The influence of initial impurities and irradiation conditions on defect production %%@ and annealing in silicon for particle detectors
Silicon detectors in particle physics experiments at the new accelerators or in space %%@ missions for physics goals will be exposed to extreme radiation conditions. The principal %%@ obstacles to long-term operation in these environments are the changes in detector %%@ parameters, consequence of the modifications in material properties after irradiation. The phenomenological model developed in the present paper is able to explain %%@ quantitatively, without free parameters, the production of primary defects in silicon %%@ after particle irradiation and their evolution toward equilibrium, for a large range of %%@ generation rates of primary defects. Vacancy-interstitial annihilation, interstitial %%@ migration to sinks, divacancy and vacancy-impurity complex ($VP$, $VO$, $V_2O$, $C_iO_i$ %%@ and $C_iC_s$) formation are taken into account. The effects of different initial impurity %%@ concentrations of phosphorus, oxygen and carbon, as well as of irradiation conditions are %%@ systematically studied. The correlation between the rate of defect production, the %%@ temperature and the time evolution of defect concentrations is also investigated. \medskip PACS: \\ 29.40 Pe: Semiconductor detectors \\ 61.80 Az: Theory and models of radiation defects\\ 61.70 At: Defect formation and annealing\\ \medskip Keywords: silicon, detectors, radiation damage, kinetics of defects, annealing %%@ processes
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\title{The influence of initial impurities and irradiation conditions on defect production %%@
and annealing in silicon for particle detectors}


\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}
Silicon detectors in particle physics experiments at the new accelerators or in space %%@
missions for physics goals will be exposed to extreme radiation conditions. The principal %%@
obstacles to long-term operation in these environments are the changes in detector %%@
parameters, consequence of the modifications in material properties after irradiation.

The phenomenological model developed in the present paper is able to explain %%@
quantitatively, without free parameters, the production of primary defects in silicon %%@
after particle irradiation and their evolution toward equilibrium, for a large range of %%@
generation rates of primary defects. Vacancy-interstitial annihilation, interstitial %%@
migration to sinks, divacancy and vacancy-impurity complex ($VP$, $VO$, $V_2O$, $C_iO_i$ %%@
and $C_iC_s$) formation are taken into account. The effects of different initial impurity %%@
concentrations of phosphorus, oxygen and carbon, as well as of irradiation conditions are %%@
systematically studied. The correlation between the rate of defect production, the %%@
temperature and the time evolution of defect concentrations is also investigated.


\medskip
\textbf{PACS}: \\
29.40 Pe: Semiconductor detectors \\
61.80 Az: Theory and models of radiation defects\\
61.70 At: Defect formation and annealing\\

\medskip
\textbf{Keywords}: silicon, detectors, radiation damage, kinetics of defects, annealing %%@
processes
\end{abstract}


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

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, the effects of irradiation conditions and various initial impurities in the %%@
starting material are discussed in the frame of the phenomenological model able to explain %%@
quantitatively defect production and evolution toward stable defects during and after %%@
irradiation in silicon. The model supposes three steps. 

In the first step, the incident particle, having kinetic energies with values in the %%@
intermediate up to high energy range, interacts with the semiconductor material. The %%@
peculiarities of the interaction mechanisms are explicitly considered for each kinetic %%@
energy. 

In the second step, the recoil nuclei resulting from these interactions lose their energy %%@
in the lattice. Their energy partition between displacements and ionisation is considered %%@
in accord with the Lindhard theory (\cite{lind} and authors' contributions \cite{anal}).

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{4} are used in relation to defects. We denote the displacement %%@
defects, vacancies and interstitials, as primary point defects, prior to any further %%@
rearrangement. After this step the concentration of primary defects is calculated. 

The mechanisms of interaction of the incident particle with the semiconductor lattice, %%@
accompanied by displacement defect production have been discussed in some papers, see, %%@
e.g. references \cite{1, 2, 3}. The incident particle produces, as a consequence of its %%@
interaction with ions of the semiconductor lattice, cascades of displacements.

