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

Подождите немного. Документ загружается.

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

450 Principles of Radiation Interaction in Matter and Detection

Equation (5.115) gives the maximum momentum to achieve particle separation. For

instance [Braunschweig, Ko enigs, Sturm and Wallraﬀ (1976)], if the uncertainty

on the time-of-ﬂight is σ(∆t) = ±250 ps and the distance of separation between

two scintillator counters is L = 5 m, Eq. (5.115) gives a maximum momentum of

∼ 1.9 GeV/c to achieve pion-kaon separation, for which

− m

= 0.224 GeV

and a maximum momentum of ∼ 3.3 GeV/c for proton–kaon separation, for which

− m

= 0.637 GeV

ToF measurement technique does not apply to highly relativistic particles since

in that case the ToF diﬀerence is close to zero and negligible compared to the

standard time resolutions of a few hundred picoseconds achieved with scintillator

counters.

In the non-relativistic limit (β ∼ 0.1, γ ∼ 1.0), Eq. (5.112) becomes (p small):

∆t =

− m

) ≡

∆m, (5.116)

or using Eq. (5.108)

∆t =

mβγc

∆m = t

∆m

. (5.117)

Consequently, for a time resolution of ∆t ∼ 200 ps and a ﬂight path L = 1 m, it

is possible to discriminate between low-energy particles to an accuracy better than

1%.

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

Chapter 6

Solid State Detectors

Solid state detectors are made from semiconductor materials. The semiconductor

detectors beneﬁt from a small energy gap between their valence and conduction

bands. Therefore, a small energy deposition can move electrons from the valence

band to the conduction band, leaving holes behind. When an electric ﬁeld is applied,

the two charge carriers (electron and hole) drift and produce a signal. Therefore, the

passage of an ionizing particle can be detected by collecting the charge carriers liber-

ated by energy deposition in the semiconductor. In this chapter, the basic principles

of operation and the main features of semiconductor detectors will be explained in

details. A speciﬁc example, the microstrip

‡

detector, will also be discussed.

The eﬃciency of silicon detectors at charged and neutral particles detection (note

that for neutron detection, the silicon detectors have to be covered by neutron con-

verters (

LiF for slow neutrons and CH

for fast neutrons), their small thickness of

a few hundreds microns, the possibility to cut silicon detectors to any size and their

low power consumption explain their large use, as dosimeters and radiation detectors

in particle physics experiments. The silicon devices features and general detection

properties of non-irradiated and irradiated silicon detectors based on the standard

planar (SP) and MESA technologies are also reviewed. The parameters evolution

under irradiation ﬂuence of these types of silicon detectors is also discussed. The

study of the peaks evolution with bias voltage can provide precise information on

the active medium structure and charge collection performances of the detector. In

the case of irradiated detectors, the study of the peak evolution with bias voltage

and irradiation ﬂuence allows the measurement of the charge collection eﬃciency

degradation in various regions of the diodes and structure alteration. A review on

induced-damage in silicon radiation detectors operated in radiation environments

was provided by Leroy and Rancoita (2007).

This chapter includes a discussion of the violation of non-ionizing energy-loss

‡

This detector was ﬁrstly designed by Heijne and colleagues at CERN in 1979 for use in high

energy particle experiments [Heijne et al. (1980)]. They made segmented Si detectors with narrow

strip pattern and matched readout electronics for the small signals. The idea of a linearly segmented

silicon diode originated in the Philips Research Laboratories in Amsterdam in around 1963 [Heijne

(2003)].

451

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

452 Principles of Radiation Interaction in Matter and Detection

(NIEL) scaling for low energy protons (. 10 MeV). Furthermore, it is treated the

detection of neutrons with a silicon detector. This detection exploits the interactions

of these neutrons with the nuclei of a converter layer adjacent to the detector diode.

6.1 Basic Principles of Operation

The semiconductor materials are characterized by a small gap between the electronic

conduction band and the valence band. In the case of silicon, an energy

= 1.12 eV

is needed to excite an electron from the valence band into the conduction band. For

comparison, E

> 5 eV for insulators and conductors have their valence and con-

duction bands in contact. The hole left by an electron in the valence band under

some excitation has a positive electric charge.

The jump of electrons from the valence into the conduction band under the

excitation generates holes moving in the valence band in a opposite direction to

that of electrons in the conduction band. Therefore, a current of holes in the valence

band is the counterpart of a current of electrons in the conduction band. Charge

carriers, electrons and holes, drifting through a semiconductor under the inﬂuence

of an electric ﬁeld, E, have a velocity given by

= µ

E, (6.1)

for electrons, and

= µ

E, (6.2)

for holes.

