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Chapter 2 – Physics of semiconductor detectors

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CHAPTER 2 PHYSICS OF SEMICONDUCTOR DETECTORS


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This chapter is meant to describe the basic physics that stands behind interaction of radiation and particles

with matter, what are its consequences and how these principles are applied in semiconductor silicon

detectors technology. In the real world, energy moving through space is identified with the name of

electromagnetic radiation, and it is characterized by quantity of energy E, speed c, frequency ν and

wavelength λ with which is moving1. Different values of energy, frequency and wavelength creates the

flavors of electromagnetic radiation, but differences between them are evident only after interaction with

matter, when they show particle-like behavior out of wave-light one. Hence in the definition of

electromagnetic radiation charged particles are included, such as alpha and beta radiation, beams of

charged particles created by accelerating machines, electromagnetic radiation or photons, and beams of

neutral particles such as neutrons.

2.1 - Electromagnetic and particulate radiation

The principal types of radiation can be first divided into two main categories: electromagnetic (X-rays,

produced outside the nucleus and γ-rays, emanated from within nuclei) and particulate (α particles,

protons, neutrons, electrons β-, positrons β+). This distinction, as already mentioned, belongs to the proper

“history” of the radiation, drawn by the history of the particle (subject connected to the concepts of energy

loss of a particle, range, interactions) and by the history of the target atoms (that leads to displacements,

recombination, ionization, excitation, radiation damage and build-up concepts). A beam of radiation that

passes through matter can lead to the complete absorption (electronic transitions and vibration-rotational

transitions), to some scattering (Rayleigh, Rutherford, Raman and Mie scattering) and/or to the passage

with no interaction. These processes can be explained in terms of interactions between particles that are

stopped or scattered. The basic effect of the interaction can be the scattering, absorption, thermal

emission, refraction, and reflection of the incoming radiation. With the absorption and emission spectra (of

molecules) it is possible to outline characteristic structures and so to identified and quantified molecules

by these ‘fingerprints’. The spectra are determined by position (wavelength) of absorption/emission line,

knowing the difference of energy levels of the transition and by strength of absorption/emission line,

knowing the probability of the transition. The most commonly used transition is the electron transition in

the atoms and vibration-rotational modes in the molecules. Moreover, a particle travelling through matter

can lose energy gradually (losing energy nearly continuously through interactions with the surrounding

material), or catastrophically (moving through with no interaction until losing all its energy in a single last

collision). Gradual energy loss is typical of charged particles, whereas photon interactions are of the "all-or-

nothing" kind.

2.2 - Photon interactions with matter

Gamma rays, x rays and light are photons with different energies. Depending on their energy and on the

nature of the material, photons interact electromagnetically with charge particles of matter, and they can

do it in three main ways: with the Photoelectric Effect (or Photoelectric Absorption), the Compton scattering

and the Pair Production. It is also important to mention the Rayleigh Scattering, which consists in the

diffusion of the photons over the electrons of atoms, without ionization or excitation of the atoms: this

process gives birth to all the colors we see.

1 correlated together by fundamental relativistic equations


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2.2.1 - Photoelectric effect

This process consists in the absorption of a photon with consequent expulsion of an electron out from the

atom hit. In order to remove a bound electron from an isolated atom a threshold energy for the photon is

needed: it’s the ionization potential, and it varies depending on what atomic shell the electron occupies. If

the energy of the photon Eγ overtakes the ionization potential (bound energy EB2), an electron will be

emitted out of the atom with energy Ee given by the following formula:

𝐸𝑒 = 𝐸𝛾 − 𝐸𝐵

It has been given a letter to name the shells (K, L, M ...) depending on the principal quantum number (n = 1,

2, 3, ...). As example, for hydrogen atom H the ionization potential from n=1 corresponds to an ultraviolet

photon, but for heavier elements the K-shell ionization shifts rapidly into the x-ray regime. The ionization

potential depends on the square of the atomic number Z of the atom ( and so from the dimension of the

atom), as given from cross-section formulas for the effect3.

For example, Figure 2.1 shows the cross-section behavior

for the plumb atom, pointing out that ionization cross

section peaks just above threshold for each shell, to then

fall rapidly at higher energy due to the difficulty in

transferring the excess photon momentum to the

nucleus. For n > 1 there are sub-shell structures (2s,

2p1/2, 2p3/2, . . .). The photoelectric effect is important

in the design of x-ray proportional counters, as an

example. In this thesis the photoelectric effect has been

used in order to prove the calibration of 3D silicon

detectors, trying to reproduce with them the

photoelectric peaks of some known gamma radioactive

elements (241Am and 109Cd).

When other atoms are present, as in molecules and solids, the electronic energy levels will be very

different, as will the photoelectric cross sections. For solids in vacuum, the thresholds can be ≈ 1 eV and it

depends on the crystalline structure and on the nature of the surface. The ionization potential in this case is

usually called work function. Photon absorption efficiencies approach 100% in the visible and ultraviolet,

but the overall device efficiencies are limited by the electron escape probabilities. In a semiconductor a

photon can be thought of as ”ionizing” an atom, producing a ”free” electron which remains in the

conduction band of the lattice. Thresholds are of order 0.1–1 eV for intrinsic semiconductors and of order

to 0.01–0.1 eV for extrinsic semiconductors. The latter photon energies correspond to infrared photons.

Photochemistry is somewhat similar in that photons produce localized ionization or electronic excitation. In

the end, the escaping electron produces a redistribution of the atomic electrons, that can lead to

Fluorescence (emission of photons) or Auger Effect (emission of characteristic X ray radiation).

2 1≤EB≤100 KeV, depending on the shell and on the atom 3 There are correct different formulations for low and high energy behaviour

Figure 2.1 - Pb photoelectric cross-section


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2.2.2 - Compton scattering

The Compton scattering takes place when a photon scatters off a free (or bound) electron, yielding a

scattered photon with a new, lower frequency and a new direction. For an unbound electron initially at

rest, it is possible to have the following equations4:

𝑕𝜈 ′ = 𝑕𝜈 1 + 𝑕𝜈

𝑚𝑒𝑐2 (1 − cos 𝜃)

−1

with hν and hν’ initial and final energy, θ photon angle change, me electron mass and c speed of light. Low

energy photons loose little energy, while high energy photons, called γ rays, loose a lot of energy. The

wavelength increases by of order 0.0024 nm, independently from the wavelength. The Compton cross

section is given by the Klein-Nishina formula[e.g.2-1]. The largest Compton scattering cross section is at

small energy, and it decreases monotonically with energy. At low energies lots of scattering events take

place, but very little energy is lost. It is a consequence that the energy absorption cross section is small at

low energy because little energy is transferred to the electron, and it rises to a peak for photon energies

around 1 MeV that declines at higher energy.

