Chapter 2 – Physics of semiconductor detectors
CHAPTER 2
PHYSICS OF SEMICONDUCTOR DETECTORS
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Chapter 2 – Physics of semiconductor detectors
This chapter is meant to describe the basic physics that stands behind interaction of radiation and particles
with matter, what are its consequences and how these principles are applied in semiconductor silicon
detectors technology. In the real world, energy moving through space is identified with the name of
electromagnetic radiation, and it is characterized by quantity of energy E, speed c, frequency ν and
wavelength λ with which is moving1. Different values of energy, frequency and wavelength creates the
flavors of electromagnetic radiation, but differences between them are evident only after interaction with
matter, when they show particle-like behavior out of wave-light one. Hence in the definition of
electromagnetic radiation charged particles are included, such as alpha and beta radiation, beams of
charged particles created by accelerating machines, electromagnetic radiation or photons, and beams of
neutral particles such as neutrons.
2.1 - Electromagnetic and particulate radiation
The principal types of radiation can be first divided into two main categories: electromagnetic (X-rays,
produced outside the nucleus and γ-rays, emanated from within nuclei) and particulate (α particles,
protons, neutrons, electrons β-, positrons β+). This distinction, as already mentioned, belongs to the proper
“history” of the radiation, drawn by the history of the particle (subject connected to the concepts of energy
loss of a particle, range, interactions) and by the history of the target atoms (that leads to displacements,
recombination, ionization, excitation, radiation damage and build-up concepts). A beam of radiation that
passes through matter can lead to the complete absorption (electronic transitions and vibration-rotational
transitions), to some scattering (Rayleigh, Rutherford, Raman and Mie scattering) and/or to the passage
with no interaction. These processes can be explained in terms of interactions between particles that are
stopped or scattered. The basic effect of the interaction can be the scattering, absorption, thermal
emission, refraction, and reflection of the incoming radiation. With the absorption and emission spectra (of
molecules) it is possible to outline characteristic structures and so to identified and quantified molecules
by these ‘fingerprints’. The spectra are determined by position (wavelength) of absorption/emission line,
knowing the difference of energy levels of the transition and by strength of absorption/emission line,
knowing the probability of the transition. The most commonly used transition is the electron transition in
the atoms and vibration-rotational modes in the molecules. Moreover, a particle travelling through matter
can lose energy gradually (losing energy nearly continuously through interactions with the surrounding
material), or catastrophically (moving through with no interaction until losing all its energy in a single last
collision). Gradual energy loss is typical of charged particles, whereas photon interactions are of the "all-ornothing" kind.
2.2 - Photon interactions with matter
Gamma rays, x rays and light are photons with different energies. Depending on their energy and on the
nature of the material, photons interact electromagnetically with charge particles of matter, and they can
do it in three main ways: with the Photoelectric Effect (or Photoelectric Absorption), the Compton scattering
and the Pair Production. It is also important to mention the Rayleigh Scattering, which consists in the
diffusion of the photons over the electrons of atoms, without ionization or excitation of the atoms: this
process gives birth to all the colors we see.
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correlated together by fundamental relativistic equations
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Chapter 2 – Physics of semiconductor detectors
2.2.1 - Photoelectric effect
This process consists in the absorption of a photon with consequent expulsion of an electron out from the
atom hit. In order to remove a bound electron from an isolated atom a threshold energy for the photon is
needed: it’s the ionization potential, and it varies depending on what atomic shell the electron occupies. If
the energy of the photon Eγ overtakes the ionization potential (bound energy EB2), an electron will be
emitted out of the atom with energy Ee given by the following formula:
It has been given a letter to name the shells (K, L, M ...) depending on the principal quantum number (n = 1,
2, 3, ...). As example, for hydrogen atom H the ionization potential from n=1 corresponds to an ultraviolet
photon, but for heavier elements the K-shell ionization shifts rapidly into the x-ray regime. The ionization
potential depends on the square of the atomic number Z of the atom ( and so from the dimension of the
atom), as given from cross-section formulas for the
effect3.
For example, Figure 2.1 shows the cross-section behavior
for the plumb atom, pointing out that ionization cross
section peaks just above threshold for each shell, to then
fall rapidly at higher energy due to the difficulty in
transferring the excess photon momentum to the nucleus.
For n > 1 there are sub-shell structures (2s, 2p1/2, 2p3/2, .
. .). The photoelectric effect is important in the design of
x-ray proportional counters, as an example. In this thesis
the photoelectric effect has been used in order to prove
the calibration of 3D silicon detectors, trying to reproduce
with them the photoelectric peaks of some known gamma
radioactive elements (241Am and 109Cd).
Figure 2.1 - Pb photoelectric cross-section
When other atoms are present, as in molecules and solids, the electronic energy levels will be very
different, as will the photoelectric cross sections. For solids in vacuum, the thresholds can be ≈ 1 eV and it
depends on the crystalline structure and on the nature of the surface. The ionization potential in this case is
usually called work function. Photon absorption efficiencies approach 100% in the visible and ultraviolet,
but the overall device efficiencies are limited by the electron escape probabilities. In a semiconductor a
photon can be thought of as ”ionizing” an atom, producing a ”free” electron which remains in the
conduction band of the lattice. Thresholds are of order 0.1–1 eV for intrinsic semiconductors and of order
to 0.01–0.1 eV for extrinsic semiconductors. The latter photon energies correspond to infrared photons.
