Radiation defects in silicon due to hadrons and leptons, their annealing and %%@ influence on detector performance
A phenomenological model was developed to explain quantitatively, without free parameters, %%@ the production of primary defects in silicon after particle irradiation, the kinetics of %%@ their evolution toward equilibrium and their influence on detector parameters. The type of %%@ the projectile particle and its energy is considered in the evaluation of the %%@ concentration of primary defects. Vacancy-interstitial annihilation, interstitial %%@ migration to sinks, vacancy - impurity complexes ($VP$, $VO$, $V_2O$), and divacancy %%@ ($V_2$) formation are taken into account in different irradiation conditions, for %%@ different concentrations of impurities in the semiconductor material, for 20 and 0 $^o$C. %%@ The model can be extended to include other vacancy and interstitial complexes. The density %%@ of the reverse current in the detector after irradiation is estimated. Comparison with %%@ experimental measurements is performed. A special application considered in the paper is %%@ the modelled case of the behaviour of silicon detectors operating in the pion field %%@ estimated for the LHC accelerator, under continuum generation and annealing. \medskip PACS: \\ 29: Experimental methods and instrumentation for elementary-particle and nuclear physics\\ 81: Materials science\\ 78: Optical properties, condensed-matter spectroscopy and other interactions of radiation %%@ and particles with condensed matter \medskip
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\begin{document}


\title{Radiation defects in silicon due to hadrons and leptons, their annealing and %%@
influence on detector performance}


\author {Ionel Lazanu\footnote{University of Bucharest, POBox MG-11, Bucharest-Magurele, %%@
Romania} \hspace{2 pt}
and Sorina Lazanu\footnote{National Institute for Materials Physics, POBox MG-7, %%@
Bucharest-Magurele, Romania}}


\date{\today}
\maketitle
\begin{abstract}
A phenomenological model was developed to explain quantitatively, without free parameters, %%@
the production of primary defects in silicon after particle irradiation, the kinetics of %%@
their evolution toward equilibrium and their influence on detector parameters. The type of %%@
the projectile particle and its energy is considered in the evaluation of the %%@
concentration of primary defects. Vacancy-interstitial annihilation, interstitial %%@
migration to sinks, vacancy - impurity complexes ($VP$, $VO$, $V_2O$), and divacancy %%@
($V_2$) formation are taken into account in different irradiation conditions, for %%@
different concentrations of impurities in the semiconductor material, for 20 and 0 $^o$C. %%@
The model can be extended to include other vacancy and interstitial complexes. The density %%@
of the reverse current in the detector after irradiation is estimated. Comparison with %%@
experimental measurements is performed. A special application considered in the paper is %%@
the modelled case of the behaviour of silicon detectors operating in the pion field %%@
estimated for the LHC accelerator, under continuum generation and annealing. 


\medskip
\textbf{PACS}: \\
29: Experimental methods and instrumentation for elementary-particle and nuclear physics\\
81: Materials science\\
78: Optical properties, condensed-matter spectroscopy and other interactions of radiation %%@
and particles with condensed matter

\medskip

\end{abstract}





\section{Introduction}

The use of silicon detectors in high radiation environments, as to be expected in future %%@
high energy accelerators, poses severe problems due to changes in the properties of the %%@
material, and consequently influences the performances of detectors.

The incident particle, hadron or lepton, interacts with the electrons and with the nuclei %%@
of the semiconductor lattice. It losses its energy in several processes, which depend on %%@
the nature of the particle and on its energy. The effect of the interaction of the %%@
incident particle with the target atomic electrons is ionisation, and the characteristic %%@
quantity for this process is the energy loss or stopping power. The nuclear interaction %%@
between the incident particle and the lattice nuclei produces bulk defects and this %%@
phenomenon is studied in the present paper. As a result of this interaction, if the %%@
primary projectile is a particle, one or more light particles are formed, and usually one %%@
(or more) heavy recoil nuclei. This recoil nucleus has charge and mass numbers equal or %%@
lower than that of the medium. After this first interaction, the recoil nucleus or nuclei %%@
are displaced from the lattice positions into interstitials. Then, the primary knock-on %%@
nucleus, if its energy is large enough, can produce the displacement of a new nucleus, and %%@
the process continues as long as the energy of the colliding nucleus is higher than the %%@
threshold for atomic displacements. We denote these displacement defects, vacancies and %%@
interstitials, as primary defects, prior to any further rearrangement. 
In silicon these defects are essentially unstable and interact via migration, %%@
recombination, annihilation or produce other defects.

