Content uploaded by F. Alexander Wolf

Author content

All content in this area was uploaded by F. Alexander Wolf on Aug 29, 2014

Content may be subject to copyright.

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

6ln(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

4GbFLΩ

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].

REFERENCES

[1] B. J. Pawlak, T. Janssens, S. Singh, I. Kuzma-Filipek,

J. Robbelein, N. E. Posthuma, J. Poortmans, F. Cristiano,

and E. M. Bazizi, “Studies of implanted boron emitters

for solar cell applications,” Progress in Photovoltaics:

Research and Applications, vol. 20, no. 1, pp. 106–110,

2012. URL: http://dx.doi.org/10.1002/pip.1106

[2] M. G. Kang, J.-H. Lee, H. Boo, S. J. Tark,

H. C. Hwang, W. J. Hwang, H. O. Kang, and

D. Kim, “Effects of annealing on ion-implanted Si for

interdigitated back contact solar cell,” Current Applied

Physics, vol. 12, no. 6, pp. 1615–1618, 2012. URL:

http://dx.doi.org/10.1016/j.cap.2012.05.035

[3] J. Benick, R. M¨

uller, N. Bateman, M. Hermle, and

S. Glunz, “Fully implanted n-type PERT solar cells,” EU

PVSEC Proceedings, vol. 27, pp. 676–679, 2012. URL:

http://dx.doi.org/10.4229/27thEUPVSEC2012-2BO.7.5

[4] W. Ho, Y.-H. Huang, W.-W. Hsu, Y.-Y. Chen, and

C. Liu, “Ion implanted boron emitter n-silicon solar

cells with wet oxide passivation,” in Photovoltaic

Specialists Conference (PVSC), 2011 37th IEEE, june

2011, pp. 001 058 –001 060. URL: http://dx.doi.org/10.

1109/PVSC.2011.6186134

[5] A. Florakis, T. Janssens, E. Rosseel, B. Douhard,

J. Delmotte, E. Cornagliotti, J. Poortmans, and

W. Vandervorst, “Simulation of the anneal of

ion implanted boron emitters and the impact on

the saturation current density,” Energy Procedia,

vol. 27, no. 0, pp. 240–246, 2012. URL:

http://dx.doi.org/10.1016/j.egypro.2012.07.058

[6] K. R. C. Mok, R. C. G. Naber, and L. K. Nanver,

“Insights to emitter saturation current densities of boron

implanted samples based on defects simulations,” AIP

Conference Proceedings, vol. 1496, no. 1, pp. 245–248,

2012. URL: http://dx.doi.org/10.1063/1.4766534

[7] J. Kr¨

ugener, F. A. Wolf, R. Peibst, F. Kiefer,

C. Sch¨

ollhorn, A. Grohe, R. Brendel, and H. J. Os-

ten, “Correlation of dislocation line densities and emit-

ter saturation current densities of ion implanted boron

emitters,” in 23rd Photovoltaic Science and Engineering

Conference (PVSEC), 2013, pp. 1–0–31.

[8] J. Kr¨

ugener, R. Peibst, F. A. Wolf, E. Bugiel, T. Ohrdes,

F. Kiefer, C. Sch¨

ollhorn, A. Grohe, R. Brendel, and

H. J. Osten, “Electrical and structural analysis of crystal

defects after high-temperature rapid thermal annealing of

highly boron ion-implanted emitters,” to be submitted to

IEEE Journal of Photovoltaics, 2014.

[9] A. H. Gencer and S. T. Dunham, “A predictive model for

transient enhanced diffusion based on evolution of {311}

defects,” Journal of Applied Physics, vol. 81, no. 2, pp.

631–636, 1997. URL: http://dx.doi.org/10.1063/1.364204

[10] N. E. B. Cowern, G. Mannino, P. A. Stolk,

F. Roozeboom, H. G. A. Huizing, J. G. M. van

Berkum, F. Cristiano, A. Claverie, and M. Jara´

ız,

“Energetics of self-interstitial clusters in Si,” Phys.

Rev. Lett., vol. 82, pp. 4460–4463, May 1999. URL:

http://dx.doi.org/10.1103/PhysRevLett.82.4460

[11] M. Law and K. Jones, “A new model for {311}defects

based on in situ measurements,” in Electron Devices

Meeting (IEDM), Technical Digest. International, 2000,

pp. 511–514. URL: http://dx.doi.org/10.1109/IEDM.

2000.904367

[12] D. Stiebel, P. Pichler, and N. E. B. Cowern, “A reduced

approach for modeling the inﬂuence of nanoclusters and

{113}defects on transient enhanced diffusion,” Applied

Physics Letters, vol. 79, no. 16, pp. 2654–2656, 2001.