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

The concentration of primary defects represents the starting point for the following step %%@
of the model, the consideration of the annealing processes, treated in the frame of the %%@
chemical rate theory. A review of previous works about the problem of the annealing of %%@
radiation induced defects in silicon can be found, e.g. in Reference \cite{5}.

Without free parameters, the model is able to predict the absolute values of the %%@
concentrations of defects and their time evolution toward stable defects, starting from %%@
the primary incident particle characterised by type and kinetic energy.

The first two steps have been treated extensively in previous papers \cite{1, 2, 3}, where %%@
the concentration of primary defects has been calculated. In this paper, the third step is %%@
discussed extensively and represents a generalisation of the previous results published in %%@
references \cite{5, 6} including also the carbon contributions to defect kinetics.


The influence and the effects of different initial impurity concentrations of phosphorus, %%@
oxygen and carbon as well as of the irradiation conditions were systematically studied. 
Vacancy-interstitial annihilation, interstitial migration to sinks, vacancy - impurity %%@
complexes ($VP$, $V_2$, $VO$, $V_2O$, $C_iO_i$, $C_iC_s$) - only the stable defects %%@
confirmed experimentally in silicon for high energy physics applications were considered. %%@
The correlation between the rate of defect productions, the temperature and the time %%@
evolution of the defect concentrations was also investigated. Some conclusions about the %%@
possibilities to obtain semiconductor materials harder to radiation are given.

\section{Production of primary defects}
The basic assumption of the present model is that the primary defects, vacancies and %%@
interstitials, are produced in equal quantities and are uniformly distributed in the %%@
material bulk. They are produced by the incoming particle, as a consequence of the %%@
subsequent collisions of the primary recoil in the lattice, or thermally (only Frenkel %%@
pairs are considered). The generation term $G$ is the sum of two components:
\begin{equation}
G=G_R+G_T
\end{equation}
where $G_R$ accounts for the generation by irradiation, and $G_T$ for thermal generation.
The concentration of the primary radiation induced defects per unit fluence $CPD$ in %%@
silicon has been calculated as the sum of the concentration of defects resulting from all %%@
interaction processes, and all characteristic mechanisms corresponding to each interaction %%@
process, using the explicit formula (see details, e.g. in references \cite{7, 8}):
\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 ($NIEL$) the rate of energy loss by %%@
displacement $(dE/dx)_{ni}$  \cite{4, toti}. 

In Figure 1, the kinetic energy dependence of $CPD$ for different particles is presented: %%@
for pions the values are calculated in accord with equation (1) and are from reference %%@
\cite{9}, and for the other particles these are evaluated from the published $NIEL$ %%@
calculations, as follows: for protons - reference \cite{3, 10}, for neutrons - reference %%@
\cite{11}, for electrons - reference \cite{3, 10} and for photons from \cite{3}.
The main source of errors in the calculated concentration of primary defects comes from %%@
the modelling of the particle - nucleus interaction and from the number and quality of the %%@
experimental data available for these processes. 

\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}

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, and %%@
annihilation or produce other defects.
In the simplifying hypothesis of random distribution of $CPD$ for all particles, used in %%@
the present paper, the identity of the particle is lost after the primary interaction, and %%@
two different particles could produce the same generation rate ($G_R$) for %%@
vacancy-interstitial pairs if the following condition:
\begin{equation}
G_R=[\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.

\section{The kinetics of radiation induced defects}
Silicon used in high energy physics detectors is n-type high resistivity (4 $\div$ 6 %%@
K$\Omega$ cm) phosphorus doped FZ material. 

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{12, 13}. Some people %%@
consider that these impurities added to the silicon bulk modify the formation of %%@
electrically active defects, thus controlling the macroscopic device parameters. %%@
Empirically, it is considered that if the 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 for vacancies, thus reducing the %%@
probability of formation of the divacancy related complexes, associated with deeper levels %%@
inside the gap.

