Chapter 2

1

SECT. A : Interaction of electromagnetic radiation with matter

2.1 - Overview

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 moving. These

quantities are all correlated together by the following equations (where h is Planck constant1 and c is the

speed of light in vacuum2):

𝜆 ∗ 𝜈 = 𝑐

𝐸 = 𝑕 ∗ 𝜈

𝐸 = 𝑕 ∗ 𝑐

𝜆

If one of these quantity is known, it is so possible to reach for all the others (the factor hc occurs so often in

atomic and nuclear physics that it can be considered as a separate constant3).

Different values of energy, frequency and wavelength creates the flavours of electromagnetic radiation, but

difference between them is evident only after the interaction with matter, when they show particle-like

behaviour out of wave-light behaviour. Hence in the definition of radiation the 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).

This chapter is meant to describe the physics that stands behind this interaction, what are the

consequences of it and how these principles are applied in semiconductor silicon detectors technology.

2.2 - 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.

1 4.13x10-18 keV/sec 2 3x108 m/sec 3 1.24 eV*µm = 1240 MeV*fm

Chapter 2

2

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.

2.3 - Photon interactions with matter

First the "all-or nothing" type interactions are considered.

2.3.1 - Attenuation coefficients

The description of the attenuation of a beam of particles, 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 attenuation coefficient4.

This 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.

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 that 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 attenuation coefficient:

4 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)

Chapter 2

3

𝜆 = 1

µ𝐿

It follows an alternative way of expressing the exponential attenuation 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 attenuation coefficient and to the mean free path by:

𝑥12

= ln(2)

µ𝐿= ln(2) ∗ 𝜆 = 0.693 𝜆

The attenuation 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

total amount of atoms ahead of its path (since they interact only with single atoms).

Therefore, it is useful to describe the attenuation process without the dependence on the density of

material, but only on the kind of material. This is obtained by introducing the mass attenuation coefficient

μm, which relates the linear attenuation coefficient to the density of the material ρ:

µ𝐿 = µ𝑚𝜌

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

whereas the linear attenuation coefficients differs greatly.

It is so possible to have a ULTERIORE definition of the attenuation 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 density5, 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 SI6 unit of density-thickness is kg*m-2, the obsolete unit g*cm-2 is still

used in the literature.

5 mass per area 6 International System of measurements

Chapter 2

4

If an absorber is made of a composite material the mass attenuation coefficient is readily calculated by

adding together the products of the mass attenuation coefficient and the proportion (α) of the mass due to

each element present in the material:

µ𝑚 𝑇𝑂𝑇𝐴𝐿 = (𝛼 µ𝑚 )

The law of attenuation always describes the attenuation 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

attenuation 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.

2.3.2 - Effects of photon interaction

Gamma rays, x rays and light are photons with different energies: depending on their energy and the

nature of the material, photons can interact in three main ways: photoelectric effect (or photoelectric

absorption), Compton scattering and pair production.

2.3.3 - Photoelectric effect

In order to remove a bound electron from an isolated atom a threshold energy is needed: it’s the ionization

potential, and it varies depending on what shell the electron occupies. It has been given a letter name to

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 following equation summarizes the

dependence of the ionization potential from the atomic number Z of the atom (so from the dimension of

the atom):

𝐸𝐾 = 𝑍2

PLOT CURVE XE

Chapter 2

5

The figure show that ionization cross section peaks just above threshold for each shell, to then fall rapidly

(≈ ν-3) at higher energy due to the difficulty in transferring the excess photon momentum to the nucleus.

For n > 1 there is subshell structure (2s, 2p1/2, 2p3/2, . . .). The photoelectric effect will be important in the

design of x-ray proportional counters.

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.

2.3.4 - 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, as shown. For an unbound electron

initially at rest, it is possible to have the following equations7:

mmm

Low energy photons lose little energy, while high energy photons, called γ rays, lose 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 following expresses Klein-Nishina formula:

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

7 h/ (mec) has units of length and equals 0.0024 nm

Chapter 2

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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.

2.3.5 - Pair production

Photons with energies in excess of 2mec2 produce electron-positron pairs, and an 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:

2.4 - Interactions of charged particles with matter

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

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

coulombic excitation. Eventually there could also be elastic collisions with a nucleus, Rutherford diffusion

and elastic collisions with an electron.

