# Study of point- and cluster-defects in radiation-damaged silicon

Elena M. Donegani1 Eckhart Fretwurst1 Erika Garutti1 Robert Klanner1 Gunnar
Lindstroem1 Ioana Pintilie2 Roxana Radu2 Joern Schwandt1 1Institute for
Experimental Physics, University of Hamburg, Luruper Chaussee 149, 22761
Hamburg, Germany
2National Institute of Material Physics, Magurele, Romania

###### Abstract

Non-ionising energy loss of radiation produces point defects and defect
clusters in silicon, which result in a significant degradation of sensor
performance. In this contribution results from TSC (Thermally Stimulated
Current) defect spectroscopy for silicon pad diodes irradiated by electrons to
fluences of a few $10^{14}$ cm-2 and energies between 3.5 and 27 MeV for
isochronal annealing between 80 and 280${}^{\,\circ}$C, are presented. A
method based on SRH (Shockley-Read-Hall) statistics is introduced, which
assumes that the ionisation energy of the defects in a cluster depends on the
fraction of occupied traps. The difference of ionisation energy of an isolated
point defect and a fully occupied cluster, $\Delta E_{a}$, is extracted from
the TSC data.

For the VOi (vacancy-oxygen interstitial) defect $\Delta E_{a}=0$ is found,
which confirms that it is a point defect, and validates the method for point
defects. For clusters made of deep acceptors the $\Delta E_{a}$ values for
different defects are determined after annealing at 80${}^{\,\circ}$C as a
function of electron energy, and for the irradiation with 15 MeV electrons as
a function of annealing temperature. For the irradiation with 3.5 MeV
electrons the value $\Delta E_{a}=0$ is found, whereas for the electron
energies of 6 to 27 MeV $\Delta E_{a}>0$. This agrees with the expected
threshold of about 5 MeV for cluster formation by electrons. The $\Delta
E_{a}$ values determined as a function of annealing temperature show that the
annealing rate is different for different defects. A naive diffusion model is
used to estimate the temperature dependencies of the diffusion of the defects
in the clusters.

###### keywords:

Silicon detectors , radiation damage , Shockley-Reed-Hall statistics , point
defects , cluster defects , defect diffusion

###### Contents

  1. 1 Introduction
  2. 2 Test Structures and Measurements
  3. 3 Analysis Method
  4. 4 Results
    1. 4.1 The VOi defect
    2. 4.2 Cluster defects
    3. 4.3 Cluster annealing
  5. 5 Conclusions
  6. 6 Acknowledgements
  7. 7 References

##  1 Introduction

Bulk-radiation damage in silicon limits the use of silicon detectors in high-
radiation environments like at the CERN-LHC or in space. Although both
microscopic and macroscopic effects of bulk damage are qualitatively
understood, in spite of claims to the contrary, a consistent quantitative
description of the data available has not yet been achieved. However, this is
required for reliably predicting the sensor performance as a function of
particle type and fluence, sensor design and operating parameters. The reason
is that the number of radiation-induced states in the silicon band gap is
large (see figure 1), their properties are frequently only poorly known, and
_effective states_ [1] have to be used in simulations because of the large
number of defects. In addition, a quantitative understanding of defect
clusters is lacking and in the TCAD simulations they are approximated by point
defects.

In this contribution we propose a simple, physics motivated parametrization of
the properties of cluster defects and apply it to spectroscopic results from
TSC (Thermally Stimulated Current) measurements. We present results for
silicon irradiated with electrons of 3.5 to 27 MeV kinetic energy [2, 3]. As
the threshold for defect cluster production is expected to be around 5 MeV for
electrons [4], these data are well suited to check the validity of the method.

In the present work single vacancy related defects are considered to be point-
like defects (e. g. the VOi defect). Such isolated point defects are produced
in silicon by low energy recoils, whereas an agglomeration of defects (or a
cluster for short) results from high energy recoils that introduce a dense
cascade of silicon atoms displaced from their original lattice position.

Figure 1: TSC spectra, $I_{TSC}(T)$, normalized to an electron fluence of
$\Phi=10^{14}$ cm-2, after irradiation by electrons with different energies
and annealing for 30 min at 80${}^{\,\circ}$C, measured at a heating rate of
0.183 K/s. The pad diodes were fabricated using standard float-zone silicon.
For the trap filling forward-current injection has been applied at $T_{0}=10$
K, so that both electron (E) and hole (H) traps contribute to $I_{TSC}(T)$.
For clarity, the individual curves are shifted vertically by 1 pA. For the
identification and labeling of the individual defects we refer to [2] and
references therein. The dark current, which starts to dominate above 200 K, is
subtracted.