The concentrations of interstitial oxygen $O_i$ and substitutional carbon $C_i$ in silicon %%@
are strongly dependent on the growth technique. In high purity Float Zone Si, oxygen %%@
interstitial concentrations are around $10^{15}$ cm$^{-3}$, while in Czochralski Si these %%@
concentrations can reach values as high as $10^{18}$ cm$^{-3}$. Because Czochralski %%@
silicon is not available in detector grade quality, an oxygenation technique developed at %%@
BNL produces Diffusion Oxygenated Float Zone in silicon, obtaining a $O_i$  concentration %%@
of the order 5x10$^{17}$ cm$^{-3}$. These materials can be enriched in substitutional %%@
carbon up to 1.8x10$^{16}$ cm$^{-3}$.

After the irradiation of silicon, the following stable defects have been identified (see %%@
References \cite{4, 14}): $Si_i$, $VP$, $VO$, $V_2$, $V_2O$, $C_iO_i$, $C_i$, $C_iC_s$.
The pre-existing thermal 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.

Vacancy-interstitial annihilation, interstitial migration to sinks, divacancy, vacancy and %%@
interstitial impurity complex formation are considered. The role of phosphorus, oxygen and %%@
carbon is taken into account, and the following stable defects :  $VP$, $VO$, $V_2$, %%@
$V_2O$, $C_iO_i$, $C_i$, $C_iC_s$ are considered. Other possible defects as $V_3O$, %%@
$V_2O_2$, $V_3O_3$ \cite{15}, are not included in the present model.

The following picture describes in terms of chemical reactions the mechanisms of %%@
production and evolution of the defects considered in the present paper:
\begin{equation}                                                        
V+I\ _{\overleftarrow{G}} ^{\underrightarrow{K_1}}\text{annihilation}
\end{equation}
\begin{equation}                                                       
I\stackrel{K_2}{\rightarrow } \text{sinks}
\end{equation}
\begin{equation}							
V+O\ _{\overleftarrow{K_4}} ^{\underrightarrow{K_3}}\ VO
\end{equation}
$VO$ is the $A$ centre.
\begin{equation}							
V+P\ _{\overleftarrow{K_5}} ^{\underrightarrow{K_3}}\ VP
\end{equation}
$VP$ is the $E$ centre.
\begin{equation}						
V+V\ _{\overleftarrow{K_6}} ^{\underrightarrow{K_3}}\ V_2
\end{equation}
\begin{equation}						
V+VO\ _{\overleftarrow{K_{7}}} ^{\underrightarrow{K_{3}}}\ V_2O
\end{equation}
\begin{equation}
I+C_s\stackrel{K_1}{\rightarrow }  C_i
\end{equation}
\begin{equation}
C_i+O_i\stackrel{K_8}{\rightarrow } C_iO_i
\end{equation}
\begin{equation}
A+I\stackrel{K_9}{\rightarrow }  O
\end{equation}
\begin{equation}
C_i+C_s\stackrel{K_8}{\rightarrow }  C_iC_s
\end{equation}
Some considerations about the determination of the reaction constants are given in %%@
references \cite{5, 6}.

The reaction constants $K_i$ (i = 1, $3 \div 9$) have the general form:
\begin{equation}
K_i=C\nu \exp \left( -E_i/k_BT\right)
\end{equation}
with $\nu$ the vibration frequency of the lattice, $E_i$ the associated activation energy %%@
and $C$ a numerical constant that accounts for the symmetry of the defect in the lattice.

The reaction constant related to the migration of interstitials to sinks could be %%@
expressed as:
\begin{equation}
K_2=\alpha \nu \lambda ^2\exp \left( -E_2/k_BT\right)                                                 
\end{equation}
with $\alpha$ the sink concentration and $\lambda$ the jump distance.

The system of coupled differential equations corresponding to the reaction scheme (4) %%@
$\div$ (13) cannot be solved analytically and a numerical procedure was used.

