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

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500 Principles of Radiation Interaction in Matter and Detection

a) b)

Fig. 6.28 Measurements with a MediPix2-USB device of ions produced from Rutherford back-scattering of ion beams on a gold foil: a) carbon

ions (large dots) from a 30 MeV carbon beam and b) oxygen ions (large dots) from a 35 MeV oxygen beam. The ions were striking the MediPix2

perpendicularly. The measurements were performed at the Van de Graaﬀ tandem accelerator of the University of Montreal [RBS (2008)]. The

threshold was set at 13.3 keV for both measurements. The bias voltage was 25 V in both cases. Small dots can be observed in these ﬁgures. Their

origin is explained by the Au X-rays (Kα and Kβ). They are generated via proton induced X-ray emission.

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Solid State Detectors 501

they deposit large amounts of energy in a very conﬁned space. Under the inﬂuence

of an electric ﬁeld, the carriers drift towards the corresponding electrode of the

device. The lateral spread of the charges are caused by several eﬀects. A plasma

eﬀect is caused by the high density of charge carriers released in a column following

the particle path of very small radius. When a heavy-charged particle enters a silicon

detector, it deposits an amount of energy so large that the conditions to create a

plasma are fulﬁlled. This condition is related to the Debye length (λ

) which can

be calculated using

²kT

, (6.114)

where ² is the silicon permittivity, k the Boltzman constant (8.617 × 10

−5

eV/K),

T the temperature, q the elementary charge, and n

the carrier concentration. The

column of carriers is considered to be in a plasma state if the Debye length is small

compared with the plasma dimensions (the plasma condition). An α -particle of

5.4 MeV (

241

Am in vacuum) of energy creates 1.5 × 10

pairs in a column 28 µm

long with an initial radius of about 1 µm [Tove and Seibt (1967)]. The carrier con-

centration will therefore be:

pairs

πr

1.5 × 10

π (1 × 10

−6

(28 × 10

−6

= 1.7 × 10

−3

The Debye length at room temperature (300 K) for this carrier concentration

[Eq. (6.114)] is 0.03 µm, which is much smaller than the initial radius of the column,

so fulﬁlling the plasma condition.

The plasma time, t

, is given by the approximate expression [Dearnaley and

Northrop (1966); Tove and Seibt (1967)]

= n

lin

(4π²)

, (6.115)

where n

lin

is the linear density of the plasma (electrons/cm). Using the dimensions

of the initial column of charges, one ﬁnds n

lin

= 5.43 × 10

[e/m]. The ambipolar

diﬀusion coeﬃcient D

can be calculated using:

2kT

+ µ

= 17.48 cm

/s (6.116)

with µ

(= 450 cm

−1

) and µ

(= 1350 cm

−1

) being the mobilities of

holes and electrons in silicon, respectively. The value for the plasma time depends

on the applied ﬁeld, E. For a ﬁeld strength of 10

V/m, one obtains t

= 2.5 µs. In

the case of MediPix2, the silicon sensitive device is 300 µm thick and has a resistiv-

ity of ∼ 5 kΩcm with a full depletion voltage of ∼ 20–25 Volts (value of the voltage

to measure all the charges created in this detector by an α-particle of 5.4 MeV). In

that case, the typical ﬁeld strength before full depletion is ∼ 10 V/300 µm giving a

plasma time t

= 22 µs. This local high concentration is the source of lateral dif-

fusion which increases initially the cluster size at low ﬁeld. Present at low ﬁeld are

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502 Principles of Radiation Interaction in Matter and Detection

Fig. 6.29 Illustration of the Charge sharing induced by 5.4 MeV α-particles from an

241

Am source

striking a MediPix2-USB detector perpendicularly. The cluster radius (in pixels) is shown as a

function of the applied bias voltage (in volts). The threshold was set at 5.6 keV. The α-source was

ﬁt into a vacuum chamber of the Van de Graaﬀ Tandem accelerator of the University of Montreal.