##  2 Test Structures and Measurements

For the studies p+n n+ pad diodes, produced on 283 $\upmu$m thick float-zone
n-type silicon with a phosphorous-doping of approximately 1012 cm-3 and an
oxygen concentration $<10^{16}$ cm-3, were used. The p+ implant of 25 mm2 area
is surrounded by a guard ring. A window in the aluminum on top of the p+
implant with the shape of a decagon and an area of 3.25 mm2 allows to inject
light through the p+ contact. The n+ back contact is covered by an aluminum
grid.

The pad diodes have been irradiated with electrons of $E_{e}=3.5$, 6, 15, and
27 MeV to fluences between 1 and $10\times 10^{14}$ cm-2. For the sample
irradiated with 15 MeV electrons, TSC measurements were performed before and
after isochronal annealing for 30 min at temperatures
$T_{ann}=80-280^{\,\circ}$C, in 20${}^{\,\circ}$C steps. At the other
energies, data were only taken before and after annealing of 30 min at
$T_{ann}=80^{\,\circ}$C. For the TSC measurements the bias was applied to the
back n+ contact, and both p+ contact and guard ring were at ground potential.

Figure 1 shows typical TSC spectra of pad diodes irradiated by electrons of
different energies after 30 min annealing at 80${}^{\,\circ}$C. Trap filling
has been applied at $T_{0}=10$ K with a forward-current of approximately 1 mA,
so that both electron- and hole-traps are filled and contribute to the TSC
spectra. The heating rate was $\beta=0.183$ K/s. For more details we refer to
[2, 5, 6]. In the following we will limit the discussion to the point defect
VOi (vacancy-oxygen interstitial) at 70 K and to the states in the $120-200$ K
region, which are known to be cluster defects and have a significant impact on
the sensor performance [7].

For the TSC measurements analysed in this paper, the diodes were cooled to
$T_{0}=10$ K at 200 V reverse bias to assure empty traps. At the temperature
$T_{0}$ the traps were filled with electrons by injecting light of 520 nm
through the 3.25 mm2 window of the p+ contact, and the current, $I_{TSC}$,
released by the traps was recorded as a function of $T$ for a constant heating
rate $\beta=0.183$ K/s. As a result of the light injection, only electron
traps are filled and only the charges released from electron traps contribute
to $I_{TSC}$ (see figure 2).

Figure 2: TSC spectra, $I_{TSC}(T)$, obtained after trap filling with light
injection (with $\lambda=520$ nm) through the 3.25 mm2 window of the p+
contact at $T_{0}=10$ K, for the same irradiation conditions and samples as
the data shown in figure 1. As a result of the trap filling by light
injection, only electron traps contribute to the TSC current. For clarity, the
individual curves are shifted vertically by 1 pA.

##  3 Analysis Method

According to SRH statistics [8, 9] the temperature dependence of the TSC
current, $I_{TSC}(T)$, from acceptor states at an energy $E_{a}$ from the
conduction band, which are filled with electrons at the temperature $T_{0}$ to
the concentration $N_{t}$, is given by:

$I_{TSC}(T)=\frac{A\cdot d\cdot q_{0}}{2}\cdot e(T)\cdot f_{t}(T)\cdot N_{t},$
(1) $e(T)=\sigma\cdot v_{th}(T)\cdot
N_{C}(T)\cdot\textrm{exp}\left(-\frac{E_{a}}{k_{B}T}\right),$ (2)
$f_{t}(T)=\textrm{exp}\left(-\frac{1}{\beta}\int_{T_{0}}^{T}e(T^{{}^{\prime}})\,\mathrm{d}T^{{}^{\prime}}\right),$
(3)

with the elementary charge $q_{0}$, the diode volume where the traps have been
filled $A\cdot d$, the emission rate $e(T)$, the ratio of filled states at the
temperature $T$ relative to $T_{0}$, $f_{t}(T)$, the thermal velocity of
electrons $v_{th}(T)$, the electron capture cross-section $\sigma$, and the
density of states at the conduction band $N_{C}(T)$. For these quantities the
default values of Synopsys TCAD [10], in particular the relation
$v_{th}(T)=\sqrt{8\cdot k_{B}T/\left(\pi\cdot m_{th}(300~{}\text{K})\right)}$
of [11], are used. The effective thermal electron mass
$m_{th}(300~{}\text{K})=0.278\cdot m_{e}$, with the Boltzmann constant $k_{B}$
and the free electron mass $m_{e}$.

For point defects a constant value for $E_{a}$ is expected. As proposed in
[12], cluster defects can be characterized by an occupation-dependent value of
$E_{a}(f_{t})$. For the dependence $E_{a}(f_{t})$, i. e. the change of the
ionisation energy with the fraction $f_{t}$ of defects filled in a cluster, a
linear dependence is assumed:

$E_{a}(f_{t})=E_{0}-\Delta E_{a}\cdot f_{t}.$ (4)

The schematic drawing of figure 3 shows the potential energies of electrons in
a cluster of 15 equally spaced electron traps on a straight line. The full
dots correspond to $f_{t}=1$, when all 15 traps are filled with electrons. The
empty circles is the situation $f_{t}\rightarrow 0$, when only a single trap
is filled and the energy is equal to the energy of the single isolated trap.
The ionisation energy $E_{a}=E_{0}-\Delta E_{a}$ is minimal for $f_{t}=1$, and
reaches its maximum $E_{a}=E_{0}$, the value for point defects, for
$f_{t}\rightarrow 0$.