The following values of the parameters have been used: $E_1$=$E_2$ = 0.4 eV, $E_3$ = 0.8 %%@
eV, $E_4$ = 1.4 eV, $E_5$ = 1.1 eV, $E_6$ = 1.3 eV, $E_7$ = 1.6 eV, $E_8$ = 0.8 eV, $E_9$   %%@
= 1.7 eV, $\nu$ = 10$^{13}$ Hz, $\lambda$ = 10$^{15}$ cm$^2$, $\alpha$ = 10$^{10}$ %%@
cm$^{-2}$, in accord with standard constants characterising the silicon material, see for %%@
example the books of Lannoo and Bourgoin \cite{4} or Damask and Dienes \cite{damask}.

\section{Results, discussion, comparison with experimental data and predictions}
The formation and time evolution of stable defects depends on various factors, e.g. the %%@
concentrations of impurities pre-existent in the sample, the rate of generation, and the %%@
temperature.

In Figures 2 a $\div$ f, and 3 a $\div$ f the formation and time evolution of the %%@
vacancy-oxygen, vacancy-phosphorus, divacancy, divacancy-oxygen, carbon interstitial - %%@
oxygen interstitial and carbon interstitial.- carbon substitutional are modelled in %%@
silicon containing different initial concentrations of phosphorus, oxygen and carbon, and %%@
for two very different rates of generation.
\begin{figure}[ht]
\centering
\includegraphics[width=.75\textwidth]{F22.EPS}
\caption{\small{Time dependence of the concentrations of a)$VO$, b)$VP$, c)$V_2$, %%@
d)$V_2O$, e)$C_iO_i$ and f) $C_iC_s$ induced in silicon with the following concentrations %%@
of impurities: (1): 10$^{14}$ P/cm$^3$, 2x10$^{15}$ O/cm$^3$, and 3x10$^{15}$ C/cm$^3$; %%@
(2):  10$^{14}$ P/cm$^3$, 10$^{16}$ O/cm$^3$, and 3x10$^{15}$ C/cm$^3$; (3):10$^{14}$ %%@
P/cm$^3$, 5x10$^{16}$ O/cm$^3$, and 3�10$^{15}$ C/cm$^3$; (4): 10$^{14}$ P/cm$^3$, %%@
4x10$^{17}$ O/cm$^3$, and 3x10$^{15}$ C/cm$^3$; (5)): 10$^{15}$ P/cm$^3$, 10$^{16}$ %%@
O/cm$^3$, and 3x10$^{15}$ C/cm$^3$ (6): 10$^{14}$ P/cm$^3$, 10$^{16}$ O/cm$^3$, and %%@
3x10$^16$ C/cm$^3$by a generation rate $G_R$ = 7x10$^8$ $VI$ pairs/cm$^3$s, at 20$^o$C.}}
\label{f2}
\end{figure}

In Figure 2, the evolution of defect concentrations during high rate irradiation ($G_R$ = %%@
7x10$^8$ $VI$ pairs/cm$^3$s) is presented. This corresponds, in the model hypothesis, to %%@
order of magnitude of the radiation levels estimated for the forward tracker region at the %%@
future LHC accelerator.

The increase of the initial oxygen concentration in silicon, from 2x10$^{15}$ O/cm$^3$ to  %%@
4x10$^{17}$ O/cm$^3$, conduces, after ten years of operation in the field characterised by %%@
a high and constant generation rate, to the increase of the concentrations of  $VO$ and %%@
$C_iO_i$ centres, and to the decrease of the concentrations of $V_2$, $VP$ and $C_iC_s$   %%@
ones. With the increase of oxygen concentration, a variation of the $V_2O$ generation rate %%@
is observed, so that, from the studied cases, the maximum concentration for this defect is %%@
obtained for 5x10$^{16}$ O/cm$^3$ initial oxygen. The increase of initial phosphorus is %%@
seen in the increase of concentration of $VP$  centres, while the increase of initial %%@
carbon concentration has important consequences on the concentrations of $C_iC_s$  %%@
centres. It is interesting to observe that in almost all cases, an equilibrium in reached %%@
between generation and annealing, and a plateau is obtained in the time dependence of the %%@
concentrations. The slowest is, in this respect, $V_2O$, that has the highest binding %%@
energy.