As a consequence of the degradation to radiation  of the semiconductor material, an %%@
increase of the reverse current due the reduction of the minority carrier lifetime, a %%@
reduction of the charge collection efficiency and a modification of the effective doping, %%@
due to the generation of trapping centres, are observed in the detector characteristics.
In this paper, for the first time, a phenomenological model was developed to explain %%@
quantitatively, without free parameters, the mechanisms of production of the primary %%@
defects during particle irradiation, the kinetics of their evolution toward stable defects %%@
and equilibrium and the influence of the defects on detector parameters. The effects of %%@
the incident particle type, of its kinetic energy and of the irradiation conditions on the %%@
concentration of defects are studied. Vacancy-interstitial annihilation, interstitial %%@
migration to sinks, vacancy - impurity complexes ($VP$, $VO$ and $V_2O$), and divacancy %%@
($V_2$) formation are considered in different irradiation conditions, for different %%@
concentrations of impurities in the semiconductor material and at different temperatures %%@
near room temperature. The model can be extended directly to include the effects of other %%@
mechanisms related to these, or other impurities in silicon, and of their interaction with %%@
the vacancy and/or interstitial. The density of the reverse current in the detector after %%@
irradiation is estimated. Comparison with experimental published data of the time %%@
evolution of the concentration of defects is performed, as well as with measurements of %%@
the density of the leakage current. For different discrepancies, some explanations are %%@
suggested. A special application considered in the paper is the simulated case of the %%@
behaviour of silicon detectors operating in the pion field simulated for the future %%@
conditions at the new LHC (Large Hadron Collider) accelerator.


\section{Production of primary defects}
A point defect in a crystal is an entity that causes an interruption in the lattice %%@
periodicity. In this paper, the terminology and definitions in agreement with M. Lannoo %%@
and J. Bourgoin \cite{1} are used in relation to defects.

The basic assumption of the present model is that vacancies and interstitials are produced %%@
in materials exposed to radiation in equal quantities, uniformly in the bulk of the %%@
sample. They are the primary radiation defects, being produced either by the incoming %%@
particle, or as a consequence of the subsequent collisions of the primary recoil in the %%@
lattice.

The concentration of the primary radiation induced defects per unit fluence (CPD) in the %%@
semiconductor material has been calculated using the explicit formula (see details, e.g. %%@
in references \cite{2,3}):
\begin{equation}
CPD	\left(E\right)= \frac{N_{Si}}{2E_{Si}} \int \sum _{i} \left( \frac{d\sigma}{d\Omega} %%@
\right)_{i,Si} L(E_{Ri})_{Si} d\Omega=\frac{1}{N_A} \frac{N_{Si}A_{Si}}{2E_{Si}} %%@
NIEL\left(E\right)
\end{equation}
where $E$ is the kinetic energy of the incident particle, $N_{Si}$ is the atomic density %%@
in silicon, $A_{Si}$ is the silicon atomic number, $E_{Si}$ - the average threshold energy %%@
for displacements in the semiconductor, $E_{Ri}$  - the recoil energy of the residual %%@
nucleus produced in interaction $i$, $L(E_{Ri})$ - the Lindhard factor that describes the %%@
partition of the recoil energy between ionisation and displacements and %%@
$(d\sigma/d\Omega)_i$ - the differential cross section of the interaction between the %%@
incident particle and the nucleus of the lattice for the process or mechanism $i$, %%@
responsible in defect production. $N_A$ is Avogadro's number. The formula gives also the %%@
relation with the non ionising energy loss ($NIEL$). It is important to observe that there %%@
exists a proportionality between the $CPD$ and $NIEL$ only for monoelement materials.

For $CPD$ produced by pions, the pion - silicon interaction has been modelled \cite{4} and %%@
the energy dependencies of the Lindhard factors have been calculated in the frame of %%@
analytical approximations for different recoils in Si \cite{5}.

The concentration of primary defects produced by protons, neutrons, electrons and photons %%@
have been obtained from the $NIEL$. The calculations of Summers and co-workers for proton %%@
and electron $NIEL$ in silicon from reference \cite{6}, the calculations of proton, %%@
electron and photon $NIEL$ of Van Ginneken \cite{7} as well as those of Ougouang for %%@
neutrons \cite{8} have been considered.