Photochemistry is somewhat similar in that photons produce localized ionization or electronic excitation. In
the end, the escaping electron produces a redistribution of the atomic electrons, that can lead to
Fluorescence (emission of photons) or Auger Effect (emission of characteristic X ray radiation).
2
3
1≤EB≤100 KeV, depending on the shell and on the atom
There are correct different formulations for low and high energy behaviour
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Chapter 2 – Physics of semiconductor detectors
2.2.2 - Compton scattering
The Compton scattering takes place when a photon scatters off a free (or bound) electron, yielding a
scattered photon with a new, lower frequency and a new direction. For an unbound electron initially at
rest, it is possible to have the following equations4:
with hν and hν’ initial and final energy, θ photon angle change, me electron mass and c speed of light. Low
energy photons loose little energy, while high energy photons, called γ rays, loose a lot of energy. The
wavelength increases by of order 0.0024 nm, independently from the wavelength. The Compton cross
section is given by the Klein-Nishina formula[e.g.2-1]. The largest Compton scattering cross section is at
small energy, and it decreases monotonically with energy. At low energies lots of scattering events take
place, but very little energy is lost. It is a consequence that the energy absorption cross section is small at
low energy because little energy is transferred to the electron, and it rises to a peak for photon energies
around 1 MeV that declines at higher energy.
2.2.3 - Pair production
Photons with energies in excess of 2mec2 produce electron-positron pairs, and interaction with a nucleus is
needed in order to balance momentum. The pair production cross section starts at 1.022 MeV for then
rising to an approximately constant value at high photon energy, in the gamma ray region of the spectrum
of electromagnetic radiation. Cross sections scale with the square of the atomic number, and complete
formulas describing the cross-section are Bethe-Heitler formulas [e.g. 2-2].
2.3 - Absorption coefficient
The description of the absorption of a beam of photons, all with the same energy and all travelling in the
same direction, is given by an exponential law:
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 absorption coefficient5 given by:
with σ cross-section, Na Avogadro’s constant, ρ density of the material, A molecular weight. The probability
of interaction of the photons is given by total cross-section of photon is given by the sum of the single
effect cross-sections, which are summarized in Figure 2.2:
4
h/ (mec) has units of length and equals 0.0024 nm
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
Figure 2.2 – Absorption coefficient for Rayleigh effect, Photoelectric effect, Compton effect and Pair Production
For the silicon (Z=14), and for energies of photons under 100 KeV, the dominant effect is the photoelectric
effect, whereas for over 10 MeV it is the pair production. Giving out some values, the attenuation
coefficient for 241Am, which decades with gamma rays of 59.5 KeV, is 0.3 cm2g-1 , and the probability of
detection for silicon detector of 300 µm depth is only of 2%. This because of the fact that cross-sections for
photons are really low, and consequently also the probability of detection are low, too. Nevertheless,
gamma sources are suited for calibrating silicon detectors because the whole photon energy can be
detected in the sensor with the assumption that electrons doesn’t escape from the detector.
The absorption law follows from the fact that, over any short distance, the probability of losing a particle
from the beam is proportional to the number of particles left into it: if particles are present in high number
many are going to be lost, but if the number left decreases the same does the rate of loss. Moreover, the
exponential attenuation law does not describe what happens to the energy carried by the photons
removed from the beam, and it is possible that some of those may be carried through the medium by other
particles, including some new photons. The average distance travelled by a photon before it is absorbed is
given by λ, the attenuation length or mean free path, that is the reciprocal of the linear absorption
coefficient:
It follows an alternative way of expressing the exponential absorption law:
The distance over which one half the initial beam is absorbed is called the half thickness,
to the linear absorption coefficient and to the mean free path by:
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and is related
Chapter 2 – Physics of semiconductor detectors
The absorption of photons depends on the total amount of material in the beam path, and not on how it is
distributed, because the probability for a photon to interact somewhere within the matter depends on the
total amount of atoms ahead of its path (since they interact only with single atoms). Therefore, it is useful
to describe the absorption process without the dependence on the density of material, but only on the kind
of material. This is obtained by introducing the mass absorption coefficient μm, which relates the linear
absorption coefficient to the density of the material ρ:
This means, for example, that the mass absorption coefficient is the same for ice, liquid water and steam
whereas the linear absorption coefficients differs greatly. It is so possible to have an ulterior definition of
the absorption law:
that states that the total attenuating effect of a slab of given type material can be described by quoting the
mass attenuation coefficient, which is characteristic of the material's chemical composition, and the photon
energy, together with the material's density and thickness. The product ρx, the areal density6, of a thickness
x of the attenuating material is also called the density-thickness, and is often quoted instead of the
geometrical thickness x. Although the SI7 unit of density-thickness is kg*m-2, the obsolete unit g*cm-2 is still
used in the literature. If an absorber is made of a composite material the mass absorption coefficient is
readily calculated by adding together the products of the mass absorption coefficient and the proportion
(α) of the mass due to each element present in the material:
The law of absorption always describes the absorption of the original radiation. If the radiation changes,
degrades in energy, it is not completely absorbed or if secondary particles are produced, then the effective
absorption decreases, and so the radiation penetrates more deeply into matter than predicted. It is also
possible to have an increasing number of particles with depth in the material: this process is called build-up,
and has to be taken into account when evaluating the effect of radiation shielding, for example.