As underlined above, vacancy-oxygen formation in oxygen enriched silicon is favoured in %%@
respect to the generation of $V_2$, $V_2O$ and $VP$. This is an important feature that %%@
could be used for detector applications, determining the decrease of the leakage current %%@
\cite{6}. At high oxygen concentrations, the concentration of $VO$ centres attains a %%@
plateau during the 10 years period considered. 
\begin{figure}[ht]
\centering
\includegraphics[width=.8\textwidth]{F33.EPS}
\caption{\small{Same as Figure 2, $G_R$ = 200 $VI$ pairs/$cm^3$s.}}
\label{f3}
\end{figure}

The other extreme situation corresponds to a generation rate $G_R$ = 200 $VI$ %%@
pairs/cm$^3$s, equivalent with a rate of defect production by protons from the cosmic ray %%@
spectra in the orbit near the Earth, at about 400 Km, as will be the position of the %%@
International Space Station. The same concentrations of $P$, $O$ and $C$ have been %%@
considered as pre-existent in silicon as in Figure 2. For this generation rate, the %%@
increase of the oxygen concentration produces the decrease of the concentration of all %%@
centres, with the exception of the $VO$ concentration, that, at these rates, it is not %%@
influenced by the oxygen content, and of the $C_iO_i$ concentration, where an increase is %%@
observed. As could be seen from Figure 3, as a consequence of the small rate of generation %%@
rate of vacancy - interstitial pairs, after ten years of operation the equilibrium between %%@
generation and annealing is not reached, the concentrations of defects being, with the %%@
exception of $VP$ (that has a relatively low binding energy), slightly increasing %%@
functions of time.

All curves, both from Fig. 2 and Fig. 3, have been calculated for 20$^o$C temperature. %%@
Thermal generation has been taken into account in both cases, although it is important %%@
only for the silicon exposed to low rates of defect production.

The influence of the generation rate of primary defects on the concentration of stable %%@
defects and on their time evolution has also been investigated. In Figure 4 a $\div$ f, %%@
the time evolution of the $VO$, $VP$, $V_2$, $V_2O$, $C_iO_i$ and $C_iC_s$ concentrations %%@
is presented for six decades of generation rates of defects ($G_R$=7x10$^5 \div $ %%@
7x10$^{10}$ $VI$ pairs/cm$^3s$), for silicon containing the following concentrations of %%@
impurities: 10$^{14}$ P/cm$^3$, 10$^{16}$ O/cm$^3$, and 3�10$^{15}$ C/cm$^3$ at  20$^o$C %%@
temperature. At small times, the curves corresponding to different generation rates are %%@
all parallel and equidistant in a log-log representation. Starting from the highest %%@
generation rates, they start to increase slower ($V_2$, $VP$), attain a plateau $V_2O$, or %%@
event start to decrease  ($VO$, $C_iO_i$, $C_iC_s$. The maximum attained can be the same, %%@
independent on the generation rate as is the case of $VO$ concentration, or could be %%@
generation dependent $C_iO_i$, $C_iC_s$.
\begin{figure}[ht]
\centering
\includegraphics[width=.8\textwidth]{F44.EPS}
\caption{\small{Time dependence of the concentrations of a)$VO$, b)$VP$, c)$V_2$, %%@
d)$V_2O$, e)$C_iO_i$ and f) $C_iC_s$ induced in silicon with:  10$^{14}$ P/cm$^3$, %%@
10$^{16}$ O/cm$^3$, and 3x10$^{15}$ C/cm$^3$ by continuous irradiation.}}
\label{f4}
\end{figure}