In Figure \ref{f1}, the dependence of the $CPD$ on the particle kinetic energy is %%@
presented: for pions, our calculations from reference \cite{9} have been used; for %%@
protons, in the energy range $10^{-3} \div 1$ MeV the calculations of Summers, in the %%@
range $1 \div 200$ MeV the average between those of Summers and of Van Ginneken, while in %%@
the range $200 \div 10000$ MeV, Van Ginneken's. The curve for electrons uses, up to 1 MeV %%@
only the values from reference \cite{6}, in the range $1 \div 200$ MeV an average between %%@
the values from references \cite{6} and \cite{7}, and after 200 MeV only from \cite{7}. %%@
The curves for photons and neutrons are calculated from Van Ginneken's \cite{7} and %%@
Ougouang's \cite{8} respectively.

The main source of errors in the calculated concentration of defects comes from the %%@
modelling of the  particle - nucleus interaction and from the number and quality of the %%@
experimental data available for these processes. 
Due to the important weight of annealing processes, as well as to their very short time %%@
scale, CPD is not a measurable physical quantity.

In silicon, vacancies and interstitials are essentially unstable and interact via %%@
migration, recombination, annihilation or produce other defects. 

\begin{figure}[ht]
\centering
\includegraphics[width=.8\textwidth, clip, angle=90]{fig1.eps}
\caption{\small{Energy dependence of the concentration of primary defects on unit fluence 
induced by protons, pions, electrons, photons and neutrons in silicon - see text for 
details.}}
\label{f1}
\end{figure}

\section{The kinetics of radiation induced defects}

In the frame of the model, equal concentrations of vacancies and interstitials are %%@
supposed to be produced by irradiation, in much greater concentrations than the %%@
corresponding thermal equilibrium values, characteristic to each temperature. Both the %%@
pre-existing defects and those produced by irradiation, as well as the impurities, are %%@
assumed to be randomly distributed in the solid. An important part of the vacancies and %%@
interstitials annihilate. The sample contains certain concentrations of impurities which %%@
can trap interstitials and vacancies respectively, and form stable defects.

In the present paper, vacancy-interstitial annihilation, interstitial migration to sinks, %%@
divacancy and vacancy impurity complex formation  ($VP$, $VO$, $V_2O$), are considered. 
The mechanisms of formation of higher order defects involving vacancy and oxygen can be %%@
added, as well as the effects of other impurities, e.g. carbon. 

This picture could be described in terms of chemical reactions by the kinetic scheme:
\begin{equation}                                                        
V+I\ _{\overleftarrow{K_1}} ^{\underrightarrow{G}}\text{annihilation}
\end{equation}
\begin{equation}                                                       
I\stackrel{K_2}{\rightarrow } \text{sinks}
\end{equation}
\begin{equation}							
V+P\ _{\overleftarrow{K_5}} ^{\underrightarrow{K_5}}\ VP
\end{equation}
$VP$ is the $E$ centre.
\begin{equation}							
V+O\ _{\overleftarrow{K_3}} ^{\underrightarrow{K_4}}\ VO
\end{equation}
$VO$ is the $A$ centre.
\begin{equation}						
V+V\ _{\overleftarrow{K_8}} ^{\underrightarrow{K_7}}\ V_2
\end{equation}
\begin{equation}						
V+A\ _{\overleftarrow{K_{10}}} ^{\underrightarrow{K_{9}}}\ V_2O
\end{equation}

The bimolecular recombination law of interstitials and vacancies is supposed to be a valid %%@
approximation for the present discussion, because at the concentrations of vacancies of %%@
interest, only a small fraction of defects anneals by correlated annihilation if their %%@
distribution is random (see the discussion in reference \cite{10}).

The multivacancy oxygen defects as, e.g. $V_3O$, $V_2O_2$, $V_3O_2$, $V_3O_3$, are not %%@
considered in the model.

The reaction constant $K_1$ (corresponding to vacancy - interstitial annihilation) is %%@
determined by the diffusion coefficient of the interstitial atom to a substitutional trap:
\begin{equation}
K_1=30\nu \exp \left( -E_{i1}/k_BT\right)
\end{equation}
where $E_{i1}$ is the activation energy of interstitial migration and $\nu$ 
the vibrational frequency. The reaction constant in process (2) is 
proportional to the sink concentration $\alpha$:

\begin{equation}
K_2=\alpha \nu \lambda ^2\exp \left( -E_{i1}/k_BT\right)                                                 
\end{equation}
with $\lambda$ the jump distance.

