Radiation Damage In Silicon Detectors

Transcription

1 UNIVERZA V LJUBLJANI FAKULTETA ZA MATEMATIKO IN FIZIKO ODDELEK ZA FIZIKO Joµzef Pulko SEMINAR Radiation Damage In Silicon Detectors MENTOR: prof. dr. Vladimir Cindro Abstract: Radiation damage in silicon detectors can roughly be divided in surface and bulk damage. The subject of this work is the bulk damage which is the limiting factor for the use of silicon detectors in the intense radiation elds close to the interaction point of high energy physics (HEP) experiments. Seminar starts after short introduction about silicon detectors, with the description of the basic radiation damage mechanism initiated by the interaction of high energy particles (hadrons, leptons, photons) with the silicon crystal and resulting in the formation of point defects and defects clusters. At the end impacts of defects on the electrical properties of silicon detectors are summarized. Keywords: Silicon detector, radiation damage, defects Ljubljana, April 2007

2 Contents Introduction 3 2 Basic features of silicon diodes 4 2. p-n junction Operation of silicon detectors Defect Generation 6 4 The NIEL scaling hypothesis 9 5 Impact of defects on detector properties 5. Change in Ne Change in leakage current Trapping of the drifting charge Conclusion 8 2

3 Introduction As the collider experiments in high energy physics go towards higher energy use of silicon detectors becomes inevitable. Their superior spatial resolution, short signal formation times and good energy resolution make them ideal for tracking ionizing particles. Besides the ability to accurately measure the momentum of energetic charge particle from bending of their trajectories in magnetic eld their most important feature is the capability of distinguishing secondary from primary vertices. Therefore there are placed as close as possible to the interaction point. Longer operation under high radiation, results in signi cant radiation damage of the detector. A careful study of radiation damage of silicon detectors is necessary. Radiation damage in silicon can be divided in damage of bulk and surface damage. The latter is related with the accumulated xed positive charge in the oxide. Fortunately, the surface damage seems to be manageable. It depends on detectors design and manufacturing, which have been studied and understood. Bulk damage is generated over whole volume of Semiconductor. My seminar is concentrated in this type of defects. Text is organized in 6 chapters. In the following chapter the basic of the detector operation along with theory of signal formation are presented. In chapter 3 the basic radiation damage mechanisms and the radiation induced defects in silicon bulk are reviewed. In chapter 4 we discuss about nonionizing energy losses in silicon. Chapter 5 illustrates in uence of defects on operating properties of the detector and, nally, summary with some conclusions is given in Chapter 6. 3

4 2 Basic features of silicon diodes Basic idea of silicon detectors is similar to the ionizing cell. Ionizing particles passing through silicon generate electron-hole pairs along their path. The number of pairs is proportional to the particle s energy loss. The creation of electron-hole pair in silicon requires a mean energy of 3:6 ev with the average energy loss in silicon of about 390 ev= m for minimum-ionizing particle. This give rise to 08 pairs per m. For a typical detector thickness of about D = 300 m, on average 3: electron-hole pairs are obtained, a signal detectable with low-noise electronics. 2. p-n junction In order to explain the operation of a p-n diode one may imagine the opposite sides of the junction originally isolated, and then brought into intimate contact. Thermal equilibrium is established as equal number of highly mobile electrons and holes, from the n-type and the p-type material, respectively recombine. A potential di erence bi prevents further charge ow. Is maintained by the static space charge build up around the junction by the ionization of the donor and acceptor atoms in the doped semiconductor. This region is e ectively depleted of all mobile charge carriers and the voltage corresponding to the potential di erence is called built-in voltage V bi. In the case of an abrupt p-n junction one side is more heavily doped that the other and overall charge neutrality then implies that the depletion region of thickness W extends much further into the less heavily doped side of the device. This is displayed in Fig. for reverse biased Figure : Schematic gure of a p + c) electron potential energy. n abrupt junction: a) electrical charge density, b) electric eld strenght, abrupt p + n diode of thickness d under the assumption of a homogeneous distribution of dopant atoms. 4