the erosion and funneling eﬀects [Tove and Seibt (1967)]. The erosion eﬀect is the

removal of the carriers located at the periphery of the column by the ﬁeld present in

the detector. The longitudinal ﬁeld forces the outer carriers towards the collecting

electrodes thus reducing the concentration of carriers in the plasma. The funneling

eﬀect [Dearnaley and Northrop (1966); Tove and Seibt (1967)] is also present as

it is caused by the alteration of the local ﬁeld within the plasma. Carriers at the

center of the column only feel the electric ﬁeld from the carriers in the plasma,

escape recombination and are funneled to the collecting electrode. Funneling and

diﬀusion reduce the carrier concentration in the plasma. As the electric ﬁeld near

the depleted region is modiﬁed, charges inside the column are pushed towards the

electrode, decreasing the charge density inside the column hence decreasing the lat-

eral spread. When the depleted region reaches the particle track, more of the charge

diﬀusing laterally is pulled by the detector electric ﬁeld to the electrode. Therefore,

the charge deposited increases at the edges of the cluster and can pass the low

threshold of the MediPix2. Finally, when all of the particle track is included in

the depleted region, the radius of the cluster decreases because of the increasing

strength of the longitudinal electric ﬁeld. An example of cluster radius (in pixels)

as function of the applied bias voltage is shown in Fig. 6.29 for 5.4 MeV α-particles

perp endicularly incident on a MediPix2 detector. The shape of the curve shown in

this ﬁgure is explained along the line developed above: i) ﬁrst increase in cluster

size: the α-particle entering from the back, this shape is mainly due to diﬀusion;

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Solid State Detectors 503

ii) ﬁrst decrease in cluster size: the funneling eﬀect sets in. The phenomenon is

important when the column approaches the depleted area. The electric ﬁeld near

the depleted region is modiﬁed. Charges inside the column are pushed towards the

electrode, decreasing the charge density inside the column hence decreasing the

lateral spread; iii) second increase in cluster size: the depleted region reaches the

alpha particle track. More of the charge diﬀusing laterally is pulled by the detector’s

electric ﬁeld to the electrode. Therefore, the charge deposited increases at the edges

of the cluster and can pass the low threshold of the Medipix; iv) second decrease in

cluster size: all of the particle track is included in the depleted region. The size of

the cluster decreases because of the increasing strength of the longitudinal electric

ﬁeld. A model describing the eﬀects of plasma, diﬀusion and funneling on the charge

collection and charge sharing can be found in [Campbell et al. (2008)].

6.6 Photovoltaic and Solar Cells

A photovoltaic cell is a device which exploits the photovoltaic eﬀect (eﬀect discov-

ered in 1839 by Alexandre-Edmond Becquerel) that converts the electromagnetic

energy from a source of light into electrical energy. When the source of light is the

sun one speaks about a solar cell. These cells have many applications in current life

and activities such as powering electronic calculators, solar modules which generate

home electricity, Earth-orbiting satellites, space stations and probes. The principle

of operation of a solar cell follows four requirements: i) photogeneration of charge

carriers (electrons and holes); ii) separation of charge carriers; iii) transp ort of elec-

trons to a conductive (metal) contact that will transmit the electricity (wire, for

instance); iv) does not require an external voltage source to operate unlike radia-

tion/particles detectors that require a dedicated power supply. The case of solar cells

will be explicitely discussed below. In order for a cell to generate power, a voltage

and a current need to be generated. Usually, a solar cell is a large area (typically

10 cm ×10 cm) silicon p−n junction (or an n−p junction, see below) made by diﬀus-

ing a n-type dopant into the p-type side of a silicon wafer. Photons (of wavelength

less than 1.13 µm corresponding to a solar energy intensity ≤ 250 W m

−2

µm

−1

the Earth’s surface) from solar radiation falling on the junction generate electron-

hole pairs in a light absorbing material contained within the junction structure.

A photon in a solar cell can generate electron-hole pairs if it has an energy

equal or larger than the band gap of the semiconductor, here silicon (band gap

of 1.12 eV). However, if the photon has an energy above the band gap, the excess

energy above the band gap is lost as heat. Photons with energy smaller than the

band gap are not absorbed and are lost. The value of the band gap determines the

possible maximum current generated by the cell. The number G of carriers generated

depends on the number of incident photons N

, on the absorption coeﬃcient α of

the material, on the incident photon wavelength λ, on the thickness x of the material

and on the number of photons N

at the surface of the material (N

= N

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504 Principles of Radiation Interaction in Matter and Detection

x = 0). The number of photons is

= N

−αx

(6.117)

and G is given by

G = −

= αN

−αx

(1 − R) = αN

(1 − R), (6.118)

where one has introduced R, the reﬂectivity at the surface of the cell (0 ≤ R ≤ 1).

