Leroy C., Rancoita P.-G. Principles Of Radiation Interaction In Matter And Detection

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470 Principles of Radiation Interaction in Matter and Detection

beta particles is shown in Fig. 6.10. The silicon detector under study is located

inside a test box with a

106

Ru (β) source ﬁtted in its cover. The source, above

the geometrical center of the detector, is collimated to a 5 mm diameter electron

beam. The temperature inside the test box is controlled by a water cooling system

allowing the selection of temperatures from 6 to 25

◦

C. Nitrogen is ﬂowing through

the box to prevent condensation on the detectors at low temperatures. The current

pulse induced by electrons in the detector is detected with a charge pre-ampliﬁer

or a current pre-ampliﬁer (according to the goal pursued). The ampliﬁcation and

shaping were ensured by a Ortec 450 research ampliﬁer for the data reported in

Figs. 6.4 and 6.5. The shaping integration and diﬀerentiation were 100 ns, with a

gain of 500. The p olarity of the input signals was negative. The output signals

were in the −3 volts range. The pulse energy was measured with a peak sensing

Lecroy ADC 2259A for a 500 ns gate triggered by the coincidence signal of two

photomultipliers detecting photons produced in a scintillator of 1 cm

area put

behind an iron absorber (≈ 0.5 mm thick) allowing to select minimum ionizing

electrons with an energy ≥ 2 MeV.

100

200

300

400

500

600

700

260 280 300 320

Temperature (K)

Holes mobility (cm

/Vs)

250

500

750

1000

1250

1500

1750

2000

2250

260 280 300 320

Temperature (K)

Electrons mobility (cm

/Vs)

Fig. 6.9 Hole (left) and electron (right) mobility (in cm

−1

) as a function of temperature

(in K), Eq. (6.70).

The setup for measuring the charge induced by the charge carriers generated

241

Am α-particles is shown in Fig. 6.11. The charge induced by the collimated

α-particles incident on the front side and on the rear side of the detector (the source

can be moved on the other side of the detector) can be measured in a way similar to

the case of β-particles. However, the setup is somewhat simpliﬁed since the charge

collection is performed in a self-trigger mode.

A summary of the characteristics of the silicon detectors used for the example is

given in Table 6.1. The current pulses induced by particles penetrating the silicon

diode are detected by a fast current ampliﬁer with an input impedance R

= 50 Ω,

providing a gain of G = 1000. The pulses are recorded by a LeCroy digital oscillo-

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Solid State Detectors 471

1 M100 k Ω

- Sourceβ

Discriminator Discriminator

Coincidence

Ω

Back

-Vb

Collimator

Computer

Absorber

PM PM

Scope

Silicon detector

Front

Fig. 6.10 Experimental setup for β-particle charge collection measurements [Roy (2000)].

scope (Fig. 6.12) used in averaging mode, to improve the signal-to-noise ratio [Roy

(1994); Leroy et al. (1997b)]. It is necessary to know the concentration of the elec-

trons and holes, as well as the electric ﬁeld at every space-time coordinate. These

quantities can be extracted from the system of partial diﬀerential equations intro-

duced above.

In the absence of an analytical solution to the system of the ﬁve partial diﬀe-

rential equations, considered above, the equations are discretized using Gummel’s

decoupling scheme [Gummel (1964)] to obtain a numerical solution [Leroy, Roy,

Casse, Glaser, Grigoriev and Lemeilleur (1999a); Roy (2000)]. The quantities of

interest are extracted by using the code MINUIT [ASG (1992)] to minimize the χ

obtained from ﬁtting the numerical solutions of Eq. (6.72) to the experimental data

obtained from the measurement of the current pulse response induced by α- and

β-particles in the silicon detectors.

Figure 6.13 allows the transport of the charge carriers to be visualized by showing

the location of the charge carriers as a function of the collection time, as well as the

corresponding signals for incident α- and β-particles. For the α-particles entering

the rear side (n

), the holes (h) drifting in the detector give the main contribution

to the induced current. For the α-particles entering the front side (junction side),

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-bo ok975x65˙n˙2nd˙Ed

472 Principles of Radiation Interaction in Matter and Detection

1 M100 k Ω

- Sourceα

Ω

Silicon detector

-Vb

Scope

ADC

Collimator

Back

Front

Fig. 6.11 Experimental setup for α-particle charge collection measurements [Roy (2000)].

the electrons (e) are the main contributors to the induced current. As hole mobility

is smaller than electron mobility, the pulse from the rear side is longer. For the

β-particles, both electrons and holes contribute signiﬁcantly to the current with a

shorter signal for electrons due to their higher mobility.