The quantity µ

and µ

are the electron and hole mobility, resp ectively. The hole

mobility (µ

= 450 cm

−1

in silicon [Caso (1998)]) is smaller than the elec-

tron mobility (µ

= 1350 cm

−1

in silicon [Caso (1998)]). The current carried

by electrons and holes in a semiconductor is determined by the mobilities. Using

Eqs. (6.1, 6.2) (e.g., see Section 1.5.2 of [Sze (1981)]), the current density, J

, for

concentrations n of electron carriers and p of hole carriers, q being the electronic

charge, is given by

= q (n µ

+ p µ

) E. (6.3)

There is equilibrium between the generation and recombination of free electrons and

holes at a given temperature. Crystals are not perfect and present defects. Various

impurities are also present in the crystal. The mean free carrier lifetime is reduced

by these defects and various impurities which act as generation or recombination

centers. There are n-type and p-type conductivity semiconductors resulting from

the introduction of electrically active donor and acceptor impurity atoms, respec-

tively. n-type and p-type semiconductors have an excess of electrons and holes,

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

Solid State Detectors 453

0-x

-qN

ρ(x)

-E(x)

-ψ(x)

front back

Fig. 6.1 Schematic representation of an unpolarized p −n junction of thickness w. (a) represents

the junction, (b) the charge distribution ρ(x), (c) the electric ﬁeld -E(x) and (d) the electrostatic

potential -ψ(x).

respectively. At temp erature T , the product of concentrations of electrons n and

holes p remains constant and is given by

np = n

int

, (6.4)

where n

int

(in cm

−3

) is the intrinsic carrier concentration. At T = 300 K (in silicon),

we have n

int

' 1.45 × 10

−3

For silicon (e.g., see Section 4.1.2 of [Neamen (2002)]), the temperature depen-

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

454 Principles of Radiation Interaction in Matter and Detection

dence of n

int

is given by

int

' 4.1056 × 10

× (k

T )

3/2

exp

−

[cm

−3

], (6.5)

where E

= 1.12 eV (e.g., Sect. 4.3.3.3) is the energy gap in silicon, as already

mentioned; k

= 8.617 ×10

−5

eV/K is the Boltzmann constant (see Appendix A.2)

and T is the temperature in kelvin. However, one needs to modify this formula

to improve the description of the experimental results. Below 700 K, the following

phenomenological function gives good agreement with the data ([Morin and Maita

(1954)], see also [Lindstroem (1991)]):

int

' 3.873 × 10

× T

3/2

exp

−

1.21

[cm

−3

]. (6.6)

At low (and very low) temperature, n

int

is better expressed by Eq. 4.146 (see

Sect. 4.3.3.3 and references therein).

6.1.1 Unpolarized p − n Junction

A p−n junction is formed when a n-type region in a silicon crystal is put adjacent to

a p-type region in the same crystal. In practice, such a junction is built by diﬀusing

acceptor impurities into a n-type silicon crystal or by diﬀusing donors into a p-type

silicon crystal. The junction can be abrupt if the passage from the n-type doping

density to p-type doping density is just a step. If the passage is through a gradual

change in doping density, the junction is linearly graded. The size of the relative

doping densities on each side of the junction dictates the type of junction. For

instance, a p

−n junction results from an acceptor density on the p-type side being

much larger than the donor density on the n-type side of the junction. A charge

depleted region occurs at the interface of the n- and p-type regions. This depleted

region is created as the result of the diﬀusion of electrons from n-type material into

p-typ e material and diﬀusion of holes from p-type to n-type material. This diﬀusion

is the consequence of the motion of carriers from regions of high concentration to

regions of low concentration. Therefore, diﬀusion is responsible for the existence

of a space-charge region with two zones: a ﬁrst zone, of non-zero electric charge,

made of ﬁlled electron acceptor sites not compensated by holes and a second zone,

again of non-zero electric charge made of positively charged empty donor sites not

compensated by electrons. If, for instance, the density of acceptors in the p-type

region is much lower than the density of donors in the n-type region, the space-

charge region extends much deep er into the p-region than into the n

-region. The

net result is the creation of a space-charge region with acceptor centers in the p-

region, ﬁlled with donor electrons from the n

-region and not compensated by

holes. This space-charge region is called the depletion region.

