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

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400 Principles of Radiation Interaction in Matter and Detection

with the conductance (i.e., the real part of the admittance) given by

G(ω) =

+ R

+ ω

+ R

)

+ R

)

+ ω

+ C

)

(4.153)

and the susceptance (i.e., the imaginary part of the admittance) given by

B(ω) = ω

+ C

) + R

+ R

)

+ ω

+ C

)

. (4.154)

As well known, the conductance is the reciprocal of the equivalent circuit resistance,

thus

r,eﬀ

(ω) =

+ R

)

+ ω

+ C

)

+ R

+ ω

+ R

)

; (4.155)

while the susceptance expresses the capacitance,

r,eﬀ

(ω) =

+ C

) + R

+ R

)

+ ω

+ C

)

, (4.156)

of the equivalent circuit multiplied by ω.

At ro om temperature, for non-irradiated and weakly doped (e.g., see Sect. 4.3.2)

devices

∗

, the reverse current is about or lower than 10 nA at full depletion voltage

and, as a consequence, R

is typically larger than about 10

MΩ, while C

is not

lower than 1 pF. Thus, even for fully depleted devices, i.e., when C

is at the mi-

nimum, we have that R

À 1/(ωC

) at frequencies above a few kHz. In addition,

since the time constant of the electrical neutral bulk

†

is the same as the dielectric

relaxation time τ

rel

(e.g., see Sections 7.6 and 7.7 of [Reitz and Milford (1970)] and

Section 6.3.4 of [Neamen (2002)])

rel

= R

(4.157)

= ε ρ

bulk

(4.158)

≈ (3.2–7.4) × 10

−9

we have that R

¿ 1/(ωC

) for frequencies . 10 MHz. When R

¿ 1/(ωC

) and

À 1/(ωC

), the equivalent circuit for the SIJD model can be approximated by

that shown in Fig. 4.29(b) and its admittance becomes:

(ω) =

1 + ω

+ j ω

1 + ω

. (4.159)

With decreasing temperature, the leakage current decreases and the condition

À 1/(ωC

) is satisﬁed at frequencies lower than a few kHz. Therefore, the equi-

valent circuit at low and very low temperatures, i.e., when the carriers start to freeze-

out (see Sect. 4.3.3.3), can be approximated with that shown in Fig. 4.29(c), whose

∗

Typically this type of device is similar to that described at page 397.

†

bulk

is the bulk resistivity [typically (3–7) kΩcm] of the device and ε is the silicon electric

permittivity (e.g., see Appendix A.2).

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Radiation Environments and Damage in Silicon Semiconductors 401

admittance is given by (e.g., see [Viswanathan, Divakaruni and Kizziar (1991); Di-

vakaruni, Prabhakar and Viswanathan (1994); Croitoru, David, Rancoita, Rattaggi

and Seidman (1997a, 1998b)]):

(ω) =

1 + ω

+ C

)

+ j ω C

1 + ω

+ C

)

1 + ω

+ C

)

. (4.160)

In the temperature range (indicated as saturation range in Fig. 4.30) within

which the carrier concentration is almost constant, we can rewrite the conductance

term of Eq. (4.160) as (e.g., see [Croitoru, David, Rancoita, Rattaggi and Seid-

man (1997a)] and also [Viswanathan, Divakaruni and Kizziar (1991); Divakaruni,

Prabhakar and Viswanathan (1994); Li (1994a)]):

(ω) = A

bulk

(d − X)

+ ω

bulk

, (4.161)

where A

is the device active area, ε is the silicon electric permittivity (e.g., see

Appendix A.2), X is the depth of the depletion region [e.g., Eq. (6.17)], d is the

thickness of the substrate, and ρ

bulk

is the resistivity

‡

of the extrinsic n-type sili-

con. It has to be noted that the resistivity depends on the temperature, because

the mobility varies (i.e., it increases with decreasing temperature) by almost two

orders of magnitude compared to that at 300 K (e.g., [Jonscher (1961); Misiakos and

Tsamakis (1994)]). In Eq. (4.161), for a test frequency of 1 MHz the term ω ε ρ

bulk

is smaller than X even in a partially depleted device. Thus, to a ﬁrst approximation,

the conductance G

is proportional to the resistivity and then has a similar tem-

perature dependence, i.e., that of 1/µ

(T ). For instance, the measured values of the

conductance (Fig. 4.30) for an MCZ-device operated at 5 V of reverse bias indicate

that

(T ) ∝

−α

where α ' 2.3 for 170 . T . 270 K and α ' 1.8 for 70 . T . 150 K [Croitoru,

David, Rancoita, Rattaggi and Seidman (1997a,b); David (1997); Croitoru, David,

Rancoita, Rattaggi and Seidman (1998b)], in agreement with that expected due

to the temperature dependence of the electron mobility found in weakly doped

silicon ([Arora, Hauser and Roulston (1982); Misiakos and Tsamakis (1994)], see

also page 409 and Section 1.2 of [M¨uller and Kamins (1977)]).