2.4 - Interactions of charged particles with matter
The most common way in which charged particles (such as electrons, protons and alpha particles) can
interact with matter is the electromagnetic interaction, that involves collisions with electrons in the
absorbing material and is the easiest mechanism to detect them. They can also interact through one of the
two kinds of nuclear interactions, the weak interaction or the strong interaction. Principally, they either
loose all their energy, or they are deflected form the original trajectory. The main process of energy loss
producing excitation and ionization is the inelastic collisions with an electron; it can also happen an inelastic
collisions with a nucleus, that leads to Bremsstrahlung and coulombian excitation. Eventually there could
also be elastic collisions with a nucleus (Rutherford diffusion) and elastic collisions with an electron.
2.4.1 - Electromagnetic interaction
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mass per area
International System of measurements
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Chapter 2 – Physics of semiconductor detectors
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. Unlike
photons, each charged particle suffers many interactions along its path before finally coming to rest, losing
only a small fraction of its energy during every interaction (for example, a typical alpha particle might make
50000 collisions before it stops). Hence the energy loss can usually be considered as a continuous process
(scattering). Although the amount of scattering at each collision may be small, the cumulative effect may
be quite a large change in the direction of travel. Occasionally an incident particle passes very near a
nucleus and then there is a single large deflection (this nuclear scattering effect is most pronounced for
light incident particles interacting with heavy target nuclei).
2.4.2 - Excitation and ionization
Electromagnetic interaction between the moving charged particle and atoms within the absorbing material
is the dominant mechanism of energy loss at low (non-relativistic) energies; it extends over some distance,
keeping not necessary for the charged particle to make a direct collision with an atom. Energy can be
transferred simply by passing close by, but only certain restricted values of energy can be exchanged. The
incident particle can transfer energy to the atom, raising it to a higher energy level (excitation) or it can
transfer enough energy to remove an electron from the atom altogether (ionization). This is the
fundamental mechanism operating for all kinds of charged particles, but there are considerable differences
in the overall patterns of energy loss and scattering between the passage of light particles (electrons and
positrons), heavy particles (muons, protons, alpha particles and light nuclei), and heavy ions (partially or
fully ionized atoms of high Z elements). Most of these differences arise from the dynamics of the collision
process: in general, when a massive particle collides with a much lighter particle, the laws of energy and
momentum conservation predict that only a small fraction of the massive particle's energy can be
transferred to the less massive particle. The actual amount of energy transferred will depend on how
closely the particles approach and from restrictions imposed by quantization of energy levels. The largest
energy transfers occur in head-on collisions.
2.4.3 - Energy loss by electrons and positrons
Concerning the electrons and positrons loss of energy, they also ionize but with several differences with
heavy particles (for example they have lower loss rates at high energies than heavier particles travelling at
the same speed). There is also a slight difference between the interactions of positrons and of electrons,
resulting in a slightly higher energy loss for the positrons. An electron is easily scattered in collisions with
other electrons because of its light mass: as a result, the final erratic path is longer than the linear
penetration (range) into the material, with greater straggling.
2.4.4 - Bremsstrahlung effect
Literally translated from German into 'braking radiation', Bremsstrahlung is an effect that occurs whenever
the speed or direction of a charged particle motion changes (when it is accelerated), and consist in the
emission of electromagnetic energy (photons) when the acceleration takes place. It is most noticeable
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Chapter 2 – Physics of semiconductor detectors
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. The kinetic energy at which energy loss by radiation equals the
energy loss by collisions is called critical energy, Ec, and is approximately
It’s also interesting the quantity called radiation length, that is the distance over which the energy of an
incident particle is reduced by a factor e-1 (0.37) due to only radiation losses.
2.4.5 - Electron-photon cascades
A high energy electron performing Bremsstrahlung results in a high energy photon as well as a high energy
electron, and a high energy photons performing pair production results in a high energy electron as well as
a high energy positron: in both cases two high energy particles are produced from a single incident particle.
It follows that the products of one of these processes can be the incident particles for the other, with the
result of a cascade of particles which increases in number while decreasing in energy per particle, until the
average kinetic energy of the electrons falls below the critical energy. The cascade is then absorbed by
ionization losses. Such cascades, or showers, can penetrate large depths of material.
2.4.6 - Stopping power
The most important way to describe the net effects of charged particle interactions with matter and the
rate of energy loss along the particle's path is with the linear stopping power Sl, also known as
(where E
is the particle 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:
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Chapter 2 – Physics of semiconductor detectors
where ρ is the density of the material and ρx is the density-thickness.
