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

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480 Principles of Radiation Interaction in Matter and Detection

carriers, one ﬁnds:

Q = Q

+ Q

(6.86)

= −Q

r,e

exp (−t/τ

)

−Q

r,h

exp (−t/τ

)

Q = Q

+ Q

(6.87)

E τ

1 − exp

−

d − x

E τ

¶¸

E τ

1 − exp

−

E τ

¶¸

Using the simpliﬁed notations,

= µ

E τ

(6.88)

and

= µ

E τ

, (6.89)

one ﬁnds

Q = Q

1 − exp

−

d − x

¶¸

1 − exp

−

¶¸¾

, (6.90)

which is the Hecht equation [Hecht (1932)]. Then, the charge collection eﬃciency,

CCE, is the ratio of the collected charge to the initial charge Q

CCE =

(6.91)

with Q

= N

q, where N

= E/E

ion

is the number of electron-hole pairs created

at the point x

by an incident particle of energy E and E

ion

is the energy needed

to create an electron-hole pair, E

ion

= 3.62 eV for silicon. Equation (6.91) can be

applied to determine the CCE of detectors illuminated by α-particles.

The range of α-particle in silicon is 25 µm for α-particle from

241

Am source and

cannot cross the whole depth of the detector which is a few hundreds µm thick

in practice. For a source conﬁned under vacuum, the α-particle incident energy is

∼ 5.45 MeV and Q

= (5.45 × 10

/3.62)1.6 × 10

−19

C ∼ 0.24 pC.

The CCE depends on Λ

, Λ

and the point x

where the charge has been

created. This point is not accessible experimentally. The location of this point be-

ing random, the width of the peak in the energy spectrum broadens. The relative

broadening is [Iwanczyk, Schnepple and Materson (1992)]:

σ(E)

2Λ

(Λ

− Λ

)

−d/Λ

− e

−d/Λ

−

−d/Λ

− 1

+ Λ

−d/Λ

− 1

´i

−

−2d/Λ

− 1

−

−2d/Λ

− 1

. (6.92)

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Solid State Detectors 481

The CCE and width improve with the increase of Λ

/d and Λ

/d. Note that

e,h

ρ j

where

and

i/d

are the electric ﬁeld, the semiconductor resistivity and

the leakage current density, respectively [Abyzov, Davydov, Kutny, Rybka, Row-

land and Smith (1999)]. To optimize the energy resolution, one should maximize

e,h

ρ j with the constraint, however, that the leakage current, i, contributes to

the noise charge

ENC

iθ

which has to be minimized [θ is the pulse shaping time of the preampliﬁer, expression

already used earlier, see Eq. (6.80)].

For minimum ionizing β-particles, which traverse the whole depth of the detec-

tor, calculations similar to those leading to Eq. (6.90) can be done and lead to a

ratio of collected charge Q

coll

to the initial charge Q

CCE

coll

+ Λ

−

1 − e

−d/Λ

−

1 − e

−d/Λ

. (6.93)

In the case of relativistic electrons (selected for instance from β-particles produced

by a

106

Ru source) traversing a silicon detector 300 µm thick, the energy loss is

about 80 keV per 300 µm and Q

= (80 × 10

/3.62)1.6 × 10

−19

C ∼ 3.5 fC, as seen

before.

6.3 Spectroscopic Characteristics of Standard Planar Detectors

Precise spectroscopy technique is a powerful tool for investigating features and

functionalities of semiconductor detectors [Casse et al. (1999a); Chren et al. (2001);

Houdayer et al. (2002)]. It provides information on the structure and charge col-

lection capabilities of the detectors. The scanning of the diode structure and its

sensitive volume by illuminating the front and back sides with heavy charged par-

ticles (alpha’s and protons) of low energy and well deﬁned range is illustrated in

Fig. 6.18. In the case of α-particles, the pulse height is given by:

= η

ion

, (6.94)

where E

is the α-particle energy deposited in silicon, q is the electronic charge, E

ion

(= 3.62 eV) is the energy necessary to produce a electron-hole pair and, ﬁnally, C

is the detector capacitance. The factor η is the charge collection eﬃciency. For high

ionizing radiation, such as α-particles, η also reﬂects possible plasma eﬀects, which

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482 Principles of Radiation Interaction in Matter and Detection

Fig. 6.18 Illustration in the schematic cross-sectional view of a detector of thickness d, of the

method for scanning the detector structure and sensitive volume (from [Leroy and Rancoita (2007)];

see also [Casse et al. (1999a)]). The thicknesses of Al contact metallisation and p

layer (front side)

or n

layer (back side) are indicated. The shaded area represents the depleted region (detector sen-

sitive volume). The detector response is measured as a function of the applied bias voltage for pro-

tons or alpha particles incident on the front or on the back side at energies T

> T

corresponding to incoming particle ranges R(T

) > R(T

) inside the detec-

tor. Bragg energy-loss distributions inside the detectors are shown for the various incident energies.

aﬀect the collection eﬃciency. For instance, in Fig. 6.18 it is shown the dependence

of the depth of the depleted (sensitive) layer on the applied bias voltage (V

) is

compared with the Bragg curve distribution of charge produced along the path of

the incoming particle.