2.2.3 - Pair production

Photons with energies in excess of 2mec2 produce electron-positron pairs, and interaction with a nucleus is

needed in order to balance momentum. The pair production cross section starts at 1.022 MeV for then

rising to an approximately constant value at high photon energy, in the gamma ray region of the spectrum

of electromagnetic radiation. Cross sections scale with the square of the atomic number, and complete

formulas describing the cross-section are Bethe-Heitler formulas [e.g. 2-2].

2.3 - Absorption coefficient

The description of the absorption of a beam of photons, all with the same energy and all travelling in the

same direction, is given by an exponential law:

𝑁 𝑥 = 𝑁0𝑒−µ𝐿𝑥

that performs the exponential decrease of the number of particles N(x) at x given depth into the material

from the initial number 𝑁0, where µL is the linear absorption coefficient5 given by:

𝜇𝐿 =𝜍𝑁𝑎𝜌

𝐴

𝜍 = 𝜍𝑝𝑕𝑜𝑡𝑜𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 + 𝜍𝑐𝑜𝑚𝑝𝑡𝑜𝑛 + 𝜍𝑝𝑎𝑖𝑟

with σ cross-section, Na Avogadro’s constant, ρ density of the material, A molecular weight. The probability

of interaction of the photons is given by total cross-section of photon is given by the sum of the single

effect cross-sections, which are summarized in Figure 2.2:

4 h/ (mec) has units of length and equals 0.0024 nm 5 It gives a measure of how fast the original photons are removed from the beam (if of high values the original photons are removed after travelling only small distances)


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For the silicon (Z=14), and for energies of photons under 100 KeV, the dominant effect is the photoelectric

effect, whereas for over 10 MeV it is the pair production. Giving out some values, the attenuation

coefficient for 241Am, which decades with gamma rays of 59.5 KeV, is 0.3 cm2g-1 , and the probability of

detection for silicon detector of 300 µm depth is only of 2%. This because of the fact that cross-sections for

photons are really low, and consequently also the probability of detection are low, too. Nevertheless,

gamma sources are suited for calibrating silicon detectors because the whole photon energy can be

detected in the sensor with the assumption that electrons doesn’t escape from the detector.

The absorption law follows from the fact that, over any short distance, the probability of losing a particle

from the beam is proportional to the number of particles left into it: if particles are present in high number

many are going to be lost, but if the number left decreases the same does the rate of loss. Moreover, the

exponential attenuation law does not describe what happens to the energy carried by the photons

removed from the beam, and it is possible that some of those may be carried through the medium by other

particles, including some new photons. The average distance travelled by a photon before it is absorbed is

given by λ, the attenuation length or mean free path, that is the reciprocal of the linear absorption

coefficient:

𝜆 = 1

µ𝐿

It follows an alternative way of expressing the exponential absorption law:

𝑁 𝑥 = 𝑁0𝑒−

µ𝐿𝜆

The distance over which one half the initial beam is absorbed is called the half thickness, 𝑥1

2

, and is related

to the linear absorption coefficient and to the mean free path by:

𝑥12

= ln(2)

µ𝐿= ln(2) ∗ 𝜆 = 0.693 𝜆

The absorption of photons depends on the total amount of material in the beam path, and not on how it is

distributed, because the probability for a photon to interact somewhere within the matter depends on the

Figure 2.2 – Absorption coefficient for Rayleigh effect, Photoelectric effect, Compton effect and Pair Production


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total amount of atoms ahead of its path (since they interact only with single atoms). Therefore, it is useful

to describe the absorption process without the dependence on the density of material, but only on the kind

of material. This is obtained by introducing the mass absorption coefficient μm, which relates the linear

absorption coefficient to the density of the material ρ:

µ𝐿 = µ𝑚𝜌

This means, for example, that the mass absorption coefficient is the same for ice, liquid water and steam

whereas the linear absorption coefficients differs greatly. It is so possible to have an ulterior definition of

the absorption law:

𝑁 𝑥 = 𝑁0𝑒−µ𝐿𝑥 = 𝑁0𝑒

−µ𝑚 𝜌𝑥

that states that the total attenuating effect of a slab of given type material can be described by quoting the

mass attenuation coefficient, which is characteristic of the material's chemical composition, and the photon

energy, together with the material's density and thickness. The product ρx, the areal density6, of a thickness

x of the attenuating material is also called the density-thickness, and is often quoted instead of the

geometrical thickness x. Although the SI7 unit of density-thickness is kg*m-2, the obsolete unit g*cm-2 is still

used in the literature. If an absorber is made of a composite material the mass absorption coefficient is

readily calculated by adding together the products of the mass absorption coefficient and the proportion

(α) of the mass due to each element present in the material:

µ𝑚 𝑇𝑂𝑇𝐴𝐿 = (𝛼 µ𝑚 )

The law of absorption always describes the absorption of the original radiation. If the radiation changes,

degrades in energy, it is not completely absorbed or if secondary particles are produced, then the effective

absorption decreases, and so the radiation penetrates more deeply into matter than predicted. It is also

possible to have an increasing number of particles with depth in the material: this process is called build-up,

and has to be taken into account when evaluating the effect of radiation shielding, for example.

2.4 - Interactions of charged particles with m atter

The most common way in which charged particles (such as electrons, protons and alpha particles) can

interact with matter is the electromagnetic interaction, that involves collisions with electrons in the

absorbing material and is the easiest mechanism to detect them. They can also interact through one of the

two kinds of nuclear interactions, the weak interaction or the strong interaction. Principally, they either

loose all their energy, or they are deflected form the original trajectory. The main process of energy loss

producing excitation and ionization is the inelastic collisions with an electron; it can also happen an inelastic

collisions with a nucleus, that leads to Bremsstrahlung and coulombian excitation. Eventually there could

also be elastic collisions with a nucleus (Rutherford diffusion) and elastic collisions with an electron.

2.4.1 - Electromagnetic interaction

Two main mechanisms characterize the electromagnetic interaction: the first is the excitation and

ionisation of atoms, and the second is the so-called Bremsstrahlung, word meant to describe the emission

6 mass per area 7 International System of measurements


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of electromagnetic radiation (photons) when a charged particle is severely accelerated (usually by

interaction with a nucleus). Moreover, there exists a third kind of interaction, producing Cherenkov

radiation, that absorbs only a small amount of energy (but it plays an important role in the detection of

very high energy charged particles). Charge, mass and speed of the incident particle as well as the atomic

numbers of the elements of the absorbing material define the contribution of each mechanism. Unlike

photons, each charged particle suffers many interactions along its path before finally coming to rest, losing

only a small fraction of its energy during every interaction (for example, a typical alpha particle might make

50000 collisions before it stops). Hence the energy loss can usually be considered as a continuous process

(scattering). Although the amount of scattering at each collision may be small, the cumulative effect may

be quite a large change in the direction of travel. Occasionally an incident particle passes very near a

nucleus and then there is a single large deflection (this nuclear scattering effect is most pronounced for

light incident particles interacting with heavy target nuclei).