2.4.7 - Energy loss by heavy particles
Charged particles lose their energy when interacting with matter; consequently they leave track of their
passages, with differences for every kind of particle. Knowing these processes has allowed to develop
detectors able to track the particles. Assuming that the speed and mass of the atomic electron is negligible
respect to that of the incoming particle, it is possible to have a classical model of energy loss of particle
through matter in the Bethe-Block formula [2-1]:
with ρ density of the material, x depth into the material, D constant, β and γ relativistic parameters of the
particle, z charge of the particle, Z atomic number of the material that is traversed, A atomic weight of the
material, me mass of the electron, c speed of light, I average ionization energy of the material, δ and C
relativistic corrections of the formula, r0 classical electron radius, NAV Avogadro’s constant.
Figure 2.3 - Energy loss of µ on Cu [2-3]
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Figure 2.5 - Energy loss for heavy charged particles in
different materials [2-3]
Chapter 2 – Physics of semiconductor detectors
Figure 2.4 - Energy loss for different particles [2-3]
Figure 2.3 show the behavior of energy loss for a µ particle in a wide set of momentum, while Figure 2.4
shows the same for different particles. Figure 2.5 shows the energy loss for heavy charged particles in
different materials, pointing out the fact that the minimum of the curve varies from 1.15 MeV of Pb to 2
MeV of He, with exception for the H2. These graphs are plots of the energy-loss rate as a function of the
kinetic energy of the incident particle. It’s important to
notice that in Figure 2.4 the stopping power is expressed
using density-thickness units. To obtain the energy loss per
path length it needs to multiply the energy loss per densitythickness, as shown in Figure 2.5, by the density of the
material. As for photon interactions, it is found that when
expressed as loss rate per density-thickness, the graph is
nearly the same for most materials. There is, however, a
small systematic variation; the energy loss is slightly lower in
materials with larger atomic numbers. At high incident
energies there is also some variation with density of the same material because a higher density of atomic
electrons protects the more distant electrons from interactions with the incident particle. This results in
lower energy loss rates for higher densities. Concerning the silicon, in Figure 2.6 it is shown its behavior.
Figure 2.6 - Energy loss in silicon
For low energies the stopping power varies approximately as the reciprocal of the particle's kinetic energy.
The rate of energy loss reaches a minimum called Minimum Ionization Point (MIP), to then start to increase
slowly with further grow in kinetic energy. Minimum ionization occurs when the particle's kinetic energy is
about 2.5 times its rest energy, and its speed is about 96% of the speed of light in vacuum. Although the
energy loss rate depends only on the charge and speed of the incident particle but not on its mass it is
convenient to use kinetic energy and mass rather than the speed. At minimum ionization the energy loss is
about 2 MeV cm2 g-1 (= 3 × 10-12 J∙m2∙kg-1 in SI units), and it slightly decreases with the increasing atomic
number of the absorbing material. Given that the minimum of the curve is quite the same for all particles in
all materials, it is of common use to define the MIP, used to quantify a detector response without the need
to refer to a specific particle, as the minimum signal that can be detected. For silicon, the <dE/dx>min≈1.66
MeV cm2 g-1. The distribution of probability for lost energy by a particle in a single hit with a material
electron follows a Landau curve, because events with high energy exchange can happen but are less
probable. Experimentally, a Gauss curve is obtained only when the depth of the material allows to have
enough hits with its atoms. For thin depts., hits with atoms are not a lot, and hits with high energy loss can
produce a tail of the distribution through high energies. For example, when a particle hits an atom it is
possible that electrons are free to move in the material, and if with enough energy they can ionize by
themselves creating secondary couples of ions and electrons. Historically these ionizing electrons are called
δ rays.
Reassuming, the charged particle leaves in the material a track formed by ion-electron couples and photons
produced by disexcitation. For every material there exist an energy value for the production of couples,
that does not depend from the particle energy, defined as:
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Chapter 2 – Physics of semiconductor detectors
with ∆E energy given by the incident particle, nT number of couples created. In the definition of nT there are
included the primary nP and secondary nS couples as:
In silicon the mean energy loss is of 1.66 MeV cm2 g-1, and the density is of 2.33 g cm-3; it means that the
loss of energy is of 390 eV/μm. Then to generate a hole-electron couple an energy of 3.6 eV is needed, and
it follows that a MIP creates ~110 couples per μm in silicon. For a thickness of 250 μm a MIP creates about
20000 hole-electrons couples, with 27000 as mean amount and 19400 as most probable value (MPV). In
Figure 2.7 this calculation is shown for different depth of silicon.
Figure 2.7 - Mean and Most Probable Value of energy loss of a MIP in
different thickness of silicon [2-3]
A massive particle that collides with an electron loses relatively small quantity of energy at each collision.
For example, a slow alpha particle hitting an electron transfers a maximum of only 0.05% of its energy to
the electron. Since head-on collisions are rare, usually the energy loss is much lower. In order to
significantly reduce the incident particle's energy many collisions are needed, so the energy loss can be
considered as a continuous process. Although the energy given to an electron may be a small fraction of
the incident energy, it may be sufficient to ionize the atom and for making the ejected electron travel some
distance away from the interaction point, leaving a trail of excited and ionized atoms of its own. These
'knock-on' electrons can leave tracks from δ rays. Mostly, however, the knock-on electrons lose their
energy within a very short distance of the interaction point.