The model also gives the values of the electrons and holes mobilities. The results

are reported in Table 6.2. The average mobilities achieved for electrons and holes

for the detectors described in Table 6.1 are: µ

= 491 ± 10 cm

−1

and µ

1267 ± 20 cm

−1

, respectively.

As above discussed, the relevant electrical characteristics of a p

− n diode

are: ﬁrst, the eﬀective concentration of dopants which deﬁnes the internal elec-

tric ﬁeld and thus determines the depletion voltage; second, the electron and hole

mobilities which inﬂuence the time needed to collect the charges; and third, the

charge trapping lifetimes which aﬀect the eﬃciency of the charge collection. Also

Table 6.2 Electron and hole mobilities of the detectors, listed

in Table 6.1, are extracted from the model ﬁtted to β and α

data [Roy (2000)].

Detector Current µ

pulse source (cm

−1

) (cm

−1

)

M4 α 503.8 ± 2.2 1278 ± 15

M18 α, β 474.4 ± 2.4 1236 ± 15

M25 α 476.0 ± 2 1308 ± 28

M35 α 472.1 ± 3 1272 ± 5

M49 β 546 ± 11 1266 ± 24

M50 β 529 ± 13 1272 ± 20

M53 β 478 ± 12 1350 ± 20

P88 α 459.1 ± 4 1222 ± 20

P189 α 480 ± 20 1340 ± 27

P304 α 495 ± 3 1124 ± 22

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Solid State Detectors 473

Fig. 6.12 The information given by a digital oscilloscope are i) A, representing the average signal

produced by the passage of a particle through the detector, as a function of time and ii) C, obtained

from the integration of A over time, is a measurement of the collected charge (for instance 170 pVs

corresponds to 3.5 fC for a 50 Ω resistance).

after irradiation, these electrical characteristics as a function of the particle ﬂu-

ence (Φ) can be extracted by using the transport model of carriers. The mobility

for electrons and holes as a function of ﬂuence can be obtained from the ﬁt of

the current pulse response to β-particles incident on detectors exposed to suc-

cessive levels of ﬂuence. The mobility, after an initial decrease, tends for Φ >

5 ×10

particles/cm

, typically, to a saturation value of µ

sat,e

≈ 1080 cm

−1

and µ

sat,h

≈ 450 cm

−1

for electrons and holes, respectively. The resulting

behavior of the mobility as a function of ﬂuence can be represented by [Leroy and

Roy (1998)]:

(Φ) = µ

sat,e

+ 272 × exp(−5.17 × 10

−14

× Φ) (6.73)

and

(Φ) = µ

sat,h

+ 33.5 × exp(−2.09 × 10

−14

× Φ). (6.74)

The eﬀective doping concentration (N

eﬀ

) is changing with ﬂuence and can be rep-

resented as a function of the irradiation ﬂuence, Φ, via

eﬀ

(Φ)| = | − N

exp(−c Φ) + b Φ|, (6.75)

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474 Principles of Radiation Interaction in Matter and Detection

-12

-10

-8

-6

-4

-2

0 2 4 6 8 10121416

Time (ns)

Signal (10E-1 Volt)

-6

-5

-4

-3

-2

-1

0 5 10 15 20 25

Time (ns)

Signal (10E-1 Volt)

-25

-20

-15

-10

-5

0 2.5 5 7.5 10 12.5 15 17.5 20

Time (ns)

Signal (10E-3 Volt)

-25

-20

-15

-10

-5

0 5 10 15 20 25 30 35

Time (ns)

Signal (10E-3 Volt)

a) b)

c) d)

Fig. 6.13 Fits (solid line) of the charge transport model to the current pulse response at Φ = 0

for an α-particle incident on the front side (a), on the rear side (b) of the M25 detector and for a

relativistic electron for the M18 (c) and M50 (d) detectors; a bias voltage V

= 160 V is applied

in all cases. The individual electron (e) and hole (h) contributions are shown [Leroy, Roy, Casse,

Glaser, Grigoriev and Lemeilleur (1999a); Roy (2000)].

where N

, N

are the concentration of acceptors and donors at Φ = 0, respectively;

c is the donor removal constant and b the acceptor creation rate.

