Modeling the Annealing of Dislocation Loops in Implanted c-Si Solar Cells

Abstract

This paper is motivated by the question of how residual implantation damage degrades solar cell performance. In order to avoid such degradation, annealing processes of implanted c-Si solar cells use high thermal budgets. Still, implantation-induced dislocation loops may survive these processes. We derive two models for the annealing kinetics of dislocation loops that are suitable for the study of high thermal budgets: a model that is able to describe the parallel ripening of faulted and perfect dislocation loops and a model that explicitly implements the conservative and nonconservative processes associated with Ostwald ripening. Both models lead to a better agreement with the experiment than what has been published before.

Full text

IEEE JOURNAL OF PHOTOVOLTAICS 1
Modeling the annealing of dislocation loops in
implanted c-Si solar cells
F. A. Wolf, A. Martinez-Limia, D. Stichtenoth, and P. Pichler, Senior Member, IEEE
Abstract—This paper is motivated by the question of how resid-
ual implantation damage degrades solar cell performance. In or-
der to avoid such degradation, annealing processes of implanted
c-Si solar cells use high thermal budgets. Still, implantation-
induced dislocation loops may survive these processes. We derive
two models for the annealing kinetics of dislocation loops that
are suitable for the study of high thermal budgets: A model
that is able to describe the parallel ripening of faulted and
perfect dislocation loops, and a model that explicitly implements
the conservative and non-conservative processes associated with
Ostwald ripening. Both models lead to better agreement with
experiment than what has been published before.
I. INTRODUCTION
STUDIES of implanted solar cells carried out within recent
years showed that significantly higher thermal budgets
than in microelectronics are needed to achieve satisfactory
cell performance [1, 2, 3, 4]. Typically, for such annealing
conditions (>950 ◦C, >10 min), the primary implant damage
evolves into a defect configuration in which only dislocation
loops survive. Such loops were found to correlate with the
emitter saturation current [5, 6] although also other hypotheses
exist [1, 6]. Recently, the causation of implantation-related
performance degradation by dislocation loops, likely decorated
with metallic impurities, could be substantiated [7, 8].
The agglomeration of implantation-induced self-interstitials
(Is) via the formation of small interstitial clusters (SMICs) and
their transformation to {311}defects and even small loops has
been addressed by a variety of models, e.g. [9, 10, 11, 12, 13,
14, 15], and is well described by them. This does not hold true
for the high thermal budgets of solar cell fabrication. As we
will show later (Sec. II), even the latest of them, the model of
Zographos et al. [15] implemented in Sentaurus Process [16],
fails then to reproduce two important qualitative aspects
(i) The predicted dissolution velocity at high temperatures
is by orders of magnitude too high.
(ii) The simulated growth of the mean loop radius does not
saturate for long annealing times at high temperatures,
as would be expected from theory and experiment.
Point (i) implies that for almost all relevant annealing condi-
tions, the model of Zographos et al. [15] predicts a complete
F. A. Wolf and A. Martinez-Limia are with Corporate Research, Robert
Bosch GmbH, Robert-Bosch-Platz 1, 70839 Gerlingen-Schillerh¨
ohe, Ger-
many.
D. Stichtenoth is with Bosch Solar Energy, Arnstadt, Germany.
P. Pichler is with the Fraunhofer Institute for Integrated Systems and Device
Technology, Schottkystrasse 10, 91058 Erlangen, Germany and with the
Chair of Electron Devices, University of Erlangen-Nuremberg, Cauerstrasse
6, 91058 Erlangen, Germany.
Manuscript received April 19, 2005; revised December 27, 2012.
dissolution of loops. Thus, it cannot be used to establish cor-
relations between processing conditions and cell performance.
Point (ii) is more subtle and of lower priority: A quantitatively
wrong prediction of the mean loop radius translates directly
into a likewise wrong prediction of the proportionally related
dislocation-line density, which directly correlates to the re-
combination activity of loops [17, 18]. Two other loop models
[13, 19] implemented in Sentaurus Process [20] suffer from
even more severe problems for annealing processes with high
thermal budgets. For an in-depth discussion of the modeling of
dislocation loops the interested reader is referred to the review
article of Claverie et al. [21] and the dissertation of one of the
authors (F.A.W) [22].