Before losing all its kinetic energy into the material, a penetrating particle follows some distance, called
range of that particle, characterized by a rise near the end of the path is due to the increased energy loss
rate at low incident energies. At very low speeds the incident particle picks up charge from the material,
becomes neutral and is then entirely absorbed by the material. Moreover, particles of the same kind with
the same initial energy have nearly the same range for a given material. There exists a final small variation
in the range, called straggling, and is due to the statistical nature of the energy loss process which consists
of a large number of individual collisions subjected to some fluctuation. In spite of that, the average range
can be used to determine the average energy of the incident particles.
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Chapter 2 – Physics of semiconductor detectors
2.5 – Physics and behavior of semiconductors
In the previous sections the principles of how particles interact with matter have been considered. From
now on the concentration will be pointed out to semiconductors, and focused on the interaction of MIP
particle with silicon sensors, looking on how these concepts are used to think and create devices capable to
detect real particles. Semiconductor devices are widely used in the electronics (power-switching devices)
because of their specific electrical conductivity, σ, which is between that of good conductors (>1020 free
electron density) and that of good insulators (<103 free electron density), and they also fit well for particle
detection because they are materials into which there exist little numbers of free charges, and particles
passing through it can easily produce a detectable quantity of them.
2.5.1 - Conduction in a solid
After Quantum Mechanics discoveries, a theory about solid state materials that includes semiconductors
has been commonly approved by the scientific community. The structure of an isolated atom shows
numerable states of the electrons surrounding the nucleus, characterized univocally by a definite energy
En8. In a solid, it is to be taken into account the entire number of the atoms that constitutes the lattice: the
interactions among the atoms and their high existing number9 make the electron states so dense to make
them forming a continuous band of allowed energy. These bands can be separated by gaps that electrons
cannot occupy, the forbidden gaps. Because of their fermionic nature10, electrons fill the states starting
from the lowest energy level available, filling up the energy bands to a maximum energy E0 :
Figure 2.8 - Questa figura non mi piace
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
8
n is a set of integer numbers
~ 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)
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Chapter 2 – Physics of semiconductor detectors
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 apply Boltzmann statistics [], which gives the density of electrons n having energy greater than E0.
From this it follows:
, where kB is the Boltzmann constant.
2.5.2 - Classification of Semiconductors
Although there is a large variety of semiconductor materials today available, there is one of them that
stands out from the group and dominates the scene: it is the silicon. Its properties are entirely well known,
it is quite easy to find and to manage practically, and – last but not least for the productive processes – not
expensive. Nonetheless, according to their chemical composition, each different kind of semiconductor can
have different properties, and so used for different specified duty in the applications.
Elementary semiconductors are located within the IV group of the Periodic Table of Elements [],and they
are the Silicon (Si), the Germanium (Ge), the grey tin (α-Sn), and Carbon (C), that can solidify in two
different structures (graphite and diamond, that is an insulator but with the same crystal structure as Si, Ge
and α-Sn).
Material
Diamond (C)
Silicon (Si)
Germanium (Ge)
Gray tin (α-Sn)
White tin (β-Sn)
Graphite (C)
Lead (Pb)
a (nm)
0.357
0.543
0.566
0.649
0.583
0.318
0.246
0.673
0.495
EG (eV)
5.48
1.12
0.664
-
EG (nm)
226
1107
1867
-
Type
indirect
indirect
indirect
-
Structure
cubic
cubic
cubic
cubic
tetragonal
-
-
-
hexagonal
-
-
-
cubic
Table 1 - Lattice constant a, energy gap EG at 300K, type of energy gap and lattice structure of group IV elements [2-4]
The main characteristic of the IV elements mentioned is that they all have the outer shell of the individual
atoms is exactly half filled, and so by sharing one of the four electrons of the outer shell with another Si
atom it is possible to obtain a three-dimensional crystal structure with no preferential direction (except for
graphite), and it is also possible to combine two of IV group semiconductors in order to form useful
compounds (such as SiC or SiGe) with new peculiarity (for example the SiC is a borderline compounds
between semiconductor and insulator, and can be useful for high temperature electronics).
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Chapter 2 – Physics of semiconductor detectors
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).
It also exists the II-IV class of semiconductors, characterized by an higher ionic bond degree total
percentage -~60%- since the respective elements differ more in the electron affinity due to their location in
the Periodic Table of Elements. Also I-VII compounds can form semiconductors, with larger energy gap.
There are other elementary semiconductors such as selenium and tellurium from group VI, the
chalcogenes, but only with two missing valence electrons to be shared with the neighboring atoms, so they
have the tendency to form chain structures.
There are also some spare compounds that works as semiconductors: they are the IV-VI compounds (PbS,
PbSe,PbTe), V-VI (B2Te3), II-V (Cd3As2, CdSb), and a number of amorphous semiconductors (the a-SI:H,
amorphous hydrogenate silicon, for example, is a mixture of Si and H). Moreover it is still possible to cite
the chalcogenide glasses (As2Te3, As2Se3, that can be used in xerography)[2-4].