Pulse-height spectra are obtained by measuring the signal of the diodes with

a standard spectroscopy system. The pulse-height spectra are recorded at various

applied bias voltages.

The variation of the peak p osition with the incoming particle energy and the

applied bias voltage provides precise information on the spatial eﬃciency of the

charge collection controlled by the electric ﬁeld conﬁguration inside the diode. This

technique requires the use of a reference detector, a silicon surface barrier detector

(SSBD) which has a thin entrance window, an excellent energy resolution and 100 %

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Solid State Detectors 483

Fig. 6.19 Spectra illustrating the response of a SP detector to alpha particles from Pu, ThC

and ThC’ sources incident on the front side of the diode for various applied reverse bias voltage:

= 0.0 (a), 8.7 (b) and 80 V (c) (from [Leroy and Rancoita (2007)]; see also [Casse et al.

(1999a)]). The shaping time was τ = 0.25 µs.

charge collection eﬃciency (CCE). The normalization of the peak position observed

with the diodes under study to that observed with the reference detector is used

to determine the CCE. Typically, the Full-Width at Half-Maximum (FWHM) at

5.486 MeV (

241

Am α-line) for the SSBD is 14 keV, giving a relative energy resolution

of a fraction of the percent for the whole spectroscopy system (including the refe-

rence detector). This very good energy resolution achieved allows the observation

of changes in pulse height resulting from changes in CCE and/or inﬂuence of the

electrode structures. From the splitting and shifts of α-peak, one can determine the

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484 Principles of Radiation Interaction in Matter and Detection

Fig. 6.20 Spectra illustrating the response of a SP detector to alpha particles from Pu, ThC

and ThC’ sources incident on the back side of the diode for applied reverse bias: V

= 62.5 (a),

100.9 (b) and 170.1 V (c) (from [Leroy and Rancoita (2007)]; see also [Casse et al. (1999a)]). The

shaping time was τ = 0.25 µs.

thicknesses of metallic electrodes and implanted layers. The capability to measure

shifts at the level of a keV is equivalent to measuring layer thicknesses of about ten

nanometers.

Some of the spectra illustrating the response of a Standard Planar (SP) detector

to α-particles impinging on the front (back) side of the diode are shown in Fig. 6.19

(Fig. 6.20) for diﬀerent values of applied reverse bias (V

= 0.0, 8.7, 80 V), with a

shaping time τ = 0.25 µs.

The understanding of these peaks behavior is presented in [Casse et al.

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Solid State Detectors 485

(1999a)]. The sensitivity of the spectroscopic method for SP detectors is illustrated

in Fig. 6.21. The response of the reference SBBD detector [Fig. 6.21 (a)] is compared

with the responses of a SP detector illuminated from the front side [Fig. 6.21 (b)]

and from the back side [Fig. 6.21 (c)]. From the splitting and shifts of the α-peak,

one can determine the thicknesses of metallic electro des and of implanted layers as

well.

6.3.1 Energy Resolution of Standard Planar Detectors

Semiconductor detectors have very good energy resolution. There are several con-

tributions to the resolution such as statistics of electron-hole formation, detector

and readout noises. If one considers only the contribution of electron-hole formation

statistics, the energy resolution of a planar silicon detector is, in standard notation:

= 2.36 × σ(E), (6.95)

where σ(E) is the standard deviation in the amount of energy deposited in the

detector by an incident monoenergetic particle. If N

e−h

is the average number of

electron-hole pairs produced by the incident particle in the material, σ

can be

written as:

= 2.36 × E

ion

e−h

, (6.96)

where E

ion

(= 3.62 eV) is the average energy to create a electron-hole pair in sili-

con. If E (expressed in eV) is the energy of the incident particle,

e−h

ion

and one ﬁnds

= 2.36 ×

ion

. (6.97)

For an incident particle of 5 MeV on silicon, σ

= 10 keV. For a 140 keV Compton-

photon emitted by a

99m

Tc source, σ

= 1.7 keV.

