UNIVERZA V LJUBLJANI

FAKULTETA ZA MATEMATIKO IN FIZIKO

ODDELEK ZA FIZIKO

Joµzef Pulko

SEMINAR

Radiation Damage In Silicon DetectorsMENTOR: 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 inthe 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 thesilicon crystal and resulting in the formation of point defects and defects clusters. At the end impacts ofdefects on the electrical properties of silicon detectors are summarized.

Keywords: Silicon detector, radiation damage, defects

Ljubljana, April 2007

Contents

1 Introduction 3

2 Basic features of silicon diodes 42.1 p-n junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Operation of silicon detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Defect Generation 6

4 The NIEL scaling hypothesis 9

5 Impact of defects on detector properties 115.1 Change in Ne¤ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Change in leakage current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 Trapping of the drifting charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Conclusion 18

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1 Introduction

As the collider experiments in high energy physics go towards higher energy use of silicon detectors becomesinevitable. Their superior spatial resolution, short signal formation times and good energy resolution makethem ideal for tracking ionizing particles. Besides the ability to accurately measure the momentum ofenergetic charge particle from bending of their trajectories in magnetic �eld their most important featureis the capability of distinguishing secondary from primary vertices. Therefore there are placed as close aspossible to the interaction point. Longer operation under high radiation, results in signi�cant radiationdamage 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 bemanageable. 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 ofdefects.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 theradiation induced defects in silicon bulk are reviewed. In chapter 4 we discuss about nonionizing energylosses 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.

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2 Basic features of silicon diodes

Basic idea of silicon detectors is similar to the ionizing cell. Ionizing particles passing through silicon generateelectron-hole pairs along their path. The number of pairs is proportional to the particle�s energy loss. Thecreation of electron-hole pair in silicon requires a mean energy of 3:6 eV with the average energy loss insilicon of about 390 eV= �m for minimum-ionizing particle. This give rise to 108 pairs per �m. For a typicaldetector thickness of about D = 300�m, on average 3:25 � 104 electron-hole pairs are obtained, a signaldetectable with low-noise electronics.

2.1 p-n junction

In order to explain the operation of a p-n diode one may imagine the opposite sides of the junction originallyisolated, and then brought into intimate contact. Thermal equilibrium is established as equal number ofhighly mobile electrons and holes, from the n-type and the p-type material, respectively recombine. Apotential di¤erence �bi prevents further charge �ow. Is maintained by the static space charge build uparound the junction by the ionization of the donor and acceptor atoms in the doped semiconductor. Thisregion is e¤ectively depleted of all mobile charge carriers and the voltage corresponding to the potentialdi¤erence is called built-in voltage Vbi. In the case of an abrupt p-n junction one side is more heavily dopedthat the other and overall charge neutrality then implies that the depletion region of thickness W extendsmuch further into the less heavily doped side of the device. This is displayed in Fig. 1 for reverse biased

Figure 1: Schematic �gure of a p+�n abrupt junction: a) electrical charge density, b) electric �eld strenght,c) electron potential energy.

abrupt p+ � n diode of thickness d under the assumption of a homogeneous distribution of dopant atoms.

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Furthermore the so-called depletion approximation is assumed to be valid which demands that the spacecharge is constant in the region 0 < x < W , although it is known that the di¤usion of electrons from then-type bulk into the depletion zone results into a smooth distribution of the charge around x = W . Theelectric �eld strength and electrochemical potential can be calculated by solving Poisson�s equation:

�d2�

dx2=�el""0

=q0Neff""0

(1)

Here Neff denotes the e¤ective doping concentration which is given by the di¤erence between the con-centration of ionized donors and acceptors in the space charge region. Furthermore ""0 stands for thepermittivity of silicon with "0 = �11:9. The �rst integration of Eq. 1 with the boundary conditionsE (x =W ) = �d�(x=W )

dx = 0 leads to an expression for the electrical �eld strength which depends linearlyon x (see Fig. 1b) and reaches the maximum �eld strength of

Em (V ) = �q0Neff""0

W (V ) (2)

at the p+�n interface (x = 0). A further integration under the boundary condition � (x =W ) = 0 leads toa parabolically function for the potential:

� (x) = � 12q0Neff

""0(x�W )2 for 0 � x �W

and W � d (3)

The corresponding electron potential energy (�e0�) is schematically displayed in Fig. 1c. There it is alsoindicated that the applied reverse bias V is equal to the di¤erence between the Fermi levels in the p+ andn region, EFp and EFn which, of course, in the case of thermal equilibrium have to be the same as theelectrochemical potential. With the condition � (x = 0) = �Vbi � V one obtains an expression for thedepletion depth:

W (V ) =q

2""0q0jNeff j (V + Vbi) for W � D. (4)

With increasing reverse bias the �eld zone expands until the back contact is reached (W = d). The corre-sponding voltage, needed to fully deplete the diode, is called depletion voltage Vdep and connected with thee¤ective doping concentration Neff by:

Vdep + Vbi =q02""0

jNeff j d2 (5)

Very often the build-in voltage Vbi is neglected since the depletion voltage is in most case more than oneorder 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 ion-ization 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 particlehas passed (tracking). The latter case is the main application of silicon detectors in high energy physics. Aminimum ionizing particle (mip) traversing a silicon layer of d = 300�m thickness deposits most probableyan energy of � 90k eV. Although the energy gap in silicon is about 1:12 eV at room temperature the requiredaverage energy to produce an electron-hole pair is � 3:6 eV. Thus most probable about 22000 electron-holepairs are created by mip (about 72 e-h per �m). If the detector is fully depleted all generated electrons andholes drift in the applied �eld with their drift velocity vdr;n and vdr;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 = q0vdr;n;pd

with vdr;n;p = �n;pE (x)� E (x) . (6)

The mobility �n;p is depending on the �eld strength E and the �eld strength itself is depending on the depthx in the detector (see Fig. 1).

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3 Defect Generation

The energy loss of an incoming particle by interaction with matter can be divided into ionizing and non-ionizing energy loss (NIEL). Due to fast recombination of charge carriers the ionizing energy loss does notlead to bulk damage. NIEL contains displacements of lattice atoms and nuclear reactions. The introductionrate of defects, resulting from nuclear reactions, is more than two orders of magnitude lower compared tointroduction 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 itslattice site resulting in silicon interstitial and a left over vacancy (Frenkel Pair). However, the primary recoilatom can only be displaced if the imparted energy is higher than the displacement threshold energy Ed ofapproximately 25 eV[11]. The energy of recoil PKA or any other residual atom resulting from a nuclearreaction can of course be much higher. Along the path of these recoils the energy loss consists of twocompeting contributions, one being due to ionization and the other caused by further displacements. Atthe end of any heavy recoil range, the nonionizing interactions are prevailing and an dense agglomerationof defects (disordered regions or clusters) is formatted as displayed in Fig. 2. Both, point defects along theparticle path and the clusters at the end of their range are, responsible for various damage e¤ects in the bulkof 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 ER of 50 k eV. The primaryrecoil energy of 50 k eV has been chosen because it is approximately the average kinetics energy that a 1M eV neutrons imparts on a PKA. The PKA releases its energy over a distance of about 1000 Å to thesilicon lattice. Approximately 37% of the recoil energy will go into ionization e¤ects and the rest can displacefurther 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 ER;max that can be imparted by a particle of mass mp

and kinetic energy Ep to the recoil atom by elastic scattering (nonrelativistic approach):

ER;max = 4EpmpmSi

(mp +mSi)2 (7)

Taking into account the displacement threshold of Ed � 25 k eV and a threshold energy of � 5 k eV for theproduction of clusters[11] one can deduce that neutrons need a kinetic energy of � 185 eV for the production

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of a Frenkel pair and more than � 35 k eV to produce a cluster. Electrons, however, need a kinetic energyEe of about 255 k eV to produce a Frenkel pair and more than � 8 M eV to produce cluster, if one takesinto account the approximate relativistic relation ER;max = 2Ee

�Ee + 2m0c

2�=�mSic

2�.