2.5.3 - Silicon
Silicon is the most widely used material in radiation detectors mostly because it is the only semiconductor
material having a native oxide with good interface properties fitting for a high integration technology, and
it is the most chosen for devices involving semiconductors. It has four valence electrons, so it can form
covalent bonds with four of neighbors atoms. When the temperature increases the electron in the covalent
bond can become free, generating holes that can afterwards be filled by absorbing other free electrons, so
effectively there is a flow of charge carriers. The effort needed to break off an electron from its covalent
bond is given by Eg (band gap energy). There exists an exponential relation between the free-electron
density ni and Eg given by the formula:
, where NC and NV are respectively the number of states in conduction and valence band. To give some
numbers, it is also possible to write it as:
For example, at T=300 K, ni = 1.45 x 1010 electrons/cm3, and at T=600 K, ni = 1.54 x 1015 electrons/cm3[2-8].
These electrons form the thermal charge carriers pairs concentration, and so an intrinsic current in silicon
material when a voltage is applied, properly connected to silicon specific natural properties.
In pure silicon at equilibrium, the number of electrons is equal to the number of holes. The silicon is so
called intrinsic and the electrons are considered as negative charge-carriers. Holes and electrons both
contribute to conduction, although holes have less mobility due to the covalent bonding. Electron-hole
11
8N atomic rule
14
Chapter 2 – Physics of semiconductor detectors
pairs are continually generated by thermal ionization and in order to preserve equilibrium previously
generated, they recombine. The intrinsic carrier concentrations ni are equal, small and highly dependent on
temperature.
2.5.4 – Doping of silicon
In order to fabricate either a silicon detector or a power-switching device, it is necessary to greatly increase
the free hole or electron population. This is achieved by deliberately doping the silicon, so by adding
specific impurities called dopants. The doped silicon is subsequently called extrinsic and as the
concentration of dopant Nc increases its resistivity ρ decreases. Pure silicon electrical properties can be
changed by doping it with group V periodic table elements such as phosphourous (P), that adds more
electrons to it (type N doping, with free negative charges), or with group III periodic table elements such as
boron (B), that adds more holes (type P doping, with free positive hole charges). A group V dopant is called
a donor, having donated an electron for conduction. The resultant electron impurity concentration is
denoted by ND (donor concentration). If silicon is doped with atoms from group III, such as B, Al, Ga or In,
which have three valence electrons, the covalent bonds in the silicon involving the dopant will have one
covalent-bonded electron missing. The impurity atom can accept an electron because of the available
thermal energy. The dopant is thus called an
acceptor, which is ionized with a net positive
charge. Silicon doped with acceptors is rich in holes
and is therefore called p-type. The resultant hole
impurity concentration is denoted by NA (acceptor
concentration).
To be pointed out that for manufactory industries
it is not easy to grow large area silicon crystals
doped with a rate of less than 10% around the
resistivity wanted. Final device electrical properties
will therefore vary widely in all lattice directions.
Tolerances better than ±1 per cent in resistivity
Figure 2.9 - Doping of silicon [2-6]
and homogeneous distribution of phosphorus can
be attained by neutron radiation, commonly called
neutron transmutation doping (NTD). The neutron irradiation flux transmutes silicon atoms first into a
silicon isotope with a short 2.62-hour half-lifetime, which then decays into phosphorus. Subsequent
annealing removes any crystal damage caused by the irradiation. Neutrons can penetrate over 100mm into
silicon, thus large silicon crystals can be processed using the NTD technique.
2.5.5 - Charge Carriers
Electrons in n-type silicon and holes in p-type are called majority carriers, while holes in n-type and
electrons in p-type are called minority carriers. The carrier concentration equilibrium can be significantly
changed by irradiation by photons, the application of an electric field or by heat. Such carrier injection
mechanisms create excess carriers. Noticeable is the fact that the product of electron and holes densities (n
and p) is always equal to the square of intrinsic electron density regardless of doping levels:
15
Chapter 2 – Physics of semiconductor detectors
It follows that, for n-type doped semiconductors:
and for p-type doped semiconductors:
2.5.6 - Charge transportation
A first mechanism of charge transportation into semiconductor is identified under the name of drift
mechanism: it simply consists in the application of an electric field at the extremity of the semiconductor,
and the charge particles will move at a velocity (vh, ve)proportional to the electric field E given (µP, µn
constants of proportionality called mobility):
In silicon, at room temperature, typical values of mobility are 1450 cm 2/V∙s for the electrons and 450
cm2/V∙s for the holes.
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.
Figure 2.10 - Diffusion into a semiconductor [2-6]
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.
Surprisingly, there exists a relation between the drift and diffusion currents, although they are totally
different.It is the Einstein’s relation
2.5.7 - PN junctions
Electrons and holes are discrete charge carriers, and the current generated by their drift and diffusion is
affected by a noise proportional to the current itself. The amplitude of this intrinsic noise depends on the
resistance of the semiconductor used (230 kΩ∙cm for silicon), and it can unfortunally be of the same order
17
Chapter 2 – Physics of semiconductor detectors
of signal generated by a particle passing through
the semiconductor material. It follows that
semiconductors as they are can’t be suitable for
particle detection, but it has been discovered that
using a pn junction can resolve this inconvenient.