URL: http://dx.doi.org/10.1063/1.1406147

[13] I. Avci, M. E. Law, E. Kuryliw, A. F. Saavedra, and

K. S. Jones, “Modeling extended defect ({311}and

dislocation) nucleation and evolution in silicon,” Journal

of Applied Physics, vol. 95, no. 5, pp. 2452–2460, 2004.

URL: http://dx.doi.org/10.1063/1.1645644

[14] C. Zechner, N. Zographos, D. Matveev, and A. Erlebach,

“Accurate and efﬁcient TCAD model for the formation

and dissolution of small interstitial clusters and {311}

IEEE JOURNAL OF PHOTOVOLTAICS 8

defects in silicon,” Materials Science and Engineering:

B, vol. 124-125, no. 0, pp. 401–403, 2005. URL:

http://dx.doi.org/10.1016/j.mseb.2005.08.010

[15] N. Zographos, C. Zechner, and I. Avci, “Efﬁcient

TCAD model for the evolution of interstitial clusters,

{311}defects, and dislocation loops in silicon,” MRS

Proceedings, vol. 994, p. 0994 F10 01, April 2007.

URL: http://dx.doi.org/10.1557/PROC-0994-F10-01

[16] SPAC, Advanced Calibration for Sentaurus Process User

Guide, Version G-2012.06, Synopsys, Inc., Mountain

View, CA, 2012.

[17] C. Donolato, “Modeling the effect of dislocations on the

minority carrier diffusion length of a semiconductor,”

Journal of Applied Physics, vol. 84, no. 5, pp. 2656–

2664, 1998. URL: http://dx.doi.org/10.1063/1.368378

[18] V. Kveder, M. Kittler, and W. Schr¨

oter, “Recombination

activity of contaminated dislocations in silicon: A

model describing electron-beam-induced current contrast

behavior,” Phys. Rev. B, vol. 63, p. 115208, Mar 2001.

URL: http://dx.doi.org/10.1103/PhysRevB.63.115208

[19] R. Y. S. Huang and R. W. Dutton, “Experimental

investigation and modeling of the role of extended

defects during thermal oxidation,” Journal of Applied

Physics, vol. 74, no. 9, pp. 5821–5827, 1993. URL:

http://dx.doi.org/10.1063/1.355306

[20] SP, Sentaurus Process User Guide, Version G-2012.06,

Synopsys, Inc., Mountain View, CA, 2012.

[21] A. Claverie, B. Colombeau, B. de Mauduit, C. Bonafos,

X. Hebras, G. Ben Assayag, and F. Cristiano, “Extended

defects in shallow implants,” Applied Physics A, vol. 76,

pp. 1025–1033, 2003. URL: http://dx.doi.org/10.1007/

s00339-002-1944-0

[22] F. A. Wolf, “Modeling of annealing processes for ion-

implanted single-crystalline silicon solar cells,” Ph.D.

dissertation, University of Erlangen-Nuremberg, 2014.

[23] C. Bonafos, D. Mathiot, and A. Claverie, “Ostwald

ripening of end-of-range defects in silicon,” Journal of

Applied Physics, vol. 83, no. 6, pp. 3008–3017, 1998.

URL: http://dx.doi.org/10.1063/1.367056

[24] M. Milosavljevi´

c, M. Lourenco, G. Shao, R. Gwilliam,

and K. Homewood, “Optimising dislocation-engineered

silicon light-emitting diodes,” Applied Physics B,

vol. 83, pp. 289–294, 2006. URL: http://dx.doi.org/10.

1007/s00340-006-2149-6

[25] B. Burton and M. V. Speight, “The coarsening and

annihilation kinetics of dislocation loop,” Philosophical

Magazine A, vol. 53, no. 3, pp. 385–402, 1986. URL:

http://dx.doi.org/10.1080/01418618608242839

[26] J. Liu, M. Law, and K. Jones, “Evolution of dislocation

loops in silicon in an inert ambient I,” Solid-State

Electronics, vol. 38, no. 7, pp. 1305–1312, 1995. URL:

http://dx.doi.org/10.1016/0038-1101(94)00257-G

[27] F. Cristiano, J. Grisolia, B. Colombeau, M. Omri,

B. de Mauduit, A. Claverie, L. F. Giles, and N. E. B.

Cowern, “Formation energies and relative stability of

perfect and faulted dislocation loops in silicon,” Journal

of Applied Physics, vol. 87, no. 12, pp. 8420–8428,

2000. URL: http://dx.doi.org/10.1063/1.373557

[28] G. Z. Pan, K. N. Tu, and A. Prussin, “Size-distribution

and annealing behavior of end-of-range dislocation

loops in silicon-implanted silicon,” Journal of Applied

Physics, vol. 81, no. 1, pp. 78–84, 1997. URL:

http://dx.doi.org/10.1063/1.364099

[29] D. J. Stowe, “An investigation of efﬁcient room

temperature luminescence from silicon which contains

dislocations,” Ph.D. dissertation, University of Oxford,

2006. URL: http://ora.ox.ac.uk/resolve/info:fedora/uuid:

9ee073b7-9e3c-4637-9ce1-62e9e4ade69d/THESIS01

[30] M. Sztucki, T. H. Metzger, I. Kegel, A. Tilke,

J. L. Rouvi`

ere, D. L¨

ubbert, J. Arthur, and J. R.

