Chapter 2 Physics of semiconductor detectors




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Chapter 2 – Physics of semiconductor detectors





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 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:



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.





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

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.

2.3.5 - Pair production

Photons with energies in excess of 2mec2 produce electron-positron pairs, and an interaction with a nucleus is needed in order to balance momentum. The pair production cross section starts at 1.022 MeV for then rising to an approximately constant value at high photon energy, in the gamma ray region of the spectrum of electromagnetic radiation. Cross sections scale with the square of the atomic number:






2.4 - Interactions of charged particles with matter

The most common way in which charged particles (such as electrons, protons and alpha and beta particles) can interact with matter is the electromagnetic interaction, that involves collisions with electrons in the absorbing material and is the easiest mechanism to detect them. They can also interact through one of the two kinds of nuclear interactions, the weak interaction or the strong interaction.

The main process of energy loss producing excitation and ionization is the inelastic collisions with an electron; it can also happen an inelastic collisions with a nucleus, that leads to Bremsstrahlung and coulombic excitation. Eventually there could also be elastic collisions with a nucleus, Rutherford diffusion and elastic collisions with an electron.


2.4.1 - 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.
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