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Defects in amorphous silicon : dynamics and role on crystallization
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Shin, Jung Hoon
(1994)
Defects in amorphous silicon : dynamics and role on crystallization.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/TW9D-5657.
Abstract
Defects play a crucial role in determining the properties of many materials of scientific and technological interest. With ion irradiation, it is possible to controllably inject defects, and thus carefully study the dynamics of defect creation and annihilation, as well as the effects such defect injection has on materials properties and phase transformations. Amorphous silicon is a model system for the study of amorphous solids characterized as continuous random networks. In hydrogenated form, it is an important material for semiconductor devices such as solar cells and thin film transistors. It is the aim of this thesis to elucidate the dynamics of defects in an amorphous silicon matrix, and the role such defects can play on crystallization of amorphous silicon.

In the first chapter, the concept of a continuous random network that characterizes amorphous silicon is presented as an introduction to amorphous silicon. Structural relaxation, or annihilation of non-equilibrium defects in an amorphous matrix, is introduced. Also developed are the concept of the activation energy spectrum theory for structural relaxation of amorphous solids and the density of relaxation states. In the second chapter, the density of relaxation states for the structural relaxation of amorphous silicon is measured by measuring changes in electrical conductivity, using ion irradiation and thermal anneal to create and annihilate defects, respectively. A new quantitative model for defect creation and annihilation, termed the generalized activation energy spectrum theory, is developed in Chapter 3, and is found to be superior to previous models in describing defect dynamics in amorphous silicon. In Chapter 4, the effect of irradiation on the crystallization of amorphous silicon is investigated. It is found that irradiation affects crystallization even when the growth kinetics of crystal grains is unaffected, and that defects injected into amorphous matrix by irradiation probably play a role in affecting the thermodynamic quantities that control nucleation. The role of defect injection in affecting the thermodynamic quantities is investigated in Chapter 5, where we estimate the change in the free energy of amorphous silicon under the irradiation conditions of Chapter 4, using the generalized activation energy theory of Chapter 3. The experimental data and its interpretation are consistent with predictions of generalized activation energy spectrum theory.
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Applied Physics
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California Institute of Technology
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Engineering and Applied Science
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Atwater, Harry Albert
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Unknown, Unknown
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1 November 1993
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Defects in Amorphous Silicon: Dynamics and Role on

Crystallization

Thesis by

Jung H. Shin

In Partial Fulfillment of the Requirements
for the Degree of

Doctor of Philosophy

California Institute of Technology

Pasadena, California

1994

(Submitted November, 1993)

Acknowledgments

First and foremost, I would like to thank my family for their loving support
and encouragement. I am here because of them.

My growth as a scientist here at Caltech would have been impossible
without the help and support of Prof. Harry A. Atwater, and I would like
to express my deepest gratitude to him. Not only did he provide invaluable
guidance, he also showed Zen-like patience and faith during the early part of
my research, when nothing seemed to be going right. I am still amazed that
I was not thrown out. Thanks.

I also would like to thank Prof. Marc A. Nicolet for allowing me to use
his implanter and other equipment in his lab. They have been invaluable in
my research. Many thanks also go to Prof. James S. Im, who not only helped
me, fresh out of college, get started on research, but also provided me with
countless midnight meals. Thanks.

I would like to express special thanks to: Jimmy Yang, who helped
me with all matters concerning computers (I still am not sure what CON-
FIG.SYS does); Peter Stolk, who provided me with his measurements of den-

sity of electronic states of amorphous silicon; to Jason Reid, who deposited

ii
diffusion barrier and silver contacts for some of the samples; to Kyong-Hee
Kim, for performing X-ray analysis of some of the samples; to Edward Ryan
and Loren Funk of High Voltage Electron Microscope/Tandem Accelerator
at Argonne National Laboratory for their expert assistance; Imran Hashim
and Ramana Murty, for their help with transmission electron microscopy;
Maggie Taylor and Ruth Brain, for proofreading this thesis; and to Carol
Garland, for her help with transmission electron microscopy, and also for
steering me toward the research area of structural relaxation.