Lee and Corbett \cite{11} argue that divacancies, vacancy-oxygen and divacancy-oxygen %%@
centres are equally probable below 350 $^o$C; thus, $K_3$, $K_5$, $K_7$ and $K_9$, that %%@
describe the formation of vacancy - impurity complexes and of divacancies, are determined %%@
by the activation energy of vacancy migration, $E_{i2}$, and are given by:
\begin{equation}
K_3=K_5=K_7=K_9=30\nu \exp \left( -E_{i2}/k_BT\right)
\end{equation}
while $K_4$, $K_6$, $K_8$ and $K_{10}$ are related to the activation energies of %%@
dissociation of the $A$, $E$, $V_2$ and $V_2O$ centres respectively.

\begin{equation}
K_4=5\nu \exp \left( -E_A/k_BT \right)					
\end{equation}
\begin{equation}
K_6=5\nu \exp \left( -E_E/k_BT \right)					
\end{equation}
\begin{equation}
K_8=5\nu \exp \left( -E_{V_2)}/{k_BT}\right)				
\end{equation}
\begin{equation}
K_{10}=5\nu \exp \left( -E_{V_2O}/{k_BT}\right)				
\end{equation}
where $E_A$, $E_E$, $E_{V_2}$ and $E_{V_2O}$ are the dissociation energies of the $A$, %%@
$E$, $V_2$ and $V_2O$ complexes respectively.

$G$ is the generation rate of vacancy-interstitial pairs, and is given by the product of %%@
$CPD$ by the irradiation flux. Thermal generation is neglected, this approximation %%@
corresponding to high irradiation fluxes. 

In the simplifying hypothesis of random distribution of $CPD$ for all particles, two %%@
different particles can produce the same generation rate for vacancy-interstitial pairs.

\begin{equation}
G=[\left(CPD\right)_{part.a}\left(E_1)\right]\cdot\Phi_{part.a}(E_1)=[\left(CPD\right)_{pa%%@
%%@
rt.b}\left(E_2)\right]\cdot\Phi_{part.2(E_2)}
\end{equation}
is fulfilled.

Here, $\Phi$ is the flux of particles ($a$) and ($b$) respectively, and $E_1$ and $E_2$ %%@
their corresponding kinetic energies.
The system of coupled differential equations corresponding to the reaction scheme (2)-(7) %%@
cannot be solved analytically. 

The following values of the parameters have been used: $E_{i1}$ = 0.4 eV, $E_{i2}$ = 0.8 %%@
eV, $E_A$ = 1.4 eV, $E_B$ = 1.1 eV, $E_{V_2}$ = 1.3 eV, $E_{V_2}$ = 1.6 eV, $E_{V_2O}$ = %%@
$10^{13}$ Hz, $\lambda$ = $10^{15}$ cm$^2$, $\alpha = 10^{10}$ cm$^{-2}$.

Defect concentrations, as well as their time evolution, have been calculated solving %%@
numerically the system of coupled differential equations.

We would like to underline the specific importance of the irradiation and annealing %%@
history (initial material parameters, type of irradiation particles, energetic source %%@
spectra, flux, irradiation temperature, measurement temperature, temperature and time %%@
between irradiation and measurement) on defect evolution.

\begin{figure}[ht]
\centering
\includegraphics[width=.8\textwidth, clip, angle=90]{fig2.eps}
\caption{\small{Time dependence of the concentrations of: a)$V_2$, b)$VO$, c)$V_2O$ and %%@
d)$VP$,
induced in silicon with 10$^{14}$ P/cm$^3$ and 5x10$^{16}$ O/cm$^3$, irradiated with 200 %%@
MeV kinetic energy pions, at a total fluence of $10^{15}$ pions/cm$^2$ in different %%@
irradiation conditions - see text.}}
\label{f2}
\end{figure}