In this paper, we extend the model of Zographos et al.
[15] to overcome problem (i) (Sec. III). We succeed in doing
so by accounting for the parallel ripening of faulted and
perfect dislocation loops. The new model is able to provide
meaningful information about the local densities and sizes of
insufficiently annealed dislocation loops [7, 8]. Subsequently,
we analyze the late stages of Ostwald ripening, which leads us
to sketch a model that is able to resolve problem (ii) (Sec. IV).
Finally, we conclude the paper (Sec. V). We start with a brief
discussion of the model of Zographos et al..
II. THE MODEL OF ZOGRAPHOS et al.
Zographos et al. [15] model the mean concentration of loops
DLand the concentration of self-interstitials (Is) CLcomprised
in these loops by
∂tCL=R311
L+RI
L,(1a)
∂tDL=e
R311
L+e
RI
L.(1b)
These distributions depend on time and position. The reaction
terms that describe the unfaulting reaction from {311}defects
to loops are given by
R311
L=k311
LC311C311 ,(2a)
e
R311
L=k311
Le
k311
LC311D311 ,(2b)
where D311 and C311 denote the concentration of {311}
defects and the concentration of Is bound in {311}defects,
in complete analogy to the meaning of DLand CL.k311
Lis
described by an Arrhenius law [20]. Zographos et al. [15] set
e
k311
L=1
2, which means that {311}defects with twice the
mean size unfault to loops.
The interaction of loops with free Is is described by
RI
L=kI
L2π2rLDI(CI−C∗
I,L)DL,(3a)
e
RI
L=−kI
L2π2rLDIC∗
I,L
DLDL
CL
=−kI
L2πDI
rLnL
C∗
I,LDL,(3b)

IEEE JOURNAL OF PHOTOVOLTAICS 2
where CIis the concentration of free Is, kI
Lis a calibration
constant and DIthe diffusion coefficient of Is. The mean loop
radius rLand the equilibrium concentration of Is in the vicinity
of a loop C∗
I,L are defined by
rL=rCL
DLnLπand C∗
I,L =C∗
Ie∆EL(rL)/kT ,(4)
where nLis the atomic areal density in the close-packed {311}
planes. For the formation energy per I in faulted dislocation
loops ∆EL(rL), Zographos et al. [15] used the definition
∆EL(rL) = γΩ
b+GbΩ
4πrL(1 −ν)ln(8rL/b).(5)
Therein, γis the stacking-fault energy per unit volume, Ωthe
atomic volume, bthe modulus of the Burger’s vector of the
loop, Gthe shear modulus of Si and νits Poisson ratio. With
these equations, the loop model is completely defined.
In the present paper, the focus is on the phase of the
annealing process in which the last step of the I cluster
nucleation process, the unfaulting reaction from {311}defects
to loops, is completed. At this point, almost all {311}defects
have vanished, so that R311
L'0and e
R311
L'0, and one is left
to discuss, instead of Eqs. (1), the equations ∂tCL=RI
Land
∂tDL=e
RI
L.
A. Dissolution velocity of dislocation loops
The first fundamental problem that arises from the model
of Zographos et al. [15] is a qualitative overestimation of
the dissolution velocity of loops. This is illustrated in Fig. 1
for the areal density of loops NL=RtW
0dx DL(x), where
we integrated over the wafer thickness tW. We empirically
checked that this problem cannot be resolved by recalibrating
the value [16] of the sole free parameter kI
Lin Eqs. (3).
Fig. 1(a) shows that the model yields satisfying results for
temperatures up to 950 ◦C. But at 1000 ◦C and 1050 ◦C,
the simulated dissolution is much too rapid. In the case of
Fig. 1(b), where many more implantation-induced Is have been
produced, the dissolution velocity is qualitatively wrong even
at 950 ◦C.