5 Furthermore the so-called depletion approximation is assumed to be valid which demands that the space charge is constant in the region 0 < x < W, although it is known that the di usion of electrons from the n-type bulk into the depletion zone results into a smooth distribution of the charge around x = W. The electric eld strength and electrochemical potential can be calculated by solving Poisson s equation: d 2 dx 2 = el "" 0 = q 0N eff "" 0 () Here N eff denotes the e ective doping concentration which is given by the di erence between the concentration of ionized donors and acceptors in the space charge region. Furthermore "" 0 stands for the permittivity of silicon with " 0 = :9. The rst integration of Eq. with the boundary conditions E (x = W ) = d(x=w ) dx = 0 leads to an expression for the electrical eld strength which depends linearly on x (see Fig. b) and reaches the maximum eld strength of E m (V ) = q 0N eff "" 0 W (V ) (2) at the p + n interface (x = 0). A further integration under the boundary condition (x = W ) = 0 leads to a parabolically function for the potential: (x) = q 0N eff 2 "" 0 (x W ) 2 for 0 x W and W d (3) The corresponding electron potential energy ( e 0 ) is schematically displayed in Fig. c. There it is also indicated that the applied reverse bias V is equal to the di erence between the Fermi levels in the p + and n region, E F p and E F n which, of course, in the case of thermal equilibrium have to be the same as the electrochemical potential. With the condition (x = 0) = V bi V one obtains an expression for the depletion depth: q W (V ) = 2""0 q 0jN eff j (V + V bi) for W D. (4) With increasing reverse bias the eld zone expands until the back contact is reached (W = d). The corresponding voltage, needed to fully deplete the diode, is called depletion voltage V dep and connected with the e ective doping concentration N eff by: V dep + V bi = q 0 2"" 0 jn eff j d 2 (5) Very often the build-in voltage V bi is neglected since the depletion voltage is in most case more than one order of magnitude higher. 2.2 Operation of silicon detectors A silicon detector is a diode operated under reverse bias with depleted zone acting as a solid state ionization chamber. If the incident particle is stopped in the detector the particle energy can be measured (spectroscopy), if the particle is traversing the detector it is only possible to say whether or not a particle has passed (tracking). The latter case is the main application of silicon detectors in high energy physics. A minimum ionizing particle (mip) traversing a silicon layer of d = 300 m thickness deposits most probabley an energy of 90k ev. Although the energy gap in silicon is about :2 ev at room temperature the required average energy to produce an electron-hole pair is 3:6 ev. Thus most probable about electron-hole pairs are created by mip (about 72 e-h per m). If the detector is fully depleted all generated electrons and holes drift in the applied eld with their drift velocity v dr;n and v dr;p in direction of the anode and cathode, respectively. The current in uenced by a single charge carrier can be described by Ramo s theorem: I = q 0 v dr;n;p d with v dr;n;p = n;p E (x) E (x). (6) The mobility n;p is depending on the eld strength E and the eld strength itself is depending on the depth x in the detector (see Fig. ). 5

6 3 Defect Generation The energy loss of an incoming particle by interaction with matter can be divided into ionizing and nonionizing energy loss (NIEL). Due to fast recombination of charge carriers the ionizing energy loss does not lead to bulk damage. NIEL contains displacements of lattice atoms and nuclear reactions. The introduction rate of defects, resulting from nuclear reactions, is more than two orders of magnitude lower compared to introduction rates of defects originating from displaced silicon atoms and thus negligible [4]. The bulk damage produced in silicon particle detectors by hadrons (neutrons, protons, pions and others) or higher energetic leptons is caused primarily by displacing a primary knock on atom (PKA) out of its lattice site resulting in silicon interstitial and a left over vacancy (Frenkel Pair). However, the primary recoil atom can only be displaced if the imparted energy is higher than the displacement threshold energy E d of approximately 25 ev[]. The energy of recoil PKA or any other residual atom resulting from a nuclear reaction can of course be much higher. Along the path of these recoils the energy loss consists of two competing contributions, one being due to ionization and the other caused by further displacements. At the end of any heavy recoil range, the nonionizing interactions are prevailing and an dense agglomeration of defects (disordered regions or clusters) is formatted as displayed in Fig. 2. Both, point defects along the particle path and the clusters at the end of their range are, responsible for various damage e ects in the bulk of the silicon detector. However, ionization losses will not lead to any relevant changes in the silicon lattice. Figure 2: Monte Carlo simulation of a recoil-atom track with a primary energy E R of 50 k ev. The primary recoil energy of 50 k ev has been chosen because it is approximately the average kinetics energy that a M ev neutrons imparts on a PKA. The PKA releases its energy over a distance of about 000 Å to the silicon lattice. Approximately 37% of the recoil energy will go into ionization e ects and the rest can displace further lattice atoms. In average 3 terminal clusters are produced with a typical diameter of about 50 Å. It is instructive to calculate the maximum energy E R;max that can be imparted by a particle of mass m p and kinetic energy E p to the recoil atom by elastic scattering (nonrelativistic approach): m p m Si E R;max = 4E p (m p + m Si ) 2 (7) Taking into account the displacement threshold of E d 25 k ev and a threshold energy of 5 k ev for the production of clusters[] one can deduce that neutrons need a kinetic energy of 85 ev for the production 6