In Figures 2a $\div$ d, the formation and time evolution of the divacancy, vacancy-oxygen, %%@
divacancy-oxygen, and vacancy-phosphorous is modelled in silicon containing the initial %%@
concentrations of impurities: 10$^{14}$ P/cm$^3$   and 5x10$^{16}$ O/cm$^3$, and %%@
irradiated with pions with about 200 MeV kinetic energy (corresponding to the in their %%@
maximum of $CPD$ in the energetic distribution), at a total fluence of 10$^{15}$ %%@
pions/cm$^2$, in different irradiation conditions. The instantaneous irradiation process %%@
is an ideal case \cite{12} where the total fluence is received by the material at time t = %%@
0, and only the relaxation process is studied. This process supposes that the annealing %%@
effects are not present during irradiation. The second case considered is: the irradiation %%@
is performed in a single pulse for a time 2x10$^4$ seconds, followed by relaxation, and %%@
the third case is a continuum irradiation process with a generation rate of defects equal %%@
with 5x10$^{10}$ pions/cm$^2$/s. In these last two cases, annealing takes place during %%@
irradiation. During and after irradiation, the temperature is kept at 293K.
As expected, after instantaneous irradiation the concentrations of defects are higher in %%@
respect with "gradual" irradiation. A special attention must be paid to the formation of %%@
$VO$ and $V_2O$ centres, for which the same concentration is attained both in the case of %%@
continuous and instantaneous irradiation.

\begin{figure}[ht]
\centering
\includegraphics[width=.8\textwidth]{fig3.eps}
\caption{\small{Time dependence of the concentrations of $VO$, $VP$, $V_2$ and $V_2O$  %%@
induced in silicon with $10^{14}$ P/cm$^3$ and 5x10$^{16}$ O/cm$^3$, irradiated with 200 %%@
MeV kinetic energy pions a total fluence of $10^{15}$ pions/cm$^2$ with the flux estimated %%@
for LHC, for 293 and 273 K.}}
\label{f3}
\end{figure}

The effect of the decrease of temperature, from 293 to 273 K during irradiation and %%@
annealing, is presented in Figures 3a and b. The material contains the same phosphorous %%@
and oxygen concentrations as in the modelled case presented in Figure 2, and was %%@
irradiated with pions of 200 MeV kinetic energy, receiving continuously a fluence of %%@
$10^{15}$ pions/cm$^2$ in ten years, in accord to the pions simulated radiation field at %%@
LHC \cite{13,14}.

The increase of temperature increases the rate of all defect formation. In the case of %%@
$VO$ and $VP$ a plateau in the time dependencies is attained. Only for $VP$  the plateau %%@
value is temperature sensitive.

\begin{figure}[ht]
\centering
\includegraphics[width=.8\textwidth]{fig5.eps}
\caption{\small{Time dependence of the concentrations of  $VO$, $VP$, $V_2$ and $V_2O$  %%@
induced in 
silicon with $10^{14}$ P/cm$^3$ and $5�10^{16}$ O/cm$^3$, by continuous irradiation with %%@
200 MeV kinetic energy pions with the flux: :a) estimated for LHC, b) 10 times the flux %%@
estimated for LHC, c) 100 times the flux estimated for LHC, at 293 K.}}
\label{f4}
\end{figure}

The rate of generation of primary defects in silicon influences the concentrations of all %%@
stable defects. In Figures 4a $\div$ c, the time evolution of the concentrations of $VO$, %%@
$VP$, $V_2$ and $V_2O$ in silicon with 10$^{14}$ P/cm$^3$ and 5x10$^{16}$ O/cm$^3$, %%@
irradiated with pions of 200 MeV with the flux estimated for $LHC$, 10 and 100 times %%@
higher respectively, is presented. It can be observed that at the $LHC$ generation rate %%@
for $CPD$, after 3x10$^7$ seconds an equilibrium is established for the $VP$ complex: its %%@
rate of formation equals its rate of dissociation - Fig. 4a. This time is shorter for %%@
higher generation rate, as can be observed in Figures 4b and 4c. The value of the plateau %%@
concentration for the vacancy-oxygen complex is attained after around the same time as the %%@
plateau for the  concentration, in conditions of the $LHC$ generation rate, and a shorter %%@
time, about 2x10$^6$ sec. for a rate hundred times higher than the $LHC$ one. For other %%@
defects, as divacancies and divacancy-oxygen, the processes to established the equilibrium %%@
are very slow.

The formation of divacancy-oxygen is delayed in respect to vacancy oxygen, and for long %%@
exposure times, the same value for the concentration is obtained.