A pn junction is the location in a doped
semiconductor where the impurity changes from
Figure2.11 - PN junction and electrical schematic [2-6]
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 so formed. In order to understand
the operation of a diode, it is necessary to study its three operation regions: equilibrium, that introduces
the depletion zone and the built-in potential, reverse bias, that introduces the junction capacitance, and
forward bias, that introduces the IV characteristics.
2.5.8 - Diffusion across the junction
Each side of the junction contains an excess of holes or electrons compared to the other side, and this
situation induces a large concentration gradient. Therefore, a diffusion current flows across the junction
from each side.
Figure 2.11 - Diffusion in a PN junction, with nn concentration of electrons on n side, pn concentration of
holes on n side, pp concentration of holes on p side, np concentration of electrons on p side [2-6]
As free electrons and holes diffuse across the junction, a region of fixed ions is left behind. This region is
known as the depletion region, and is particularly attractive for particle detection purposes, because any
charges created by a passing-through particle are going to be swept out by the electric field generated in
this zone, and can be detected by electronics connected to the junction.
Figure 2.12 – Creation of the depletion zone [2-6]
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Chapter 2 – Physics of semiconductor detectors
So the fixed ions in depletion region create an electric field that results in a drift current; at equilibrium, the
drift current flowing in one direction cancels out the diffusion current flowing in the opposite direction,
creating a net current of zero.
;
Because of the junction there exists a built-in potential:
that is caused by the capture of electrons by holes on the n-side and the diffusing electrons which fill holes
in the p-side, and is due to different concentration of holes and electrons in both doped materials that start
diffusing. This is added to the contact potential VC, that is the potential difference across the junction (for
silicon is about 0.7 V).
2.5.9 – Biasing the junction with forward bias
There are two ways for biasing the junction: one the is direct, the
other is the reverse way. When the n-type region of a diode is at a
lower potential than the p-type region, the diode is in forward bias.
This situation leads to shorten the depletion width and decrease
the built-in potential. Under this condition minority carriers in each
region increase, and diffusion currents also increase to supply
them.
Minority charge profile should not be constant along the x-axis, in
order to have a concentration gradient and so diffusion current:
recombination of the minority carriers with the majority carriers accounts for the dropping of minority
carriers as they go deep into the P or N region.
Figure 2.13 - Forward bias [2-6]
2.5.10 – Biasing the junction with reverse biasing
Opposite to the precedent situation, when the n-type region of a
diode is connected to a higher potential than the p-type region
the diode is under reverse bias, which results in wider depletion
region and larger built-in potential across the junction. This is
important for having the as wider depleted zone as possible in
order to increase the sensible zone of creation of charges and
consequently of particles detection. Varying the value of VR it is
so possible to vary also the width of the depletion zone W,
following the law [2-9]:
, where xn and xp are the parts of depletion zone on the n and p
Figure 1.14 - Reverse biasing [2-6]
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Chapter 2 – Physics of semiconductor detectors
side respectively, ε0 and εSi absolute and relative to silicon dielectric constants. In silicon sensors the
junction is usually realized by a shallow and highly doped (NA> 1018 cm-3) p+-implant in a low-doped
(ND≈1012 cm-3) bulk material, therefore the term 1/NA can be neglected, meaning that the space charge
region is reaching much deeper into the lower doped side of the junction. Moreover, also the built-in
voltage can be neglected because it is small compared to typical operation voltages (0.5V compared to
50V). This leads to:
The depletion zone width increases so with the applied voltage, and reaches a maximum at which the
junction breaks down and becomes conductive (breakdown zone). This is also the point at which the
electric field reaches its maximum value:
It also affects the capacitance value, fact that leads to identify the PN junction as with the same behavior of
a voltage dependent capacitor with its capacitance described by the following equation:
with
An useful application of this statement is to use the junction to form a LC oscillator circuit, that varies the
frequency by changing the VR (and so changing the capacitance).
2.5.11 - IV characteristics of PN junction
The current and voltage relationship of a PN junction is
exponential in forward bias region, and relatively constant in
reverse bias region.
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. When a large reverse
Figure 2 - IV characteristic [2-6]
bias voltage is applied, breakdown occurs and an enormous
current flows through the junction. There exist two kinds of reverse breakdown: Zener and Avalanche
breakdown. The first is a result of the large electric field inside the depletion region that breaks electrons or
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Chapter 2 – Physics of semiconductor detectors
holes off their covalent bonds, while the second is a result of electrons or holes colliding with the fixed ions
inside the depletion region.
2.6 – Semiconductor silicon detectors
In principle a semiconductor detector behaves like a ionization chamber, with a simple configuration made
by an absorbing medium, in the case the semiconductor, connected to two electrodes. The electrodes are
themselves connected to an external bias supply, that creates the electric field through the pn junction, and
when a particle passes to the material and generates charges particles12 this electric field drifts the
generated charges to the respective electrodes producing the outgoing signal. Thinking of the radiation in
terms of photons, as for first example, shows that the basis of photon detectors have to be found through
variations on the photoelectric effect, the Compton scattering, and the pair production.