I would also like to express my deep thanks to people who have pro-
vided me with friendship and good times while here at Caltech: Dr. Shouleh
Nikzad, Dr. Byungwoo Park, Dr. Cho Jen Tsai, Dr. Channing Ahn, Selmer

Wong, Heather Frase, Gang He, and Kirill Shcheglov. Thanks.

Abstract

Defects play a crucial role in determining the properties of many materials
of scientific and technological interest. With ion irradiation, it is possible to
controllably inject defects, and thus carefully study the dynamics of defect
creation and annihilation, as well as the effects such defect injection has
on materials properties and phase transformations. Amorphous silicon is a
model system for the study of amorphous solids characterized as continuous
random networks. In hydrogenated form, it is an important material for
semiconductor devices such as solar cells and thin film transistors. It is the
aim of this thesis to elucidate the dynamics of defects in an amorphous silicon
matrix, and the role such defects can play on crystallization of amorphous
silicon,

and the density of relaxation states. In the second chapter, the density of re-

iv
laxation states for the structural relaxation of amorphous silicon is measured
by measuring changes in electrical conductivity, using ion irradiation and
thermal anneal to create and annihilate defects, respectively. A new quan-
titative model for defect creation and annihilation, termed the generalized
activation energy spectrum theory, is developed in Chapter 3, and is found
to be superior to previous models in describing defect dynamics in amor-
phous silicon. In Chapter 4, the effect of irradiation on the crystallization
of amorphous silicon is investigated. It is found that irradiation affects crys-
tallization even when the growth kinetics of crystal grains is unaffected, and
that defects injected into amorphous matrix by irradiation probably play a
role in affecting the thermodynamic quantities that control nucleation. The
role of defect injection in affecting the thermodynamic quantities is inves-
tigated in Chapter 5, where we estimate the change in the free energy of
amorphous silicon under the irradiation conditions of Chapter 4, using the
generalized activation energy theory of Chapter 3. The experimental data
and its interpretation are consistent with predictions of generalized activation

energy spectrum theory.

List of Publications

Parts of this thesis have been, or will be published under the following titles.
1) J. H. Shin, J. S. Im, and H. A. Atwater; “Dynamics of Change in Elec-
trical Conductivity of Ion Irradiated Amorphous Silicon.”; Phase Formation
and Modification by Beam-Solid Interactions, edited by G. S. Was, D. M.
Follstaedt, and L. E. Rehn, Mat. Res. Soc. Symp. Proc. 235, p. 21 (1992).

2) J. H. Shin and H. A. Atwater; “In situ Analysis of Irradiation-Induced
Crystal Nucleation in Amorphous Silicon: A Microscope for Thermodynamic
Processes in Nucleation.”; Nucl. Inst. Meth. B 80 973 (1993).

3) J. H. Shin and H. A. Atwater; “Ion Irradiated Amorphous Silicon: A
Model Approach to Dynamics of Defect Creation and Annihilation.”; Ther-
modynamics and Kinetics of Phase Transformation in Thin Films, edited by
M. Atzmon, A. Greer, J. Harper, and M. Libera, Mat. Res. Soc. Symp. Proc.
311.

4) J. H. Shin and H. A. Atwater; “Activation Energy Spectrum and Struc-
tural Relaxation Dynamics of Amorphous Silicon.”; Phys. Rev. B 48 5964
(1993).

5) J. H. Shin and H. A. Atwater; “Generalized Activation Energy Spec-

trum Theory: A New Approach to Modelling Structural Relaxation in Amor-

phous Solids.”; submitted to Phys. Rev. Lett.