When the carriers freeze-out, the resistivity increases again because the sili-

con substrate starts to become similar to an insulator. Furthermore, the function

(ω, T )/ω has a maximum for

(T ) =

ερ

bulk

(T )

X(T )

. (4.162)

In the freeze-out range of temperature, the free-carrier concentration depends on

both the compensation level and the energy level of the dopant atoms. Thus, it

‡

The resistivity is expressed as ρ

bulk

' 1/(e µ

), where µ

and n

are the mobility and the

concentration of electrons, respectively.

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402 Principles of Radiation Interaction in Matter and Detection

is possible to determine the energy level of the dopant atoms by means of the

measurement of ω

(T ) as a function of T ([Viswanathan, Divakaruni and Kizziar

(1991); Divakaruni, Prabhakar and Viswanathan (1994)] and references therein). It

has to be added that the depth of the depleted layer (X) may depend on the

temperature because i) fewer ionized dopants are available to create the space-charge

region and ii) the carrier concentrations in the p

- and n-region exhibit diﬀerent

freeze-out ranges of temperature due to the large diﬀerence between their dopant

concentrations [see Sect. 4.3.3.3 and references therein]. Furthermore, the resistance

of the bulk silicon increases with decreasing temperature so that ωR

) À 1

at low frequency. Therefore as experimentally observed

(e.g., page 397, see also [Li,

Chen and Kraner (1991); Divakaruni, Prabhakar and Viswanathan (1994); Croitoru,

David, Rancoita, Rattaggi and Seidman (1997a); Croitoru, Rancoita, Rattaggi,

Rossi and Seidman (1997)] and references therein), from Eq. (4.160) the capacitance

of the equivalent circuit becomes independent of frequency and approaches

r,eﬀ

(ω) '

+ C

= C

geo

where

geo

εA

is the total capacitance of the medium, for R

so large that it can be ignored in the

equivalent circuit shown in Fig. 4.29(c).

The reverse-biased capacitance (C

) of a p

−n junction depends on the deple-

tion depth X [see Eq. (6.35) and discussion in Sect. 6.1.3] which, in turn, depends

[Eq. (6.17)] on the applied reverse bias (V

) and on the donor (i.e., a shallow impu-

rity) concentration (N

), thus, by combining Eqs. (6.17, 6.35) we obtain:

= A

eεN

2(V

+ V

)

, (4.163)

where V

is the built-in voltage [Eq. (6.12)]. In the substrate, the presence of trap-

ping or deep centers results in a frequency dependence of the capacitance related

to the energy levels of the centers because, at the edge of the depletion region,

the ionization state of these centers may become unable to follow the variation of

the applied voltage over the full spectrum of frequencies. For instance in the Sah-

Reddi model [Sah and Reddi (1964)], if a junction contains deep impurities, the

reverse-biased capacitance (C

d,t

) can be written as

d,t

= C

1 −

+ V

(ω)

¸¾

−

, (4.164)

where C

is given by Eq. (4.163), N

(ω) is the frequency-dependent concentra-

tion of deep impurities, φ

is the energy diﬀerence b etween the Fermi level (of

At 18 K in the devices mentioned at page 397, the capacitance is independent of the applied

reverse bias at a frequency of 300 Hz.