B. Late stages of Ostwald ripening
The other fundamental problem of the model of Zographos
et al. [15] concerns the time evolution of the mean loop radius
rL. From Eq. (4), the time evolution of rLis obtained using
Eqs. (1) and Eqs. (3) as
drL
dt =kI
LπDI
nL
CI.(6)
Now consider the case CI'C∗
I, i.e. the time during annealing
at which the number of loops has considerably decreased. The
remaining ones are so large that they can gain relatively few
energy by exchanging Is. In this situation, Eq. (6) predicts a
linearly diverging time evolution of rLwhile, in reality, the
loop radius growth is much weaker [25, 23]. As soon as the
surface and other sinks for Is become dominant, loops finally
start shrinking again, although slowly, as Ostwald ripening
ensures that rather the number of loops decreases, than their
0 1 2 3 45678
time (min)
109
1010
1011
density of loops
N
L (cm-2 )
(a)
900 C
950 C
1000 C
1050 C
0 10 20 30 40 50 60
time (min)
109
1010
1011
density of loops
N
L (cm-2 )
(b)
950 C
Fig. 1. Time evolution of the areal loop density NLduring annealing. (a)
Rapid thermal annealing (RTA) after a Ge implant of 2·1015 cm−2at 150 keV,
experimental data from [23]. (b) RTA after a B implant of 1·1015 cm−2at
30 keV, experimental data from [24]. Symbols depict experimental results,
lines simulation results. Simulations were done with the model of Zographos
et al. [15] in the implementation of [16].
size. The described phenomenon is illustrated in Fig. 2. For all
the experiments shown, temperatures are sufficiently high and
annealing times sufficiently long so that Ostwald ripening is in
its final stage. In all panels of Fig. 2, the experiments shown
indicate a saturation of the loop radius with time. By contrast,
the solid lines calculated using the model of Zographos et al.
[15] display a diverging radius evolution.
III. PAR AL LE L RI PE NI NG O F FAULTED AND PERFEC T
LO OP S
The model of Zographos et al. [15] describes faulted loops
(FLs) and makes correct predictions for experiments with low
thermal budgets, while failing for high temperatures and ex-
tended annealing times. This failure can be explained with the
following experimental observation. For low thermal budgets,
mainly FLs are observed [27, 28, 29, 30]. For high thermal
budgets, during which loops undergo strong Ostwald ripening
and grow considerably, almost only perfect loops (PLs) are
observed [31, 32, 28, 33, 34, 29]. This behavior is due to
the well-known fact that for large loop sizes, perfect loops are
more stable than faulted loops. Our model extends Zographos’
model to describe not only FLs, but to also include PLs. While
modeling FLs is still necessary to correctly describe the early
stages of a high-thermal-budget process, modeling the different
reaction dynamics of PLs allows to describe the late stages.
Hence, our new model needs to physically describe both types
of loops.

IEEE JOURNAL OF PHOTOVOLTAICS 3
0 20 40 60 80 100 120
time (min)
0
10
20
30
40
50
60
70
mean loop radius r (nm)
(a) 800 C
900 C
1000 C
0 10 20 30 40 50 60
time (min)
0
10
20
30
40
50
60
70
mean loop radius r (nm)
(b)
950 C
Fig. 2. Time evolution of the mean loop radius rLduring annealing. (a) RTA
after an Si implant of 1·1015 cm−2at 50 keV, experimental data from [26]. (b)
RTA after a B implant of 1·1015 cm−2at 30 keV, experimental data from [24].
Symbols depict experimental results, lines simulation results. Solid lines were
calculated with the model of Zographos et al. [15] in the implementation of
[16]. Dashed lines were calculated with the same parameters using the model
presented in Sec. IV.
For intermediate thermal budgets, a transition from the state
with more faulted loops to the state with more perfect loops
takes place. This transition has traditionally been assumed
to result from an unfaulting reaction [35], meaning that a
faulted loop, in our cases always a Frank partial dislocation,
reacts with a Shockley partial dislocation to produce a perfect
dislocation loop. In contradiction to that, several authors have
observed independent time evolutions of the FL and the PL
ensembles, clearly identifying two different Ostwald ripening
mechanisms [28, 27, 29]. This observation suggests that a
second explanation for the FL-PL transistion is more likely:
Rather than by a direct unfaulting reaction, the transition
occurs as the two ensembles of different loop types exchange
Is among each other. Our model is based on this second
explanation.