A p − n junction based solar cell consists of a thick n-type crystal covered by

a thin p-type layer which will be exposed to sunlight. An external electrical circuit

with a load resistance R

through which the electrons will ﬂow is connected across

the junction. In order to achieve the so-called forward bias mode condition, the n-

side is connected to the positive terminal of the circuit and the p-side is connected

to the negative terminal. After passing through the load, the electrons annihilate

with holes at the rear contact (p-side), so completing the circuit.

Nowadays due to their greater radiation hardness, solar cells based on n − p

junctions are used rather than p −n junctions [van Lint, Flanahan, Leadon, Naber

and Rogers (1980)]. A schematic view of such a cell is shown in Fig. 6.30. The cell is

covered with an antireﬂection layer to maximize the incoming light absorption (R

as small as possible in Eq. (6.118). fraction of 1 micron) which is an emitter region,

an n − p junction layer and a back junction layer (a p-type silicon layer) which

is a base region. Two electrical contact layers made of metal are needed to allow

the electric current to ﬂow into and out of the cell. The contact layer of the cell,

facing incident photons, must present a widely spaced metallic grid pattern to allow

light to enter the cell. It also allows reduction of the internal resistance. The back

metal contact layer covers the entire back surface of the cell structure. An exter-

nal load (resistance R

) is connected to the cell. With this structure, the electrons

generated in the cell by the conversion of the incoming photons with suﬃciently

high energy will be freed in the p-type layer and the electric ﬁeld will send the

electrons to the n-side of the junction and the hole to the p-side causing disruption

of the electrical neutrality. Then, the electrons cross the now thin depletion region

and ﬂowing through a connection wire invade the whole n-type material and reach

the negative battery terminal. The external circuit, with the load resistance R

provides a path through which the electrons can return to the other layer and a

current ﬂow is produced that will continue as long as the light strikes the cell. The

depth of the junction is chosen so that all the carriers are generated in the base

region and none in the emitter region or front face, in general. This is true for light

with large wavelength, i.e. typically higher than 0.6 µm. For light of lower wave-

length (∼ 0.4 µm, i.e., blue light) corresponding to small absorption coeﬃcients

(0.12 (0.8) µm for λ = 0.4 (0.5) µm), carriers are generated in the front face and

diﬀuse to the surface of the cell where they recombine and do not contribute to the

cell current. The product of the voltage V , given by the cell electric ﬁeld and the

current I, caused by the electrons ﬂow is the power W of the cell. The voltage given

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Solid State Detectors 505

by the cell electric ﬁeld is low, typically 0.5–0.6 volt. For instance, a 10 ×10 cm

cell

exposed to direct sunlight with clear skies produces a voltage of ∼ 0.5 volt and a

current of ∼ 2.4 A, giving a power of ∼ 1.2 W. Then, modules composed of several

cells have to be used in order to increase the power production. Cells are connected

together in series and encapsulated into modules. Usually, a module contains 36 cells

to generate an output voltage of 12–18 volts, typically. These modules can be used

singly or connected in series and parallel into arrays with larger current and voltage

output. The sunlight illumination varies during the day due to the variation of the

sunrays incidence on the device and depends on meteorogical conditions. Then, the

modules must be integrated with a charge storage system (battery) and with com-

ponents for power regulation. The battery is used for storing the charge generated

during sunny periods and the power regulating components ensure that the power

supply is regular and minimize the sensitivity to variations in solar irradiation.

For a large part, we follow now a development that can be found in [van

Lint, Flanahan, Leadon, Naber and Rogers (1980); Wahab (1999); Van Zeghbroeck

(2007)], for instance. When the light is striking the cell, a photocurrent I

due to

light generated electron-hole pairs is produced opposite to the diode current I

. One

can express the diode current I

as [Wahab (1999); Van Zeghbro eck (2007)]

= I

/kT

− 1

, (6.119)

where I

(1 pA, typically) and V

are the saturation current of the diode and the vol-

tage across the load (or the voltage applied to the junction if the internal resistance

is zero), respectively. Therefore, in the presence of light when the load resistance

= 0, I is the diﬀerence

I = I

− I

. (6.120)

The short-circuit current, I

, is the current at zero voltage and is equal to I

. Then,

it can be calculated from Eqs. (6.120), (6.119) with

hν

(1 − R)