2.4.1 - Electro magnetic 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

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.

Individual interactions - scattering

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. 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).

Stopping power

Chapter 2

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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's 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.

ADD BETHE BLOCK

Excitation and ionization(riformula e connetti all’altro scritto)

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

transfered simply by passing close by, but only certain restricted values of energy can be transferred. 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 (ionisation). 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.

Energy loss by heavy particles

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

Chapter 2

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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 called delta rays. Mostly, however, the knock-on electrons lose their

energy within a very short distance of the interaction point.

The energy dependence of the rate of energy loss (stopping power) by excitation and ionization of heavy

particles for some typical materials is shown in figure 4.16. This graph is a plot of the energy-loss rate as a

function of the kinetic energy of the incident particle. Note that the stopping power is expressed using

density-thickness units. To obtain the energy loss per path length you would need to multiply the energy

loss per density-thickness (shown on the graph) 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. The diagram shows the rate of energy loss for the extreme cases of carbon (Z = 6)

and lead (Z = 82). 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.

Figure 4.16 metti quela con tutti gli elementi Vercelin

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 0.2 MeV *(kg *m-2)-1 (= 3 × 10-12 J*m2*kg-1 in SI units), and it slightly decreases with the increasing

atomic number of the absorbing material.

Chapter 2

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

range of that particle. Energy loss along the path is shown in figure 4.17. The 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.

Particles of the same kind with the same initial energy have nearly the same range for a given material. The

number of particles as a function of distance along the path is shown in figure 4.18. The final small variation

in the range is 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.

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.

Chapter 2

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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

(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

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.

del review physics p 267

Chapter 2

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Chapter 2

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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.

INTERACTIONS OF NEUTRAL PARTICLES WITH MATTER

Neutral particles (such as the neutron) can interact with matter only through the nuclear interactions. They

have to suffer nuclear interactions which produces charged particles before their presence can be

detected.

Chapter 2

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Physics of semiconductors

This chapter is meant as an exhaustive introduction about the physical principles that stand behind the

behaviour of devices made from semiconductor materials. Such 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).

Conduction in a solid

SemiconductorsBasics.pdf

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 (see figure

10.1).

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

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)

Chapter 2

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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

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 applies Boltzmann statistics [], which gives the density of electrons having energy greater than E0 (i.e.

n(E > E0) = e� E0 kBT ; kB = 1:3807 _ 10�23J/K, where kB is the Boltzmann constant).

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).

TABLE GROUP IV ELEMENTS (nella didascalia commenta anche gli altri elementi)

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 preferencial 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

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).

11 8N atomic rule

Chapter 2

15

TABELLA 1.4 http://www.pdf-search-engine.com/semiconductor-an-introduction-pdf.html

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.

TABELLA 5

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).

Silicon

The silicon is the most important (because the most exploited in applications) of the 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 (bandgap energy). There exists an exponential

relation between the free-electron density and Eg given by the formula:

𝑛𝑖 = 5.2 ∗ 1015𝑇32 𝑒−

𝐸𝑔

2𝑘𝑇 [ 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠

𝑐𝑚3]

For example, at T=300 K, ni = 1.08 x 1010 electrons/cm2, and at T=600 K, ni = 1.54 x 1015 electrons/cm2.

In pure silicon at equilibrium, the number of electrons is equal to the number of holes. The silicon is 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 being generated by thermal ionization and in order to preserve equilibrium previously

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

temperature.

Doping

DEF INTRINSIC ESTRINSIC SEMICOND

In order to fabricate a power-switching device, it is necessary to increase greatly the free hole or electron

population. This is achieved by deliberately doping the silicon, by adding specific impurities called dopants.

The doped silicon is subsequently called extrinsic and as the concentration of dopant Nc increases, the

resistivity ρ decreases.

Chapter 2

16

Pure silicon can be doped with other elements in order to change its electrical properties: it can be doped

with P (phosphourous), and so it will have more electrons (type N doping, with free negative charges), or

with B (boron), and so it will have 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 - the 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 ionised 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 - the acceptor

concentration.

To be pointed out that for manufactury 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.