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Radiation Environments and Damage in Silicon Semiconductors 403

Fig. 4.30 Conductance per unit of active area measured at 1 and 0.5 MHz as a function of the

temperature (in 1000/T with T in Kelvin) for a non-irradiated sample with n-type MCZ substrate

and operated at a reverse bias of 5 V (adapted from [Leroy and Rancoita (2007)]). The lines

are to guide the eye. Within the saturation range of temperature the concentration of carriers is

almost constant (i.e., almost all the dopant atoms are ionized), but depends on the temperature in

the freeze-out range (see also [Croitoru, David, Rancoita, Rattaggi and Seidman (1997a, 1998b)]

and [David (1997)]).

the silicon substrate) and the level of the deep center divided by the electronic

charge e. This model has been extended to treat more than one deep center for

non-irradiated (e.g., see [Beguwala and Crowell (1974)] and references therein) and

irradiated (e.g., see [Li (1994a)] and references therein) devices. Investigations of

the frequency and temperature-dependence of the junction capacitance allow the

determination of these deep center levels (e.g., see Section 8.3 of [Milnes (1973)],

also [Sah and Reddi (1964); Schultz (1971); Beguwala and Crowell (1974)] and refe-

rences therein). The mechanism of the repetitive ﬁlling and emptying of deep levels

in the depletion region of a junction is also utilized in the deep-level transient spec-

troscopy (DLTS, e.g., see Sect. 4.2.2) [Lang (1974)]. The DLTS exploits the thermal

discharging of the occupied traps which results from capacitance transients induced

by a pulsed bias.

Systematic investigations of the admittance dep endence on temperature were

carried out for fast-neutron ﬂuences up to ≈ 10

n/cm

(e.g., [Li (1994a); Croitoru,

Gambirasio, Rancoita and Seidman (1996); Croitoru, Rancoita, Rattaggi, Rossi and

Seidman (1997); Croitoru, David, Rancoita, Rattaggi and Seidman (1998a)] and re-

ferences therein) and for temperatures down to 10 K (e.g., [Croitoru, Rancoita,

Rattaggi, Rossi and Seidman (1997); Croitoru, David, Rancoita, Rattaggi and Sei-

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404 Principles of Radiation Interaction in Matter and Detection

Fig. 4.31 Values of T

ins

(in K) when the conductance (2) and the capacitance (•) of the samples

irradiated with fast-neutron ﬂuences up to ≈ 10

n/cm

become independent of the test frequency

(adapted from [Leroy and Rancoita (2007)]; see also [David (1997)]). The lines are to guide the

eye.

dman (1998a)] and references therein).

For fast-neutron ﬂuences . 10

n/cm

, a quasi non-irradiated model by means

of Eq. (4.160) can approximately describe the equivalent circuit behavior of the ir-

radiated samples with decreasing temperature. For instance, the conductance exhi-

bits an approximate ω

dependence [e.g., see Eq. (4.161)], like the response of the

equivalent circuit of non-irradiated devices outside the freeze-out range of tempe-

rature [Croitoru, David, Rancoita, Rattaggi and Seidman (1998a)]. However, the

non-freeze-out range of temperature was found to depend on ﬂuence [Croitoru,

David, Rancoita, Rattaggi and Seidman (1998a)]. Moreover, as for non-irradiated

devices (Fig. 4.30), the conductance reaches a minimum and then increases again

with decreasing temperature. For a ﬂuence of ≈ 10

(10

) n/cm

the minimum

occurs at ≈ 70 (150) K [David (1997); Croitoru, David, Rancoita, Rattaggi and

Seidman (1998a)]. In addition, at room temperature [Li (1994a)], the capacitance

behavior of devices irradiated with fast-neutron ﬂuences . 3.2 × 10

n/cm

is de-

scribed by Eq. (4.154). In this latter equation, the depletion region capacitance has

to account for the trapping centers generated by irradiation, but treated similarly

to deep defects [e.g., see Eq. (4.164)] in the framework of the Sah–Reddi model [Sah

and Reddi (1964)].

A progressive departure from the quasi non-irradiated model description ap-

pears for ﬂuences above ≈ 4 × 10

n/cm

[Croitoru, David, Rancoita, Rattaggi

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Radiation Environments and Damage in Silicon Semiconductors 405

and Seidman (1998a)]. For instance, the dependence of the conductance on ω goes

as ω

with m < 2 [David (1997); Croitoru, David, Rancoita, Rattaggi and Sei-

dman (1998a)]. Furthermore (e.g., see Sect. 4.3.4) for f = 10 kHz [Croitoru, Ran-

coita, Rattaggi, Rossi and Seidman (1997)], the capacitance becomes independent

of the applied voltage at a temperature larger than the one corresponding to a non-

irradiated device (e.g., see page 397). More generally, it has been demonstrated that,

within the experimental errors [David (1997)], the conductance and the capacitance