The temperature is another important factor determining the time evolution of defects. In %%@
Figure 5 a $\div$ f and 6 a $\div$ f, the effect of the temperature is studied for the %%@
same generation rates of primary defects as in Figures 2 and 3 respectively, for five %%@
temperatures: 20$^o$C, 10$^o$C, 0$^o$C, -10$^o$C, and -20$^o$C. The concentrations of %%@
pre-existing impurities in silicon are as follows: 10$^{14}$ P/cm$^3$, 10$^{16}$ O/cm$^3$, %%@
and 3x10$^{15}$ C/cm$^3$. The decrease of the temperature decreases the generation rate of %%@
stable defects, with the exception of $VP$, where the highest values correspond to the %%@
lowest temperature. The maximum values of the $VO$  concentration are temperature %%@
independent; the values of the concentration on the plateau decrease with the decrease of %%@
temperature for $C_iO_i$ and $C_iC_s$ and increase for $V_2O$.
\begin{figure}[ht]
\centering
\includegraphics[width=.8\textwidth]{F55.EPS}
\caption{\small{Time dependence of the concentrations of a)$VO$, b)$VP$, c)$V_2$, %%@
d)$V_2O$, e)$C_iO_i$ and f) $C_iC_s$ induced in silicon with:  10$^{14}$ P/cm$^3$, %%@
10$^{16}$ O/cm$^3$, and 3x10$^{15}$ C/cm$^3$ by continuous irradiation, in the same %%@
conditions as in Figure 2, at 20$^o$C, 10$^o$C, 0$^o$C, -10$^o$C, and -20$^o$C.}}
\label{f5}
\end{figure}

For silicon exposed to low rates of defect production the same amount of time (Figure 6), %%@
the process of defect production in slowed down. The most unexpected time dependence is %%@
for $V_2$, that has the highest values at the lowest temperature.
\begin{figure}[ht]
\centering
\includegraphics[width=.8\textwidth]{F66.EPS}
\caption{\small{Time dependence of the concentrations of a)$VO$, b)$VP$, c)$V_2$, %%@
d)$V_2O$, e)$C_iO_i$ and f) $C_iC_s$ induced in silicon with:  10$^{14}$ P/cm$^3$, %%@
10$^{16}$ O/cm$^3$, and 3x10$^{15}$ C/cm$^3$ by continuous irradiation, in the same %%@
conditions as in Figure 3, at 20$^o$C, 10$^o$C, 0$^o$C, -10$^o$C, and -20$^o$C.}}
\label{f6}
\end{figure}

In a previous paper \cite{6}, we demonstrated in concrete cases the importance of the %%@
sequence of irradiation process, considering that the same total fluence can be attained %%@
in different situations: the ideal case of instantaneous irradiation, irradiation in a %%@
single pulse followed by relaxation, and respectively continuos irradiation process. As %%@
expected, after instantaneous irradiation the concentrations of defects are higher in %%@
respect with "gradual" irradiation.

After this analysis, 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 must be to underline.

The model predictions have been compared with experimental measurements. A difficulty in %%@
this comparison is the insufficient information in published papers regarding the %%@
characterisation of silicon, and on the irradiation parameters and conditions for most of %%@
the data.

It was underlined in the literature \cite{16} that the ratio of $VO$ to $VP$  centres in %%@
electron irradiated silicon is proportional to the ratio between the concentrations of %%@
oxygen and phosphorus in the sample. For electron irradiation, in Ref. \cite{17} a linear %%@
dependence of the $V_2$  versus $VO$  centre concentration has been found experimentally. %%@
In the present paper, the ratio of concentrations of  $V_2$ to $VO$  centres and $VO$  to   %%@
$VP$ ones has been calculated in the frame of the model, for the material with the %%@
characteristics specified in Ref. \cite{17}, and irradiated with 12 MeV  electrons, in the %%@
conditions of the above mentioned article. The time dependence of these two ratios is %%@
represented in Figure 7. Annealing is considered both during and after irradiation. It %%@
could be seen that for the ratio of $V_2$  and $VO$  concentrations the curves %%@
corresponding to different irradiation fluences are parallel, while the ratio of $VO$  to   %%@
$VP$ concentrations is fluence independent, in the interval 2x10$^{13} \div$ 5.5x10$^{14}$ %%@
cm$^{-2}$, in good agreement with the experimental evidence. The ratio between $V_2$  and %%@
$VO$  concentrations is determined by the generation of primary defects by irradiation, %%@
while the ratio between $VO$  and  $VP$ concentrations is determined by the concentrations %%@
of oxygen and phosphorus in silicon. 
\begin{figure}[ht]
\centering
\includegraphics[width=.6\textwidth]{F77.EPS}
\caption{\small{Time dependence for a) $V_2/VO$  and b) $VO/VP$  concentrations calculated %%@
for silicon with 1.4x10$^{14}$ P/cm$^3$, 5x10$^{17}$ O/cm$^3$, and 3x10$^{15}$ C/cm$^3$  %%@
by 12 MeV electron irradiation, with the flux 5.8x10$^{10}$ e/cm$^2$s, up to the fluences: %%@
2x10$^{13}$ e/cm$^2$, 4x10$^{13}$ e/cm$^2$, 8x10$^{13}$ e/cm$^2$, 2x10$^{14}$ e/cm$^2$  %%@
and 5.5x10$^{14}$ e/cm$^2$, followed by relaxation (see reference \cite{17}.}}
\label{f7}
\end{figure}