Figure 3: Schematic presentation of the potential energy of the electrons in a
cluster consisting of 15 point defects equally spaced on a straight line.
$E_{V}$ denotes the edge of the valence band and $E_{C}$ of the conduction
band. The filled dots show the ionisation energies for the individual defects,
when all traps are filled. The electron occupying the central defect has the
lowest ionisation energy, $E_{0}-\Delta E_{a}$. It will discharge first, when
the sample is heated in the TSC measurement. The open circles correspond to
the ionisation energy when only one state is filled with an electron. Its
value is $E_{0}$, the ionisation energy of the point defects, which build the
cluster. Figure 4: Simulated dependence of $E_{a}(f_{t})$ for different
cluster topologies. Line (25): 25 equally spaced point charges on a line, Loop
(25): 25 equally spaced point charges on a circle, and Sphere (19): 19 point
charges in a sphere on an equally spaced 3D-grid.

Figure 4 shows the results of simple, electrostatic simulations of the
dependence of $E_{a}(f_{t})$. For _Line_ (25) the cluster is assumed to be a
straight line of 25 equally spaced point defects, fixed in space. The energy
difference between the conduction band and a single point defect is $E_{0}$.
In the initial state, $f_{t}=1$, each point defect is occupied by an electron
and negatively charged. The point defect with the highest (negative) potential
energy, $-\Delta E_{a}$, and thus the lowest $E_{a}$ value,
$E_{a}=E_{0}-\Delta E_{a}$, is discharged first, when the temperature is
raised in the TSC measurement. Next, the point defect with the lowest $E_{a}$
value for the new charge configuration is found and discharged. This chain is
continued, until all defects are discharged. The cluster _Loop_ (25) consists
of 25 equally spaced point defects on a circle, and _Sphere_ (19) of 19 point
defects in a sphere on a 3-D Cartesian grid. Figure 4 shows that to a good
approximation for all three topologies, $E_{a}(f_{t})$ is linear in $f_{t}$.
Loop clusters are discussed in [12], which actually triggered our studies.

The data analysis consists in subtracting from the measured TSC current the
dark current, obtained from the TSC measurement without trap filling. Then
$\chi^{2}$-fits of eq. 1 for different $T$ intervals are performed. For the
errors 1 % times the measured currents have been assumed, which takes into
account the uncertainty of the current measurement as well as the accuracy of
the temperature regulation of the TSC setup. The free parameters of the model
are $N_{t}$, $E_{0}$, $\Delta E_{a}$ and $\sigma$ for each defect, where
$N_{t}$ is the density of point defects per volume, which are filled in the
experiment at $T_{0}$, i. e. the product of cluster density times the number
of filled point defects in a cluster.

##  4 Results

###  4.1 The VOi defect

As a first step, the vacancy-oxygen (VOi) defect located at approximately
$T=70$ K is investigated. The VOi is known to be a point defect. According to
reference [13] it is an electron trap with an energy $E_{0}=176$ meV from the
conduction band and an electron capture cross-section $\sigma=79\times
10^{-16}$ cm-2. The values in reference [14] are $E_{0}=169$ meV and
$\sigma=(100\pm 10)\times 10^{-16}$ cm-2. In figure 5 the TSC data after
electron irradiation with $E_{e}=27$ MeV to the fluence $\Phi=4.3\times
10^{14}$ cm-2 and 30 min annealing at $80^{\,\circ}$C are shown as dots. The
fit results are shown as a solid line, and the parameters are given in table
1. For the assumed 1 % uncertainty for the $I_{TSC}$ measurement, a
$\chi^{2}=25$ for $NDF=47$ degrees of freedom is found. We note that the
differences _fit – measured_ are dominated by the systematics of the
temperature regulation and not by the statistical fluctuations of the current
measurement. The values $E_{0}=166.1\pm 0.7$ meV and $\sigma=(20\pm 2)\times
10^{-16}$ cm-2 (statistical errors only) differ significantly from the values
given in [13]. If $E_{0}=176$ meV is assumed, the $\chi^{2}$ increases to 61
and $\sigma=80\times 10^{-16}$ cm2 is obtained. However, the value of $\Delta
E_{a}=0.88\pm 0.03$ meV is incompatible with zero, the value expected for a
point defect. For the remaining curves of figure 5 the value $E_{0}=166.1$ meV
is assumed and $\Delta E_{a}$ is fixed to the values given in the insert.
Values significantly larger than zero are incompatible with the data. We
conclude that $\Delta E_{a}$ values, which differ significantly from zero are
excluded.