2.4.2 - Excitation and ionization

Electromagnetic interaction between the moving charged particle and atoms within the absorbing material

is the dominant mechanism of energy loss at low (non-relativistic) energies; it extends over some distance,

keeping not necessary for the charged particle to make a direct collision with an atom. Energy can be

transferred simply by passing close by, but only certain restricted values of energy can be exchanged. The

incident particle can transfer energy to the atom, raising it to a higher energy level (excitation) or it can

transfer enough energy to remove an electron from the atom altogether (ionization). This is the

fundamental mechanism operating for all kinds of charged particles, but there are considerable differences

in the overall patterns of energy loss and scattering between the passage of light particles (electrons and

positrons), heavy particles (muons, protons, alpha particles and light nuclei), and heavy ions (partially or

fully ionized atoms of high Z elements). Most of these differences arise from the dynamics of the collision

process: in general, when a massive particle collides with a much lighter particle, the laws of energy and

momentum conservation predict that only a small fraction of the massive particle's energy can be

transferred to the less massive particle. The actual amount of energy transferred will depend on how

closely the particles approach and from restrictions imposed by quantization of energy levels. The largest

energy transfers occur in head-on collisions.

2.4.3 - Energy loss by electrons and positrons

Concerning the electrons and positrons loss of energy, they also ionize but with several differences with

heavy particles (for example they have lower loss rates at high energies than heavier particles travelling at

the same speed). There is also a slight difference between the interactions of positrons and of electrons,

resulting in a slightly higher energy loss for the positrons. An electron is easily scattered in collisions with

other electrons because of its light mass: as a result, the final erratic path is longer than the linear

penetration (range) into the material, with greater straggling.

2.4.4 - Bremsstrahlung effect

Literally translated from German into 'braking radiation', Bremsstrahlung is an effect that occurs whenever

the speed or direction of a charged particle motion changes (when it is accelerated), and consist in the

emission of electromagnetic energy (photons) when the acceleration takes place. It is most noticeable

when the incident particle is accelerated strongly by the electric field of a nucleus in the absorbing material.

Since the effect is much stronger for lighter particles, it is much more important for beta particles


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(electrons and positrons) than for protons, alpha particles, and heavier nuclei (but it happens also for

them). Radiation loss starts to become important only at particle energies well above the minimum

ionisation energy (at particle energies below about 1 MeV the energy loss due to radiation is very small and

can be neglected). At relativistic energies the ratio of loss rate by radiation to loss rate by ionization is

approximately proportional to the product of the particle's kinetic energy and the atomic number of the

absorber. So the ratio of stopping powers is:

𝑆𝑙 𝑟𝑎𝑑

𝑆𝑚 𝑖𝑜𝑛 =

1

𝐸′ 𝑍𝐸

where E is the particle's kinetic energy, Z is the mean atomic number of the absorber and E' is a

proportionality constant; E' ≈ 800 MeV. The kinetic energy at which energy loss by radiation equals the

energy loss by collisions is called critical energy, Ec, and is approximately

𝐸𝑐 ≈𝐸′

𝑍≈

800 𝑀𝑒𝑉

𝑍

It’s also interesting the quantity called radiation length, that is the distance over which the energy of an

incident particle is reduced by a factor e-1 (0.37) due to only radiation losses.

2.4.5 - Electron-photon cascades

A high energy electron performing Bremsstrahlung results in a high energy photon as well as a high energy

electron, and a high energy photons performing pair production results in a high energy electron as well as

a high energy positron: in both cases two high energy particles are produced from a single incident particle.

It follows that the products of one of these processes can be the incident particles for the other, with the

result of a cascade of particles which increases in number while decreasing in energy per particle, until the

average kinetic energy of the electrons falls below the critical energy. The cascade is then absorbed by

ionization losses. Such cascades, or showers, can penetrate large depths of material.

2.4.6 - Stopping power

The most important way to describe the net effects of charged particle interactions with matter and the

rate of energy loss along the particle's path is with the linear stopping power Sl, also known as 𝑑𝐸

𝑑𝑥 (where E

is the particle energy and x is the distance travelled):

𝑆𝑙 = − 𝑑𝐸

𝑑𝑥

commonly measured in MeV * m-1. It depends on the charged particle's energy, on the density of electrons

within the material, and hence on the atomic numbers of the atoms. So a more fundamental way of

describing the rate of energy loss is to specify the rate in terms of the density thickness, rather than the

geometrical length of the path, so energy loss rates are often given as the quantity called the mass stopping

power:

𝑆𝑚 = − 𝑑𝐸

𝑑(𝜌𝑥)= −

1

𝜌 𝑑𝐸

𝑑𝑥

where ρ is the density of the material and ρx is the density-thickness.


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2.4.7 - Energy loss by heavy particles

Charged particles lose their energy when interacting with matter; consequently they leave track of their

passages, with differences for every kind of particle. Knowing these processes has allowed to develop

detectors able to track the particles. Assuming that the speed and mass of the atomic electron is negligible

respect to that of the incoming particle, it is possible to have a classical model of energy loss of particle

through matter in the Bethe-Block formula [2-1]:

𝑑𝐸

𝑑𝜉=

1

𝜌

𝑑𝐸

𝑑𝑥= 𝐷

1

𝛽2𝑧2

𝑍

𝐴{ln

2𝑚𝑒𝛾2𝛽2𝑐2

𝐼 − 𝛽2 −

𝛿

2−

𝐶

𝑍}

𝐷 = 4𝜋𝑟02𝑚𝑒𝑐

2𝑁𝐴𝑉 = 0,307 𝑀𝑒𝑉 𝑐𝑚2

with ρ density of the material, x depth into the material, D constant, β and γ relativistic parameters of the

particle, z charge of the particle, Z atomic number of the material that is traversed, A atomic weight of the

material, me mass of the electron, c speed of light, I average ionization energy of the material, δ and C

relativistic corrections of the formula, r0 classical electron radius, NAV Avogadro’s constant.

Figure 2.3 - Energy loss of µ on Cu [2-3]

Figure 2.4 - Energy loss for different particles [2-3] Figure 2.5 - Energy loss for heavy charged particles in

different materials [2-3]


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Figure 2.3 show the behavior of energy loss for a µ particle in a wide set of momentum, while Figure 2.4

shows the same for different particles. Figure 2.5 shows the energy loss for heavy charged particles in

different materials, pointing out the fact that the minimum of the curve varies from 1.15 MeV of Pb to 2

MeV of He, with exception for the H2. These graphs are plots of the energy-loss rate as a function of the

kinetic energy of the incident particle. It’s important to notice that in Figure 2.4 the stopping power is

expressed using density-thickness units. To obtain the energy loss per path length it needs to multiply the

energy loss per density-thickness, as shown in Figure 2.5, by the density of the material. As for photon

interactions, it is found that when expressed as loss rate per

density-thickness, the graph is nearly the same for most

materials. There is, however, a small systematic variation;

the energy loss is slightly lower in materials with larger

atomic numbers. At high incident energies there is also

some variation with density of the same material because a

higher density of atomic electrons protects the more distant

electrons from interactions with the incident particle. This

results in lower energy loss rates for higher densities.