In addition, the author also contributed to:

6) J. S. Im, J. H. Shin, and H. A. Atwater, “Suppression of Crystal
Nucleation in Amorphous Silicon Thin Films by High Energy Ion Irradiation
at Intermediate Temperatures.”; edited by H. A. Atwater, F. A. Houle, D.
H. Lowndes, Mat. Res. Soc. Symp. Proc. 235, p. 357 (1991).

7) J. S. Im, J. H. Shin, and H. A. Atwater; “Suppression of Nucleation
during Crystallization of Amorphous Thin Si Films.”; Appl. Phys. Lett. 59

2314 (1991).

Contents

Acknowledgments i
Abstract ill
List of Publications Vv

1 Defects in Amorphous Silicon and Activation Energy Spec-

trum Theory - 1
1.1 Introduction... 2... 0.0.00... ...0...0-.-2.. 1
1.2 Defects in Amorphous Silicon .................. 3
1.3 Activation Energy Spectrum Theory .............. 4
2 The Activation Energy Spectrum of Amorphous Silicon 18
2.1 Introduction... . 2... 0000000000. eee ee. 18
2.2 Electron Transport in Amorphous Semiconductors... ... . 20

vil

2.3 Experimental Procedure ..............0000004
2.4 Experimental Conductivity Measurements ...... Lea.
2.4.1 Temperature Dependent Conductivity of Relaxed Amor-
phous Silicon . 2... .

2.4.2 Variation of Room Temperature Conductivity with An-
neal Temperature... 2.0... 0.0.0.0 000
2.4.3 In Situ Measurements of Conductivity .........
2.5 Activation Energy Spectrum ...............000.
2.6 Discussion... .. . ee

2.0 Conclusion... . 0.0.0 eee

Generalized Activation Energy Spectrum Theory

3.1 Introduction... 2.2... 0.0... ee

3.2 Generalized Activation Energy Spectrum Theory .......

3.3 Defect Dynamics of Amorphous Silicon: Modelling and Ex-
periments 2... ee
3.3.1 Modelling Defect Dynamics ...............
3.3.2 Experiments... .......0.0.0.0 00000004 eae

3.4 Results and Discussion ..............-0.00408%

3.4.1 Evolution of Defect Density vs. Irradiation Dose... . 74
3.4.2 Evolution of Defect Density During Anneal. .... . . 84
3.4.3 In Situ Measurements of Conductivity ......... 89
3.5 Conclusion... 2.2... ee eee 93

Modification of Crystallization Dynamics of Amorphous Sil-

icon by Irradiation 97
4.1 Introduction... 2... . 0.00.00... 00000- 208% 97
4.2 Classical Theory of Nucleation... ............... 101

4.2.1 Formulation of the Theory ................ 101

4.2.2 Solutions to the Problem. ................ 106
4.3 Experiments... 2.2.2.0... 0000.00.00. 000-0 ec ee 114
4.4 Results and Discussion ...............0.-2-.. 117
4.5 Conclusion... 2... ee ee 134

Estimate of Change in the Free Energy of Crystallization

Using Generalized Activation Energy Theory 139
3.1 Introduction... 2... 0.0.0.0. ee eee. 139
9.2 Estimates of Change in the Free Energy ............ 140

5.2.1 Estimating the Defect Population ............ 140

5.38 Conclusion. ............., re
Conclusion
A Growth Rate and Interface Rearrangement
B Code for Simulation of Thermal Anneal

C Code for Simulation of Irradiation

148

150

155

164

List of Figures

2.1

2.2

2.3

2.4

2.9

2.6

2.7

2.8

Electronic density of states of amorphous silicon ........
Schematic of the experimental setup used for in situ conduc-
tivity measurements. .............0 00000 cee
Temperature dependence of the electrical conductivity of re-
laxeda-Si . 2. ee ee
Room temperature conductivity of a-Si as a function of the
anneal temperature... 2... ee ee ee
dn situ measurements of the conductivity during and after ir-
radiation. 2... ee eee,
In situ measurements of the conductivity at 773 K.......