of the irradiated devices become independent of the frequency

∗

and of the applied

reverse-bias at a temperature T

ins

(see Fig. 4.31), which increases with increasing

ﬂuence (i.e., with the concentration of the created complex-defects, see Sect. 4.2.2,

which are electrically active): T

ins

is ≈ 190 K for a ﬂuence of 7.8 × 10

n/cm

. A

general model, which has been proposed to account for the behavior of G(ω) and

C(ω) at these neutron ﬂuences, assumes that i) the irradiation changes the value

of the resistivity in localized volumes inside the silicon and that ii) in these lat-

ter regions the medium can have a diﬀerent frequency dependence with respect to

that in the undamaged regions (e.g., [Croitoru, David, Rancoita, Rattaggi and Sei-

dman (1998a)], see also [Croitoru, Dahan, Rancoita, Rattaggi, Rossi and Seidman

(1997)]).

4.3.5 Resistivity, Hall Coeﬃcient and Hall Mobility at Large

Displacement Damage

In Sections 4.3.2–4.3.4 some of the main distinctive characteristics of non-irradiated

and irradiated silicon semiconductors were apparent from their simple transport

properties. Additional investigations were carried out to determine further proper-

ties: for example the Hall coeﬃcient shows whether the charges are transported by

positive or negative carriers in extrinsic semiconductors. The Hall coeﬃcient is of

primary importance since, except for a numerical factor (the so-called Hall factor

or scattering factor) of the order of unity, it is the reciprocal of the carrier density

and electronic charge

. Thus, it provides an immediate indication of the type and

concentration of dopants in the sample and can be combined with the conducti-

vity

to determine the carrier mobility (the so-called Hall mobility). However, in

materials partly compensated by impurities of the opposite type, the simple mea-

surement of the Hall coeﬃcient may not be easily interpreted to determine the type

of majority-carriers.

The Hall scattering factor is indicated as r

) for electrons (holes) and ex-

presses the eﬀect of the magneto-resistance: it accounts for the energy dependence

∗

The range of employed frequencies were 10 ≤ f ≤ 10

kHz.

For a general introduction to the Hall eﬀect, the reader may see, for instance, [Putley (1960)],

Section 2 of [Blood and Orton (1978)], Chapter 5 of [Smith (1978)], Chapter 3 in [Blood and Orton

(1992)] and [Popovic (2004)]; for the Hall factor and its dependence on temperature, the reader

may read, for instance, [Norton, Braggins and Levinstein (1973); Kirnas, Kurilo, Litovchenko,

Lutsyak and Nitsovich (1974)] and Section 2.6.2 of [Blood and Orton (1978)].

The conductivity is the reciprocal of the resistivity.

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406 Principles of Radiation Interaction in Matter and Detection

Fig. 4.32 Absolute value of the Hall coeﬃcient, in cm

/C, as a function of the temperature, in

1000/T with T in K, for two non-irradiated n-type samples (from [Leroy and Rancoita (2007)]) with

resistivities of ≈ 2.7 (¥) and ≈ 6 (•) kΩcm at room temperature (see [Croitoru, Gubbini, Rancoita,

Rattaggi and Seidman (1999a,b)], respectively). The temperature range is 25 ≤ T ≤ 350 K. The

lines are to guide the eye.

of the relaxation time with respect to the carrier motion in presence of a magnetic

ﬁeld (e.g., see Chapter 5 of [Smith (1978)]). The values of the relaxation time

for

electrons and holes are not necessarily the same and may depend on the tempera-

ture, the energy of the carrier and the shape of the constant-energy surfaces in the

k-vector

∗∗

space. Experimental determinations and theoretical predictions of the

Hall factor as a function of temperature and dopant concentration under the low

magnetic-ﬁeld approximation

††

can be found, for instance, in [Messier and Merlo-

Flores (1963); Norton, Braggins and Levinstein (1973); Rode (1973); Dmitrenko et

al. (1974); Kirnas, Kurilo, Litovchenko, Lutsyak and Nitsovich (1974); Blood and

Orton (1978)]: at room temperature, in weakly doped silicon semiconductors, r

and

are ≈ 1.15 and 0.80, respectively (e.g., see Chapter 3 of [Blood and Orton (1992)],

[Mangiagalli, Levalois, Marie, Rancoita and Rattaggi (1999)] and, also, [Norton,

Braggins and Levinstein (1973); Rode (1973); Kirnas, Kurilo, Litovchenko, Lutsyak

and Nitsovich (1974)]).

