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 constant 1 and c is the speed of light in vacuum 2 ): ∗ = = ∗ = ∗ 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 constant 3 ). 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 3x10 8 m/sec 3 1.24 eV*μm = 1240 MeV*fm
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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.
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
6
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
7
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
8
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
9
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
10
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
11
Chapter 2
12
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
13
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
14
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
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,