A. Energy of faulted and perfect dislocation loops
The energy associated with a dislocation loop consists of the
elastic energy of the surrounding strain fields, its core energy
and a potential stacking-fault energy. The faulted loops most
frequently observed in Si are Frank partial loops, which lie
in {111}planes and have a Burger’s vector bFL =a
3[111]
with a=5.43 ˚
A denoting the size of the silicon unit cell.
Perfect loops in fcc lattices also lie in {111}planes but have
a Burger’s vector of bPL =a
2[110]. Their respective elastic
energies are [35, 27]
Eel
FL =rLGb2
FL
2(1 −ν)ln(r1/r0), b2
FL =a2
3,(7a)
Eel
PL =rLGb2
PL
2(1 −ν)1−ν
6ln(r1/r0), b2
PL =a2
2.(7b)
Linear elasticity theory is only valid at some distance away
from the core. This distance is characterized by r0for which
we assume r0=bFL/4[27]. For r1, roughly characterizing the
extent of the lattice distortions, we assume r1= 2rL[36, 35].
The energy Ecore stored in the core can only be accessed
by atomistic calculations, albeit with a low precision. The
stacking-fault energy of faulted loops is γ'70 mJ/m2per
atomic volume. Adding these terms to Eqs. (7a) and (7b), one
obtains
EFL =γπr2
L+rLGb2
FL
2(1 −ν)(ln(8rL/bFL)−1 + AFL ),(8a)
EPL =rLGb2
PL
2(1 −ν)1−ν
6(ln(8rL/bFL)−1 + APL ).(8b)
The constants AFL,PL have been introduced to represent
the core energy via Ecore
FL =AFL rLGb2
FL
2(1−ν)and Ecore
PL =
APL rLGb2
PL
2(1−ν)1−ν
6, respectively. Their values will be discussed
in more detail below. The constant −1was introduced in
the last expressions in brackets to account for a frequent
convention in the literature, ensuring a stress-free boundary at
the upper integration limit for the elastic energy r1[37, 38].
Although the functional form of the expressions (8) is
undebated, the numerical values for the integration boundaries
r1,r0and the core energy Ecore are only roughly known [39].
If, in addition, the loop is not ciruclar but shaped like an n-
sided regular polygon, which can be a reasonable assumption
for PLs, a further constant has to be added to AFL,PL [40].
One should therefore be satisfied with specifying a numerical
range of meaningful values for these parameters. Surveying the
literature, one finds the following values (We abbreviate AFL,PL
with AL.): Some authors choose AL= 0 [15, 23, 19, 38, 41]
and others AL= 1 [27, 35] and all of these either choose
r0=b/4or r0=b. Again others employ a very different
expression AL=2ν−1
4ν−4[42, 13]. Having made the choice
for r0=b/4and r1= 2rLin Eqs. (8), all uncertainty is
accommodated in AL. Assuming that the true value of r0is
in the range r0∈[b/4,4b][43, 35, 44] and that the true
core-energy-related component of ALis in the range [0,1],
one obtains the following range of meaningful values for
AL∈[−2.77,1].
B. Energy per self-interstitial in a dislocation loop
The energy necessary to incorporate one I in a loop is
∆EL=dE
dN =1
nL2πrL
dE
drL. Using Eq. (8), we obtain
∆EFL =γΩ/bFL +GbFLΩ
4πrL(1 −ν)(ln(8rL/bFL ) + AFL),(9a)
∆EPL =6−ν
4GbFLΩ
4πrL(1 −ν)(ln(8rL/bFL ) + APL).(9b)
The result of Eqs. (9) is shown in Fig. 3(a) for AFL = 0 and
APL = 0. The stability inversion occurs for loop sizes around

IEEE JOURNAL OF PHOTOVOLTAICS 4
0 10 20 30 40 50
loop radius rL (nm)
0.0
0.1
0.2
0.3
formation energy per atom (eV)
(a)
EFL
EPL
0 10 20 30 40 50
loop radius rL (nm)
100
101
102
I supersaturation at loop C *
L,I/C *
I
(b)
C*
FL,I 850 C
C*
PL,I 850 C
C*
FL,I 1050 C
C*
PL,I 1050 C
Fig. 3. Comparison of FL and PL formation energy (a) and I supersaturation
in the vicinity of a loop (b). Parameters are given in the text.