1 − e

−αx

where P

is the incident optical power and is typically 100 mW cm

−2

for sun-

light (average wavelength of about 570 nm). The value I

∼ 25 mA/cm

is found

(for R

= 0). This value is consistent with the more precise calculation [van Lint,

Flanahan, Leadon, Naber and Rogers (1980)] where a cell of 2 cm

area was consi-

dered. The exact calculation of this current has been done in [van Lint, Flanahan,

Leadon, Naber and Rogers (1980)] taking into account the dependence on the ir-

radiating light wavelength (from 0.5 up to 0.9 µm as the major contribution to the

light spectrum). When the load resistance, R

, is increased an increasing forward

bias V

is applied to the junction and the current is reduced as the p otential barrier

is lowered. If now R

→ ∞, one has an open circuit. If there is a load resistance

connected (in series) with the solar cell, a voltage V

= I R

is generated across

the load and then

I = I

/kT

− I

. (6.121)

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506 Principles of Radiation Interaction in Matter and Detection

For the open circuit, I = 0 and V

(I = 0) = V

, so that

I = I

/kT

− I

= 0. (6.122)

Thus,

+ I

∼

. (6.123)

From Eq. (6.123) one ﬁnds the corresponding value of V

to be

∼ 0.62 V (6.124)

= 25 mA for an area of 1 cm

, I

= 1.0 × 10

−12

A, kT /q = 0.0259 V).

The power generated by the solar cell is

P = V

I = V

q V

/kT

− 1

− V

. (6.125)

The optimal power release of a solar cell can be calculated from the condition

dP/dV

= 0. If V

and I

are the voltage and current corresponding to the optimal

power, respectively:

= I

/kT

− 1

− I

+ I

q V

/kT

= 0. (6.126)

From which we can extract the voltage corresponding to the maximum power, i.e.,

+ I

1 + qV

/(kT )

. (6.127)

Using Eq. (6.124) and the approximation therein, one ﬁnds the transcendental equa-

tion which has to be solved by iterations

∼ V

−

1 +

. (6.128)

With a starting value V = 0.5 V in Eq. (6.128), one ﬁnds a value of V

converging

to 0.540 (V

= 0.50, 0.542, 0.540, 0.540). The value of I

is obtained from

= I

1 −

(6.129)

= 24 mA (6.130)

using V

= 0.540 V.

Solar cells are characterized by several eﬃciency factors.

• The energy conversion eﬃciency which is deﬁned as

η =

. (6.131)

The ratio η represents the percentage of power converted from absorbed

light into electrical energy and collected once the solar cell is connected

to an electrical circuit as described above. η is the ratio of the maximum

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Solid State Detectors 507

Fig. 6.30 A standard schematic view of an n − p solar cell. The cell is covered with an antireﬂection layer. The photovoltaic eﬀect occurs in three

energy-conversion layers: a top junction layer (a n-type silicon layer), an absorption layer (an p − n junction) and a back junction layer (a p-type

silicon layer). Two electrical contact metallic layers allow the electric current to ﬂow into and out of the cell. The front face contact layer present

a metallic grid pattern to allow light to enter the cell. The metal back contact layer cover the entire back surface of the cell structure. An external

load (load resistivity R

) is connected to the cell.

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508 Principles of Radiation Interaction in Matter and Detection

power P

= V

×I

to the input light irradiance P

expressed in W m

−2

(in STC conditions, i.e., at a temperature of 25

◦

C and an irradiance of

1 kW m

−2

with an air mass spectrum 1.5 - technically called AM1.5) [ASTM

(2007)] and the surface area of the solar cell expressed in m

. In the example

considered above, one ﬁnds (for P

= 100 mW cm

−2

, V

= 0.54 V and

= 25 mA) η ∼ 13%.