Figure 5: The measured TSC current after light injection at $T_{0}=10$ K (dots) for the point-like vacancy-oxygen defect (VOi) after electron irradiation with $E_{e}=27$ MeV to the fluence $\Phi=4.3\times 10^{14}$ cm-2. For the fit (solid line) in the range $T=64.5$ K to 74.0 K, the activation energy $E_{0}$, $N_{t}$, $\sigma$, and $\Delta E_{a}$ are free parameters. The values $E_{0}=166.1$ meV and $\Delta E_{a}=-0.1$ meV are found. For the other fits $E_{0}$ is set to 166.1 meV, $\Delta E_{a}$ to the values given in the insert, and $N_{t}$ and $\sigma$ are free parameters. $E_{0}$ [meV] |  $\Delta E_{a}$ [meV] |  $N_{t}/10^{12}$ [cm-3] |  $\sigma/10^{-16}$ [cm2] | $\chi^{2}/NDF$
---|---|---|---|---
$166.1\pm 0.7$ | $-0.1\pm 0.1$ | $3.44\pm 0.01$ | $19.7\pm 2.2$ | 25/47
176 | $0.88\pm 0.03$ | $3.43\pm 0.01$ | $80.0\pm 0.2$ | 61/48
Table 1: Results of the fits to $I_{TSC}$ in the temperature range 64.5 to
74.0 K for the diode irradiated to the fluence of $\Phi=4.3\times 10^{14}$
cm-2 by electrons with $E_{e}=27$ MeV. The first row gives the results for the
fit with $E_{0}$ left free, and the second line the fit with $E_{0}$
constrained to the value from reference [13].

The data after irradiation by electrons with energies $E_{e}$ between 3.5 and
15 MeV and after 30 min annealing at 80${}^{\,\circ}$C, have also been
analysed. In all cases, $\Delta E_{a}$ values close to zero and consistent
values for $E_{0}$ and $\sigma$ are obtained. This confirms that VOi is a
point defect, and validates the analysis method for an isolated point defect.

###  4.2 Cluster defects

Next, the region $T=120-200$ K is investigated, where overlapping defect
clusters are located, which are responsible for part of the generation
current, charge trapping and changes in the effective space-charge density
[3]. The TSC spectra for the irradiations at the different $E_{e}$ values
after 30 min annealing at 80${}^{\,\circ}$C are shown in fig. 6.

(a)

(b)

(c)

(d)

Figure 6:  TSC spectra after 30 min annealing at $80^{\,\circ}$C with the fits
described in the text for the temperature interval $T=144.1-184.6$ K and
irradiations with electrons of energies (a) 3.5 MeV, (b) 6 MeV, (c) 15 MeV,
and (d) 27 MeV.

To describe the data, four defects have been assumed: the di-vacancy V2, the
tri-vacancy V3 and two additional defects, E148 and E205a. The values of
$E_{0}$ from the literature are given in table 2. Using eq. 1 the sums of the
contributions of all four defects have been fitted to the $I_{TSC}$
measurements. In the fit, the literature values for $E_{0}$, a $\Delta E_{a}$
value common for the two defects E148 and V3, and another $\Delta E_{a}$ value
common for E205a and V2, have been assumed. Other assumptions, e. g. a common
$\Delta E_{a}$ value for all 4 states, gave qualitatively similar results.
Without such constraints, which are only poorly motivated, the fits do not
converge to stable results. Figure 6 compares the fit results to the data, and
table 2 shows as an example the parameters derived for the $E_{e}=15$ MeV
data. The TSC spectra are well described by the fits. It has been verified
that they cannot be described with acceptable trap parameters if $\Delta
E_{a}=0$ is assumed, and conclude that the model with an occupation-dependent
ionisation energy provides a good description of the TSC spectra of the
cluster defects.

Defect |  $E_{0}$ [meV] | Ref. |  $\Delta E_{a}$ [meV] |  $N_{t}$/$10^{12}$ [cm${}^{-3}]$ |  $\sigma$/$10^{-16}$ [cm2]
---|---|---|---|---|---
E148 | 359 | [5, 15] | 4.3 | 0.33 | 4.6
E205a | 393 | [5, 13] | 7.6 | 0.56 | 6.1
V2($-/0$) | 424 | [13] | 7.6 | 2.44 | 7.0
V3($-/0$) | 456 | [3, 13] | 4.3 | 0.65 | 9.7
Table 2: Parameters for the four electron traps in the $T=120-200$ K region:
$E_{0}$ is the energy distance of the isolated point defects from the
conduction band taken from the reference given in the third column. $\Delta
E_{a}$, $N_{t}$ and $\sigma$ are the cluster-related energy shifts, the
density of traps filled at $T_{0}$ and the electron cross-sections obtained
from the fit for $T=144.1-184.6$ K of eq. 1 to the data for $E_{e}=15$ MeV
after 30 min annealing at $80^{\,\circ}$C. The statistical errors are about
0.1 meV for $\Delta E_{a}$ and at the few % level for $N_{t}$ and $\sigma$.
The systematic uncertainties, e. g. from the choice of the fit interval, are
significantly larger.