Our estimations are also in agreement with the measurements presented in reference %%@
\cite{18}, after electron irradiation, where defect concentrations are presented as a %%@
function of the time after irradiation. In Figure 8, both measured and calculated %%@
dependencies of the $VO$  and $VP$  concentrations are given. The irradiation was %%@
performed with 2.5 MeV  electrons, up to a fluence of 3x10$^{16}$ e/cm$^2$. A good %%@
agreement can be observed for these concentrations. The dependencies put in evidence the %%@
important role played by the carbon-related defects. The relative values are imposed by %%@
the arbitrary units of experimental data.
\begin{figure}[ht]
\centering
\includegraphics[width=.6\textwidth]{F88.EPS}
\caption{\small{Time dependence of $VO$ and $VP$  concentrations after electron %%@
irradiation: crosses - experimental data from reference \cite{18}]; continuous line - %%@
present model calculations; dashed line - without the consideration of carbon contribution %%@
to defect formation.}}
\label{f8}
\end{figure}

Also, a good agreement has been obtained between the absolute values of concentrations of %%@
$VP+V_2$  and $C_iC_s$ predicted by the model, and the experimental results after   %%@
neutron irradiation at a total fluence of 5.67x10$^{13}$ cm$^{-2}$, reported in reference %%@
\cite{19}. The calculated 1.5x10$^{13}$ cm$^{-3}$  and 4.1x10$^{12}$ cm$^{-3}$ %%@
concentrations for $VP+V_2$  and $C_iC_s$ respectively, are in accord with the values of %%@
1.1x10$^{13}$ cm$^{-3}$ and 3.8x10$^{12}$ cm$^{-3}$, measured experimentally. For the $VO$ %%@
concentration, a poorer concordance has been obtained.

\section{Summary}
A phenomenological model that describes silicon degradation due to irradiation from the %%@
point of view of the kinetics of produced defects toward equilibrium was developed. 

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

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 ($VP$, $VO$, $V_2O$, %%@
$C_iO_i$, $C_iC_s$), and divacancy formation were considered in different irradiation %%@
conditions, for different concentrations of impurities in the initial semiconductor, in %%@
the temperature range $- 20 \div + 20 ^oC$. 

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 are favoured in respect to the other stable defects, %%@
so, for detector applications it is expected that the leakage current decreases after %%@
irradiation. At high oxygen concentrations, this defect saturates starting from low %%@
fluences at high generation rates of defects.



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\bibitem{3}	A. Van Ginneken, Preprint Fermi National Accelerator Laboratory, FN-522, 1989 

\bibitem{5} 	S. Lazanu, I. Lazanu, Nucl. Instr. Meth. Phys. Res. {\bf B 183} (2001) %%@
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\bibitem{damask}	A. C. Damask, G. J. Dienes, Points Defects in Metals, Gordon and %%@
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\end{thebibliography}

\end{document}

physics0208027.extracted_bib
\begin{thebibliography}{99}

\bibitem{lind}  J. Lindhard, V. Nielsen, M. Scharff, P. V. Thomsen, Mat. Phys. Medd. Dan %%@
Vid. Sesk. 33 (1963) 1.
	