Patel, “X-ray analysis of temperature induced defect

structures in boron implanted silicon,” Journal of

Applied Physics, vol. 92, no. 7, pp. 3694–3703, 2002.

URL: http://dx.doi.org/10.1063/1.1505982

[31] K. S. Jones, S. Prussin, and E. R. Weber, “A

systematic analysis of defects in ion-implanted silicon,”

Applied Physics A: Materials Science And Processing,

vol. 45, pp. 1–34, 1988. URL: http://dx.doi.org/10.1007/

BF00618760

[32] J. J. Comer, “Electron microscope study of stackingfault

formation in boron implanted silicon,” Radiation

Effects, vol. 36, no. 1-2, pp. 57–61, 1978. URL:

http://dx.doi.org/10.1080/00337577808233171

[33] Y. L. Huang, M. Seibt, and B. Plikat, “Nonconservative

Ostwald ripening of dislocation loops in silicon,” Applied

Physics Letters, vol. 73, no. 20, pp. 2956–2958, 1998.

URL: http://dx.doi.org/10.1063/1.122642

[34] M. Milosavljevi´

c, G. Shao, M. A. Lourenco, R. M.

Gwilliam, and K. P. Homewood, “Engineering of

boron-induced dislocation loops for efﬁcient room-

temperature silicon light-emitting diodes,” Journal of

Applied Physics, vol. 97, no. 7, p. 073512, 2005. URL:

http://dx.doi.org/10.1063/1.1866492

[35] D. Hull and D. J. Bacon, Introduction to Dislocations,

5th ed. Amsterdam: Butterworth-Heinemann, 2011.

[36] G. Schoeck and W. A. Tiller, “On dislocation formation

by vacancy condensation,” Philosophical Magazine,

vol. 5, no. 49, pp. 43–63, 1960. URL: http://dx.doi.org/

10.1080/14786436008241199

[37] N. Martsinovich, “Theory of defects arising from

hydrogen in silicon and diamond,” Ph.D. dissertation,

University of Sussex, 2004. URL: http://www.lifesci.

sussex.ac.uk/research/tc/thesis

[38] F. Kroupa, “Circular edge dislocation loop,”

Cechoslovackij ﬁziceskij zurnal B, vol. 10, pp. 284–293,

1960. URL: http://dx.doi.org/10.1007/BF02033533

[39] R. LeSar, “Ambiguities in the calculation of dislocation

self energies,” physica status solidi (b), vol. 241, no. 13,

pp. 2875–2880, 2004. URL: http://dx.doi.org/10.1002/

pssb.200302054

[40] D. J. Bacon and A. G. Crocker, “The elastic

energies of symmetrical dislocation loops,” Philosophical

Magazine, vol. 12, no. 115, pp. 195–198, 1965. URL:

http://dx.doi.org/10.1080/14786436508224960

[41] W. Cai and C. R. Weinberger, “Energy of a prismatic

dislocation loop in an elastic cylinder,” Mathematics and

IEEE JOURNAL OF PHOTOVOLTAICS 9

Mechanics of Solids, vol. 14, no. 1-2, pp. 192–206, 2009.

URL: http://dx.doi.org/10.1177/1081286508092611

[42] C. D. Meekison, “A model of diffusion-controlled

annealing of dislocation loops in pre-amorphized

silicon,” Philosophical Magazine A, vol. 69, no. 2,

pp. 379–396, 1994. URL: http://dx.doi.org/10.1080/

01418619408244350

[43] H. F¨

oll, Defects in Crystals. Germany: Lecture Notes,

University of Kiel, 2012. URL: http://www.tf.uni- kiel.

de/matwis/amat/def en/

[44] D. B. Holt and B. G. Yacobi, Extended Defects in Semi-

conductors — Electronic Properties, Device Effects and

Structures, 1st ed. Cambridge: Cambridge University

Press, 2007.

[45] C. Bonafos, A. Martinez, M. Faye, C. Bergaud,

D. Mathiot, and A. Claverie, “Transient enhanced

diffusion of dopant in preamorphised Si: The role of EOR

defects,” Nuclear Instruments and Methods in Physics

Research Section B, vol. 106, no. 1-4, pp. 222–226, 1995.

URL: http://dx.doi.org/10.1016/0168-583X(95)00707-5

[46] I. Lifshitz and V. Slyozov, “The kinetics of precipitation

from supersaturated solid solutions,” Journal of Physics

and Chemistry of Solids, vol. 19, no. 1-2, pp. 35–

50, 1961. URL: http://dx.doi.org/10.1016/0022-3697(61)

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.