Charge Carriers

Chapter 2

17

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.

The product of electron and holes densities 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

𝑀𝑖𝑛𝑜𝑟𝑖𝑡𝑦 𝑐𝑎𝑟𝑟𝑖𝑒𝑟𝑠 𝑝 ≈ 𝑛𝑖

2

𝑁𝐷

and for p-type doped semiconductors:

𝑀𝑎𝑗𝑜𝑟𝑖𝑡𝑦 𝑐𝑎𝑟𝑟𝑖𝑒𝑟𝑠 𝑝 ≈ 𝑁𝐴

𝑀𝑖𝑛𝑜𝑟𝑖𝑡𝑦 𝑐𝑎𝑟𝑟𝑖𝑒𝑟𝑠 𝑛 ≈ 𝑛𝑖

2

𝑁𝐴

Charge transportation

A first mechanism of charge transportation into semiconductor can be focused in the so called drift

mechanism: it is simply the application of an electric field at the extremity of the semiconductor, and the

charge particles will move at a velocity proportional to the electric field given:

𝑣𝐻 = µ𝑃𝐸

𝑣𝑒 = −µ𝑛𝐸

The current is calculated as shown in the following formula:

𝐼 = −𝑣𝑊𝑕𝑛𝑞

In general the drift current is expressed as:

Chapter 2

18

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.

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, as shown in the plot:

Chapter 2

19

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, NONOSTANTE they are totally

different: it is the Einstein’s relation

𝐷

µ=

𝑘𝑇

𝑞

PN junctions

A pn junction is the location in a 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 semiconductor, a PN junction (or diode)

is 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.

Chapter 2

20

Processes forming p-n junctions

CHP1 BARRY

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.

The fixed ions in depletion region create an electric field that results in a drift current.

Chapter 2

21

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. The figure shows the charge profile of the PN junction.

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

It exist a built-in potential because of the junction:

Reverse biasing

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 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 electric field across the junction.

Varying the value of VR it is consequently possible to vary also the width of the depletion zone, changing

also the capacitance value: this leads to identify the PN junction as with the same behavior of a voltage

dependent capacitor. Its capacitance is described by the following equation:

Chapter 2

22

𝐶𝑗 = 𝐶𝑗0

1 +𝑉𝑅

𝑉0

with 𝐶𝑗0 = 𝜀𝑆𝑖 𝑞 𝑁𝐴𝑁𝐷

2 𝑁𝐴 + 𝑁𝐷 𝑉0

A 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).

Forward bias

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 electric field.

Under forward bias, minority carriers in each region increase due to the lowering of built-in field/potential.

Therefore, diffusion currents increase to supply these minority carriers.

TANTE EQ

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.

Chapter 2

23

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.

Constant-voltage diode model and reverse breakdown

Diode operates as an open circuit if VD< VD,on and a constant voltage source of VD,on if VD tends to

exceed VD,on.

When a large reverse bias voltage is applied, breakdown

occurs and an enormous current flows through the

diode.

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,

whilethe second is a result of electrons or holes colliding with the fixed ions inside the depletion region.

Chapter 2

24

Text Book: Fundamentals of Semiconductor Physics and Devices, First Edition

Author: Donald A. Neamen

Publishing: McGraw-Hill Company

Dear Fabio,

The materials are from the text book: Fundamentals of Microelectronics. The author is Behzad

Razavi. I just used the teaching slides provided by the author. I think you had better refer to the text

book. The details of the text book are as follows:

Book title: Fundamentals of Microelectronics

Author: Behzad Razavi

Publication company: John Wiley & Sons, Inc.

Year: 2008

Hopefully the information will be useful for you.