With the displacement of a big number of silicon atoms from their lattice sites the damage process has notended. Interstitials and vacancies are very mobile in the silicon lattice at temperature above 150K. Thereforea part of Frenkel pairs produced at room temperature annihilate and no damage remains. Simulations haveshown that this is the case for about 60% of the overall produced Frenkel pairs and can reach in the disorderedregions 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 newcon�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 interstitialsdi¤using throughout the crystal. The most frequent reactions within the clusters, where the defect densityis high, belong to the group B. Possible reactions of both groups are listed in Fig. 3 and the most relevantdefect 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 andvacancies throughout the crystal. Most frequent reactions during a primary cascade are gathered in groupB. Indexes i and s stand for interstitial and substitutional.

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

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4 The NIEL scaling hypothesis

Charged hadrons interact with silicon primarily by the Coulomb interaction at lower energies. Thus a big partof 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 1:8 M eV alsonuclear reactions. Hence the question arise how the radiation damage produced not only by di¤erent kindof particles but also, depending on the particle energy, by di¤erent kind of interactions can be scaled withrespect to the radiation induced changes observed in the material. The answer is found in the so-called NonIonizing 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 thespatial distribution of the introduced displacement defects in one PKA cascade, and irrespective of thevarious annealing sequences taking place after the initial damage event.In each interaction leading to displacement damage a PKA with speci�c recoil energy ER is produced.

The portion of recoil energy that is deposited in form of displacement damage is depending on the recoilenergy itself and can analytically be calculated by the so-called Linhard partition function P (ER)[12]. Withthe help of the partition function the NIEL can be calculated and is expressed by the displacement damagecross section1

D (E) =X�

�� (E)

EmaxRZ0

fv (E;ER)P (ER) dER (8)

Here the index � indicates all possible interactions between the incoming particle with energy E and thesilicon atoms in the crystal leading to displacements in the lattice. �� is the cross section correspondingto the reaction with index � and f� (E;ER) gives the probability for the generation of a PKA with recoilenergy ER by a particle with energy E undergoing the indicated reaction �. The integration is done overall possible recoil energies ER and below the displacement threshold the partition function is set to zeroP (ER < Ed) = 0. Fig. 5 shows the displacement damage cross sections for neutrons, protons, pions andelectrons in an energy range from 10 G eV down to some m eV for the thermal neutrons. A thoroughdiscussion of these functions can be found in[13].The total displacement-damage energy per unit volume deposited in the silicon crystal can be written as

"d = NSitirr

1Z0

d�

dED (E) dE (9)

where tirr denotes the irradiation time, � (E) the �ux of incoming particles andNSi space density of the targetnuclei. 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 1 M eV neutrons is taken as thereference point. The standard value of 1 M eV neutrons NIEL is 95 M eVmb[14]. Irradiation with particleA with a spectral distribution d�A

dE and cut-o¤s Emin and Emax would cause the same damage as 1 M eVneutrons if

"d = NSitirr

Zd�AdE

DA (E) dE = Nsi�eqDn (1M eV) (10)

where �eq denotes the equivalent integrated �ux (�uence) of 1 M eV neutrons which would have caused thesame damage as the �uence

�A = tirr

Zd�AdE

dE (11)

of particles actually applied. It is possible to de�ne the hardness factor �A allowing to compare the damage

1Also called damage function and related to NIEL by D (E) = ANA

dEdx(E) jnon ion izing .The NIEL value ca also be referred

to as the displacement-KERMA (Kinetic Energy Released to Matter).

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Figure 5: Displacement damage function D (E) normalized to 95 M eVmb for neutrons, protons, pions andelectrons. Due to normalization to 95 M eVmb the ordinate represents the damage equivalent to 1 M eVneutrons. 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

�A =1

Dn (1 M eV)

Zd�AdE DA (E) dEZ

d�AdE dE

(12)

It follows from here that�eq = �A�A (13)