7 of a Frenkel pair and more than 35 k ev to produce a cluster. Electrons, however, need a kinetic energy E e of about 255 k ev to produce a Frenkel pair and more than 8 M ev to produce cluster, if one takes into account the approximate relativistic relation E R;max = 2E e E e + 2m 0 c 2 = m Si c 2. With the displacement of a big number of silicon atoms from their lattice sites the damage process has not ended. Interstitials and vacancies are very mobile in the silicon lattice at temperature above 50 K. Therefore a part of Frenkel pairs produced at room temperature annihilate and no damage remains. Simulations have shown that this is the case for about 60% of the overall produced Frenkel pairs and can reach in the disordered regions between 75% and 95%[5]. The remaining vacancies and interstitials migrate through the silicon lattice and perform numerous reactions with each other and the impurity atoms existent in the silicon (P, B, C, O). So they form new con guration of defects which can be stable at room temperature. The defects produced by such reactions (point defects) and the defects within the clusters are the real damage of silicon bulk material. When only a Frenkel pair is created only reactions with existing defects are possible. Thus reactions of the defects can be divided into two groups. In the group A are reactions of vacancies and interstitials di using throughout the crystal. The most frequent reactions within the clusters, where the defect density is high, belong to the group B. Possible reactions of both groups are listed in Fig. 3 and the most relevant defect con gurations are shown schematically in Fig. 4 Figure 3: Survey of possible defect reactions. Group A reactions are caused by di usion of interstitials and vacancies throughout the crystal. Most frequent reactions during a primary cascade are gathered in group B. Indexes i and s stand for interstitial and substitutional. 7

8 Figure 4: Various possible defect con gurations. Simple defects are: a.) vacancy V, b.) interstitial silicon atom I, c.) interstitial impurity atom, d.) substitutional impurity atom (e.g. phosphorus as donor). Examples of defect complexes are: e.) close pair I-V, f.) divacancy V-V, g.) substitutional impurity atom and vacancy (e.g. VP complex), h.) interstitial impurity atom and vacancy (e.g. VO complex) 8