The shap e of the electrostatic potential, Ψ, the electric ﬁeld and the width of

the depletion zone of a junction can be obtained by solving the Poisson equation

= −

ρ(x)

. (6.7)

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

Solid State Detectors 455

-qN

0-x

ρ(x)

-E(x)

-ψ(x)

front back

Fig. 6.2 Schematic representation of a p −n junction of thickness w. The applied bias voltage V

creates a depletion depth X. (a) represents the junction, (b) the charge distribution ρ(x), (c) the

electric ﬁeld -E(x) and (d) the electrostatic potential -ψ(x).

In practice, the lateral spatial extension of the charges represents a few µm

which

is negligible with respect to the area of the detectors and Eq. (6.7) can be safely

solved in one-dimension. In Eq. (6.7) (see Appendix A.2),

ε = ε

= 1.054 pF/cm

is the silicon electric permittivity, ε

and ε

are the dielectric constant

‡‡

of silicon

and the permittivity of free space, respectively. An abrupt junction is assumed with

‡‡

The dielectric constant is also referred to as relative permittivity.

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

456 Principles of Radiation Interaction in Matter and Detection

a charge density ρ(x) in the depletion zone approximately given by:

ρ(x) =







for 0 ≤ x ≤ x

−qN

for − x

≤ x ≤ 0,

(6.8)

where x

and x

are the depletion length on n-side and p-side, respectively, as

deﬁned in Fig. 6.1(a). N

et N

are the donor (n-type region) and acceptor (p-

type region) impurity concentrations on each side of the junction, respectively. The

charge density is zero outside the depletion region [Fig. 6.1(b)]. The absence of net

total charge in the depletion region is reﬂected by

= N

. (6.9)

The electric ﬁeld E resulting from the charges separation is obtained after inte-

gration of Eq. (6.7) with

E(x

) = E(−x

) = 0

as boundary condition [Fig. 6.1(c)], i.e.,

E(x) = −

dΨ







(x) = q (N

/ε)(x − x

) for 0 ≤ x ≤ x

(x) = −q (N

/ε)(x + x

) for − x

≤ x ≤ 0.

(6.10)

The thicknesses of n

and p

zones have been neglected.

If no external voltage is applied, E is only due to the diﬀerent concentrations of

electrons and holes at the junction. Diﬀusion will bring n -type material electrons

into the p-type material and holes in opposite direction. The ionized dopants are

ﬁxed charges which generate an electric ﬁeld slowing down the diﬀusion process

since this ﬁeld pushes electrons back to the n-type side and holes back to the p-type

side, until a dynamical equilibrium is reached. This equilibrium is reached when the

carrier ﬂux due to the electric ﬁeld counterbalances the carrier ﬂux due to diﬀusion,

the electron and hole ﬂux summing to zero, separately.

The free charge carrier depleted region or simply called the depletion zone built

above can serve as particle detector. Free charges can be generated in excess of the

equilibrium in the depletion region by ionizing particles traversing the diode. An

energy of E

ion

= 3.62 eV is required to produce an electron–hole pair in silicon. The

diﬀerence between this ionizing energy and the band-gap energy (3.62 eV–1.12 eV) is

spent on other excitations in the silicon lattice. The charges produced by ionization

in the depletion zone are separated and induce an electron and a hole signal. The

interest of the junction structure for particle detection is the possibility to polarize it

by applying a bias voltage. The voltage drop across the depletion zone, i.e., the elec-

trostatic potential is calculated by integrating Eq. (6.10). Deﬁning the integration

constants

Ψ(−x

) = Ψ

and Ψ(x

) = Ψ

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

Solid State Detectors 457

one obtains the electrostatic potential shown in Fig. 6.1(d):

Ψ(x) =







(x) = Ψ

− q [N

/(2ε)](x − x

)

for 0 ≤ x ≤ x

(x) = Ψ

+ q [N

/(2ε)](x + x

)

for − x

≤ x ≤ 0.

(6.11)

The contact potential or built-in voltage, V

(≈ 0.3–0.6 V for silicon at T =

300 K), is deﬁned by:

= −

E(x) dx

= Ψ

− Ψ

int

, (6.12)

with T expressed in kelvin (K).