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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 defectlevels Et for the di¤erent kind of defects are indicated by the short solid lines. The ionization energy �Etneeded to e.g. emit an electron into the conduction band corresponds to the distance between the conductionband edge EC and the defect level position (�Et = EC � Et). Acceptors are defects that are negativelycharged when occupied with an electron while donors are defects that are neutral when occupied with anelectron. In thermal equilibrium the charge state of defects is ruled by the Fermi levels. If the Fermi levelis located above the defect level, acceptors are negatively charged and donors are neutral; if it is below thedefect level, acceptors are neutral and donors are positively charged. This is indicated by the (�/�/+)-signsin the �gure. Some defects have more than one level in the band gap. As an example the levels of thethermal double donor (TDD) and the amphoteric divacancy (VV) are shown. An amphoteric defect is adefect with acceptor and donor level. In the space charge region the occupation with charge carriers is ruledby the emission coe¢ cients of the defects. Therefore, usually levels in the upper half of the band gap arenot occupied by an electron while the levels in the lower half are occupied with electrons. This means forexample that the defects V Oi (acceptor in upper half of band gap) and CiOi (donor in lower half) have noin�uence on the depletion voltage of the detector since they are neutral in the space charge region. However,BS (acceptor in lower half) and PS (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 levelsin the forbiden band gap.

In non-irradiated silicon the density of deep level defects is far below the density of shallow dopantswhich determine to a large part the electrical behavior of silicon. In these situation the deep level defectscan be considered as a disturbance to the semiconductor which properties remain basically intact. Theconcentration of the deep level defects can exceed the concentration of shallow dopant density in irradiatedsilicon. The result is a drastic change of silicon properties. Each defect can have several charge states. Inthe simplest case, like for shallow dopants, a donor can assume two charge states, neutral with the electronloosely bound to the donor site and positively ionized. The acceptor may be neutral or negatively ionized. Ageneral type of defect is much more complex. It may be a complex structure of missing silicon atoms in thelattice and impurity atoms, capable of switching between several chemical binding structures and by thatbetween the charge states. Changing from one state to another may be accomplished by thermal excitation.

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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 anelectron 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 = Ec �E2) or emission of a hole with energy E2 �E. The opposite transition requires electron emissino(E = E � Ec) or hole capture.As will be shown later the emission and capture processes are related to each other so that a (non-

degenerate) defect is characterized by the following properties:

� k energy levels Et; k describing the energy involved in changing the charge state.

� k + 1 charge states Qt; 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 chargestates, 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) =1

1 + exp�E�EFkBT

� (14)

where EF is the Fermi level, E the defect energy level. An occupied simple donor in this nomenclature isneutral while a occupied simple acceptor is negatively charged. In the following the short hand notation

�(E) = exp

�E � EikBT

�(15)

will be used, with Ei the Fermi level for the intrinsic silicon (p = n = ni) derived from the charge neutralitycondition

Ei =EV + EC

2+1

2kBT ln

�NVNC

�(16)

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where the bottom of the conduction band is denoted with EC , the top of the valence band with EV and thedensity of states in conduction and valence band with NC and NV . Electron and hole concentration in anysilicon material are thus given as

n = NC exp

�EC � EFkBT

�= ni exp

�EF � EikBT

�= ni�F (17)

p = NV exp

�EF � EVkBT

�= ni exp

�Ei � EFkBT

�=ni�F

The Fermi level is found from the requirement of overall charge neutralityXdonors

N jt

h1� F

�Ejt

�i�

Xacceptors

N jt F�Ejt

�+ND �NA � n+ p = 0 (18)

where the Nt denotes the concentration of deep defects. Also the complete ionization of shallow dopantsis assumed. Although the thermal equilibrium occupation probabilities are completely described by Fermifunction, this is the result of a continuous change of the charge state of individual defects. Thermal equilib-rium thus allows us to �nd the relations between the capture and emission processes. Considering a singledefect level in thermal equilibrium the rate of electron capture has to be equal to the rate of electron emis-sion. An analog relation holds for holes (see Fig. 6). This follows from the requirements that the averageoccupation probability of defects does not change and there is no net �ow of electrons between the valenceand conduction bands. With the introduction of the capture coe¢ cients

cn = vthe�te (19)

cp = vthh�th (20)

for the product of thermal velocity and capture cross section and the emission probabilities "n and "p onegets:

ncnNt (1� F (Et)) = NtF (Et) "n (21)

pcpNtF (Et) = NtF (1� F (Et)) "p (22)

F (Et) =1

1 + exp�Et�EFkBT

� = 1

1 + �t�F

(23)

from which the electron capture (also called recombination) and emission (also called generation) probabilitiesof a simple defect are obtained as