Concerning the silicon, in Figure 2.6 it is shown its behavior.

For low energies the stopping power varies approximately as the reciprocal of the particle's kinetic energy.

The rate of energy loss reaches a minimum called Minimum Ionization Point (MIP), to then start to increase

slowly with further grow in kinetic energy. Minimum ionization occurs when the particle's kinetic energy is

about 2.5 times its rest energy, and its speed is about 96% of the speed of light in vacuum. Although the

energy loss rate depends only on the charge and speed of the incident particle but not on its mass it is

convenient to use kinetic energy and mass rather than the speed. At minimum ionization the energy loss is

about 2 MeV cm2 g-1 (= 3 × 10-12 J∙m2∙kg-1 in SI units), and it slightly decreases with the increasing atomic

number of the absorbing material. Given that the minimum of the curve is quite the same for all particles in

all materials, it is of common use to define the MIP, used to quantify a detector response without the need

to refer to a specific particle, as the minimum signal that can be detected. For silicon, the <dE/dx>min≈1.66

MeV cm2 g-1. The distribution of probability for lost energy by a particle in a single hit with a material

electron follows a Landau curve, because events with high energy exchange can happen but are less

probable. Experimentally, a Gauss curve is obtained only when the depth of the material allows to have

enough hits with its atoms. For thin depts., hits with atoms are not a lot, and hits with high energy loss can

produce a tail of the distribution through high energies. For example, when a particle hits an atom it is

possible that electrons are free to move in the material, and if with enough energy they can ionize by

themselves creating secondary couples of ions and electrons. Historically these ionizing electrons are called

δ rays.

Reassuming, the charged particle leaves in the material a track formed by ion-electron couples and photons

produced by disexcitation. For every material there exist an energy value for the production of couples,

that does not depend from the particle energy, defined as:

𝑊 = 𝑛𝑇 Δ𝐸

Figure 2.6 - Energy loss in silicon


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with ∆E energy given by the incident particle, nT number of couples created. In the definition of nT there are

included the primary nP and secondary nS couples as:

𝑛𝑇 = 𝑛𝑃 + 𝑛𝑆

In silicon the mean energy loss is of 1.66 MeV cm2 g-1, and the density is of 2.33 g cm-3; it means that the

loss of energy is of 390 eV/μm. Then to generate a hole-electron couple an energy of 3.6 eV is needed, and

it follows that a MIP creates ~110 couples per μm in silicon. For a thickness of 250 μm a MIP creates about

20000 hole-electrons couples, with 27000 as mean amount and 19400 as most probable value (MPV). In

Figure 2.7 this calculation is shown for different depth of silicon.

A massive particle that collides with an electron loses relatively small quantity of energy at each collision.

For example, a slow alpha particle hitting an electron transfers a maximum of only 0.05% of its energy to

the electron. Since head-on collisions are rare, usually the energy loss is much lower. In order to

significantly reduce the incident particle's energy many collisions are needed, so the energy loss can be

considered as a continuous process. Although the energy given to an electron may be a small fraction of

the incident energy, it may be sufficient to ionize the atom and for making the ejected electron travel some

distance away from the interaction point, leaving a trail of excited and ionized atoms of its own. These

'knock-on' electrons can leave tracks from δ rays. Mostly, however, the knock-on electrons lose their

energy within a very short distance of the interaction point.

Before losing all its kinetic energy into the material, a penetrating particle follows some distance, called

range of that particle, characterized by a rise near the end of the path is due to the increased energy loss

rate at low incident energies. At very low speeds the incident particle picks up charge from the material,

becomes neutral and is then entirely absorbed by the material. Moreover, particles of the same kind with

the same initial energy have nearly the same range for a given material. There exists a final small variation

in the range, called straggling, and is due to the statistical nature of the energy loss process which consists

of a large number of individual collisions subjected to some fluctuation. In spite of that, the average range

can be used to determine the average energy of the incident particles.

Figure 2.7 - Mean and Most Probable Value of energy loss of a MIP in different thickness of silicon [2-3]


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2.5 – Physics and behavior of semiconductors

In the previous sections the principles of how particles interact with matter have been considered. From

now on the concentration will be pointed out to semiconductors, and focused on the interaction of MIP

particle with silicon sensors, looking on how these concepts are used to think and create devices capable to

detect real particles. Semiconductor devices are widely used in the electronics (power-switching devices)

because of their specific electrical conductivity, σ, which is between that of good conductors (>1020 free

electron density) and that of good insulators (<103 free electron density), and they also fit well for particle

detection because they are materials into which there exist little numbers of free charges, and particles

passing through it can easily produce a detectable quantity of them.

2.5.1 - Conduction in a solid

After Quantum Mechanics discoveries, a theory about solid state materials that includes semiconductors

has been commonly approved by the scientific community. The structure of an isolated atom shows

numerable states of the electrons surrounding the nucleus, characterized univocally by a definite energy

En8. In a solid, it is to be taken into account the entire number of the atoms that constitutes the lattice: the

interactions among the atoms and their high existing number9 make the electron states so dense to make

them forming a continuous band of allowed energy. These bands can be separated by gaps that electrons

cannot occupy, the forbidden gaps. Because of their fermionic nature10, electrons fill the states starting

from the lowest energy level available, filling up the energy bands to a maximum energy E0 :

Qualitatively, there are two possible configurations: one with the last band partially filled, and the other

with the last band completely filled. The partially filled (or empty) band is called conduction band, while the

band below it is referred to as valence band. Because of the thermal energy available at the absolute

temperature T, some higher energy levels are populated. In the case of a partially filled band, the solid is a

conductor, because when an electric field is applied the electrons can freely change states in the

conduction band . In the case of completely filled bands, the gap width between the valence and the

conduction band can make the solid an insulator (Φ ~10eV) or a semiconductor (Φ ~1eV). In fact, the

8 n is a set of integer numbers

9 ~ 1022 atoms/cm3 10 In Quantum Mechanics the fermions belongs to one of the two fundamental particle classes (fermions and bosons). Fermions distinguish from bosons for the fact that they obey to Pauli’s Exclusion Pinciple, that states that a single quantic estate cannot be occupied by more than one fermion (while the bosons are free to largely crowd the same quantic state)

Figure 2.8 - Questa figura non mi piace


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thermal energy available at T 300K, is sufficient to bring some electrons into the conduction band if the

gap is of the order of 1eV. To calculate the number of electrons with an energy above a given value E0 , one

must apply Boltzmann statistics [], which gives the density of electrons n having energy greater than E0.