30 nm, which is similar to the results of [27] and [29]. The pa-
rameters used for this calculation are given by γ= 70 mJ/m2,
Ω = 2 ×10−23 cm3,G= 63.28 ×105N/cm2,ν= 0.28, and
bFL = 0.3135 nm. With that, Esf =γΩ/bFL = 0.0279 eV and
GbFLΩ
4π(1−ν)=0.274 eV nm. The even more interesting quantity is
given by C∗
L,I =C∗
Ie∆E(rL)/kT , see Eq. (4) and Fig. 3(b).
Here, the stability inversion of Fig. 3 is amplified to an
exponential behavior. C∗
L,I is responsible for the dissolution
of loops due to interactions with external sinks or the other
loop ensemble.
C. Summary of model equations
The equations for the parallel evolution of ensembles of
FLs and PLs read, in complete consistence with the model
presentation for only faulted loops in Sec. II,
∂tCFL =R311
FL +RI
FL,(10a)
∂tDFL =e
R311
FL +e
RI
FL,(10b)
∂tCPL =R311
PL +RI
PL,(10c)
∂tDPL =e
R311
PL +e
RI
PL.(10d)
where the functional forms of the terms R311
FL ,e
R311
FL ,R311
PL and
e
R311
PL have been defined in Eqs. (2) for a general loop. All of
these terms contain either the reaction rate k311
FL or k311
PL , which
determine the unfaulting rate of {311}defects to loops. In
agreement with the experimental observation that PLs are only
observed for higher temperatures, we choose k311
FL to have a
lower activation energy than k311
PL . The terms e
R311
FL and e
R311
PL
contain furthermore the factors g
k311
FL and g
k311
PL , respectively. In
agreement with the experimental observation of Stowe [29],
that perfect loops nucleate at a larger mean radius than faulted
loops, we choose g
k311
PL = 0.05 in comparison to g
k311
FL = 0.5[15]
for faulted loops. Finally, it should be kept in mind that the
expressions of the formation energies per I Eqs. (9) to be used
for C∗
I,L are different for faulted and perfect loops.
As the result of a calibration based on a variety of ex-
periments, we set k311
FL = 9 ·10−5·e−4.2eV/kT cm3/s for
faulted loops while keeping g
k311
FL ,kI
FL and AFL = 0 as in
Sentaurus Process [16]. For perfect loops, we set k311
PL = 4 ·
105·e−6eV/kT cm3/s, g
k311
PL = 0.05,kI
PL = 1.5·10−7·e1.5eV/kT ,
and APL =−2.65.
D. Comparison with the experiment
We note that for low temperatures or short annealing times,
the model extension presented gives virtually the same results
as the original model of Zographos et al. [15]. Thus, we only
discuss high temperatures and extended times. The implanta-
tion conditions for these experiments were modeled with the
native Monte Carlo implantation simulator of Sentaurus Pro-
cess, using the default parameters [20]. Sentaurus Process then
accounts for amorphization and solid-phase epitaxial regrowth
by setting the concentration of implantation-induced Is and
Vs in amorphized regions to zero. Only Is that remain behind
the amorph-crystalline boundary can therefore contribute to
the formation of dislocation loops. In Fig. 4, simulations for
1000 ◦C are compared to experiments. At this temperature,
perfect loops become relevant for the implantation conditions
of [28] and they dominate the total density of loops for times
exceeding about 4 min. Due to the stability inversion, PLs
dissolve slowlier and have a larger mean radius than FLs.
Fig. 5 provides a further example for different implantation
conditions. Fig. 6 compares our model with experiments for
a wide range of temperatures, showing good agreement in
contrast to the results obtained with the model of Zographos
et al. shown in Fig. 1.
While the preceding experimental conditions comprised the
well-studied conditions of amorphizing Si and Ge implants,
for solar cells, we are also interested in non-amorphizing B
implants. For the example of a B implant of 1·1015 cm−2at
30 keV annealed for 15 min at 900 ◦C, Fig. 7 shows simula-
tions of the as-implanted and annealed boron concentrations.