Conductivity transients for in situ measurements .......

Density of the defect states of amorphous silicon. .......

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

4.1

4.2

4.3

xii

Schematic of experimental setup for the second sets of in sity

measurement of conductivity. .................. 73
Defect population vs. dose... .......0202-2--..., 75
Defect population vs. dpa ...................., 77
Cascade generated by a 190 KeV Neion............. 79
Cascade generated by a 600 KeV Xeion............. 80
Time evolution of the defect population. ............ 82

Time evolution of the low activation energy defect population 83

N(Q) after isochronal anneals .. 2... 0.0.00... 85

In situ measurement of decay transient following irradiations

at different ion fluxes... ...........000000... 91
The “cross-over” effect, shown in detail. ............ 92
Free energy of formation for silicon... ............. 111

Experimental setup for nucleation enhancement, performed at
Caltech 2... ee 116
Comparison of grain morphology of grains grown under ther-

mal and irradiation-enhanced growth conditions ........ 119

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

0.1

Grain density vs. time, performed at Caltech. ......... 120
Growth rate of nucleated grains ..............-.. 121
In situ measurements of the nucleation rate for samples for
which steady state nucleation under irradiation is observed. . . 123
In situ measurements of the nucleation rate for samples for

which irradiation is terminated before steady state nucleation

under irradiation is observed. .................. 124
Comparison of the free energies of formation .......... 128
Comparison of the calculated cluster size distribution .... . 129

Comparison of the gradient of calculated cluster size distribution 130
Schematic description of evolution of the cluster size distri-
bution function and the nucleation rate for samples for which
steady state nucleation under irradiation is not observed. . . . 132
Schematic description of evolution of the cluster size distri-
bution function and the nucleation rate for samples for which

steady state nucleation under irradiation is observed. .... . 133

Effect. of changing Qmin on steady state defect concentration

under irradiation... 2.0.0... ee ee et 143

XIV
5.2 Effect of changing the time step on steady state defect con-

centration under irradiation. Qmin =0.6eV........... 144

A.1 Schematic of activation energy for interfacial jump

List of Tables

3.1

4.1

4.2

5.1

5.2

Theories discussed in this paper, along with significant as-

sumptions associated with them................04.

Comparison of the irradiation enhanced and the thermal val-
ues of the nucleation rate and the incubation time. ......

Calculated changes in thermodynamic parameters controlling

nucleation... 2... 2... ee

Effect of changing Qmin on steady state defect concentration

under irradiation. .........0-..0 000 ee eee eee eee

Effect of changing timestep on steady state defect concentra-

tion under irradiation. Qmin = 0.6eV. ..........004.%

Chapter 1

Defects in Amorphous Silicon
and Activation Energy
Spectrum Theory

1.1 Introduction

Amorphous silicon (a-Si) is a metastable phase of silicon which undergoes
a first-order phase transition to either crystalline or liquid form at elevated
temperatures. Therefore, it is distinct from traditional “glasses,” which are
characterized as configurationally frozen liquids. It is produced most com-
monly through either low-temperature ion irradiation of crystalline silicon
(c-Si), or low temperature vapor deposition of a silicon thin film. It is also
possible to produce a thin layer of a-Si by rapid solidification following pulsed

laser melting of crystalline silicon, if the quenching rate is fast enough. In its

Chapter 1 D)

hydrogenated form, it is used extensively as the material for solar cells and
for thin film transistors used to control active matrix displays. However, this
thesis will address exclusively the pure, unhydrogenated form of amorphous
silicon.