From this it follows:

𝑛 𝐸 > 𝐸0 = 𝑒−

𝐸0𝑘𝐵𝑇

𝑘𝐵 = 1.3807 ∙ 10−23 𝐽

𝐾

, where kB is the Boltzmann constant.

2.5.2 - Classification of Semiconductors

Although there is a large variety of semiconductor materials today available, there is one of them that

stands out from the group and dominates the scene: it is the silicon. Its properties are entirely well known,

it is quite easy to find and to manage practically, and – last but not least for the productive processes – not

expensive. Nonetheless, according to their chemical composition, each different kind of semiconductor can

have different properties, and so used for different specified duty in the applications.

Elementary semiconductors are located within the IV group of the Periodic Table of Elements [],and they

are the Silicon (Si), the Germanium (Ge), the grey tin (α-Sn), and Carbon (C), that can solidify in two

different structures (graphite and diamond, that is an insulator but with the same crystal structure as Si, Ge

and α-Sn).

Material a (nm) EG (eV) EG (nm) Type Structure

Diamond (C) 0.357 5.48 226 indirect cubic Silicon (Si) 0.543 1.12 1107 indirect cubic

Germanium (Ge) 0.566 0.664 1867 indirect cubic

Gray tin (α-Sn) 0.649 - - - cubic White tin (β-Sn) 0.583

0.318 - - - tetragonal

Graphite (C) 0.246 0.673

- - - hexagonal

Lead (Pb) 0.495 - - - cubic

Table 1 - Lattice constant a, energy gap EG at 300K, type of energy gap and lattice structure of group IV elements [2-4]

The main characteristic of the IV elements mentioned is that they all have the outer shell of the individual

atoms is exactly half filled, and so by sharing one of the four electrons of the outer shell with another Si

atom it is possible to obtain a three-dimensional crystal structure with no preferential direction (except for

graphite), and it is also possible to combine two of IV group semiconductors in order to form useful

compounds (such as SiC or SiGe) with new peculiarity (for example the SiC is a borderline compounds

between semiconductor and insulator, and can be useful for high temperature electronics).

By completing the outer shell by sharing electrons with other atoms can be obtained also with other

compounding11, so obtaining compounds that are semiconductors, too. Elements of group III (II) can so be

11 8N atomic rule


14

combined with elements of group V (VI), with covalent bonds (but, in contrast with IV group ones, they

show also a certain degree -~30%- of ionic bonds). Most of the III-V semiconductors exist in the so-called

zincblende structure (cubic lattice), and some in the wurtzite structure (hexagonal lattice); GaAs and GaN

are the most known and most utilized of them (optical application, because they are direct

semiconductors).

It also exists the II-IV class of semiconductors, characterized by an higher ionic bond degree total

percentage -~60%- since the respective elements differ more in the electron affinity due to their location in

the Periodic Table of Elements. Also I-VII compounds can form semiconductors, with larger energy gap.

There are other elementary semiconductors such as selenium and tellurium from group VI, the

chalcogenes, but only with two missing valence electrons to be shared with the neighboring atoms, so they

have the tendency to form chain structures.

There are also some spare compounds that works as semiconductors: they are the IV-VI compounds (PbS,

PbSe,PbTe), V-VI (B2Te3), II-V (Cd3As2, CdSb), and a number of amorphous semiconductors (the a-SI:H,

amorphous hydrogenate silicon, for example, is a mixture of Si and H). Moreover it is still possible to cite

the chalcogenide glasses (As2Te3, As2Se3, that can be used in xerography)[2-4].

2.5.3 - Silicon

Silicon is the most widely used material in radiation detectors mostly because it is the only semiconductor

material having a native oxide with good interface properties fitting for a high integration technology, and

it is the most chosen for devices involving semiconductors. It has four valence electrons, so it can form

covalent bonds with four of neighbors atoms. When the temperature increases the electron in the covalent

bond can become free, generating holes that can afterwards be filled by absorbing other free electrons, so

effectively there is a flow of charge carriers. The effort needed to break off an electron from its covalent

bond is given by Eg (band gap energy). There exists an exponential relation between the free-electron

density ni and Eg given by the formula:

𝑛𝑖 = 𝑁𝐶𝑁𝑉 𝑒−

𝐸𝑔

2𝑘𝐵𝑇

, where NC and NV are respectively the number of states in conduction and valence band. To give some

numbers, it is also possible to write it as:

𝑛𝑖 = 5.2 ∗ 1015𝑇32 𝑒−

𝐸𝑔

2𝑘𝑇 [ 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠

𝑐𝑚3]

For example, at T=300 K, ni = 1.45 x 1010 electrons/cm3, and at T=600 K, ni = 1.54 x 1015 electrons/cm3[2-8].

These electrons form the thermal charge carriers pairs concentration, and so an intrinsic current in silicon

material when a voltage is applied, properly connected to silicon specific natural properties.

In pure silicon at equilibrium, the number of electrons is equal to the number of holes. The silicon is so

called intrinsic and the electrons are considered as negative charge-carriers. Holes and electrons both

contribute to conduction, although holes have less mobility due to the covalent bonding. Electron-hole

pairs are continually generated by thermal ionization and in order to preserve equilibrium previously

generated, they recombine. The intrinsic carrier concentrations ni are equal, small and highly dependent on

temperature.


15

2.5.4 – Doping of silicon

In order to fabricate either a silicon detector or a power-switching device, it is necessary to greatly increase

the free hole or electron population. This is achieved by deliberately doping the silicon, so by adding

specific impurities called dopants. The doped silicon is subsequently called extrinsic and as the

concentration of dopant Nc increases its resistivity ρ decreases. Pure silicon electrical properties can be

changed by doping it with group V periodic table elements such as phosphourous (P), that adds more

electrons to it (type N doping, with free negative charges), or with group III periodic table elements such as

boron (B), that adds more holes (type P doping, with free positive hole charges). A group V dopant is called

a donor, having donated an electron for conduction. The resultant electron impurity concentration is

denoted by ND (donor concentration). If silicon is doped with atoms from group III, such as B, Al, Ga or In,

which have three valence electrons, the covalent bonds in the silicon involving the dopant will have one

covalent-bonded electron missing. The impurity atom can accept an electron because of the available

thermal energy. The dopant is thus called an acceptor, which is ionized with a net positive charge. Silicon

doped with acceptors is rich in holes and is therefore called p-type. The resultant hole impurity

concentration is denoted by NA (acceptor concentration).