Sincerely,

Jen-Shiun Chiang

Professor

Department of Electrical Engineering

Tamkang University

Tamsui, Taipei, Taiwan ----- Original Message -----

From: Fabio Rivero

To: chiang@ee.tku.edu.tw

Sent: Wednesday, August 12, 2009 6:18 PM

Subject: reference request

Goodmorning Dr. Chang, sorry if I disturb you, I am an italian student of Biomedical Physics working on my master theses about 3D Silicon Detectors at Cern. Looking for references on the subject "physics of semiconductors" in the internet, I went through your slides on this web-page: http://www.ee.tku.edu.tw/~chiang/courses/electronics-1/electronics.htm "ch02.pdf". I found that some pictures you put into are really good at explaining the semiconductors behaviour and characteristics, so I want to ask you if I can borrow some of them for my thesis (citing your slides in my theses references) or knowing the reference from which you took them. I would also be glad if you have other interesting links or references on the subject. Thank you very much,

Chapter 2

25

Detectors

The study of physical laws that stand behind interaction between radiation and matter leads to the fact

that to study electromagnetic radiation it is necessary for the radiation to interact in some fashion with

some physical ”detector”. Electromagnetic radiation can be thought of in terms of waves or of particles, but

there is no strict dividing line between these views since electromagnetic radiation always retains both

particle and wave characteristics. The particle viewpoint is more useful when at high frequencies, where

the photons are energetic, whereas the wave viewpoint is more useful when at low frequencies. To

determine which viewpoint may be more useful it is to be considered the average photon occupation

number of the modes of the radiation field.

Thinking of the radiation in terms of photons, as for first example, shows that the basis of photon

detectors have to be found through the variations on the photoelectric effect, the Compton scattering, and

the pair production.

Solid State Detectors

Silicon is characterized by a unit cell of crystalline with face-centered cubic structure (like the one of

diamond) in which each atom has the four nearest neighbors joined together in a tetrahedral configuration

by a single covalent bond containing two electrons, one and one given by both of each atom.

It is useful to represent this three-dimensional construction in two dimensions with the Si atoms forming a

regular square grid. It is limited by the fact that it does not accurately represent the atomic arrangement far

beyond that of nearest neighbors, and any particular atom is not simultaneously the nearest neighbors of

another atom.

FIGURE

Highly purified silicon is known as an intrinsic semiconductor and is a poor conductor of electricity since the

electrons are tied up in valence bonds. However, if one of those bonds is broken by absorption of a photon

or by thermal excitation, then an electron is raised in energy into the conduction band, leaving behind a

hole. Both the electron and hole are mobile charge carriers, although they may have very different

mobilities.

An extrinsic semiconductor is formed if one of the silicon atoms is replaced by an impurity atom with a

different number of valence electrons. For example, arsenic atoms have five valence electrons. So when an

arsenic atom is placed in a silicon crystal, there is an extra electron not tied up in the valence bonding,

which therefore is a free carrier. This is known as N-type doping, since it provides excess negative charge

carriers. Boron has three valence electrons, so a boron impurity leaves a hole as a free carrier. This is P-type

doping since the hole carries positive charge. Consider what would happen if a piece of P-type Si and a

piece of N-type Si were joined together. At the boundary there would be a high concentration of free

electrons on the N side, and some of these electrons would diffuse across the boundary into the P side.

There would likewise be a high concentration of holes on the P side, and some of these would diffuse

Chapter 2

26

across the boundary to the N side. The donor and acceptor atoms would be left behind with static positive

and negative charges, which would set up an electrostatic field. After enough carriers had diffused across,

this field would prevent further diffusion. The region around the boundary would be depleted of the

majority charge carriers (electrons on the N side and holes on the P side) and is appropriately called the

depletion region. By applying reverse bias (positive voltage to the N side and negative voltage to the P side)

one can increase the size of the depletion region.

Silicon Diode Detectors

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). Imagine

disconnecting the bias. If photons are absorbed, let’s say, within the depletion region, electron-hole pairs

are produced. The electrostatic field within the depletion region will sweep the electrons to the N side and

the holes to the P side, decreasing the amount of stored charge. After some time one could reapply the

bias, restoring the original charge, 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. In

thermodynamic equilibrium at a temperature T, the uncertainty in the stored charge (the charge

fluctuations at fixed voltage) is given by

(#Q)2 = kTC (5-11)

This is known as kTC noise. For a temperature of 150 K and a capacitance of 1 pF,

#Q " 280 e− (5-12)

which is relatively high if one wants to detect individual photons. In order to be limited by photon statistics

rather than kTC noise, one would need of order 105 photons ('105 " 300). There is also dark noise from

thermally activated leakage currents (which depend exponentially on temperature).