9 4 The NIEL scaling hypothesis Charged hadrons interact with silicon primarily by the Coulomb interaction at lower energies. Thus a big part of the particle energy is lost due to ionization of lattice atoms which is fully reversible in silicon. Neutrons, however, interact only with the nucleus. The main reactions are elastic scattering and above :8 M ev also nuclear reactions. Hence the question arise how the radiation damage produced not only by di erent kind of particles but also, depending on the particle energy, by di erent kind of interactions can be scaled with respect to the radiation induced changes observed in the material. The answer is found in the so-called Non Ionizing Energy Loss (NIEL) hypotheses. The basic assumption of the NIEL hypothesis is that any displacement-damage induced change in the material scales linearly with the amount of energy imparted in displacing collisions, irrespective of the spatial distribution of the introduced displacement defects in one PKA cascade, and irrespective of the various annealing sequences taking place after the initial damage event. In each interaction leading to displacement damage a PKA with speci c recoil energy E R is produced. The portion of recoil energy that is deposited in form of displacement damage is depending on the recoil energy itself and can analytically be calculated by the so-called Linhard partition function P (E R )[2]. With the help of the partition function the NIEL can be calculated and is expressed by the displacement damage cross section D (E) = X (E) E max ZR 0 f v (E; E R ) P (E R ) de R (8) Here the index indicates all possible interactions between the incoming particle with energy E and the silicon atoms in the crystal leading to displacements in the lattice. is the cross section corresponding to the reaction with index and f (E; E R ) gives the probability for the generation of a PKA with recoil energy E R by a particle with energy E undergoing the indicated reaction. The integration is done over all possible recoil energies E R and below the displacement threshold the partition function is set to zero P (E R < E d ) = 0. Fig. 5 shows the displacement damage cross sections for neutrons, protons, pions and electrons in an energy range from 0 G ev down to some m ev for the thermal neutrons. A thorough discussion of these functions can be found in[3]. The total displacement-damage energy per unit volume deposited in the silicon crystal can be written as " d = N Si t irr Z 0 d D (E) de (9) de where t irr denotes the irradiation time, (E) the ux of incoming particles and N Si space density of the target nuclei. The damage caused by di erent particles is usually compared to the damage caused by neutrons. Since the damage function depends on neutron energy Fig. 5, the NIEL of M ev neutrons is taken as the reference point. The standard value of M ev neutrons NIEL is 95 M evmb[4]. Irradiation with particle A with a spectral distribution d A and cut-o s E min and E max would cause the same damage as M ev neutrons if de " d = N Si t irr Z da de D A (E) de = N si eq D n (M ev) (0) where eq denotes the equivalent integrated ux ( uence) of M ev neutrons which would have caused the same damage as the uence Z da A = t irr de () de of particles actually applied. It is possible to de ne the hardness factor A allowing to compare the damage Also called damage function and related to NIEL by D (E) = A N A de dx (E) j non ionizing.the NIEL value ca also be referred to as the displacement-kerma (Kinetic Energy Released to Matter). 9

10 Figure 5: Displacement damage function D (E) normalized to 95 M evmb for neutrons, protons, pions and electrons. Due to normalization to 95 M evmb the ordinate represents the damage equivalent to M ev neutrons. The insert displays the zoomed part of the gure. e ciency of di erent radiation sources with di erent particles and individual energy spectra as A Z d A de D A (E) de A = Z (2) D n ( M ev) d A de de It follows from here that eq = A A (3) 0

11 5 Impact of defects on detector properties Defects with the levels in the forbidden gap can capture and emit electrons and holes. In Fig. 6 the defect levels E t for the di erent kind of defects are indicated by the short solid lines. The ionization energy E t needed to e.g. emit an electron into the conduction band corresponds to the distance between the conduction band edge E C and the defect level position (E t = E C E t ). Acceptors are defects that are negatively charged when occupied with an electron while donors are defects that are neutral when occupied with an electron. In thermal equilibrium the charge state of defects is ruled by the Fermi levels. If the Fermi level is located above the defect level, acceptors are negatively charged and donors are neutral; if it is below the defect level, acceptors are neutral and donors are positively charged. This is indicated by the ( //+)-signs in the gure. Some defects have more than one level in the band gap. As an example the levels of the thermal double donor (TDD) and the amphoteric divacancy (VV) are shown. An amphoteric defect is a defect with acceptor and donor level. In the space charge region the occupation with charge carriers is ruled by the emission coe cients of the defects. Therefore, usually levels in the upper half of the band gap are not occupied by an electron while the levels in the lower half are occupied with electrons. This means for example that the defects V O i (acceptor in upper half of band gap) and C i O i (donor in lower half) have no in uence on the depletion voltage of the detector since they are neutral in the space charge region. However, B S (acceptor in lower half) and P S (donor in upper half) are ionized and therefore introduce negative, respectively positive, space charge. Figure 6: Schematic representation of the possible charge states of acceptors, donors and amphoteric levels in the forbiden band gap. In non-irradiated silicon the density of deep level defects is far below the density of shallow dopants which determine to a large part the electrical behavior of silicon. In these situation the deep level defects can be considered as a disturbance to the semiconductor which properties remain basically intact. The concentration of the deep level defects can exceed the concentration of shallow dopant density in irradiated silicon. The result is a drastic change of silicon properties. Each defect can have several charge states. In the simplest case, like for shallow dopants, a donor can assume two charge states, neutral with the electron loosely bound to the donor site and positively ionized. The acceptor may be neutral or negatively ionized. A general type of defect is much more complex. It may be a complex structure of missing silicon atoms in the lattice and impurity atoms, capable of switching between several chemical binding structures and by that between the charge states. Changing from one state to another may be accomplished by thermal excitation.