To be pointed out that for manufactory industries

it is not easy to grow large area silicon crystals

doped with a rate of less than 10% around the

resistivity wanted. Final device electrical

properties will therefore vary widely in all lattice

directions. Tolerances better than ±1 per cent in

resistivity and homogeneous distribution of

phosphorus can be attained by neutron radiation,

commonly called neutron transmutation doping

(NTD). The neutron irradiation flux transmutes

silicon atoms first into a silicon isotope with a

short 2.62-hour half-lifetime, which then decays

into phosphorus. Subsequent annealing removes

any crystal damage caused by the irradiation.

Neutrons can penetrate over 100mm into silicon, thus large silicon crystals can be processed using the NTD

technique.

2.5.5 - Charge Carriers

Electrons in n-type silicon and holes in p-type are called majority carriers, while holes in n-type and

electrons in p-type are called minority carriers. The carrier concentration equilibrium can be significantly

changed by irradiation by photons, the application of an electric field or by heat. Such carrier injection

mechanisms create excess carriers. Noticeable is the fact that the product of electron and holes densities (n

and p) is always equal to the square of intrinsic electron density regardless of doping levels:

𝑛𝑝 = 𝑛𝑖2

It follows that, for n-type doped semiconductors:

𝑀𝑎𝑗𝑜𝑟𝑖𝑡𝑦 𝑐𝑎𝑟𝑟𝑖𝑒𝑟𝑠 𝑛 ≈ 𝑁𝐷

Figure 2.9 - Doping of silicon [2-6]


16

𝑀𝑖𝑛𝑜𝑟𝑖𝑡𝑦 𝑐𝑎𝑟𝑟𝑖𝑒𝑟𝑠 𝑝 ≈ 𝑛𝑖

2

𝑁𝐷

and for p-type doped semiconductors:

𝑀𝑎𝑗𝑜𝑟𝑖𝑡𝑦 𝑐𝑎𝑟𝑟𝑖𝑒𝑟𝑠 𝑝 ≈ 𝑁𝐴

𝑀𝑖𝑛𝑜𝑟𝑖𝑡𝑦 𝑐𝑎𝑟𝑟𝑖𝑒𝑟𝑠 𝑛 ≈ 𝑛𝑖

2

𝑁𝐴

2.5.6 - Charge transportation

A first mechanism of charge transportation into semiconductor is identified under the name of drift

mechanism: it simply consists in the application of an electric field at the extremity of the semiconductor,

and the charge particles will move at a velocity (vh, ve)proportional to the electric field E given (µP, µn

constants of proportionality called mobility):

𝑣𝑕 = µ𝑃𝐸

𝑣𝑒 = −µ𝑛𝐸

In silicon, at room temperature, typical values of mobility are 1450 cm2/V∙s for the electrons and 450

cm2/V∙s for the holes.

The current is calculated as shown in the following formula:

𝐼 = −𝑣𝑊𝑕𝑛𝑞

In general the drift current is expressed as:

𝐽𝑛 = µ𝑛 𝐸 𝑛 𝑞

while the total current is the sum of the current given by holes and electrons drifts:

𝐽𝑇𝑂𝑇 = µ𝑛 𝐸 𝑛 𝑞 + µ𝑝 𝐸 𝑝 𝑞 = µ𝑛 𝑛 + µ𝑝 𝑝 𝐸 𝑞

It is important to underline that in reality the velocity does not increase linearly with electric field, but it

saturates at a critical value. The following equation expresses the velocity saturation:

µ = µ0

1 + 𝑏𝐸

𝑣𝑆𝐴𝑇 = µ0

𝑏

𝑣 = µ0

1 +µ0 𝐸𝑣𝑆𝐴𝑇

𝐸

A second charge transportation mechanism is the diffusion, that is given by the fact that charged particles

move into the semiconductors from a region of high concentration to a region of low concentration.


17

Diffusion current is proportional to the gradient of charge (dn/dx) along the direction of current flow, as

shown in the following equation:

𝐼 = 𝐴 𝑞 𝐷𝑛 𝑑𝑛

𝑑𝑥

𝐽𝑛 = 𝑞 𝐷𝑛 𝑑𝑛

𝑑𝑥

𝐽𝑝 = −𝑞 𝐷𝑝 𝑑𝑝

𝑑𝑥

𝐽𝑇𝑂𝑇 = 𝑞 (𝐷𝑛 𝑑𝑛

𝑑𝑥− 𝐷𝑝

𝑑𝑝

𝑑𝑥)

It is important to say that a linear charge density profile means constant diffusion current, whereas

nonlinear charge density profile means varying diffusion current.

𝐿𝑖𝑛𝑒𝑎𝑟: 𝐽𝑛 = 𝑞 𝐷𝑛 𝑑𝑛

𝑑𝑥= −𝑞𝐷𝑛

𝑁

𝐿

𝑁𝑜𝑛 − 𝐿𝑖𝑛𝑒𝑎𝑟: 𝐽𝑛 = 𝑞 𝐷𝑛 𝑑𝑛

𝑑𝑥=

−𝑞𝐷𝑛 𝑁

𝐿𝑑 𝑒

−𝑥𝐿𝑑

Surprisingly, there exists a relation between the drift and diffusion currents, although they are totally

different.It is the Einstein’s relation

𝐷

µ=

𝑘𝑇

𝑞

2.5.7 - PN junctions

Electrons and holes are discrete charge carriers, and the current generated by their drift and diffusion is

affected by a noise proportional to the current itself. The amplitude of this intrinsic noise depends on the

resistance of the semiconductor used (230 kΩ∙cm for silicon), and it can unfortunally be of the same order

of signal generated by a particle passing through the semiconductor material. It follows that

semiconductors as they are can’t be suitable for particle detection, but it has been discovered that using a

pn junction can resolve this inconvenient.

A pn junction is the location in a doped semiconductor where the impurity changes from p to n while the

monocrystalline lattice continues undisturbed. A bipolar diode is thus created, which forms the basis of any

bipolar semiconductor device. When N-type and P-type dopants are introduced side-by-side in a

Figure 2.10 - Diffusion into a semiconductor [2-6]


18

semiconductor, a PN junction (or diode) is so

formed. In order to understand the operation of

a diode, it is necessary to study its three

operation regions: equilibrium, that introduces

the depletion zone and the built-in potential,

reverse bias, that introduces the junction

capacitance, and forward bias, that introduces

the IV characteristics.

2.5.8 - Diffusion across the junction

Each side of the junction contains an excess of holes or electrons compared to the other side, and this

situation induces a large concentration gradient. Therefore, a diffusion current flows across the junction

from each side.

As free electrons and holes diffuse across the junction, a region of fixed ions is left behind. This region is

known as the depletion region, and is particularly attractive for particle detection purposes, because any

charges created by a passing-through particle are going to be swept out by the electric field generated in

this zone, and can be detected by electronics connected to the junction.