The other contributions to the energy resolution depend on the particular read-

out electronic chain used for the measurements. For instance, for an α-particle

emitted by an

241

Am source, E = 5.486 MeV, the application of Eq. (6.97) gives

= 10.5 keV, i.e., a relative energy resolution of 0.2%. However, the measure-

ment performed with a standard spectroscopy system composed of a charge-sensitive

preampliﬁer, linear shaping ampliﬁer and multichannel analyzer rather gives for the

same alpha line: σ

= 14 keV or a relative energy resolution of 0.3%. The diﬀerence

between the measurement and the value found through Eq. (6.97) is explained by

the readout chain noise not accounted in the equation.

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486 Principles of Radiation Interaction in Matter and Detection

Fig. 6.21 Illustration of the sensitivity of the spectroscopic method (from [Leroy and Rancoita

(2007)]; see also [Casse et al. (1999a)]). The response of a reference silicon surface barrier detector

(SSBD) (a) is compared to the response of a SP detector illuminated (b) on the front side and

particles entering through non-sensitive layers on the detector surface. ∆

: the diﬀerence of the

thicknesses of layers due to diﬀerent losses in the Au reference detector electrode and p

layer of

the measured detector (' 20 keV), ∆

: thickness of front Al electrode (' 86 keV), ∆

: backside

layer thickness compared to p

(' 33 keV), ∆

: backside Al grid thickness (' 122 keV). The

thicknesses are expressed in terms of the corresponding energy-losses.

6.4 Microstrip Detectors

The division of the p-side of a silicon diode into an array of parallel narrow strips

provides a position sensitive detector. The carriers (electrons and holes) created

around the incident particle track are basically conﬁned in a tube of ≈ 1 µm of

diameter around the track. The charge created drifts under the electric ﬁeld. In a

n-type silicon with p

-strips, the holes will move to the diode strips and electrons

to the backplane (or ground plane), see Fig. 6.22. For instance, for a 300 µm thick

detector under a bias voltage of 150 Volts, the electric ﬁeld is 5.0 kV/cm and the drift

velocities are 6.75×10

cm/s and 2.25×10

cm/s for electron and hole, respectively.

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Solid State Detectors 487

The electrons created closer to the diode strips will take more time to migrate to

the backplane. The same is true for the holes created closer to the backplane which

will take more time to reach the junction side. During the drift time, the narrow

tube of charge carriers broadens as a result of multiple collisions and the carrier

distribution obeys a Gaussian law:

√

4πD

exp

−

, (6.98)

where dN/N is the fraction of charge carriers in the element dx at a distance x from

the origin of charge location after a time t. D

(i = e, h) is the diﬀusion coeﬃcient

for the i-carrier and is given by Eq. (6.62).

At room temperature, Eq. (6.62) gives D

= 34 .9 cm

/s and D

= 11 .6 cm

using µ

= 1350 cm

−1

and µ

= 450 cm

−1

, respectively. The root mean

square in Eq. (6.98), i.e.,

√

is the same for electrons and holes since the drift

time t ∼ 1/µ. The diﬀusion of charge carriers obviously aﬀects the spatial resolution

of the microstrip detectors.

Statistical ﬂuctuations of the energy loss is another eﬀect to be considered with

that respect. The statistical nature of the ionization process on the passage of a fast

charged particle through matter results in large ﬂuctuations of energy losses in the

silicon detector, which are usually very thin compared to the particle range. One

has already seen in Sect. 2.1.2 that these ﬂuctuations can be calculated using Lan-

dau and Vavilov distributions (Vavilov provides a more accurate solution by in-

troducing the kinematical limit on the maximum transferable energy in a single

collision). Binding of electrons in silicon atoms broadens the theoretical Landau

distribution of the energy-loss measured in the silicon detector. The correct distri-

bution of energy loss is obtained from the convolution of a Landau distribution and

a Gaussian corresponding to a correction for the binding energies of electrons in

silicon and taking into account the electronics noise [Shulek et al. (1966); Hancock,

James, Movchet, Rancoita and Van Rossum (1983, 1984); Caccia et al. (1987)]. The

production of energetic secondary electrons contributes to the energy loss in silicon,

in addition to the creation of electron–hole pairs. Then, displacement of the charge

distribution is due to secondary electrons ejected with an energy larger than T ,

which is the kinetic energy imparted to the electrons ejected from the collision of

the incoming particle with atomic electrons [Damerell (1984)]. The maximal kinetic

energy which can be transferred from a collision of an incoming particle (not an

electron) of mass m and energy E with an atomic electron of mass m

is [Eq. (1.28)

on page 10]:

max

= 2m

1 +

+ 2γ

−1

+ m

+ 2m

E/c

)

, (6.99)

where the incoming particle momentum is

p = mγβc, with γ =

and β =

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488 Principles of Radiation Interaction in Matter and Detection

(v is the incoming particle velocity). In the case of an incoming electron (see

Sect. 2.1.6.1), since the outgoing electron of higher energy is considered to be the

primary electron, the maximum energy transferred is

of the incoming electron

kinetic energy.

p strip

aluminium strip

n backplane

n−type bulk

Fig. 6.22 Standard representation of a n-type strip detector; p is the strip pitch, d is the detector

thickness and w is the width of the p

implant.