6.1.6 Leakage or Reverse Current

For an ideal unpolarized p−n junction, diﬀusion of majority carrier cancels the drift

of minority carriers and no net current occurs through the junction. The situation

changes when the junction gets polarized (reversed mode): a leakage or reversed

current (I

) is observed. The leakage current can be categorized into two types:

the bulk and the surface leakage currents. The bulk leakage current has several

sources. The electron-hole pairs current (i

) produced by thermal generation in the

depleted region is one of the sources. These pairs are produced by recombination

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Solid State Detectors 475

0 20 40 60 80 100 120

Voltage (volts)

Leakage current (nA)

Fig. 6.14 The leakage current as a function of the applied voltage. The curve shows a

√

V de-

pendence [Roy (2000)].

and trapping centers present in the depleted region. This current is proportional to

the active volume of the detector, AX [A is the detector area and X the depletion

depth as given by Eq. (6.22)], and to the intrinsic carrier concentration (n

int

√

np), and inversely proportional to the mean carrier lifetime, τ, which can vary

from ∼ 100 ns to a few ms dep ending on the quality or condition of the silicon

detector. Then, this current is expressed as :

= qAX

int

2τ

. (6.76)

The dependence on X means that this current is depending on the square root

of the applied bias voltage (e.g., see Fig. 6.14). Another source of I

is the current

resulting from the motion of electrons from p-region into n-side across the junction

and holes moving from n-region into p-side across the junction. In fact, the bulk

leakage current is a few nA cm

−2

, for good quality detectors. This current is in-

creasing up to several tens of µA and even more for irradiated detectors. The bulk

leakage current depends on the temp erature (Fig. 6.15) as:

= T

exp

−

, (6.77)

where E

is deﬁned in Eq. (6.5). This strong dependence on the operating tempera-

ture makes necessary to use a renormalization of the measured current. If T = 20

◦

≈ 293 K is taken as the normalization temperature of reference and T

the tempe-

rature at which the leakage current measurement has been performed, the necessary

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476 Principles of Radiation Interaction in Matter and Detection

2.5

7.5

12.5

17.5

0 5 10 15 20 25 30 35

Temperature (C)

Leakage current (nA)

Fig. 6.15 The leakage current as a function of temperature (in

◦

C). The curve represents a ﬁt of

Eq. (6.77) to the data points.

temperature correction to the leakage current is done using the equation:

(293) = I(T

)

293

exp

−E

Á·

293

−

¶¸¾

. (6.78)

The surface leakage current is resulting from several sources, such as manufac-

turing processes and mishandling of detectors (scratches, craters of saliva,. . . ). The

surface leakage current does not have dependence on the square root of the applied

bias voltage. However, deviations from the expected behavior are extremely diﬃcult

to apprehend in practice.

6.1.7 Noise Characterization of Silicon Detectors

The noise is a relevant parameter for the detector performance study. The noise

is obtained as the root mean square (rms) of the pedestal distribution ﬁtted to a

Gaussian function, although this is not always the case since it happens that some

noise sources can give non-Gaussian contribution. The pedestal can be measured

with a setup conﬁguration where the detector signal is delayed out of the trigger

gate. The noise is dominated by the preampliﬁer noise, controlled by the capaci-

tance at the input of the instrument. Below depletion, the observed noise (EN C

)

versus voltage is proportional to 1/

√

V , i.e., it is directly proportional to the diode

capacitance. The detection system noise dependence on capacitance can b e mea-

sured by replacing the silicon detectors by capacitances of known value, up to an

external capacitance of 180 pF, which covers adequately the capacitance range of the

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Solid State Detectors 477

detectors normally used (thickness ≤ 300 µm). The noise performance of the charge

collection system used in a previous example (Fig. 6.10), expressed in equivalent

noise charge ENC, can be represented as a function of capacitance as:

ENC

= 674 + 3.3 × C [pF], (6.79)

where C is the capacitance seen at the input of the preampliﬁer. Noise, as a function

of the shaping time (θ) analysis, suggests the presence of two components, named

as parallel and series noises [Gatti and Manfredi (1986); Leroy et al. (1997a)].