So the fixed ions in depletion region create an electric field that results in a drift current; at equilibrium, the

drift current flowing in one direction cancels out the diffusion current flowing in the opposite direction,

creating a net current of zero.

𝐼𝑑𝑟𝑖𝑓𝑡 ,𝑝 = 𝐼𝑑𝑖𝑓𝑓 ,𝑝 ; 𝐼𝑑𝑟𝑖𝑓𝑡 ,𝑛 = 𝐼𝑑𝑖𝑓𝑓 ,𝑛

Because of the junction there exists a built-in potential:

Figure 2.11 - Diffusion in a PN junction, with nn concentration of electrons on n side, pn concentration of holes on n side, pp concentration of holes on p side, np concentration of electrons on p side [2-6]

Figure 2.12 – Creation of the depletion zone [2-6]

Figure2.11 - PN junction and electrical schematic [2-6]


19

𝑉0 = 𝑘𝑇

𝑞ln

𝑁𝐴𝑁𝐷

𝑛𝑖2

that is caused by the capture of electrons by holes on the n-side and the diffusing electrons which fill holes

in the p-side, and is due to different concentration of holes and electrons in both doped materials that start

diffusing. This is added to the contact potential VC, that is the potential difference across the junction (for

silicon is about 0.7 V).

2.5.9 – Biasing the junction with forward bias

There are two ways for biasing the junction: one the is direct, the

other is the reverse way. When the n-type region of a diode is at a

lower potential than the p-type region, the diode is in forward bias.

This situation leads to shorten the depletion width and decrease the

built-in potential. Under this condition minority carriers in each

region increase, and diffusion currents also increase to supply them.

Minority charge profile should not be constant along the x-axis, in

order to have a concentration gradient and so diffusion current:

recombination of the minority carriers with the majority carriers

accounts for the dropping of minority carriers as they go deep into the P or N region.

2.5.10 – Biasing the junction with reverse biasing

Opposite to the precedent situation, when the n-type region of a diode is connected to a higher potential

than the p-type region the diode is under reverse bias, which results in wider depletion region and larger

built-in potential across the junction. This is important for having the as wider depleted zone as possible in

order to increase the sensible zone of creation of charges and

consequently of particles detection. Varying the value of VR it is so

possible to vary also the width of the depletion zone W, following

the law [2-9]:

𝑊 = 𝑥𝑛 + 𝑥𝑝 = 2휀0휀𝑆𝑖

𝑒

1

𝑁𝐴+

1

𝑁𝐷 𝑉0 + 𝑉𝑅

, where xn and xp are the parts of depletion zone on the n and p

side respectively, ε0 and εSi absolute and relative to silicon dielectric

constants. In silicon sensors the junction is usually realized by a

shallow and highly doped (NA> 1018 cm-3) p+-implant in a low-

doped (ND≈1012 cm-3) bulk material, therefore the term 1/NA can

be neglected, meaning that the space charge region is reaching

much deeper into the lower doped side of the junction. Moreover,

also the built-in voltage can be neglected because it is small compared to typical operation voltages (0.5V

compared to 50V). This leads to:

𝑊 ≈ 𝑥𝑛 ≈ 2휀0휀𝑆𝑖𝑒𝑁𝐷

𝑉𝑅

Figure 2.13 - Forward bias [2-6]

Figure 1.14 - Reverse biasing [2-6]


20

The depletion zone width increases so with the applied voltage, and reaches a maximum at which the

junction breaks down and becomes conductive (breakdown zone). This is also the point at which the

electric field reaches its maximum value:

𝐸𝑚𝑎𝑥 =2𝑉

𝑊≈

2𝑒𝑁𝐷

휀0휀𝑆𝑖𝑉𝑅

It also affects the capacitance value, fact that leads to identify the PN junction as with the same behavior of

a voltage dependent capacitor with its capacitance described by the following equation:

𝐶𝑗 = 𝐶𝑗0

1 +𝑉𝑅

𝑉0

with 𝐶𝑗0 = 휀𝑆𝑖 𝑞 𝑁𝐴𝑁𝐷

2 𝑁𝐴 + 𝑁𝐷 𝑉0

An useful application of this statement is to use the junction to form a LC oscillator circuit, that varies the

frequency by changing the VR (and so changing the capacitance).

2.5.11 - IV characteristics of PN junction

The current and voltage relationship of a PN junction is exponential in forward bias region, and relatively

constant in reverse bias region.

𝐼𝐷 = 𝐼𝑆 (𝑒𝑉𝐷𝑉𝑇 − 1)

Junction currents are proportional to the junction’s cross-

section area; so two PN junctions put in parallel are

effectively one PN junction with twice the cross-section

area, and hence twice the current. When a large reverse bias

voltage is applied, breakdown occurs and an enormous

current flows through the junction. There exist two kinds of

reverse breakdown: Zener and Avalanche breakdown. The

first is a result of the large electric field inside the depletion region that breaks electrons or holes off their

covalent bonds, while the second is a result of electrons or holes colliding with the fixed ions inside the

depletion region.

Figure 2 - IV characteristic [2-6]


21

2.6 – Semiconductor silicon detectors

In principle a semiconductor detector behaves like a ionization chamber, with a simple configuration made

by an absorbing medium, in the case the semiconductor, connected to two electrodes. The electrodes are

themselves connected to an external bias supply, that creates the electric field through the pn junction, and

when a particle passes to the material and generates charges particles12 this electric field drifts the

generated charges to the respective electrodes producing the outgoing signal. Thinking of the radiation in

terms of photons, as for first example, shows that the basis of photon detectors have to be found through

variations on the photoelectric effect, the Compton scattering, and the pair production.

By applying reverse bias to a PN junction, one is effectively storing charge on the equivalent of a parallel

plate capacitor (the depletion region is an insulator and the P and N regions are conductors). Disconnecting

the bias, if it happens that photons are absorbed within the depletion region, electron-hole pairs are

produced, and the electrostatic field within the depletion region sweeps the electrons to the N side and the

holes to the P side, decreasing the amount of stored charge. Re-applying the bias after some time, the

original charge is restored, and the current would reveal how many photons had been absorbed in the

depletion region. The sensitivity of this technique is limited by thermodynamic fluctuations, as also

specified in paragraph 2.5.1. Under conditions of thermodynamic equilibrium at a temperature T, the

uncertainty in the stored charge (the charge fluctuations at fixed voltage) is given by:

< 𝜍𝑄 >2= 𝑘𝐵𝑇𝐶

This is known as kBTC noise. To give some numbers, for a temperature of 150 K and a capacitance of 1 pF,

<σQ>2 is about 280 e- [], which is relatively high if one wants to detect individual photons. In order to be

limited to photon statistics rather than kBTC noise, one would need values of order 105 photons.