It also compares simulated and experimentally measured val-
ues for the depth dependence of the dislocation-line density
ρL= 2πrLDL. While the experimental values correspond to a
mean value for FLs and PLs, the dislocation-line densities for
both types are resolved in the simulations.
Fig. 8(a) and (b) show the time evolution of loop densities
during RTA and furnace anneals after B implantation. Our
model reasonably compares with experiment, in contrast to
the model of Zographos et al. [15], see Fig. 1.
IV. MOD EL IN G TH E SATU RATION OF OSTWALD RIPENING
While the model of the preceding section for the first time
enables the simulation of dislocation loops in the parameter
regime of solar cell processes, this section is devoted to a more
subtle and weaker effect that arises for these processes. In

IEEE JOURNAL OF PHOTOVOLTAICS 5
0 1 2 3 45678
time (min)
109
1010
1011
1012
density of loops
N
L (cm-2 )
(a)
FL
PL
sum
0 1 2 3 45678
time (min)
0
5
10
15
20
25
30
35
40
mean loop radius r (nm)
(b)
FL
PL
mean
Fig. 4. Loop evolution during RTA annealing at 1000 ◦C following a Si
implant of 1·1016 cm−2at 50 keV. Experimental data from [28].
0510 15 20
time (min)
108
109
1010
1011
density of loops
N
L (cm-2 )
(a)
FL
PL
sum
0510 15 20
time (min)
0
10
20
30
40
50
mean loop radius r (nm)
(b)
FL
PL
mean
Fig. 5. Loop evolution during RTA annealing at 1000 ◦C following a Ge
implant of 2·1015 cm−2at 150 keV. Experimental data from [45].
Fig. 2 and the respective discussion, we showed that the model
of Zographos et al. predicts a linearly diverging loop radius
while experiments indicate a much weaker increase if not
reduction. As the mean loop radius enters the disloction line
density, which directly relates to the recombination activity
0 1 2 3 45678
time (min)
109
1010
1011
density of loops
N
L (cm-2 )
(a)
900 C
950 C
1000 C
1050 C
0 20 40 60 80 100 120
time (min)
108
109
1010
1011
1012
density of loops
N
L (cm-2 )
(b)
800 C
900 C
1000 C
Fig. 6. Time evolution of the areal loop density NL(sum of FL and PL
densities) during annealing. (a) Rapid thermal annealing (RTA) after an Ge
implant of 2·1015 cm−2at 150 keV, experimental data from [23]. Compare
this to Fig. 1. (b) RTA after an Si implant of 1·1015 cm−2at 50 keV,
experimental data from [26].
0 50 100 150 200 250 300 350 400
depth (nm)
1017
1018
1019
1020
1021
CB (cm-3 )
(b)
B
Bimpl
108
109
1010
1011
L (cm-2 )
FL
PL
exp
Fig. 7. The as-implanted and annealed B profile are shown on the left y-
axis. The dislocation-line densities for perfect (PL) and faulted loops (FL)
are shown on the right y-axis, together with experimental data for the mean
dislocation-line density. For this experiment a B dose of 1·1015 cm−2was
implanted at 30 keV and annealed at 900 ◦C for 15min. With this low thermal
budget, FLs can be seen to be still strongly dominant in the simulation. Only
for higher thermal budgets, as shown in Fig. 8(a), PLs start to dominate.
Experimental data from [29, p. 130].
of loops, this model artifact should reduce the quantitative
predicitive power of the model of Zographos et al.. We sketch
an idea that overcomes this artifact at the expense of an only
slightly more complicated model definition.
