PHYSICS OF SEMICONDUCTOR DETECTORS

Transcription

1 CHAPTER 2 PHYSICS OF SEMICONDUCTOR DETECTORS 1

2 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 moving 1. 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 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 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. 1 correlated together by fundamental relativistic equations 2

3 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 E γ overtakes the ionization potential (bound energy E B 2 ), an electron will be emitted out of the atom with energy E e given by the following formula: E e = E γ E B 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 effect 3. 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 ( 241 Am and 109 Cd). Figure 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 ev for intrinsic semiconductors and of order to 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 1 E B 100 KeV, depending on the shell and on the atom 3 There are correct different formulations for low and high energy behaviour 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. For an unbound electron initially at rest, it is possible to have the following equations 4 : ν = ν ν m e c 2 (1 cos θ) with hν and hν initial and final energy, θ photon angle change, m e 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 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 Pair production Photons with energies in excess of 2m e c 2 produce electron-positron pairs, and interaction with a nucleus is needed in order to balance momentum. The pair production cross section starts at 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] 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: N x = N 0 e µ Lx that performs the exponential decrease of the number of particles N(x) at x given depth into the material from the initial number N 0, where µ L is the linear absorption coefficient 5 given by: μ L = ςn aρ A ς = ς potoelectric + ς compton + ς pair with σ cross-section, N a 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/ (m e c) has units of length and equals nm 5 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) 4

5 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 241 Am, which decades with gamma rays of 59.5 KeV, is 0.3 cm 2 g -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: λ = 1 µ L It follows an alternative way of expressing the exponential absorption law: N x = N 0 e µ L λ The distance over which one half the initial beam is absorbed is called the half thickness, x1, and is related 2 to the linear absorption coefficient and to the mean free path by: x1 = ln(2) = ln(2) λ = λ 2 µ L 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 5

6 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 μ m, which relates the linear absorption coefficient to the density of the material ρ: µ L = µ m 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: N x = N 0 e µ Lx = N 0 e µ m ρx that states that the total attenuating effect of a slab of given type material can be described by quoting the mass attenuation coefficient, which is characteristic of the material's chemical composition, and the photon energy, together with the material's density and thickness. The product ρx, the areal density 6, 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 SI 7 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: µ m TOTAL = (α µ m ) 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 Interactions of charged particles with m atter 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 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 6 mass per area 7 International System of measurements 6

7 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 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) 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 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 7

8 (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: S l rad S m ion = 1 E ZE 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, E c, and is approximately E c E Z 800 MeV Z 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 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 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 S l, also known as de dx (where E is the particle energy and x is the distance travelled): S l = de dx commonly measured in MeV * m -1. It depends on the charged particle's energy, on the density of electrons within the material, and hence on the atomic numbers of the atoms. So a more fundamental way of describing the rate of energy loss is to specify the rate in terms of the density thickness, rather than the geometrical length of the path, so energy loss rates are often given as the quantity called the mass stopping power: S m = de d(ρx) = 1 de ρ dx where ρ is the density of the material and ρx is the density-thickness. 8

9 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]: de dξ = 1 de ρ dx = D 1 Z β 2 z2 A {ln 2m eγ 2 β 2 c 2 I β 2 δ 2 C Z } D = 4πr 0 2 m e c 2 N AV = 0,307 MeV cm 2 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, m e mass of the electron, c speed of light, I average ionization energy of the material, δ and C relativistic corrections of the formula, r 0 classical electron radius, N AV Avogadro s constant. Figure Energy loss of µ on Cu [2-3] Figure Energy loss for different particles [2-3] Figure Energy loss for heavy charged particles in different materials [2-3] 9

10 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 H 2. 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 density-thickness, 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 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 cm 2 g -1 (= J m 2 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 cm 2 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: W = n T ΔE 10

11 with E energy given by the incident particle, n T number of couples created. In the definition of n T there are included the primary n P and secondary n S couples as: n T = n P + n S In silicon the mean energy loss is of 1.66 MeV cm 2 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 hole-electrons couples, with as mean amount and as most probable value (MPV). In Figure 2.7 this calculation is shown for different depth of silicon. Figure 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. 11

12 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 (>10 20 free electron density) and that of good insulators (<10 3 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 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 E n 8. 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 number 9 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 nature 10, electrons fill the states starting from the lowest energy level available, filling up the energy bands to a maximum energy E 0 : Figure 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 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 8 n is a set of integer numbers 9 ~ atoms/cm 3 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) 12

