property is measured at several bias
voltages. The spectrometer is irradiated from the negative
electrode by 122 keV photons emitted by Co57. The detector
thickness is 0.4 cm. Figure 3 exhibits the effect of the applied bias voltage on the low-energy tailing, the resolution
and the charge collection efficiency. As the applied bias volt-
age increases the electron and hole drift lengths increase, the
resolution improves and the charge collection efficiency increases.
In summary, the pulse height spectrum of gamma-ray
spectrometers is calculated as a function of photon energy,
electron and hole mobility-lifetime products, and applied
electric field. The pulse height spectrum is obtained by a
single numerical integration of an analytical expression. The
model considers the effect of the variance of the collected
charge and ignores the effect of additional noise sources such
as detector dark noise9,10 and the noise contributed by the
readout electronics.11 The model assumes that the shaping
time is long compared to the charge collection time and
hence does not consider the effect of the shaping time upon
pulse height spectra, which may be significant if the shaping
time is comparable with the rise time of the pulse. The model
presented here and the calculated spectra are very useful to
gain a physical insight of the parameters that limit the performance of state-of-the art room temperature gamma-ray
spectrometers based on wide band gap semiconductors.
The research is partially supported by the international
atomic energy agency, Vienna. The support of Dr. A. Aret is
gratefully acknowledged.
1
Semiconductors and Semimetals, edited by T. E. Schlesinger and R. B.
James ~Academic, New York, 1995!.
2
K. B. Parnham, Nucl. Instrum. Methods Phys. Res. A 377, 487 ~1996!.
3
J. F. Butler, Nucl. Instrum. Methods Phys. Res. A 396, 427 ~1997!.
4
H. H. Barret, J. D. Eskin, H. B. Barber, Phys. Rev. Lett. 75, 156 ~1995!.
5
P. N. Luke, IEEE Trans. Nucl. Sci. 42, 207 ~1995!.
6
P. N. Luke, Appl. Phys. Lett. 65, 2884 ~1994!.
7
Y. Nemirovsky, J. Appl. Phys. 85, 8 ~1999!.
8
W. B. Davenport, Probability and Random Processes ~McGraw-Hill, London, 1970!.
9
Y. Nemirovsky, G. Asa, A. Ruzin, J. Gorelik, and R. Sudharsanan, J.
Electron. Mater. 27, 807 ~1998!.
10
Y. Nemirovsky, G. Asa, C. G. Jakobson, A. Ruzin, and J. Gorelik, J.
Electron. Mater. 27, 800 ~1998!.
11
Semiconductors and Semimetals, edited by J. S. Iwanczyk and B. E. Patt
~Academic, New York, 1995!, Vol. 43.
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