12 Electron-hole pair production energy for silicon is calculated to 3.69 eV [2-7], and is lower than the one for ionizing a gas, ~30eV

Figure 3 - Example of silicon detector geometry


22

2.6.1 - Photon detection

Coming to specific particle detection, it is possible that a photon interacts with a semiconductor and

creates charge when its energy is higher than the energy gap of silicon (1.12eV) and corresponds to a λ of

1.12μm (infrared region). If a photon has a wavelength longer than 1.02 μm, it will cross the silicon sensor

without being attenuated. For indirect bandgap semiconductor such as germanium and silicon the

absorption of a photon is made possible only with the involving of a phonon, that gives the additional

momentum necessary to the electron to jump to the conduction band. Indirect bandgap semiconductors

are characterized by an absorption coefficient growing gradually with the photon energy; when the photon

energy is high enough to allow the direct transition from the valence to the conduction band, phonons are

no longer required for the excitation, and the absorption coefficient saturates. For direct bandgap

semiconductors, such as GaAs, the coefficient grows for energies nearby the energy gap value, since the

transition does not require an extra particle like the phonons in order to conserve momentum.

2.6.2 – Charged Particle detection

The mean energy transferred per unit path length by charged particle passing through matter follows the

Bethe-Bloch formula, depending on parameters related to both the incident particle and to the physical

properties of the absorbed material. For high energy particles, the mean energy transferred to the matter

reaches a minimum, which is nearly the same for protons, electrons and pions and remain constant for

higher energies; particles having energies high enough to reach this minimum are knows as Minimum

Ionizing Particle (MIP). The typical energy spectrum of a MIP particle crossing a semiconductor material

follows a Landau distribution, characterized by evident asymmetric shape given by the long tail for high

energies losses due to high energy recoil electrons (δ rays). Due to the asymmetry, the Most Probable Value

(MPV) of energy loss, corresponding to the peak, differs from the mean energy lost, which is shifted at

higher energies.

2.6.3 - Functionality of silicon detectors

Detectors are reverse biased at the p+ side with negative voltage high enough to fully deplete the sensor.

Full depletion voltage can be calculated with:

𝑉𝑑𝑒𝑝𝑙 = 𝑑2𝑒

2휀0휀𝑆𝑖

𝑁𝐴𝑁𝐷

𝑁𝐴 + 𝑁𝐷

, where d is the thickness of the detector.

The electric field in the depletion zone can be calculated as:

𝐸 𝑥 = − 𝑉 + 𝑉𝑑𝑒𝑝𝑙

𝑑−

2𝑥𝑉𝑑𝑒𝑝𝑙

𝑑2

, where x is the depth in the detector.

Holes and electrons created by the passage of a mip through the depleted zone move respectively towards

p+ and n+ sides, driven by the electric field as:

𝑣𝑒,𝑕 = ∓𝜇𝑛 ,𝑝 𝐸(𝑥)


23

The depth reached by a charge carrier in function of the time is calculated with:

𝑥𝑒,𝑕 =𝑑 (𝑉 + 𝑉𝑑𝑒𝑝𝑙 )

2𝑉𝑑𝑒𝑝𝑙+ 𝑥0 −

𝑑(𝑉 + 𝑉𝑑𝑒𝑝𝑙 )

2𝑉𝑑𝑒𝑝𝑙 𝑒

∓2𝜇𝑛 ,𝑝𝑉𝑑𝑒𝑝𝑙 𝑡

𝑑2

, with x0 position at t=0.

Related velocity can be calculated with:

𝑑𝑥𝑒,𝑕

𝑑𝑡= ±𝜇𝑛 ,𝑝

2𝑉𝑑𝑒𝑝𝑙 𝑥0

𝑑2−

𝑉 + 𝑉𝑑𝑒𝑝𝑙

𝑑 𝑒

∓2𝜇𝑛 ,𝑝𝑉𝑑𝑒𝑝𝑙 𝑡

𝑑2

The electron is stopped at the xe(te)=d, and the hole at xh(th)=0, having drift times of:

𝑡𝑒 =𝑑2

2𝜇𝑛𝑉𝑑𝑒𝑝𝑙𝑙𝑛

𝑉 + 𝑉𝑑𝑒𝑝𝑙

𝑉 − 𝑉𝑑𝑒𝑝𝑙(1 −

𝑥0

𝑑

2𝑉𝑑𝑒𝑝𝑙

𝑉 + 𝑉𝑑𝑒𝑝𝑙)

𝑡𝑕 =𝑑2

2𝜇𝑝𝑉𝑑𝑒𝑝𝑙𝑙𝑛(1 −

𝑥0

𝑑

2𝑉𝑑𝑒𝑝𝑙

𝑉 + 𝑉𝑑𝑒𝑝𝑙)

The current induced by a moving charge q, that can be measured by a charge sensitive preamplifier, is:

𝑖 𝑡 =𝑞

𝑑

𝑑𝑥

𝑑𝑡=

𝑞

𝑑 −

𝑑𝑥𝑒

𝑑𝑡+

𝑑𝑥𝑕

𝑑𝑡 =

=𝑞

𝑑2 2𝑉𝑑𝑒𝑝𝑙

𝑥0

𝑑− (𝑉 + 𝑉𝑑𝑒𝑝𝑙 ) 𝜇𝑛𝑒

−2𝜇𝑛𝑉𝑑𝑒𝑝𝑙 𝑡0(𝑡𝑒−𝑡)

𝑑2 + 𝜇𝑝𝑒−2𝜇𝑝 𝑉𝑑𝑒𝑝𝑙 𝑡0(𝑡𝑕−𝑡)

𝑑2


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References to this chapter

[2-1]: W. R. Leo, Techniques for Nuclear and Particle Physics Experiments, Springer, Berlin, 1994,2.107

[2-2]: H.A. Bethe, W. Heitler, On the stopping of fast particles and the creation of positive electrons, Proc.

Royal Soc. London A 146 p.83, 1934

[2-3]: Physics Letters B, Review of Particle Physics, July 2008

[2-4]: http://www.pdf-search-engine.com/semiconductor-an-introduction-pdf.html

[2-5]: Donald A. Neamen, Fundamentals of Semiconductor Physics and Devices, McGraw-Hill Company

[2-6]: Behzad Razavi, Fundamentals of Microelectronics, John Wiley & Sons, Inc

[2-7]: R. C. Alig, S. Bloom, C. W. Struck, Scattering by ionization and phonon emission in semiconductors,

Physical Review B 22 p. 5565, 1980

[2-8]: S.M. Sze, Physics of Semiconductor Devices 2nd edn. ,Wiley, New York, 1981

[2-9]: Rossi

http://www.pdf-search-engine.com/semiconductor-an-introduction-pdf.html

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