The effect of oxygen in irradiated silicon has been a subject of intensive studies in %%@
remote past. In the last decade a lot of studies have been performed to investigate the %%@
influence of different impurities, especially oxygen and carbon, as possible ways to %%@
enhance the radiation hardness of silicon for detectors in the future generation of %%@
experiments in high energy physics - see, e.g. references \cite{15,16}. These impurities %%@
added to the silicon bulk modify the formation of electrically active defects, thus %%@
controlling the macroscopic device parameters. If silicon is enriched in oxygen, the %%@
capture of radiation-generated vacancies is favoured by the production of the %%@
pseudo-acceptor complex vacancy-oxygen. Interstitial oxygen acts as a sink of vacancies, %%@
thus reducing the probability of formation of the divacancy related complexes, associated %%@
with deeper levels inside the gap. For this purpose, in the model, the effects of the %%@
initial oxygen concentration in silicon was studied. In Figures 5 a, b, c, d the time %%@
dependencies of $V_2$, $VO$, $V_2O$ and $VP$ are presented, for silicon containing  %%@
10$^{15}$, 10$^{16}$, 10$^{17}$, and 10$^{18}$ atoms/cm$^3$ initial oxygen concentrations.

\begin{figure}[ht]
\centering
\includegraphics[width=.8\textwidth,, clip, angle=90]{fig4.eps}
\caption{\small{Effect of oxygen doping concentration on the time dependence of the 
concentrations of: a)$V_2$, b)$VO$, c)$V_2O$ and d)$VP$, induced in silicon with $10^{14}$ %%@
P/cm$^3$ irradiated with 200 MeV kinetic energy pions at total fluence of 10$^{14}$ %%@
pions/cm$^2$ in one pulse.}}
\label{f5}
\end{figure}

One can observe that vacancy-oxygen formation in oxygen enriched silicon is favoured in %%@
respect to the generation of $V_2$, $V_2O$ and $VP$, confirming the considered hypothesis, %%@
so, for detector applications the leakage current is decreased. At high oxygen %%@
concentrations, the concentration of $VO$ centres saturates starting from low fluences.

A difficulty in the comparison of model predictions with experimental data is the %%@
insufficient information in published papers regarding the characterisation of silicon, %%@
and on the irradiation (flux, temperature during irradiation and measurement, irradiation %%@
time, time and temperature between irradiation and measurement) for most of the data.

\begin{figure}[ht]
\centering
\includegraphics[width=.8\textwidth]{fig6.eps}
\caption{\small{Time dependence of a)$VP$ and b)$V_2$ concentrations after electron %%@
irradiation:
points: experimental data from reference \cite {17} and dashed line: model calculations.}}
\label{f6}
\end{figure}

For electron irradiation, our simulations are in agreement with the measurements presented %%@
in reference \cite{17}, where defect concentrations are presented as a function of the %%@
time after irradiation. In Figure 6, both measured and calculated dependencies of the $VP$ %%@
and $V_2$ concentrations are given. The irradiation was performed with 2.5 MeV electrons, %%@
up to a fluence of 10$^{16}$ cm$^{-2}$. A good agreement can be observed for the %%@
concentration of $VP$, see Figure 6a, while for the divacancy, Figure 6b, the experimental %%@
data attain a plateau faster, and at smaller values than the calculations. The relative %%@
values are imposed by the arbitrary units of experimental data.

A good agreement has also been obtained for hadron irradiation. For example, the sum of %%@
the calculated $VP$ and $V_2$ concentrations (8x10$^{12}$ cm$^{-3}$) induced in silicon by %%@
5.67x10$^{13}$ cm$^{-2}$ 1 MeV neutrons, are in accord with the experimental value of %%@
11.2x10$^{12}$ cm$^{-3}$ reported in reference \cite{18}.


\section{Correlation with detector parameters}
It is well known that the dark current in a $p-n$  junction is composed by three different %%@
terms: the diffusion current, caused by the diffusion of the minority charge carriers %%@
inside the depleted region; the generation current, created by the presence of lattice %%@
defects inside the bulk of the detector; and surface and perimetral currents, dependent on %%@
the environmental conditions of the surface and the perimeter of the diode. The appearance %%@
of the defects after irradiation corresponds therefore in an increase of the leakage %%@
current of the detector by its generational term.