The probability to eject secondary electrons (δ-rays) of kinetic energy higher or

equal to T is decreasing by about 4 orders of magnitude as T increases from 100 eV

to 1 MeV (see Sect. 2.1.2.1). However, these high energy electrons, although their

numb er decreases with increasing energy, have an increasing range in silicon. Each

ejected electron releases an increasing number of secondaries in silicon with the

result [Damerell (1984)] that the mean charge released in silicon increases linearly

with ln(T ).

As seen before, a relativistic electron releases about 80 keV in a silicon thickness

of 300 µm, which corresponds to 22000 electrons released around its track. If in

addition, a δ-ray of 50 keV is produced and drifts perpendicularly to the track, its

range will be about 16 µm in silicon and, therefore, 50 keV/3.62 eV ∼ 13800 addi-

tional electrons will be produced along this ray track and will shift (see [Damerell

(1984)]) the centroid of the charge distribution, assuming the secondary charge is

produced halfway from the track, by an amount ∆ of

∆ =

22000 × 0 + 13800 × 8

22000 + 13800

∼ 3.6 µm. (6.100)

From Eq. (2.38) on page 55 (see also Fig. 10 in [Damerell (1984)]), it is seen that

about 4×10

−4

electrons with an energy greater or equal to 50 keV are ejected per mi-

crometer of track. In the case of a silicon detector 300 µm thick, there is then a 11%

probability of producing a δ-ray which shifts the centroid of the charge distribution

by about 3 µm. The numb er of electron–hole pairs produced in a silicon detector

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Solid State Detectors 489

will increase with its thickness. However, the probability to produce δ-rays with

higher range in silicon will also increase and aﬀect the position resolution. Thinner

the detector is, lower will be the probability that such a δ-ray exists and greater

will be the spatial resolution of the detector; although there is clearly a practical

limit in lowering the thickness of the silicon detector [Manfredi and Ragusa (1985)].

The value of the strip pitch, “p” (see Fig. 6.22), inﬂuences largely the design

and performances of the microstrip detectors. Intuitively, one would choose it as

small as possible in order to have a dense array of strips and the signal spread over

several strips. This would allow one to reconstruct the charge distribution and ﬁnd

the impact point of the incoming particle by calculating the center of gravity of the

charge distributed over the strips. However, to have the signal really shared by sev-

eral strips, one should use pitches of the order of a few µm since measured F W HM

of charge distribution is of the order of 10 µm [Belau et al. (1983)]. This is practically

impossible to achieve. In practice, the readout pitch is 20−25 µm. The charge will

be collected on one strip, the strip number giving the track position. The position

can be measured more precisely for tracks generating signals on two strips. In the

case of tracks of particles traversing the detector in between two consecutive strips,

the position is calculated again by taking the center of gravity of the charge distri-

bution or inferred from the shape of the charge distribution. The precision on the

measurement of the position, when several strips share the charge, depends of the

relative position of the track with respect to the strips involved since the strip closer

to the track will show a larger signal than the strip farther. Then, the precision of

the measurement depends on the noise which is expected to be relatively larger for

the smaller signal. For a readout pitch of 25 µm, the precision of localization is a

few microns.

Most of the time, ﬁnancial costs, geometrical and mechanical constraints may

lead to a reduction in the number readout channels. It means that the space between

two consecutive active (readout) strips is increased, which means an increase of the

eﬀective pitch value, and, consequently, a decrease of the number of events detected

by two strips. To remedy this problem, diodes are introduced between the readout

strips [England et al. (1981); K¨otz et al. (1985)]. The charges collected on those

intermediate strips are transferred by capacitive coupling to the readout strips. Such

a procedure feeds up the number of events detected on the two readout strips.

More generally, the interplay between the dimension of the pitch, the strip width

and the thickness of the diode determines the operational features of the microstrip

detector as it inﬂuences the noise of the electronics readout. As we already did in

this section, it is standard to approximate a silicon microstrip by a planar dio de of

thickness d and width w. Neglecting the built-in voltage V

in Eqs. (6.22, 6.24), the

depth, x, of the depleted layer is given as a function of the bias voltage V

x =

2²V

q |N

eﬀ

, (6.101)