12 Figure 7: A schematic view of carrier capture and emission processes for a defect with multiple charge states. A simple defect has only one energy level and two charge states. If this involves a change of the charge state of the defect, it is accompanied by emission or capture of an electron or hole. An example of defects with four charge states and three energy levels is shown in Fig.??. Changing e.g. from charge state zero to the singly negatively charged state is accomplished by capture of an electron (E = E c E 2 ) or emission of a hole with energy E 2 E. The opposite transition requires electron emissino (E = E E c ) or hole capture. As will be shown later the emission and capture processes are related to each other so that a (nondegenerate) defect is characterized by the following properties: k energy levels E t ; k describing the energy involved in changing the charge state. k + charge states Q t ; l(l = 0; k) of the defect, ordered from most positive to most negative k electron capture cross sections t;le k hole capture cross sections t;lh In the most common case simple donors and acceptors, which have only one energy level and two charge states, are completely characterized by their energy level and two cross sections. In thermal equilibrium the electron occupation probability of a state and therefore also of simple defect states is described by the Fermi function F (E) = + exp E EF (4) where E F is the Fermi level, E the defect energy level. An occupied simple donor in this nomenclature is neutral while a occupied simple acceptor is negatively charged. In the following the short hand notation E Ei (E) = exp (5) will be used, with E i the Fermi level for the intrinsic silicon (p = n = n i ) derived from the charge neutrality condition E i = E V + E C k NV BT ln (6) 2 N C

13 where the bottom of the conduction band is denoted with E C, the top of the valence band with E V and the density of states in conduction and valence band with N C and N V. Electron and hole concentration in any silicon material are thus given as EC E F EF E i n = N C exp = n i exp = n i F (7) EF E V Ei E F p = N V exp = n i exp = n i F The Fermi level is found from the requirement of overall charge neutrality X h i X N j t F E j t N j t F E j t + N D N A n + p = 0 (8) donors acceptors where the N t denotes the concentration of deep defects. Also the complete ionization of shallow dopants is assumed. Although the thermal equilibrium occupation probabilities are completely described by Fermi function, this is the result of a continuous change of the charge state of individual defects. Thermal equilibrium thus allows us to nd the relations between the capture and emission processes. Considering a single defect level in thermal equilibrium the rate of electron capture has to be equal to the rate of electron emission. An analog relation holds for holes (see Fig. 6). This follows from the requirements that the average occupation probability of defects does not change and there is no net ow of electrons between the valence and conduction bands. With the introduction of the capture coe cients c n = v the te (9) c p = v thh th (20) for the product of thermal velocity and capture cross section and the emission probabilities " n and " p one gets: nc n N t ( F (E t )) = N t F (E t ) " n (2) pc p N t F (E t ) = N t F ( F (E t )) " p (22) F (E t ) = = + exp Et E F + (23) t F from which the electron capture (also called recombination) and emission (also called generation) probabilities of a simple defect are obtained as n = nc n, c p = pc p (24) c e n = Et E F Et E i n = nc n exp = n i c n exp = n i c n e t (25) e p = EF E t Ei E t p = pc p exp = n i c p exp = n ic p (26) e t c is the mean time it takes until an unoccupied defect changes its charge state by electron capture, p c, n e and p e are de ned in an analogous way. These relations (Eqs. 24, Eqs. 25, 26) are also valid in nonequilibrium situations. As emission probabilities are related to capture cross sections, simple defects will be fully described by the energy level E t and electron and hole capture cross sections. Deep levels in uence detector operation in three ways:. N eff increases with irradiation 2. leakage current increases with irradiation 3. irradiation creates trapping centers, where drifting charge can be trapped. 3