\bibitem{anal}	S. Lazanu, I. Lazanu, Nucl. Instr. Meth.Phys. Res. A 462, (1999) 530.

\bibitem{4}	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{1}	I. Lazanu, S. Lazanu, U. Biggeri, E. Borchi, M. Bruzzi, Nucl. Instr. %%@
Meth.Phys. Res. {\bf A 388} (1997) 370. 

\bibitem{2}	I. Lazanu, S. Lazanu, Nucl. Instr. Meth.Phys. Res. {\bf A 432}, (1999) 374. 

\bibitem{3}	A. Van Ginneken, Preprint Fermi National Accelerator Laboratory, FN-522, 1989 

\bibitem{5} 	S. Lazanu, I. Lazanu, Nucl. Instr. Meth. Phys. Res. {\bf B 183} (2001) %%@
383.

\bibitem{6}  	I. Lazanu, S. Lazanu, {\em''Radiation defects in silicon due to hadrons %%@
and leptons, their annealing and influence on detector performance}, in press to Physica %%@
Scripta. 

\bibitem{7}		I. Lazanu, S. Lazanu, U. Biggeri, E. Borchi, M. Bruzzi, Nucl. Phys. {\bf %%@
61B} (1998) 409c.

\bibitem{8}		S. Lazanu, I. Lazanu, U. Biggeri, S. Sciortino, Nucl. Instr. Meth.Phys. %%@
Res. {\bf A 413} (1998) 242. 

\bibitem{toti}	E. A. Burke, IEEE Trans. Nucl. Sci., NS-33, 6 (1986) 1276-1281; G. P. %%@
Summers, E. A. Burke, C. J. Dale, E. A. Wolicki, P. W. Marshall, M. A. Gehlhauser,  IEEE %%@
Trans. Nucl. Sci., NS-34, 6 (1987) 1134-1139; C. J. Dale, P. W. Marshall, E. A. Burke,G. %%@
P. Summers, E. A. Wolicki,  IEEE Trans. Nucl. Sci., NS-35, 6 (1988) 1208-1214],

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

\bibitem{10}	G. P. Summers, E. A. Burke, Ph.. Shapiro, S. R. Messenger, R. J. Walters, %%@
IEEE Trans. Nuclear Science, {\bf NS 40} (1993) 1372.

\bibitem{11} 	A. M. Ougouag, J. G. Williams, M. B. Danjaji, S.-Y. Yang, IEEE Trans. %%@
Nuclear Science, {\bf NS-37} (1990) 2219.

\bibitem{12} 	M. Moll, E. Fretwurst, G. Lindstrom, Nucl. Instrum. Meth. Phys. Res. A 439 %%@
(2000) 282. 

\bibitem{13} 	B. C. Mac Evoy, A. Santacchia, G. Hall, Physica B 273-274 (1999) 1054.

\bibitem{14}	M. Bruzzi, IEEE Trans. Nuclear Science, {\bf NS-48} (2001) 960.

\bibitem{15} 	Y. H. Lee, J. W. Corbett, Phys. Rev. B {\bf 13} (1976) 2653.

\bibitem{damask}	A. C. Damask, G. J. Dienes, Points Defects in Metals, Gordon and %%@
Breach, New York, 1963
\bibitem{16}	Z. Su, A. Husain, J.W.Farmer, J. Appl. Phys. {\bf 67} (1990) 1903.

\bibitem{17}	S. D. Brotherton and P. Bradley, J. Appl. Phys. {\bf 53} (1982) 5720.

\bibitem{18}	L. W. Song, B. W. Benson, G. D. Watkins, Phys. Rev. {\bf B 33} (1986) %%@
1452. 

\bibitem{19} 	U. Biggeri, E. Borchi, M. Bruzzi, Z. Li, S. Lazanu, IEEE Trans. Nucl. Sci. %%@
{\bf NS 41}, (1994)  964.



\end{thebibliography}