The depletion depths of the n-type and p-typ e zones are calculated by imposing

the continuity of the potential at x = 0 [Ψ

(0) = Ψ

(0)] and using Eq. (6.9):

2εV

−1

(6.13)

and

2εV

−1

. (6.14)

Then, the total depth of the depletion zone is given by:

X = x

+ x

2εV

. (6.15)

6.1.2 Polarized p − n Junction

Without external polarization, the depletion depth is typically a few microns which

limits the capability of the junction for particle detection. Therefore, an external

voltage is applied between the n- and p-regions: −V

< 0 on the p-side of the

junction, the boundary conditions get modiﬁed into

Ψ(−x

) = Ψ

− V

and Ψ(x

) = Ψ

[Fig. 6.2(d)].

As a consequence, V

is replaced by V

+ V

in Eqs. (6.13, 6.14). Equation (6.15)

becomes now

X = x

+ x

2ε(V

+ V

)

. (6.16)

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

458 Principles of Radiation Interaction in Matter and Detection

The junction is said to be reverse biased and the depth of the depletion zone

grows for increasing V

until the depletion depth reach the total thickness of the

detector (w).

In most applications, n-type detectors and p

−n junctions are used (N

¿ N

)

and from Eq. (6.9) (neutrality equation), one ﬁnds x

¿ x

. Therefore, Eq. (6.16)

is replaced by:

X ≈ x

≈

2ε

+ V

). (6.17)

The voltage to be applied in order to fully deplete the detector is called the full

depletion voltage, V

, i.e., V

= V

at X = w. From Eq. (6.17) where X = w, the

full depletion voltage is given by:

2ε

− V

. (6.18)

The silicon detector resistivity is given by

ρ =

µq |N

eﬀ

, (6.19)

where µ is the electron (hole) mobility for a n-type (p-type) detector as given in

Eqs. (6.1, 6.2) and N

eﬀ

is the eﬀective dopant concentration. The use of N

eﬀ

allows

one to write equations valid for both n-type and p-type detectors. It is also intended

to reﬂect the fact that silicon usually contains n-type and p-type impurities. This

eﬀective dopant concentration, N

eﬀ

, is deﬁned as:

eﬀ

= N

− N

, (6.20)

thus

eﬀ

| = |N

− N

|. (6.21)

As will be seen later, N

eﬀ

can be measured from capacitance–reverse bias (C–

V ) measurements. Using resistivity and eﬀective doping concentration, Eq. (6.17)

becomes:

X ≈

2ε (V

+ V

)

q|N

eﬀ

, (6.22)

X ≈ 0.53

ρ (V

+ V

) [µm] (6.23)

with ρ and (V

) expressed in Ωcm and volts, respectively. Then, the bias voltage

at full depletion is given by:

q|N

eﬀ

2ε

− V

, (6.24)

2εµρ

− V

. (6.25)

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

Solid State Detectors 459

As shown by Eq. (6.25), the value of V

depends on the detector resistivity. High

resistivity detectors have a low full depletion voltage while low resistivity detectors

have a high full depletion voltage.

As already mentioned, in most applications of particle physics, n-type detectors

and p

− n junction are used. A p

− n junction, results from an acceptor density

on the p-type side being much larger than the donor density on the n-type side

of the junction. Thus, in practical cases the contact potential V

can be neglected

with respect to the full depletion voltage [see Eq. (6.12)] in Eqs. (6.22, 6.24). Using

Eq. (6.19), one can rewrite Eq. (6.22) as

X ' 0.53

ρV

[µm], (6.26)

if ρ is expressed in Ω cm and V

in volts. Equation (6.25) can be rewritten as

2εµρ

. (6.27)

Equation (6.27) shows that high resistivity detectors have low full depletion bias

voltage while low resistivity detectors have high full depletion bias voltage.

6.1.3 Capacitance

Silicon detectors used in experiments can be considered as parallel plate capaci-

tors. Their capacitance C, in the parallel plate capacitor approximation, is given

C = ε

, (6.28)

where A is the detector area. This approximation reﬂects the behavior of the space-

charge with voltage in the junction transition region. In fact, in the transition

zone of a junction are interfaced two regions of space-charges of equal value but of

opposite sign. Using previous notations, one region, located between −x

et 0, has

a total charge of −AqN

while the other region located between 0 et x

has a

total charge of AqN

. The depths x

and x

depend on the applied voltage via

the equations Eqs. (6.13, 6.14) [where V

is replaced by V

+ V

2ε(V

+ V

)

−1

(6.29)

and

2ε(V

+ V

)

−1

. (6.30)

Application of an increasing voltage, V (the built-in voltage is included), at the

borders of the junction increases the amount of electrons from the n-zone (neutraliz-

ing the positive space-charge) and the amount of holes from the p-zone (neutralizing