During annealing, Is will be exchanged among the extended
defects as well as between them and the surface or other
sources and sinks. The former processes is conservative and
leads always to an increase of the mean loop radius rL
of the ensemble. The latter process is non-conservative and

IEEE JOURNAL OF PHOTOVOLTAICS 6
0 10 20 30 40 50 60
time (min)
108
109
1010
1011
density of loops
N
L (cm-2 )
(a) FL
PL
sum
40 20 0 20 40 60
time (min)
108
109
1010
1011
1012
density of loops
N
L (cm-2 )
(b) 900 C
1000 C
Fig. 8. Time evolution of the areal loop density NLduring annealing. (a) RTA
at 950 ◦C after a B implant of 1·1015 cm−2at 30 keV, experimental data from
[24]. Compare this to Fig. 1. (b) Loop evolution during furnace annealing,
ramp-up at 10 K/min starting at 600 ◦C. B implant of 1·1016 cm−2at
30 keV. Experimental data from [29]. Negative times refer to ramp-up and
preannealing phases. Symbols depict experimental results, lines simulation
results. If not specified otherwise, data points refer to the sum of FL and PL
densities.
increases or decreases rL. Although the distinction of these two
processes has been acknowledged for a long time [25], it has
never explicitly been taken into account by process simulation
models. By explicitly accounting for these processes, we
obtain a new loop model that gives rise to different loop-loop
and loop-I reaction dynamics than the model of Zographos et
al..
A. Model equation ansatz
We make the following ansatz for the time evolution of the
mean loop radius rL
drL
dt =fnc(rL) + fc(rL),(11)
where fnc and fcare functions of rLthat correspond to
the non-conservative and conservative contributions to loop
growth. The natural choice for fnc is known to be [19]
fnc(rL) = kL
nL
(CI−C∗
I,L(rL)),(12)
where kLis a reaction rate. In the conservative regime, when
no other comparable sinks or sources but the loop ensemble
itself are present, fnc should be zero. This holds indeed true
as then CI'C∗
I,L(rL).
For the conservative contribution to loop growth fc(rL), we
make the ansatz
fc(rL) = kL
nL
(C∗
I,L(rL−σL)−C∗
I,L(rL+σL)) (13)
0 10 20 30 40 50
mean loop radius r (nm)
10-1
100
101
102
C*
L,I/C *
I and fc/(C *
IkL/nL)
C*
FL,I 850 C
fc 850 C
C*
FL,I 1050 C
fc 1050 C
Fig. 9. Comparison of the quantities C∗
I,L and fc∝(C∗
I,L(rL−σL)−
C∗
I,L(rL+σL)), see Eq. (13), which govern the non-conservative and the
conservative processes during the evolution of the loop ensemble, respectively.
Here, σL
rL= 0.4.
where σLhas the meaning of the variance of the loop-size
distribution. The choice is motivated as follows. Ostwald
ripening is due to the conservative exchange of Is from small
loops to large loops. In analogy to the ansatz of Burton
and Speights [25], who described the evolution of a loop
with radius rLas a result of the interaction with the mean I
concentration C∗
I,L(rL), Eq. (13) describes the radius evolution
of a loop with radius rL+σLas a result of the interaction
with the emitted I concentration of a smaller loop with radius
rL−σL. As the simulation cannot account for all processes
that occur between loops of the full loop-size distribution, we
assume that the exchange of Is between loops of size rL−σL
and rL+σLshould be a representative approximation for these
processes. Note that also for the choice of fnc, we neglected
the details of the loop-size distribution.
In the early stages of the annealing process, when many
small loops are present, the conservative exchange of Is (13)
will govern the loop evolution, while, at a stage at which loops
have become large, the non-conservative contributions (12)
may become more important. Our choice for fcreflects this
transition: fctakes high values for low values of rLand tends
to zero for high values of rL, i.e. in the late-stage regime.
This behavior is obvious in Fig. 9. Furthermore, fcreflects
the fact that a loop-size distribution with a large size variance
σLdisplays faster Ostwald ripening than a distribution with a
small variance.
We note that we always choose σL∝rL. As the size
distribution function of loops, normalized to the mean loop
radius rL, is time-independent for conservative Ostwald ripen-
ing [46, 25] as well as in the early stage when the conservative
exchange of Is dominates, the normalized variance σL/rLof
the distribution must be time-independent, too. In the late-
stage phase, fc'0holds and, by that, the value of σL
becomes unimportant. An additional equation for the time
evolution of σL, which corresponds to the third moment of the
loop-size distribution, is therefore unnecessary. Consequently,
the preceding definition of a two-moment model automatically
corresponds to a three-moment model.