In order to investigate the dependence of the cluster formation on the
electron energy, $E_{e}$, figure 7 shows $\Delta E_{a}(E_{e})$ for the V3/E148
and the V2/E205a clusters obtained from the fits described above. The $\Delta
E_{a}$ values for $E_{e}=3.5$ MeV are compatible with zero. For higher
electron energies the $\Delta E_{a}$ values differ significantly from zero and
the energy dependence is as expected for a threshold of cluster formation in
the region $E_{e}=3.5-6$ MeV.

(a)

(b)

Figure 7:  $\Delta E_{a}$ after 30 min annealing at $80^{\,\circ}$C as a
function of electron energy, $E_{e}$, for (a) the E148/V3-, and (b) the
E205a/V2-defect.

The phenomenological function with the free parameters $B$, $\gamma_{e}$ and
$E_{th}$,

$\Delta
E_{a}(E_{e})=B\cdot\left[1-\mathrm{exp}\left(\frac{E_{e}-E_{th}}{\gamma_{e}}\right)\right],$
(5)

is fit to the $\Delta E_{a}(E_{e})$ data. For the threshold values
$E_{th}^{\mathrm{V}_{3}/\mathrm{E148}}=3.0\pm 0.5$ MeV and
$E_{th}^{\mathrm{V}_{2}/\mathrm{E205a}}=3.7\pm 0.5$ MeV are obtained.

We note that for the models presented in section 3, a value of $\Delta
E_{a}=7.5$ meV corresponds to a spacing of about 100 nm of the point defects,
which are filled at $T_{0}=10$ K in the clusters, to be compared to the
silicon-lattice constant of 0.543 nm at 300 K. The values are compatible with
simulations of electron-induced damage in silicon, which take into account the
Mott cross-section, the vacancy and interstitial generation, as well as their
diffusion and recombination [16].

###  4.3 Cluster annealing

In the following, the analysis of the data for the irradiation with $E_{e}=15$
MeV for the conditions _as irradiated_ and after isochronal annealing for
$T_{ann}=80-280^{\,\circ}$C in 20${}^{\,\circ}$C steps and 30 min annealing
time, is presented. The temperature history for _as irradiated_ is only
approximately known: After the 15 MeV irradiation at Dresden-Rossendorf the
diode had been at room temperature for about eight hours during the transport
to Hamburg, where it has been stored in the freezer at $-25^{\,\circ}$C. In
the following this condition is labeled $T_{ann}=20^{\,\circ}$C. In figure 8
we show as examples the comparison of the fit results with the measurements
for _as irradiated_ and after the isochronal annealing steps of
80${}^{\,\circ}$C, 180${}^{\,\circ}$C and 280${}^{\,\circ}$C. For the fits the
assumptions discussed in section 4.2 are made. For all $T_{ann}$ values the
data are well described by the model.

(a)

(b)

(c)

(d)

Figure 8: Measured TSC spectra, $I_{TSC}(T)$, and results of the fits assuming
four defect clusters in the $120-200$ K region for the diode irradiated by 15
MeV electrons to a fluence of 2.6$\cdot$1014 cm-2 for (a) _as irradiated_ ,
and after 30 min annealing at (b) 80${}^{\,\circ}$C, (c) 180${}^{\,\circ}$C,
and (d) 280${}^{\,\circ}$C.

The results for $N_{t}$ of the four defects, and the cross-sections for
electrons, $\sigma$, as a function of $T_{ann}$ are shown in figure 9. As
discussed in section 3, $N_{t}$ is not the density of filled clusters, but the
product of the cluster density times the number of filled traps in the
cluster. We observe that with increasing annealing temperature $N_{t}$ remains
constant for V2, increases for E148, and vanishes for V3 and for E205a. More
details on the annealing behavior of $V_{3}$ and E205a can be found in [5]. As
expected the electron cross-sections do not depend on $T_{ann}$.

(a)

(b)

Figure 9:  (a) Densities of traps filled at $T_{0}$, $N_{t}$, and (b) electron
cross-sections, $\sigma$, for the diodes irradiated by 15 MeV electrons to a
fluence of $2.6\times 10^{14}$ cm-2 as a function of the annealing
temperature, $T_{ann}$.