Inside the depleted zone, $n, p << n_i$ ($n_i$ is the intrinsic free carrier %%@
concentration), each defect with a bulk concentration $N_T$ causes a generation current %%@
per unit of volume of the form \cite{19}:

\begin{equation}
I=qU=q<v_t>n_t \frac{\sigma_n \sigma_p N_T}{\sigma_n \gamma_n %%@
e^{\left(E_t-E_i\right)/k_BT}+\sigma_p \gamma_p e^{\left(E_i-E_t\right)/k_BT}}
\end{equation}
where $\gamma _n$ and $gamma _{p}$ are degeneration factors, $\sigma _n$ ($\sigma _p$) are %%@
the cross sections for majority (minority) carriers of the trap, $E_i=(E_C-E_V)/2$ and %%@
$<v_t>$ is the average between electron and hole thermal velocities.
In the case of $E$ and $A$ centres and $V_2^-$  and $V_2^(--)$  defects, the current %%@
concentration can be expressed in the simple form:
\begin{equation}
I=qU=q<v_t>n_t \frac{\sigma_n}{\sigma_p}N_Te^{\left(E_t-E_i\right)/k_BT}
\end{equation}

The primary effect in the recombination process is the change the charge state of the %%@
defect. The different charge states of the same deep centre may have different barriers %%@
for migration or for reacting with other centres. Thus, carrier capture can either enhance %%@
or retard defect migration or particular defect reactions. As a characteristics for %%@
detectors (as diode junction), the defect kinetics is dependent to the reverse - bias %%@
voltage during the irradiation \cite{20}.

The comparison between theoretical and experimental generation current densities after %%@
irradiation shows a general accord between experiment and the model results for the lepton %%@
irradiation and large discrepancies for the hadron case.

There could be several reasons for the observed discrepancies.

The model hypothesis of defects distributed randomly in semiconductors exclude the %%@
possibility of cluster defects. For this case, other mechanisms of defect formation are %%@
necessary, which suppose different reaction rates and correlation between the constituent %%@
defects of the cluster.

In the Shockley-Read-Hall model used for the calculation of the reverse current, each %%@
defect has one level in the gap, and the defect levels are uncoupled, thus the current is %%@
simply the sum of the contributions of different defects. In fact, the defects could have %%@
more levels, and charge states, as is the case of the divacancy, and  also could be %%@
coupled, as in the case of clusters. As shown in the literature \cite{21,22}, both cases %%@
can produces modifications of the generation rate.

Also the multivacancy oxygen defects as, e.g. $V_3O$, $V_2O_2$,  $V_3O_2$,  $V_3O_3$, are %%@
not considered in the model.

\begin{figure}[ht]
\centering
\includegraphics[width=.8\textwidth, clip, angle=90]{fig7.eps}
\caption{\small{Time dependence of the reverse current after 200 MeV kinetic energy pions 
irradiation, with the rate estimated for LHC, at 293K, for silicon containing: %%@
a)5x10$^{16}$ cm$^{-3}$ and b)$10^{16}$ cm$^{-3}$ oxygen.}}
\label{f7}
\end{figure}

A model estimation of the time dependence of the leakage current, in conditions of %%@
continuous irradiations with pions of 200 MeV kinetic energy, in the conditions of the %%@
$LHC$ \cite{13,14} and at 293K is presented in Figure 7, for two concentrations of oxygen %%@
in silicon: 5x10$^{16}$ cm$^{-3}$ and 10$^{16}$ cm$^{-3}$ respectively. As underlined %%@
before, oxygen incorporation in silicon has beneficial effects, decreasing the reverse %%@
current. This conclusion is valid in the hypothesis of random distribution of defects %%@
inside the depleted zone of the p-n junction. These values are probably underestimated.


\section{Summary}
A phenomenological model that describes silicon degradation due to irradiation, the %%@
kinetics of defects toward equilibrium, and the influence on the reverse current of %%@
detectors was developed. 

The production of primary defects (vacancies and interstitials) in the silicon bulk was %%@
considered in the frame of the Lindhard theory, and considering the peculiarities of the %%@
particle - silicon nuclei interaction.

The mechanisms of formation of stable defects and their evolution toward equilibrium was %%@
modelled, and the concentrations of defects were calculated solving numerically the system %%@
of coupled differential equations for these processes. Vacancy-interstitial annihilation, %%@
interstitial migration to sinks, vacancy-impurities complexes ($VP$, $VO$ and $V_2O$), and %%@
divacancy formation were considered in different irradiation conditions, for different %%@
concentrations of impurities in the initial semiconductor material and at different %%@
temperatures of irradiation. The calculated results suggest the importance of the %%@
conditions of irradiation, temperature and annealing history. The model supports the %%@
experimental studies performed to investigate the influence of oxygen in the enhancement %%@
of the radiation hardness of silicon for detectors. The $VO$ defects in oxygen enriched %%@
silicon is favoured in respect to the other stable defects, so, for detector applications %%@
it is expected that the leakage current decreases after irradiation. The second result in %%@
the model is that at high oxygen concentrations, this defect saturates starting from low %%@
fluences. 