IEEE JOURNAL OF PHOTOVOLTAICS 7
B. Summary of model equations and comparison to experi-
ment
Before, we obtained the time evolution of rLin Eq. (6)
from Eqs. (1) with RI
Land e
RI
Ldefined in Eqs. (3). Now we
use the expression for rLin Eq. (11) and the expression of
RI
Lof Eqs. (3) to obtain an expression for e
RI
L. Adopting the
notation of [15], we set the constant kL=πkI
LDIand obtain
RI
L=kI
L2π2rLDI(CI−C∗
I,L(rL))DL,(14a)
e
RI
L=−kI
L2πDI
rLnL
(C∗
I,L(rL−σL)−C∗
I,L(rL+σL))DL.(14b)
This is to be compared with the model equations of [15] given
in Eqs. (3). The difference between the model we propose and
the one of [15] is that instead of the term (C∗
I,L(rL−σL)−
C∗
I,L(rL+σL)) in the equation for e
RI
L, [15] employ C∗
I,L(rL).
We have already compared these terms in Fig. 9.
We compare the results of our model (14) with the results
of Zographos’ model in Fig. 2. Our model clearly allows to
identify the early and late stages of Ostwald ripening via the
observation of two time-scales in the reaction dynamics: In the
early stage, when the conservative exchange of Is dominates,
the loop radius increases strongly ∝√t[25], while for
long times where the non-conservative exchange dominates,
it increases only weakly or even saturates. Zographos’ model,
by contrast, leads to a strong loop-radius growth also for long
times when considering high temperatures.
For the calculation of Fig. 2 we used σFL/rFL = 0.4with
all other parameters as in the preceding section. Only for k311
FL
the original value of [16] was used.
V. CONCLUSION
We presented a dislocation-loop model for the concurrent
growth of faulted and perfect dislocation loops and sketched
a model that reproduces the saturation of Ostwald ripening.
In comparison to established work, both approaches improve
the agreement with experiments particularly for high thermal
budgets. The model has already proven its viability for solar-
cell processing conditions [7, 8].
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90054-3
F. Alexander Wolf obtained a M. Sc. in Physics in 2011 from the University
of Augsburg. At the time of this work, he was a PhD student at Bosch
Corporate Research, Gerlingen, Germany. He now is with the Ludwigs-
Maximilians-Universit¨
at (LMU), Munich.
Alberto Martinez-Limia obtained his diploma in Nuclear Physics in 1994
from the Institute of Nuclear Sciences and Technology of Havana and his
PhD in 2002 from the Institute of Theoretical Chemistry of the University
of Erlangen-Nuremberg. Later he worked in several academic and scientific
organizations: the University of South Carolina, the University of Chemnitz
and the Fraunhofer Institute IISB in Erlangen. He investigated topics related
to material modeling and process simulation. In October 2008 he joined the
Robert Bosch GmbH as a Research and Development scientist.
Daniel Stichtenoth studied physics at the University of G¨
ottingen, Germany
and Uppsala, Sweden. He received his diploma degree in physics in 2005 and
his Dr. rer nat in 2008 from the University of G¨
ottingen, Germany. Since 2009
he is with BOSCH Solar Energy AG, Arnstadt, Germany and works in the
field of silicon solar cell development. His main focus is on the interaction
of silicon with high-efficiency silicon solar cell processes.
Peter Pichler obtained the Dip.-Ing. degree in Electrical Engineering in
1982 and the Dr. techn. degree 1985 both from the Technical University
of Vienna. Since 1986 he has been Group Manager at Fraunhofer IISB,
responsible now for the doping and device simulation activities. In 2004
he obtained the venia legendis from the University of Erlangen-Nuremberg.
Dr. Pichler contributed to various European projects. coordinated the EC
projects RAPID, FRENDTECH, and ATOMICS on diffusion and activation
phenomena in silicon and silicon-based materials, and coordinates the ICT
Project ATEMOX on the modeling of leakage currents and technologies for
low-leakage ultrashallow junctions. He is the author or coauthor of some
120 publications in international journals and conference proceedings, and
author of the book “Intrinsic Point Defects, Impurities, and Their Diffusion
in Silicon” published by Springer Wien-New York.