14 5. Change in Ne Deep levels contribute to the e ective space charge. The space charge density of a single defect type is given by Q t = e 0 N t ( P t ) for donors (27) Q t = e 0 N t P t for acceptors (28) Irradiation of silicon produces many di erent defects. The sum of Q t =e 0 over all defects gives the e ective dopant concentration N eff as N eff = X donors N t ( P t ) X acceptors N t P t + N D N A (29) In the depleted region donors in the upper half and acceptors in the lower half of the band gap are ionized in the space charge region at room temperature while donors in the lower half and acceptors in the upper half are not ionized[2]. The irradiation of silicon produces more electrically active primary acceptors than donors that are electrically active. If n-type silicon is irradiated jn eff j decreases initially with uence up to the inversion point where N eff 0. At this point, silicon bulk undergoes type inversion from n-type to p-type (negative space charge) under reverse bias. After that, jn eff j increases with uence and by that also V F D. Measurements of N eff time development after irradiation (Fig. 9) show that in the beginning the electrically active defects decay into non-active (annealing). After around 0 days at room temperature the concentration of electrically active defects starts rising again due to electrically non-active defects turning into electrically active ones (reverse annealing). The initial slope of N eff rise due to reverse annealing is found to scale linearly with uence indicating that reverse annealing is a rst order process[]. The open symbols in Fig. 8 indicate the results for the standard FZ silicon. No di erence is observed between the data obtained after pion, proton of neutron irradiation. Compared to the standard silicon (open symbols) the oxygen enriched silicon ( lled symbols) shows an improved radiation hardness after neutrons as well as after charged hadrons irradiation. However, the improvement after charged hadron irradiation is much more pronounced. 5.2 Change in leakage current Bulk current through the depleted region comes from two main contributions: di usion of charge carriers from the non-depleted region (di usion current) and generation of carriers in the depleted region (generation current). The generation current represents the dominant contribution to the leakage current in highly irradiated and even in most of the non-irradiated silicon detectors. Defects close to the middle of the band gap are e ciently electron-hole pair generation centers and thus responsible for the leakage current. First consider the situation under the approximation that the space charge region is completely depleted of charge carriers. This is a good assumption for a reversely biased detector with low leakage current. Thus the capture processes can be neglected. Occupation probability of a defect and its carrier generation rate are determined by considering emission processes only. Capture cross-sections appear indirectly with the use of the relation between emission and capture processes (Eqs. 25, 26) derived in thermal equilibrium. The average occupation probability of a defect P t (E t ) is determined by the requirement of equal electron G n and hole G p generation rates: G n = N t P t n = N t ( P t )p = G p (30) It is reasonable to assume that the capture coe cients for electrons and holes are of the same order of magnitude[]. With this assumption defects with energy levels more than few times above the intrinsic level E i are expected to be in the unoccupied (more positive) state and those below E i in the occupied (more negative state). The intrinsic level qualitatively plays a similar role in the space charge region as the Fermi level did in the thermal equilibrium case. 4

15 Figure 8: Dependence of N eff on the accumulated M ev neutron equivalent uence eq for standard and oxygen enriched FZ silicon irradiated with reactor neutrons, 23 G ev protons and 92 M ev pions. Figure 9: Schematic plot of time development of Ne for inverted silicon material. All three phases are shown with introduction rates for the relevant defects. Note that reverse annealing is shown in logarithmic time scale. 5