13 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 E 0, one must apply Boltzmann statistics [], which gives the density of electrons n having energy greater than E 0. From this it follows: n E > E 0 = e E 0 k B T, where k B is the Boltzmann constant Classification of Semiconductors k B = J K 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 a (nm) E G (ev) E G (nm) Type Structure Diamond (C) indirect cubic Silicon (Si) indirect cubic Germanium (Ge) indirect cubic Gray tin (α-sn) cubic White tin (β-sn) tetragonal Graphite (C) hexagonal Lead (Pb) cubic Table 1 - Lattice constant a, energy gap E G 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). By completing the outer shell by sharing electrons with other atoms can be obtained also with other compounding 11, so obtaining compounds that are semiconductors, too. Elements of group III (II) can so be 11 8N atomic rule 13

14 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 (B 2 Te 3 ), II-V (Cd 3 As 2, 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 (As 2 Te 3, As 2 Se 3, that can be used in xerography)[2-4] 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 E g (band gap energy). There exists an exponential relation between the free-electron density n i and E g given by the formula: n i = N C N V e E g 2k B T, where N C and N V are respectively the number of states in conduction and valence band. To give some numbers, it is also possible to write it as: n i = T 3 2 e E g 2kT [ electrons cm 3 ] For example, at T=300 K, n i = 1.45 x electrons/cm 3, and at T=600 K, n i = 1.54 x electrons/cm 3 [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 pairs are continually generated by thermal ionization and in order to preserve equilibrium previously generated, they recombine. The intrinsic carrier concentrations n i are equal, small and highly dependent on temperature. 14

15 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 N D (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 N A (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 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 Figure Doping of silicon [2-6] 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. 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: It follows that, for n-type doped semiconductors: np = n i 2 Majority carriers n N D 15

16 Minority carriers p n i 2 N D and for p-type doped semiconductors: Majority carriers p N A Minority carriers n n i 2 N A 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 (v h, v e )proportional to the electric field E given (µ P, µ n constants of proportionality called mobility): v = µ P E v e = µ n E In silicon, at room temperature, typical values of mobility are 1450 cm 2 /V s for the electrons and 450 cm 2 /V s for the holes. The current is calculated as shown in the following formula: In general the drift current is expressed as: I = vwnq J n = µ n E n q while the total current is the sum of the current given by holes and electrons drifts: J TOT = µ n E n q + µ p E p q = µ n n + µ p p E q It is important to underline that in reality the velocity does not increase linearly with electric field, but it saturates at a critical value. The following equation expresses the velocity saturation: v = µ = µ be v SAT = µ 0 b µ µ E 0 E v SAT 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. 16

17 Figure 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: I = A q D n dn dx J n = q D n dn dx J p = q D p dp dx J TOT = q (D n dn dx D p dp dx ) It is important to say that a linear charge density profile means constant diffusion current, whereas nonlinear charge density profile means varying diffusion current. Linear: J n = q D n dn dx = qd n N L Non Linear: J n = q D n dn dx = qd n N L d Surprisingly, there exists a relation between the drift and diffusion currents, although they are totally different.it is the Einstein s relation PN junctions D µ = kt q 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 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 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 e x L d 17

18 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 Figure PN junction and electrical schematic [2-6] capacitance, and forward bias, that introduces the IV characteristics 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 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] 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. I drift,p = I diff,p ; I drift,n = I diff,n Because of the junction there exists a built-in potential: 18

19 V 0 = kt q ln N AN D n i 2 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 V C, that is the potential difference across the junction (for silicon is about 0.7 V) 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 Figure Forward bias [2-6] 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 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 V R it is so possible to vary also the width of the depletion zone W, following the law [2-9]: W = x n + x p = 2ε 0 ε Si e 1 N A + 1 N D V 0 + V R, where x n and x p are the parts of depletion zone on the n and p side respectively, ε 0 and ε Si 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 lowdoped (ND 1012 cm-3) bulk material, therefore the term 1/NA can be neglected, meaning that the space charge region is reaching Figure Reverse biasing [2-6] 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: W x n 2ε 0 ε Si en D V R 19

20 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: E max = 2V W 2eN D ε 0 ε Si V R 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: C j = C j0 1 + V R V0 with C j0 = ε Si q N A N D 2 N A + N D V 0 An useful application of this statement is to use the junction to form a LC oscillator circuit, that varies the frequency by changing the V R (and so changing the capacitance) 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. V D I D = I S (e VT 1) 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 bias voltage is applied, breakdown occurs and an enormous Figure 2 - IV characteristic [2-6] 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 holes off their covalent bonds, while the second is a result of electrons or holes colliding with the fixed ions inside the depletion region. 20

21 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 particles 12 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 Under conditions of thermodynamic equilibrium at a temperature T, the uncertainty in the stored charge (the charge fluctuations at fixed voltage) is given by: < ς Q > 2 = k B TC This is known as k B TC noise. To give some numbers, for a temperature of 150 K and a capacitance of 1 pf, <σ Q > 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 k B TC noise, one would need values of order 10 5 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 21