Figure 3 - Example of silicon detector geometry
By applying reverse bias to a PN junction, one is effectively storing charge on the equivalent of a parallel
plate capacitor (the depletion region is an insulator and the P and N regions are conductors). Disconnecting
the bias, if it happens that photons are absorbed within the depletion region, electron-hole pairs are
produced, and the electrostatic field within the depletion region sweeps the electrons to the N side and the
holes to the P side, decreasing the amount of stored charge. Re-applying the bias after some time, the
original charge is restored, and the current would reveal how many photons had been absorbed in the
depletion region. The sensitivity of this technique is limited by thermodynamic fluctuations, as also
specified in paragraph 2.5.1. Under conditions of thermodynamic equilibrium at a temperature T, the
uncertainty in the stored charge (the charge fluctuations at fixed voltage) is given by:
This is known as kBTC noise. To give some numbers, for a temperature of 150 K and a capacitance of 1 pF,
<σQ>2 is about 280 e- [], which is relatively high if one wants to detect individual photons. In order to be
limited to photon statistics rather than kBTC noise, one would need values of order 105 photons.
12
Electron-hole pair production energy for silicon is calculated to 3.69 eV [2-7], and is lower than the one for ionizing a
gas, ~30eV
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Chapter 2 – Physics of semiconductor detectors
2.6.1 - Photon detection
Coming to specific particle detection, it is possible that a photon interacts with a semiconductor and
creates charge when its energy is higher than the energy gap of silicon (1.12eV) and corresponds to a λ of
1.12μm (infrared region). If a photon has a wavelength longer than 1.02 μm, it will cross the silicon sensor
without being attenuated. For indirect bandgap semiconductor such as germanium and silicon the
absorption of a photon is made possible only with the involving of a phonon, that gives the additional
momentum necessary to the electron to jump to the conduction band. Indirect bandgap semiconductors
are characterized by an absorption coefficient growing gradually with the photon energy; when the photon
energy is high enough to allow the direct transition from the valence to the conduction band, phonons are
no longer required for the excitation, and the absorption coefficient saturates. For direct bandgap
semiconductors, such as GaAs, the coefficient grows for energies nearby the energy gap value, since the
transition does not require an extra particle like the phonons in order to conserve momentum.
2.6.2 – Charged Particle detection
The mean energy transferred per unit path length by charged particle passing through matter follows the
Bethe-Bloch formula, depending on parameters related to both the incident particle and to the physical
properties of the absorbed material. For high energy particles, the mean energy transferred to the matter
reaches a minimum, which is nearly the same for protons, electrons and pions and remain constant for
higher energies; particles having energies high enough to reach this minimum are knows as Minimum
Ionizing Particle (MIP). The typical energy spectrum of a MIP particle crossing a semiconductor material
follows a Landau distribution, characterized by evident asymmetric shape given by the long tail for high
energies losses due to high energy recoil electrons (δ rays). Due to the asymmetry, the Most Probable Value
(MPV) of energy loss, corresponding to the peak, differs from the mean energy lost, which is shifted at
higher energies.
2.6.3 - Functionality of silicon detectors
Detectors are reverse biased at the p+ side with negative voltage high enough to fully deplete the sensor.
Full depletion voltage can be calculated with:
, where d is the thickness of the detector.
The electric field in the depletion zone can be calculated as:
, where x is the depth in the detector.
Holes and electrons created by the passage of a mip through the depleted zone move respectively towards
p+ and n+ sides, driven by the electric field as:
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Chapter 2 – Physics of semiconductor detectors
The depth reached by a charge carrier in function of the time is calculated with:
, with x0 position at t=0.
Related velocity can be calculated with:
The electron is stopped at the xe(te)=d, and the hole at xh(th)=0, having drift times of:
The current induced by a moving charge q, that can be measured by a charge sensitive preamplifier, is:
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Chapter 2 – Physics of semiconductor detectors
References to this chapter
[2-1]: W. R. Leo, Techniques for Nuclear and Particle Physics Experiments, Springer, Berlin, 1994,2.107
[2-2]: H.A. Bethe, W. Heitler, On the stopping of fast particles and the creation of positive electrons, Proc.
Royal Soc. London A 146 p.83, 1934
[2-3]: Physics Letters B, Review of Particle Physics, July 2008
[2-4]: http://www.pdf-search-engine.com/semiconductor-an-introduction-pdf.html
[2-5]: Donald A. Neamen, Fundamentals of Semiconductor Physics and Devices, McGraw-Hill Company
[2-6]: Behzad Razavi, Fundamentals of Microelectronics, John Wiley & Sons, Inc
[2-7]: R. C. Alig, S. Bloom, C. W. Struck, Scattering by ionization and phonon emission in semiconductors,
Physical Review B 22 p. 5565, 1980
[2-8]: S.M. Sze, Physics of Semiconductor Devices 2nd edn. ,Wiley, New York, 1981
[2-9]: Rossi
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