16 The generation current depends on the pair generation rate and generation volume. According to the Eq. 30 the charge generation accomplished by alternate emission of electrons and holes in the space charge region can be calculated as c p c n G = G n = G p = N t n c n t + cp i = N t P t c n n i t (3) t Only defects centers whose energy E t are close to the intrinsic Fermi level E i contribute signi cantly to the generation rate, and thus to the leakage current. The electron-hole generation rate on defects and their contribution to the generation current is maximal if the energy level of defects and intrinsic level are equal E t = E i exp Et E i thus giving rise to the current. The generated electron-hole pairs are immediately separated by the electric eld, I g = e 0 ws X t G t (32) where S is the area and w is thickness of the totally depleted detector and G t generation rate of electron-hole pairs for a trap in the space charge region. Since w (U) / p U also bulk generation current is proportional to p U as long as the diode is not fully depleted. In Fig. 0 the uence dependence of the increase in leakage current normalized to volume I=V is shown. Each point corresponds to an individual detector irradiated with fast neutrons in a single exposure to the given uence. The measured increase in current was observed to be proportional to uence and can thus be described by I = eq V where the proportional factor is called current related damage rate. Figure 0: Fluence dependence of leakage current for silicon detectors produced by various process technologies from di erent silicon materials. 5.3 Trapping of the drifting charge The levels in the band gap act as traps for the drifting charge. Each level can trap both electrons and holes and by that the defect changes its charge state. For example a simple donor can trap holes if it is occupied and electrons if it is empty. The analogue holds for an acceptor. Since in the SCR both acceptors and donors above the intrinsic level are predominantly empty they mainly trap electrons (electron traps). In the same 6

17 way acceptors and donors below the intrinsic level mainly trap holes (hole traps). For the calculation of carrier trapping probabilities a similar consideration as in Eq. 24 is used with the concentration of defects replacing the concentration of free carriers. The trapping probability is here de ned as: t tr e = c n ( P t ) N t ;electrons (33) t tr h = c p P t N t ;holes (34) The trapping time t tr h represents the mean time that a free carrier spends in the space charge region before it is trapped by trap t. To get the e ective trapping probability =t effe;h for electrons and holes one has to sum over the trapping probabilities of all defects t effe;h = X t t tr e;h = X t N t P e;h te;h v the;h (35) where P e t = P t and P h t = P t. At a given time after the irradiation concentration Nt of a general defect formed either directly by irradiation or by primary defect decay or reactions will be given by t N t = g t eq f t (36) where g t is the creation amplitude and f t 2 [0; ] describes the evolution of the defect with time. For the defects constant in time f t =. Using the relation Eq. 36, Eq. 35 can be rewritten as t effe;h X = eq g t f t P e;h te;h v the;h (37) t If the traps are constant in time or created with a rst order process, f t does not depend on uence. Hence, the e ective trapping probability at a given temperature and time after irradiation can be parameterized as t t effe;h = e;h (t; T ) eq (38) Analogous to leakage current damage constant, e;h can be called the e ective electron or hole trapping damage constant. Figure : Fluence dependence of e ective trapping probability for electrons and holes for neutron irradiated samples[]. 7

18 6 Conclusion Microscopic picture, that would explain behavior of how defect are related with radiation induced changes in the leakage current, the e ective doping concentration and the charge collection e ciency is not yet complete clear. The bulk damage in uences detector operation in three main ways: The increase of the leakage current results in increased noise and contributes to higher power consumption and therefore heat. Silicon detector becomes less e cient. contribute to the signal. A part of the drifting charge is trapped and thus does not At operation temperatures of LHC detectors the n type silicon bulk undergoes type inversion and becomes e ectively p type under bias. Further irradiation increases the e ective negative dopant concentration and by that the operation voltage. The radiation induced changes of the macroscopic silicon detector properties - leakage current, e ective doping concentration and charge collection e ciency - are caused by radiation induced electrical active microscopic defects. Therefore, a more fundamental understanding of the macroscopic radiation damage can only be achieved by studying the microscopic defects, their reaction and annealing kinetics, and especially their relation to the macroscopic damage parameters. The result of such investigations can then be used to improve the radiation hardness of the silicon starting material by defect engineering. In other words: Based on the knowledge about the defect kinetics and the relation between the defects and macroscopic material parameters the defect kinetics has to be in uenced in such a way that less macroscopic damage is produced. One possibility of defect engineering is the enrichment of the starting material with certain impurities leading to a reduced introduction of the defects having a detrimental e ect on the detector performance. 8

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