The results for $\Delta E_{a}$ as a function of $T_{ann}$ for the traps
V2/E205a and V3/E148 are shown in figure 10. For V2/E205a the value of $\Delta
E_{a}$ remains nearly constant ($\approx 7.5$ meV) for the annealing at
$80^{\,\circ}$C, then decreases to $\approx 2$ meV above $250^{\,\circ}$C.

For V3/E148 the decrease of $\Delta E_{a}$ with $T_{ann}$ is significantly
faster, and a constant value of $\approx 1$ meV is reached at
$T_{ann}=140^{\,\circ}$C. For the description of $\Delta E_{a}(T_{ann})$ two
models are introduced: a _first order decay model_ , and a _diffusion model_.

(a)

(b)

Figure 10:  Dependence of $\Delta E_{a}(T_{ann})$ for the diode irradiated by
15 MeV electrons to a fluence of $2.6\times 10^{14}$ cm-2 for _as irradiated_
, plotted at $T_{ann}=20^{\,\circ}$C, and after isothermal annealing at the
temperature $T_{ann}$ for 30 min. The points with error bars are the data, the
solid lines the fits by the _first order decay model_ , and the dashed lines
the fits by the _diffusion model_ for (a) V2/E205a and (b) V3/E148.

The _first order decay model_ uses the following formula for $\Delta E_{a}$
after the $i$-th annealing step

$\Delta E_{a}(T_{ann}^{(i)})=\Delta E_{a}(T_{ann}^{(i-1)})\cdot
e^{-k(T_{ann}^{(i)})\cdot t_{ann}}\hskip 14.22636pt\mathrm{with}\hskip
14.22636ptk(T_{ann})=k_{0}\cdot e^{-E_{A}/(k_{B}T_{ann})},$ (6)

with the annealing time $t_{ann}=30$ min. The first order process is described
by the frequency factor $k_{0}$ and the activation energy $E_{A}$. For $\Delta
E_{a}(T_{ann}^{(0)})$ the _as irradiated_ data assigned to
$T_{ann}=20^{\,\circ}$C are used. The results of the fits to the data are
shown in figure 10 and in table 3. The model provides an acceptable
description of the data.

The _diffusion model_ follows a similar approach as the model discussed in
section 3, which was used to derive the linear dependence of the ionisation
energy on the fraction of occupied defects. Each cluster is assumed to consist
of $n_{clu}$ point defects, which diffuse during the annealing steps. For the
temperature dependence of the diffusion constant the standard parametrisation

$D(T)=D_{0}\cdot e^{-E_{D}/(k_{B}T)}$ (7)

is used. The activation energy for the diffusion of the point defects in the
cluster is $E_{D}$, and $D_{0}$ a constant. For the fit $n$ clusters with
$n_{clu}$ point defects each, are generated. The position of the $j$-th point
defect in the $k$-th cluster after the $(i-1)$-st annealing step is denoted by
$\vec{x}_{j,k}^{~{}(i-1)}$. The position after the $i$-th annealing step is
$\vec{x}_{j,k}^{~{}(i)}=\vec{x}_{j,k}^{~{}(i-1)}+\vec{\delta x}$, where
$\vec{\delta x}$ is a vector of three Gauss-distributed random numbers with
width $\sigma_{D}(T_{ann}^{(i)})=\sqrt{2~{}D(T_{ann}^{(i)})\cdot t_{ann}}$.
Assuming that all point defects are occupied by electrons, the individual
potential energies $V_{j,k}^{~{}(i)}$ are calculated. The biggest
$|V_{j,k}^{~{}(i)}|$ is assumed to be the $\Delta E_{a}$ value of the $k$-th
cluster, and the average $\Delta E_{a}$ of the $n$ clusters is fitted to the
data. The initial positions of the point charges in the
$\vec{x}_{j,k}^{~{}(0)}$ are generated as $n_{clu}$ points spaced by $d_{0}$
on a straight line each smeared with a vector of three Gauss-distributed
random numbers with width $\sigma_{0}$. For a given number of clusters, $n$,
and cluster size, $n_{clu}$, the free parameters of the fit are $\sigma_{0}$,
$d_{0}$, $D_{0}$ and $E_{D}$. For the fits to the data shown in figure 10,
$n=1000$ and $n_{clu}=15$ are used. Fits have also been performed for other
$n_{clu}$ values. The $\chi^{2}$ values obtained depend only weakly on
$n_{clu}$ and there is a shallow minimum around $n_{clu}=15$. The parameters
obtained from the fit are given in table 4, where the diffusion constants at
423.15 K, $D_{423.15\mathrm{K}}=D_{0}\cdot e^{-E_{D}/(k_{B}\cdot
423.15\,\mathrm{K})}$, instead of $D_{0}$ are shown. Using the value of $D$ at
the average annealing temperature 423.15 K has the advantage that
$D_{423.15\mathrm{K}}$ and $E_{D}$ are essentially uncorrelated. In addition,
it is more easily interpreted than $D_{0}$, which corresponds to the value of
$D$ for $T_{ann}\rightarrow\infty$.