Charge-Coupled Devices (CCD)

Instead of a PN junction, another way of obtaining a silicon detector is to start with a P-type substrate of

silicon, growing on the surface an insulating oxide (SiO2) and then depositing small and thin (semi-

transparent) metallic electrodes on top of it. Each electrode defines a MOS (metal-oxide-Si) capacitor.

Positive bias applied to the electrodes creates depletion regions12 serving as storage regions for electrons13.

A charge-coupled device (CCD) is so obtained, consisting of a 2 dimensional array of such pixels.

The detection characteristics of CCD arrays depends on the method of illumination and certain physical

characteristics of the manufacturing. In all cases photons enter the semiconductor and are absorbed in or

near the depletion region. If the absorption occurs inside the depletion region, the electron is drawn

towards the positively charged electrode and trapped by the oxide, and the hole is expelled from the

depletion region, while if the absorption occurs outside the depletion region it is necessary for the electron

to diffuse to the boundary of the depletion region before the electron recombines14.

12 depleted of holes, the majority carriers in P-type silicon 13 minority carriers 14 it would reduce the detection efficiency

Chapter 2

27

Some CCDs are front illuminated, meaning that the radiation passes through the semi-transparent

electrode. These are typically thick CCDs which have enhanced response in the red portion of the spectrum

since the thick device is able to contain several optical absorption lengths, even at long wavelengths where

the absorption length is typically longest. Quantum efficiencies are typically of order 70%. Thinned CCD’s

have been etched away from the underside, the back, and are typically back illuminated. They have poorer

red response because the thickness of the remaining material is only of the same order as the absorption

length in the red, while they have better blue response and higher peak quantum efficiencies, typically

nearly 90%.

The name CCDs is given by their method of signal readout . CCD’s are essentially shift registers, which

preserve the integrity of the trapped charge bundles with charge-transfer efficiencies of order )CT ( 0.99999

per shift as the packets are shifted across the device in a ”bucket brigade” technique to a readout amplifier.

The readout generally takes place after the exposure and can be in the form of, for example, sequential

readout of the final column of the CCD followed by a single step of all rows over by one column to

repopulate the final column. This is iterated until the entire device is read out. An alternative technique is

used in the Sloan Digital Sky Survey (SDSS) in which the telescope is stationary and the stars drift across the

focal plane and the CDD (drift scan) in synchronism with the rate of charge packet shift across the CCD15.

Advantages of CCD detectors for astronomy include high quantum efficiency, good linearity, low readout

noise (( 3 e− RMS), and large numbers of pixels (the SLOAN CCD camera has a total of 120 Megapixels,

enough to do simultaneous multi-color photometry over wide fields). Electron storage capacity can be of

order 105 − 106 electrons per pixel, which gives a large, but limited dynamic range. Image defects can be

caused by cosmic ray hits and by spillover from overfull packets due to bright sources. One does not

typically use CCD’s for high-speed photometry since the readout takes time. Careful data reduction

techniques for CCD’s include measurement of dark frames (which need to be subtracted) and ”flat fields”

(images under uniform illumination) by which the images are divided to obtain gain-corrected images.

The above discussion is somewhat oversimplified. There are varieties of semiconductor manufacturing

processes and varieties of readout techniques for CCD’s.

As an example, many Hubble Space Telescope instruments have used CCD’s: these include the STIS (Space

Telescope Imaging Spectrograph), scheduled for repair in 2009 during Servicing Mission SM4, the ACS

(Advanced Camera for Surveys), also scheduled for repair, and the ultraviolet-visible channel of WFC3

(Wide Field Camera 3), to be newly installed during SM4.

15

Chapter 2

28

INTERACTIONS OF CHARGED PARTICLES WITH MATTER

Charged particles, such as electrons, protons and alpha particles, can interact with matter

electromagnetically or through weak or strong nuclear interactions. The electromagnetic interaction is by

far the most common, and it involves collisions with electrons in the absorbing material . Neutral particles

such as the neutron can interact only through the nuclear interactions (Rutherford scattering). Thus

charged particles can be detected directly by their electromagnetic interactions whereas neutral particles

have to suffer nuclear interactions that produce charged particles before their presence can be detected.