Chapter 2 – Physics of semiconductor detectors
CHAPTER 2
PHYSICS OF SEMICONDUCTOR DETECTORS
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Chapter 2 – Physics of semiconductor detectors
2.1 – Interaction of electromagnetic radiation with matter
2.1.1 - 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):
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 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.
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
1
4.13x10-18 keV/sec
3x108 m/sec
3
1.24 eV*µm = 1240 MeV*fm
2
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Chapter 2 – Physics of semiconductor detectors
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:
that performs the exponential decrease of the number of particles N(x) at x given depth into the material
from the initial number , 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)
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Chapter 2 – Physics of semiconductor detectors
It follows an alternative way of expressing the exponential attenuation law:
The distance over which one half the initial beam is absorbed is called the half thickness,
and is related
to the linear attenuation coefficient and to the mean free path by:
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:
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:
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
6
mass per area
International System of measurements
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Chapter 2 – Physics of semiconductor detectors
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:
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):
PLOT CURVE XE
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Chapter 2 – Physics of semiconductor detectors
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[metto solo una referenza e non tutta la
formulaccia o no?]:
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.
7
h/ (mec) has units of length and equals 0.0024 nm
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Chapter 2 – Physics of semiconductor detectors
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.
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Chapter 2 – Physics of semiconductor detectors
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
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
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
(where E
is the particle's energy and x is the distance travelled):
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:
where ρ is the density of the material and ρx is the density-thickness.
ADD BETHE BLOCK
P12 jorne knetter 20-24
27-35: silicon detectors
Fino a 47 interessa
P 46 strip e pixel
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Chapter 2 – Physics of semiconductor detectors
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
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.
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Chapter 2 – Physics of semiconductor detectors
FIGURE DA COMMENTARE COL PHYSICS REVIEW E DATA BOOKLET p267
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Chapter 2 – Physics of semiconductor detectors
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.
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.
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Chapter 2 – Physics of semiconductor detectors
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.
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.
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Chapter 2 – Physics of semiconductor detectors
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.
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Chapter 2 – Physics of semiconductor detectors
INTERACTIONS OF NEUTRAL PARTICLES WITH MATTER (cenni?)
Neutral particles, such as the neutron, can interact with matter only through the nuclear interactions, that
produce charged particles allowing to detect their presence. Uncharged particles are not subject to
coulomb interaction8 , but loose the energy through strong interaction with the nuclei of the material. In
case of semiconductors, the particle knocks and then shift the nuclei from the original lattice position
creating vacancies or interstitial defects.
8
At the first order
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Chapter 2 – Physics of semiconductor detectors
2.2 – Physics and behavior 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).
2.2.1 - 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
En9. 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 number10 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 nature11, 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
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
9
n is a set of integer numbers
~ 1022 atoms/cm3
11
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)
10
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Chapter 2 – Physics of semiconductor detectors
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
compounding12, 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).
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
12
8N atomic rule
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Chapter 2 – Physics of semiconductor detectors
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
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 that fit for a high integration technology, and
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 (bandgap energy). There exists an exponential relation between the free-electron
density and Eg given by the formula:
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.
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
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Chapter 2 – Physics of semiconductor detectors
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
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:
It follows that, for n-type doped semiconductors:
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Chapter 2 – Physics of semiconductor detectors
and for p-type doped semiconductors:
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:
The current is calculated as shown in the following formula:
In general the drift current is expressed as:
while the total current is the sum of the current given by holes and electrons drifts:
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:
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Chapter 2 – Physics of semiconductor detectors
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:
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:
Surprisingly, there exists a relation between the drift and diffusion currents, NONOSTANTE they are totally
different: it is the Einstein’s relation
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Chapter 2 – Physics of semiconductor detectors
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.
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.
21
Chapter 2 – Physics of semiconductor detectors
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.
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.
;
It exist a built-in potential because of the junction:
22
Chapter 2 – Physics of semiconductor detectors
Reverse biasing
There are two ways for biasing the junction: one the is direct, the other is the reverse way. When the Ntype 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:
with
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.
23
Chapter 2 – Physics of semiconductor detectors
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 xaxis, 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.
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.
Junction currents are proportional to the junction’s crosssection 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
24
Chapter 2 – Physics of semiconductor detectors
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.
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
25
Chapter 2 – Physics of semiconductor detectors
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,
26
Chapter 2 – Physics of semiconductor detectors
2.3 – 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 particles this electric field drifts the generated
charges to the respective electrodes producing the outgoing signal.
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
27
Chapter 2 – Physics of semiconductor detectors
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
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 (semitransparent) metallic electrodes on top of it. Each electrode defines a MOS (metal-oxide-Si) capacitor.
28
Chapter 2 – Physics of semiconductor detectors
Positive bias applied to the electrodes creates depletion regions13 serving as storage regions for electrons14.
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 recombines15.
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 CCD16.
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
13
depleted of holes, the majority carriers in P-type silicon
minority carriers
15
it would reduce the detection efficiency
14
16
29
Chapter 2 – Physics of semiconductor detectors
(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.
Photon detection (zoboli questi due)
A photon can interact with a semiconductor and create charge when its energy is higher than the energy
gap of the material. For Silicon this value is 1.12eV and corresponds to a λ=1.12μm in the 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 have the 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. In all cases the incident electromagnetic radiation is attenuated
in the matter by an exponential decay described by this equation [gia’ scritta, metti il num]:
I(x) = I0 ・ e−_x (1.2)
where I0 is the initial beam intensity, x is the depth and α the absorption coefficient.
Particle detection (zoboli)
A charged particle passing through matter interacts with the atomic electrons through Coulomb forces. The
energy lost by the particle, known also as Ionizing Energy Loss (IEL), depends on its energy and on material
properties (such as density and atomic number). Electrons, for instance, due to their low mass, can be
easily deflected, whereas protons or heavy ions proceed straight in the matter.
Protons, pions, muons and electrons primary loose energy due to inelastic collision with the atomic
electrons of the material in small amount at every collision, and part of this energy creates free charge in
the material. The mean energy transferred per unit path length follows the Bethe-Bloch formula,
depending on parameters related to both the incident particle and to the physical properties of the
absorbed material [8].
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 as shown in Fig. 1.6a.
Particles having energies high enough to reach this minimum are knows as Minimum Ionizing Particle (MIP).
Fig. 1.6b plots the energy deposition of a MIP particle in a silicon detector.
30
Chapter 2 – Physics of semiconductor detectors
Figure 1.6: RIFO a) Energy loss as a function of energy for pions[3]. b) Typical Landau distribution energy
from a semiconductor detector due to a MIP particle.
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. In
Silicon, the most probable energy loss of a MIP corresponds to a most probable value of 80 electron/hole
pairs generated per micrometer. For a 300μm thick detector, the most probable collected charge is
therefore 24000el. (3.5fC). The energy range involved in the ATLAS experiment are well above this limit so
particles are always supposed to release charge in a MIP like fashion.
31