We note that, whereas the values of the parameters given in table 4 change
significantly if the fit region is changed, the dependencies of $D(T_{ann})$
and $\sigma(T_{ann})$, shown in figure 11, are much less affected. The values
of the diffusion activation energy $E_{D}\approx 0.42$ eV for V2/E205a is
similar to the value given in [17] for the vacancies V0 and V-1, and
$E_{D}\approx 0.15$ eV for V3/E148 similar to the interstitial Si${}_{i}^{0}$.

Defect |  $k_{0}$ [1/s] |  $E_{A}$ [eV] | $\chi^{2}/NDF$ | Fit range [∘C]
---|---|---|---|---
V2/E205a | $1.6\times 10^{-3}$ | 0.113 | 15.6/10 | $20-280$
V2/E205a | $42\times 10^{-3}$ | 0.225 | 1.2/8 | $20-240$
V3/E148 | $7.2\times 10^{-6}$ | $-0.105$ | 7.9/10 | $20-280$
V3/E148 | $3.6\times 10^{-4}$ | 0.014 | 2.3/3 | $20-140$
Table 3:  Parameters obtained for the fit of the _first order decay model_ , equation 6, to the $\Delta E_{a}(T_{ann})$ results shown in figure 10. The last column gives the $T_{ann}$ range of the fit Defect |  $D_{423.15\mathrm{K}}$ [nm2/s] |  $E_{D}$ [eV] |  $d_{0}$ [nm] |  $\sigma_{0}$ [nm] | $\chi^{2}/NDF$ |  $T_{ann}$ [∘C]
---|---|---|---|---|---|---
V2/E205a | 6.3 | 0.423 | 27 | 159 | 6.4/8 | $20-280$
V3/E148 | 60 | 0.146 | 27 | 200 | 5.7/8 | $20-280$
Table 4:  Parameters obtained for the _diffusion model_ described in the text,
to the $\Delta E_{a}(T_{ann})$ results shown in figure 10.

(a)

(b)

Figure 11:  Results from the fits of the _diffusion model_ for clusters
containing $n_{clu}=15$ point defects as a function of the temperature
$T_{ann}$ of 30 min isochronal annealing for the diode irradiated to a fluence
of $2.6\times 10^{14}$ cm-2. (a) Diffusion constant, and (b) $\sigma$
parameter of the Gaussian describing the diffusion of the point defects in the
clusters.

As a pictorial presentation of the diffusion of the point defects in the
clusters, figure 12 shows for the V2/E205a cluster the $x-z$ distribution of
the superposition of 20 cluster with 15 point defects each, using the
parameters of table 4 for the conditions _as irradiated_ , and after the
annealing steps $T_{ann}=80^{\,\circ}$, $180^{\,\circ}$ and
280${}^{\,\circ}$C. The point defects of the initial cluster, which are
separated by the distance $d_{0}$, are generated along the $z$ direction. As
expected from figure 11(b), hardly any diffusion is observed at
$T_{ann}=80^{\,\circ}$C, the width increases by about a factor 3 at
180${}^{\,\circ}$C, and the clusters are essentially dissociated at
280${}^{\,\circ}$C. We note that for $N_{t}=2.5\times 10^{12}$ cm-3, shown in
figure 9(a), and 15 point defects in each cluster, the average distance
between clusters is $\approx 2\,\upmu$m.

(a)

(b)

(c)

(d)

Figure 12: Diffusion of the point charges in the V2/E205a cluster simulated
using the parameters of table 4 for (a) _as irradiated_ , and after 30 min
annealing at (b) 80${}^{\,\circ}$C, (c) 180${}^{\,\circ}$C, and (d)
280${}^{\,\circ}$C. To illustrate the diffusion, for the presentation 20
clusters of 15 point charges each are superposed.

We now summarise the results of the two models used to describe the annealing
measurements: Both models provide a description of the dependence $\Delta
E_{a}(T_{ann})$ for the V2/E205a and the V3/E148 clusters. Whereas the _first
order decay model_ is a purely phenomenological parametrisation, the
_diffusion model_ is physics based and gives, within the assumptions made to
determine the $\Delta E_{a}$ values and the limitation of this simple model,
insight into the cluster annealing. The diffusion-activation energy, $E_{D}$,
is quite different for the two cluster defects: $\approx 0.4$ eV for V2/E205a
and $\approx 0.15$ eV for V3/E148. At an annealing temperature of
80${}^{\,\circ}$C the diffusion parameter, $D$, for V2/E205a is about a factor
50 smaller than for V3/E148. However at 280${}^{\,\circ}$C, the highest
annealing temperature investigated, they are quite similar. As a consequence,
the diffusion of the point defects in the V3/E148 cluster becomes significant
only for temperatures above 150${}^{\,\circ}$C, whereas for the V2/E205a
cluster it is already significant at room temperature. At the highest
annealing temperatures the spread of the points in the cluster approaches the
average distance between the clusters.