For a partly compensated silicon semiconductor, under low magnetic-ﬁeld appro-

The relaxation time expresses the mean free time between carrier collisions.

∗∗

The wave-vector, k, is the momentum of the particle divided by ~.

††

The low magnetic-ﬁeld approximation (e.g., Section 2.2 of [Blood and Orton (1978)] and, also,

Section 5.3.4 of [Smith (1978)]) can be expressed by the condition µ

B ¿ 1 where µ

and B are

the carrier mobility and the magnetic-ﬁeld, respectively.

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Radiation Environments and Damage in Silicon Semiconductors 407

ximation, the Hall coeﬃcient is given by {see Equation (147) at page 128 of [Smith

(1978)]}:

= −

n − r

(b n + p )

(4.165)

= −

n − r

(µ

n + µ

= eρ

p − r

, (4.166)

where

b =

(µ

) is the mobility

∗

, sometimes referred to as conductivity-mobility, of electrons

(holes); n (p) is the concentration

‡‡

of free electrons (holes) and, ﬁnally, the silicon

resistivity ρ is

ρ =

e(µ

n + µ

(4.167)

eµ

(b n + p)

. (4.168)

For instance, at room temperature in an (almost) intrinsic and non-irradiated silicon

semiconductor b is ≈ 3 and the Hall coeﬃcient [computed using Eq. (4.165)] is

negative, i.e., the current is mainly carried by electrons. For extrinsic silicon under

low magnetic-ﬁeld, Eq. (4.165) can be rewritten as

H,e

= −

n e

for n À p (4.169)

and as

H,h

p e

for p À b n and p À b

n. (4.170)

The Hall coeﬃcient can be experimentally determined by measuring the so-called

Hall voltage (due to the Hall eﬀect), when a magnetic-ﬁeld is applied to a semicon-

ductor carrying a current (e.g., see [Putley (1960); Popovic (2004)]).

The Hall mobility, µ

, is deﬁned as the product of the Hall coeﬃcient and the

conductivity ℘:

= |R

|℘

, (4.171)

where

ρ =

℘

∗

The dependence of the mobility on the temperature and concentration is discussed in Sect. 7.1.5

(see [Arora, Hauser and Roulston (1982)] and references therein).

‡‡

At thermal equilibrium we have np = n

int

(e.g., see Sect. 6.1).

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408 Principles of Radiation Interaction in Matter and Detection

Fig. 4.33 Resistivity, in kΩcm, as a function of the temperature, in 1000/T with T in K, for

two non-irradiated samples (from [Leroy and Rancoita (2007)]) with resistivities of ≈ 2.7 (¥) and

≈ 6 (•) kΩcm at room temperature (see [Croitoru, Gubbini, Rancoita, Rattaggi and Seidman

(1999a,b)], respectively). The temperature range is 25 ≤ T ≤ 350 K. The lines are to guide the

eye.

is the resistivity. It can be determined when combined measurements of the Hall

coeﬃcient and resistivity (see later) are performed on the same sample. For an

extrinsic silicon semiconductor, under low magnetic-ﬁeld, from Eqs. (4.169–4.171)

we have:

H,e

= r

for n À p, (4.172)

in which ρ

is the resistivity of the n-type silicon and

H,h

= r

for p À b n and p À b

n, (4.173)

where ρ

is the resistivity of the p-type silicon.

The temperature dependence of the Hall coeﬃcient [Eq. (4.169)] of non-

irradiated n-type silicon samples with resistivities of ≈ 2.7 [Croitoru, Gubbini,

Rancoita, Rattaggi and Seidman (1999b)] and ≈ 6 kΩcm [Croitoru, Gubbini, Ran-

coita, Rattaggi and Seidman (1999a)] is shown in Fig. 4.32. Above room tempera-

ture (i.e., 1000/T . 3.33), the rapid decrease of the Hall coeﬃcient with increasing

temperature results from the increase of the intrinsic carrier concentration [e.g., see

Eq. (4.146)]. R

H,e

is almost constant up to 1000/T ≈ 20 (18) for the sample with

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Radiation Environments and Damage in Silicon Semiconductors 409

resistivity of ≈ 6 kΩcm (≈ 2.7 kΩcm) resistivity (Fig. 4.32). Below these tempera-

tures, the carriers start to freeze-out (e.g., see Sect. 4.3.3.3) and the Hall coeﬃcient

increases, as expected.