Most of the model calculations simulates some of the pion field estimated at the new LHC %%@
accelerator, where the silicon detector will operate under continuum generation and %%@
annealing.

The density of the reverse current in detectors after irradiation is estimated, compared %%@
with experimental available data and for discrepancies some explanations are suggested.


\section{Acknowledgements}

The authors are very grateful to Professor Gh. Ciobanu from the Bucharest University for %%@
helpful discussions during the course of this work.


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\end{thebibliography}

\end{document}

hep-ph0203230.extracted_bib
\begin{thebibliography}{99}
\bibitem{1}	M. Lannoo, J. Bourgoin, {\em''Point Defects in 
Semiconductors''}, Springer Series in Solid State Science 2, Eds. M. Cardona, P. Fulde, %%@
H.-J. Queisser, Springer-Verlag 1981.

\bibitem{2}	I. Lazanu, S. Lazanu, U. Biggeri, E. Borchi, M. Bruzzi, Nucl. Phys. {\bf 61B}, %%@
409c (1998).


\bibitem{3}	S. Lazanu, S. Lazanu, U. Biggeri, S. Sciortino, Nucl. Instr. Meth.Phys. Res. %%@
{\bf A 413}, 242 (1998). 


\bibitem{4}	I. Lazanu, S. Lazanu, U. Biggeri, E. Borchi, M. Bruzzi, Nucl. Instr. %%@
Meth.Phys. Res. {\bf A 388}, 370 (1997). 


\bibitem{5} 	S. Lazanu, I. Lazanu, Nucl. Instr. Meth.Phys. Res. {\bf A 462} 530 (2001).


\bibitem{6}	G. P. Summers, E. A. Burke, Ph.. Shapiro, S. R. Messenger, R. J. Walters, IEEE %%@
Trans. Nuclear Science, {\bf NS 40}, :1372 (1993)


\bibitem{7}	A. Van Ginneken, Preprint Fermi National Accelerator Laboratory, FN-522, 1989 


\bibitem{8} 	A. M. Ougouag, J. G. Williams, M. B. Danjaji, S.-Y. Yang, IEEE Trans. %%@
Nuclear Science, {\bf NS-37}, 2219 (1990).


\bibitem{9} 	S. Lazanu, I. Lazanu, Nucl. Instr. Meth. Phys. Res. {\bf A 419}, 570 %%@
(1998).


\bibitem{10}	A. C. Damask and G. J. Dienes, {\em''Point Defects in Metals''}, Gordon %%@
and Breach, New York 1963.

\bibitem{11} 	Y. H. Lee, J. W. Corbett, Phys. Rev. B 13: 2653, (1976).

\bibitem{12} 	S. Lazanu, I. Lazanu, Nucl. Instr. Meth. Phys. Res. B 183, 383 (2001).

\bibitem{13} 	P. A. Aarnio, M. Huhtinen, Nucl. Instr. Meth. Phys. Res. A 336, 98 (1993).  

\bibitem{14} 	CMS Collaboration, CMS Technical Proposal, CERN/LHCC 94-38, 1994  

\bibitem{15} 	M. Moll, E. Fretwurst, G. Lindstrom, Nucl. Instrum. Meth. Phys. Res. A %%@
439, 282 (2000). 

\bibitem{16} 	B. C. Mac Evoy, A. Santacchia, G. Hall, Physica B 273-274, 1054 (1999).

\bibitem{17}	L. W. Song, B. W. Benson, G. D. Watkins, Phys. Rev. B 33, 1452 (1986). 

\bibitem{18} 	U. Biggeri, E. Borchi, M. Bruzzi, Z. Li, S. Lazanu, IEEE Trans. Nucl. Sci. %%@
NS 41, 964 (1994).

\bibitem{19} 	E. Borchi, M. Bruzzi, M. S. Mazzoni, M. S., Nucl. Instrum. Meth. Phys. %%@
Res. A 310, 273 (1991). 

\bibitem{20} 	T. S. Pantelides, editor {\em''Deep Centers in Semiconductors''}, Gordon %%@
and Breach Science Publishers, Second Edition, 1992, p.65

\bibitem{21} 	G. Lutz, Nucl. Instrum. Meth. Phys. Res. A 377, 324 (1996). 

\bibitem{22} 	S. C. Choo, Phys. Rev. B 1, 687 (1970).
 

	
\end{thebibliography}