##  5 Conclusions

In this paper a new method of analyzing TSC (Thermally Stimulated Current)
spectra of radiation-damaged silicon diodes is presented. It is based on the
Shockley-Read-Hall statistics and can be used to describe both point and
cluster defects. An ionisation energy, $E_{a}$, which depends on $f_{t}$, the
fraction of filled point defects in the cluster, is introduced. Simple
electrostatic model calculations show that a linear dependence
$E_{a}(f_{t})=E_{0}-\Delta E_{a}\cdot f_{t}$ appears to be a reasonable
approximation, where $E_{0}$ is the ionisation energy of the point defect
which makes up the cluster, and $\Delta E_{a}$ the difference in ionisation
energy between $f_{t}\rightarrow 0$ and $f_{t}=1$. The method is applied to
the analysis of TSC spectra from n-type silicon diodes irradiated by electrons
in the energy range between 3.5 and 27 MeV. For the filling of the traps the
the p+ implants of the diodes are exposed to green light, so that only
acceptor traps contribute to the TSC current.

As a first step the VOi defect (an electron trap with $E_{0}\approx 0.17$ eV,
which is known to be a point defect) is analysed for the diode irradiated by
27 MeV electrons to a fluence of $4.3\times 10^{14}$ cm-2. The value found for
$\Delta E_{a}$ is compatible with zero. This confirms that VOi is a point
defect, and also demonstrates the validity of the method for point defects.

Next, the radiation-induced deep acceptor states with energies between 0.35
and 0.46 eV from the conduction band are investigated. They are responsible
for part of the generation current, for charge trapping and for changes of the
effective space-charge density with irradiation. In the analysis it is assumed
that four defects, E148, E205a, V2, and V3 contribute to the TSC current in
this energy region. For the fits to the TSC spectra, the ionisation energies
from the literature and common $\Delta E_{a}$ values for V2 and E205a as well
as for V3 and E148 had to be assumed. Without these or similar not too well
justified assumptions, the fits do not converge to stable values.

To study the dependence of the cluster formation on electron energy, the data
for $E_{e}=3.5$, 6, 15, and 27 MeV after annealing for 30 min at
80${}^{\,\circ}$C are analysed. For the lowest electron energy $\Delta
E_{a}=0$, whereas for the higher electron energies positive $\Delta E_{a}$
values are found. After a rapid increase with electron energy, a tendency
towards a saturation of $\Delta E_{a}$ is observed. This is taken as evidence
for cluster formation above a threshold, which is somewhere between
$E_{e}=3.5-6$ MeV.

To study the annealing of the clusters, the data for the irradiation with 15
MeV electrons and the conditions _as irradiated_ and after isochronal
annealing for 30 min between $T_{ann}=80-280^{\,\circ}$C are analysed with the
assumptions discussed above. For both V2/E205 and V3/E148, $\Delta E_{a}$ is
found to decrease with $T_{ann}$, however the decrease for V3/E148 is
significantly faster than for V2/E205. Two models, a _first order decay model_
and a _diffusion model_ , provide adequate descriptions of the data. Using the
latter model the $T_{ann}$ dependence of the diffusion parameters and of the
spatial spread of the point defects in the cluster is estimated. It is found,
that at $T_{ann}=80^{\,\circ}$C the diffusion constant for V2/E205a is about a
factor 50 smaller than for V3/E148, whereas at $T_{ann}=280^{\,\circ}$C, they
are quite similar. For the diffusion-activation energy, $E_{D}$, a value of
0.42 eV is determined for the V2/E205 cluster, and 0.15 eV for the V3/148
cluster. For the highest $T_{ann}$ values, the spread of the point defects in
the clusters approaches the average distance between the clusters.

##  6 Acknowledgements

This work has been performed in the framework of the CERN-RD50 collaboration
and PNII-ID-PCE-2011-3 Nr. 72/5.10.2011, and partially funded by CiS, the
German BMBF and the Helmholtz Alliance Physics at the Terascale. I. Pintilie
and R. Radu gratefully acknowledge partial funding from the Romanian core
program PN18-11 (funded by ANCSI). R. Radu was also partially supported by the
Marie Curie Initial Training Network MC-PAD and the fellowship A127881 of the
German Academic Exchange Service (DAAD). We also thank the teams at the
irradiation facilities at the Belarusian State University at Minsk, the PTB at
Braunschweig, and the ELBE accelerator at Dresden-Rossendorf, where the
irradiations have been performed. We are grateful to P. Buhmann and M. Matysek
for maintaining the measurement infrastructure of the Hamburg Detector
Laboratory, where the measurements were performed, in an excellent shape.

##  7 References

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