To make a Hall measurement, the sample is typically prepared in a bridge form

(e.g., see Section 5.2 of [Bar-Lev (1993)]) or in a rectangular bar with side arms

(e.g., see Section 2.3 of [Blo od and Orton (1978)]) or in an arbitrary form by means

of van der Paw’s method [van der Pauw (1958)]. Thus, the same sample may be

used for conductivity measurements. In Fig. 4.33, the resistivity dependence on

temperature is shown for non-irradiated samples

∗

with high- (≈ 6 kΩcm [Croitoru,

Gubbini, Rancoita, Rattaggi and Seidman (1999a)]) and low-resistivity (≈ 2.7 kΩcm

[Croitoru, Gubbini, Rancoita, Rattaggi and Seidman (1999b)]). The data exhibit

an overall agreement with the predicted resistivity-dependence on temperature, as

discussed in Sects. 4.3.3.3 and 4.3.4. For instance, using the data shown in Fig. 4.33

obtained for a sample with a resistivity of ≈ 6 kΩcm [Croitoru, Gubbini, Rancoita,

Rattaggi and Seidman (1999a)] in the temperature range 190 ≤ T ≤ 300 K, the

temperature dependence of the electron-mobility is given by:

= µ

0,e

−α

[cm

−1

], (4.174)

where µ

0,e

= (8.5 ± 0.2) ×10

, T is the temperature (in Kelvin) and the coeﬃcient

α = 2.31±0.02 is in agreement with that determined by conductance measurements

(e.g., see page 401) and by other authors (e.g., see [Arora, Hauser and Roulston

(1982); Misiakos and Tsamakis (1994)]).

It has to be noted that the combined measurements of the Hall coeﬃcient and re-

sistivity allow one to determine both the mobility and the free carrier concentration

in extrinsic semiconductors [e.g., see Eqs. (4.169, 4.170, 4.172, 4.173)].

Measurements of resistivity, Hall coeﬃcient and Hall mobility were carried out

after irradiation

∗∗

with diﬀerent types of particles up to large ﬂuences and as a

function of temperature. At room temperature the resistivity of the irradiated sam-

ples of extrinsic (p- and n-type) silicon has been observed to increase with respect

to that before irradiation

and to become even larger than that of the intrinsic

silicon (e.g., see [Borchi and Bruzzi (1994); Li (1994b); Croitoru, Dahan, Rancoita,

Rattaggi, Rossi and Seidman (1997, 1998); Mangiagalli, Levalois, Marie, Rancoita

∗

For these samples, the dependence of Hall coeﬃcients on temperature is that shown in Fig. 4.32.

∗∗

The reader can see, e.g., [Konozenko, Semenyuk and Khivrich (1969); Lugakov, Lukashevich

and Shusha (1982); Borchi and Bruzzi (1994); Li (1994b); Biggeri, Borchi, Bruzzi and Lazanu

(1995); Biggeri, Borchi, Bruzzi, Pirollo, Sciortino, Lazanu and Li (1995); Croitoru, Dahan, Ran-

coita, Rattaggi, Rossi and Seidman (1997, 1998); Mangiagalli, Levalois, Marie, Rancoita and

Rattaggi (1998); Croitoru, Gubbini, Rancoita, Rattaggi and Seidman (1999a,b); Mangiagalli, Lev-

alois, Marie, Rancoita and Rattaggi (1999); Pirollo et al. (1999); Consolandi, Pensotti, Rancoita

and Tacconi (2008)].

One can see, e.g., [Borchi and Bruzzi (1994); Li (1994b); Biggeri, Borchi, Bruzzi and Lazanu

(1995); Biggeri, Borchi, Bruzzi, Pirollo, Sciortino, Lazanu and Li (1995); Croitoru, Dahan, Ran-

coita, Rattaggi, Rossi and Seidman (1997, 1998); Mangiagalli, Levalois, Marie, Rancoita and

Rattaggi (1998); Croitoru, Gubbini, Rancoita, Rattaggi and Seidman (1999a,b); Mangiagalli, Lev-

alois, Marie, Rancoita and Rattaggi (1999); Pirollo et al. (1999)], Section 2 in Chapter II of Part II

of [Vavilov and Ukhin (1977)], Section 5.13 of [Messenger and Ash (1992)] and references therein.