1

�nc= ncn,

1

�pc= pcp (24)

en =1

�ne= ncn exp

�Et � EFkBT

�= nicn exp

�Et � EikBT

�= nicn�t (25)

ep =1

�pe= pcp exp

�EF � EtkBT

�= nicp exp

�Ei � EtkBT

�=nicp�t

(26)

� c is the mean time it takes until an unoccupied defect changes its charge state by electron capture, �pc ,�ne and �

pe are de�ned in an analogous way. These relations (Eqs. 24, Eqs. 25, 26) are also valid in non-

equilibrium situations. As emission probabilities are related to capture cross sections, simple defects will befully described by the energy level Et and electron and hole capture cross sections.Deep levels in�uence detector operation in three ways:

1. Neff increases with irradiation

2. leakage current increases with irradiation

3. irradiation creates trapping centers, where drifting charge can be trapped.

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5.1 Change in Ne¤

Deep levels contribute to the e¤ective space charge. The space charge density of a single defect type is givenby

Qt = e0Nt (1� Pt) for donors (27)

Qt = �e0NtPt for acceptors (28)

Irradiation of silicon produces many di¤erent defects. The sum of Qt=e0 over all defects gives the e¤ectivedopant concentration Neff as

Neff =Xdonors

Nt (1� Pt)�X

acceptors

NtPt +ND �NA (29)

In the depleted region donors in the upper half and acceptors in the lower half of the band gap are ionizedin the space charge region at room temperature while donors in the lower half and acceptors in the upperhalf are not ionized[2]. The irradiation of silicon produces more electrically active primary acceptors thandonors that are electrically active. If n-type silicon is irradiated jNeff j decreases initially with �uence upto the inversion point where Neff � 0. At this point, silicon bulk undergoes type inversion from n-type top-type (negative space charge) under reverse bias. After that, jNeff j increases with �uence and by thatalso VFD.Measurements of Neff time development after irradiation (Fig. 9) show that in the beginning the

electrically active defects decay into non-active (annealing). After around 10 days at room temperature theconcentration of electrically active defects starts rising again due to electrically non-active defects turninginto electrically active ones (reverse annealing). The initial slope of Neff rise due to reverse annealing isfound to scale linearly with �uence indicating that reverse annealing is a �rst order process[1].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 (opensymbols) the oxygen enriched silicon (�lled symbols) shows an improved radiation hardness after neutronsas well as after charged hadrons irradiation. However, the improvement after charged hadron irradiation ismuch more pronounced.

5.2 Change in leakage current

Bulk current through the depleted region comes from two main contributions: di¤usion of charge carriersfrom the non-depleted region (di¤usion current) and generation of carriers in the depleted region (generationcurrent). The generation current represents the dominant contribution to the leakage current in highlyirradiated and even in most of the non-irradiated silicon detectors. Defects close to the middle of the bandgap 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. Thusthe capture processes can be neglected. Occupation probability of a defect and its carrier generation rateare determined by considering emission processes only. Capture cross-sections appear indirectly with the useof the relation between emission and capture processes (Eqs. 25, 26) derived in thermal equilibrium. Theaverage occupation probability of a defect Pt(Et) is determined by the requirement of equal electron Gn andhole Gp generation rates:

Gn = NtPtn = Nt(1� Pt)p = Gp (30)

It is reasonable to assume that the capture coe¢ cients for electrons and holes are of the same order ofmagnitude[1]. With this assumption defects with energy levels more than few times kBT above the intrinsiclevel Ei are expected to be in the unoccupied (more positive) state and those below Ei in the occupied (morenegative state). The intrinsic level qualitatively plays a similar role in the space charge region as the Fermilevel did in the thermal equilibrium case.

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Figure 8: Dependence of Neff on the accumulated 1 M eV neutron equivalent �uence �eq for standard andoxygen enriched FZ silicon irradiated with reactor neutrons, 23 G eV protons and 192 M eV pions.

Figure 9: Schematic plot of time development of Ne¤ for inverted silicon material. All three phases areshown with introduction rates for the relevant defects. Note that reverse annealing is shown in logarithmictime scale.