According to this model, the parallel noise ENC

arises from noise sources

in parallel with the detector at the preampliﬁer input. It depends on the leakage

current and, at a given shaping time, it can be calculated (in terms of number of

electrons e

−

) according to:

ENC

−

, (6.80)

where I

is the leakage current, θ is the shaping time and q is the electron

charge. ENC

decreases with temperature as it depends on the leakage current

[Eq. (6.77)]. The series component ENC

derives from the detector impedance

at the input of the preampliﬁer. This series noise component behaves as 1/θ

1/2

for

short shaping times. However, for shaping times θ ≥ 70 ns, the series component is

practically constant [Leroy et al. (1997a)]. The total noise for a detector is then

obtained by summing quadratically the two noise components:

ENC

= ENC

+ ENC

. (6.81)

The study of the noise of a standard planar detector has shown that the diode

capacitance is the main contributor to the series noise. The series noise also repre-

sents the total noise of a detector with low leakage current (I

≈ 0). An example of

the noise of a standard planar detector (area of 1 cm

and thickness of 300 µm) is

shown in Fig. 6.16 as a function of the applied voltage [Leroy (2004a)]. The noise,

expressed in ENC, has been measured at T = 7

◦

C. This low temperature mini-

mizes the contribution of ENC

, which depends on I

, and hence decreases with

T . The data show that the noise follows the diode capacitance dep endence on the

voltage, in agreement with Eqs. (6.28, 6.79). For applied voltages above the deple-

tion, the series noise reaches a constant value, which corresponds to the minimum

level of noise due to the readout electronics: in the example shown in Fig. 6.16, this

is about 1000 electrons.

6.2 Charge Collection Eﬃciency and Hecht Equation

We consider the inﬂuence of trapping and recombination on charges produced by

ionizing particles in planar (anode A and cathode C are parallel planes as shown in

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478 Principles of Radiation Interaction in Matter and Detection

500

1000

1500

2000

2500

3000

0 20 40 60 80 100 120 140 160 180 200

Voltage (Volts)

ENC (Number of electrons)

Fig. 6.16 Measured noise, expressed in equivalent noise charge ENC - number of electrons -

[Eq. (6.81)], as a function of the applied voltage for a standard planar detector of 1 cm

area and

300 µm thickness. The measurement has been performed at T = 7

◦

C [Leroy (2004a)].

Fig. 6.17) silicon detectors of the type discussed above. Trapping and recombination

are related to defects, such as impurities, or vacancies present in the semiconductor

crystal lattice as a result of the manufacturing processes or after exposure of the

detectors to high levels of irradiation ﬂuences (as large as 10

–10

neutrons or

protons per cm

). Therefore, trapping and recombination are responsible for loss of

charge during the motion of the charge from the point it was created towards its

point of collection (anode for electrons and cathode for holes) under a voltage, V ,

applied across the detector.

The theorem of Ramo for a parallel plate capacitor (the case of a planar silicon

detector) states that

= −

, (6.82)

is the width of the depleted zone. For a totally depleted detector w

= d, where

d is the detector thickness. This implies a current I

induced by the charge i that

is described by Eq. (6.83)

= −

q v

. (6.83)

With v

= µ

E is the velocity of the charge carrier i (i = e, h). E is the electric ﬁeld

taken uniform for our purpose: E = V/d, V is the applied bias voltage. In presence

of trapping, the current corresponding to the carrier i takes the following form:

(t) = I

0,i

(t) exp

−

= −Q

exp

−

, (6.84)

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Solid State Detectors 479

Cathode(C) Anode(A)

Fig. 6.17 A planar silicon detector with parallel electrodes. The electron and hole created at a

point x

by an incoming particle are drifting to the anode (A) and cathode (C), respectively, under

the ﬁeld generated by the voltage applied across the detector.

where I

0,i

(t) is the current induced by charge i in absence of trapping, τ

is the time

the charge carrier i survives before being trapped or recombined (i = e, h). One

ﬁnds that the charge, Q

, induced by the charge carrier i on the electrodes is given

(t) dt = −Q

exp

−

dt, (6.85)

where t

is the trapping time, i.e.,

= t

r,e

= τ

d − x

for electrons and

= t

r,h

= τ

for holes. In Eq. (6.85), one takes x = 0 for the front electrode (p

−n junction) and

x = d for the back electrode (n −n

junction) and x = x

, the position at which an

electron-hole pair has been created. The electrons travel toward the back-electrode

(anode) over a distance d−x

and holes travels toward the front-electrode (cathode)

over a distance x

. Taking Eq. (6.85) and adding the contributions of both charge