• The Fill Factor F F which is deﬁned as the ratio of the maximum power of

the solar cell to the product of the open-circuit voltage, V

, and the short

circuit current, I

∼ I

, i.e.,

F F =

. (6.132)

In the example considered above, one ﬁnds F F ∼ 87% for V

= 0.62 V,

= 0.54 V and I

∼ I

∼ 24 mA). The energy conversion eﬃciency and

the Fill Factor are related through:

η =

F F

. (6.133)

• The quantum eﬃciency QE of the solar cell is the eﬃciency of conversion

of the incident photons (of wavelength λ) into electric charge:

QE =

hν

Isc(λ)

. (6.134)

6.7 Neutrons Detection with Silicon Detectors

Silicon detectors are pratically 100% eﬃcient for charged particles detection. This

detection is done via direct ionization (see Sects. 6.1.4, 6.1.5). For instance, the

charge collected in a silicon detector illuminated with 5.5 MeV α-particles from a

241

Am-source is

Q = 1.6 × 10

−19

C × (5.5 × 10

eV/3.62 eV) = 240 fC,

as an α -particle deposits all its energy within a depth of ∼ 25 µm of a silicon

detector 300 µm thick. This can be compared to the charge of Q = 3.5 fC collected

in the same detector for a mip

†

which will deposit ∼ 80 keV. If the detector is in

goo d operational condition (not damaged by irradiation or wrong manipulation)

this charge will be collected at full depletion voltage. The collection of 100% of

the charge will still be possible with a damaged silicon detector provided the bias

voltage is increased beyond the value of the full depletion voltage determined for

the undamaged detector.

Silicon detectors can detect neutral particles via the measurement of indirect

ionization (for photons) or secondary radiation (for neutrons). The detection ef-

ﬁciency is much lower than the one obtained for charged particles in direct ioni-

zation (see for instance [Sadrozinski (2000)]). For instance, photons are detected

†

The mip is a minimum ionizing particle and was deﬁned at page 47.

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Solid State Detectors 509

Table 6.3 A few examples of neutron reactions with the nuclei of several conver-

ters. The probability of these reactions to occur is depending on their energy threshold

as compared to the energy of the incoming particle. For

B(n,α)

Li reaction 93.7% of

the reaction leave

Li in its ﬁrst excited state, which rapidly de-excites to the ground

state (in ∼ 100 fs) by emission of a 480 keV γ (I). The remaining 6.3% of the re-

actions leave

Li in its ground state (II). The Q-values are from [NNDC (2008c)].

Reaction Q-value σ

(MeV) (barn)

Li→α(2.05 MeV)+

H(2.71 MeV) 4.76 940.

B→α(1.47 MeV)+

Li(0.84 MeV)+ γ(0.48 MeV) (I) 2.79 3571.

B→α(1.78 MeV)+

Li(1.01 MeV) (II) 2.79 269.

113

Cd→

114

Cd + γ(0.56 MeV) + conversion e

−

9.0

155

Gd→

156

Gd + γ(0.09, 0.20, 0.30 MeV) + conversion e

−

8.5

157

Gd→

158

Gd + γ(0.08, 0.18, 0.28 MeV) + conversion e

−

7.9

: for 0.025 eV (thermal) neutrons.

: 93.7%.

: 6.3%.

through electrons emitted via photoelectric eﬀect (Sect. 2.3.1), Compton scattering

(Sect. 2.3.2), and pair production (Sect. 2.3.3). The relative importance of these

processes of interaction strongly depends on the material and photon energy. For

instance, for silicon, the photoelectric eﬀect is dominant at photon energy, typically

lower than 60 keV. Over the energy region from 80 keV up to a few MeV the domi-

nant pro cess is the Compton scattering. At photon energy higher than 1.022 MeV,

pair production rapidly increases. The detection eﬃciency is low as the photon

interaction cross sections are themselves low. For instance at 500 keV, assuming a

silicon detector d = 300 µm thick, fully depleted, the probability, P

, for Compton

interaction is

= 1 − exp [−ρ d σ

)] = 0.5%

(ρ = 4.2 × 10

atoms/cm

, σ

) ∼ 4 barns/atoms). The photoelectric eﬀect

(photo-absorption) at that energy is less than 1% of the Compton scattering.

Then, silicon detectors can also be used when low photon detection eﬃciency is

acceptable or when lower energy X-rays (e.g., ≤ 60 keV) have to be measured as it

is the case for X-rays imaging, for instance (see Sect. 6.5.1).

6.7.1 Principles of Neutron Detection with Silicon Detectors

The detection of neutrons with a silicon detector exploits the interactions of these

neutrons with the nuclei of a converter layer adjacent to the detector diode, covering

partially or fully the area of the detector active face (Fig. 6.31). Secondary heavy

charged particles are produced in the converter material either via nuclear reactions

or as energetic recoils from elastic scattering of neutrons on nuclei. These secondary

heavy charged particles generated in the converter must have a range larger than

the distance between the interaction location in the converter and the converter