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The generation current depends on the pair generation rate and generation volume. According to theEq. 30 the charge generation accomplished by alternate emission of electrons and holes in the space chargeregion can be calculated as

G = Gn = Gp = Ntcpcn

cn�t +cp�t

ni = NtPtcnni�t (31)

Only defects centers whose energy Et are close to the intrinsic Fermi level Ei contribute signi�cantly tothe generation rate, and thus to the leakage current. The electron-hole generation rate on defects and theircontribution to the generation current is maximal if the energy level of defects and intrinsic level are equal

Et = Ei exp��Et�Ei

kBT

�. The generated electron-hole pairs are immediately separated by the electric �eld,

thus giving rise to the currentIg = e0wS

Xt

Gt (32)

where S is the area and w is thickness of the totally depleted detector and Gt generation rate of electron-holepairs for a trap in the space charge region. Since w (U) /

pU also bulk generation current is proportional

topU as long as the diode is not fully depleted.In Fig. 10 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 thegiven �uence. The measured increase in current was observed to be proportional to �uence and can thus bedescribed by

�I = ��eqV

where the proportional factor � is called current related damage rate.

Figure 10: Fluence dependence of leakage current for silicon detectors produced by various process technolo-gies 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 holesand by that the defect changes its charge state. For example a simple donor can trap holes if it is occupiedand electrons if it is empty. The analogue holds for an acceptor. Since in the SCR both acceptors and donorsabove the intrinsic level are predominantly empty they mainly trap electrons (electron traps). In the same

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way acceptors and donors below the intrinsic level mainly trap holes (hole traps). For the calculation ofcarrier trapping probabilities a similar consideration as in Eq. 24 is used with the concentration of defectsreplacing the concentration of free carriers. The trapping probability is here de�ned as:

1

� ttre= cn (1� Pt)Nt ;electrons (33)

1

� ttrh= cpPtNt ;holes (34)

The trapping time � ttrh represents the mean time that a free carrier spends in the space charge regionbefore it is trapped by trap t. To get the e¤ective trapping probability 1=teffe;h for electrons and holes onehas to sum over the trapping probabilities of all defects

1

teffe;h=Xt

1

� ttre;h=Xt

Nt

�1� P e;ht

��te;hvthe;h (35)

where P et = Pt and Pht = 1� Pt. 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

Nt = gt�eqft (36)

where gt is the creation amplitude and ft 2 [0; 1] describes the evolution of the defect with time. For thedefects constant in time ft = 1.

Using the relation Eq. 36, Eq. 35 can be rewritten as

1

teffe;h= �eq

Xt

gtft

�1� P e;ht

��te;hvthe;h (37)

If the traps are constant in time or created with a �rst order process, ft does not depend on �uence. Hence,the e¤ective trapping probability at a given temperature and time after irradiation can be parameterized as

1

teffe;h= �e;h (t; T ) �eq (38)

Analogous to leakage current damage constant �, �e;h can be called the e¤ective electron or hole trappingdamage constant.

Figure 11: Fluence dependence of e¤ective trapping probability for electrons and holes for neutron irradiatedsamples[1].

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6 Conclusion

Microscopic picture, that would explain behavior of how defect are related with radiation induced changes inthe leakage current, the e¤ective doping concentration and the charge collection e¢ ciency is not yet completeclear.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 consump-tion and therefore heat.

� Silicon detector becomes less e¢ cient. A part of the drifting charge is trapped and thus does notcontribute to the signal.

� At operation temperatures of LHC detectors the n type silicon bulk undergoes type inversion andbecomes e¤ectively p type under bias. Further irradiation increases the e¤ective negative dopantconcentration and by that the operation voltage.

The radiation induced changes of the macroscopic silicon detector properties - leakage current, e¤ectivedoping concentration and charge collection e¢ ciency - are caused by radiation induced electrical activemicroscopic defects. Therefore, a more fundamental understanding of the macroscopic radiation damage canonly be achieved by studying the microscopic defects, their reaction and annealing kinetics, and especiallytheir relation to the macroscopic damage parameters. The result of such investigations can then be used toimprove the radiation hardness of the silicon starting material by defect engineering. In other words: Basedon the knowledge about the defect kinetics and the relation between the defects and macroscopic materialparameters 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 leadingto a reduced introduction of the defects having a detrimental e¤ect on the detector performance.

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