Amorphous silicon is modelled as a continuous random network (CRN)
of silicon atoms, in which all silicon atoms have four covalently bonded near-
est neighbor atoms in a roughly tetrahedral arrangement. Consequently, the
short range order of amorphous silicon is very similar to that of crystalline
silicon. However, slight variations in bond angle and, to a much lesser de-
gree, bond length, destroy any long-range order. One result of the lack of
long range order is the presence of rings other than six membered rings in the
network, which is not allowed in crystalline silicon. The validity of the CRN
model for the structure of amorphous silicon, constructed either by ball-and-
stick method [1] or by computer simulation [2], is convincingly demonstrated
by the close agreement between the predicted and measured radial distribu-
tion functions of a-Si. On the other hand, the microcrystalline model of a-Si
(3, 4] fails to give satisfactory agreement with experiments, and is no longer

considered a viable candidate for modelling a-Si (5, 6].

Chapter 1 3

1.2 Defects in Amorphous Silicon

Such continuous random networks, however, represent an “ideal” a-Si, since
they do not contain any defects. “Real” amorphous silicon produced experi-
mentally, however, contains a very large number of defects, up to © 1 atomic
percent [7]. Experimentally produced a-Si is 1.8 +0.1% less dense than c-Si
[8], while “ideal” a-Si (continuous random network of silicon atoms without
any defect) is found to be 4% more dense than c-Si {2]. Prolonged anneals
at, elevated temperatures can be employed to reduce the defect concentration
of a-Si, but only up to a factor of about 2-5 [9, 10]. This annihilation of
non-equilibrium defects, while still maintaining the amorphous phase, is de-
noted by the term structural relaxation. With ion irradiation, it is possible
to controllably introduce non-equilibrium defects into a-Si. This reverses the
effects of a thermal anneal, and is known as structural unrelaxation. The
defect concentration does not increase indefinitely, however, and saturates at
~ 1 atomic percent at room temperature. The exact nature of these defects
in a-Si is still somewhat unclear. However, the general consensus is that
they are point defects. Electron paramagnetic resonance measurements have

revealed the presence of dangling bonds in a-Si [11]. Méssbauer spectroscopy

Chapter 1 4

and positron annihilation showed the presence of vacancies in a-Si [12, 13].
Differential scanning calorimetry (DSC) of irradiated a-Si and c-Si provided
further proof that structural relaxation in a-Si is associated with annihila-
tion of point defects [7]. The point defects in a-Si are topological defects in
CRN, and are analogous to point defects in c-Si. They are distinct from the
macroscopic voids that are sometimes present in vapor deposited a-Si.
Given the high concentration of defects in a-Si, it is not surprising that
many properties of a-Si (e. g. , refractive index [14], spin density [11], viscosity
[15], enthalpy [16, 17], vibrational properties [18, 19], diffusivity [9, 10, 20],
shear viscosity [21], and electrical conductivity [22, 23]) are dependent upon
the extent of structural relaxation, or the defect concentration. The effects
can be quite dramatic. For example, irradiating well-relaxed a-Si to defect
saturation at room temperature nearly doubles the free energy difference
between a-Si and c-Si, and increases the electrical conductivity of a-Si by

nearly 4 orders of magnitude.
1.3 Activation Energy Spectrum Theory

As is true with other amorphous solids, the structural relaxation kinetics (de-

fect annihilation kinetics) of a-Si is not characterized by a single activation

Chapter 1 5

energy. In such cases, the activation energy spectrum theory is often invoked.
The activation energy spectrum theory was originally developed in the study
of structural relaxation in metallic glasses, but is applicable to amorphous
solids in general. In studies of metallic glasses, it was shown to be successful
in describing reversible and irreversible property changes, and the “In(t)” be-
havior that characterizes many relaxation processes under isothermal anneal
[24, 25]. It also has been invoked in studies of structural relaxation in a-Si
(7, 26, 27].

As treated by Narayanaswami [28] and Primak [29], and further developed
by Gibbs et al. [30], the main assumption of the activation energy spectrum
theory is that the activation energies of processes (i.e., any thermally acti-
vated transport or rearrangement of single atoms or groups of atoms) that
contribute to structural relaxation are continuously distributed in activation —
energy. Therefore, the defect population is described by N (Q), such that
N(Q)dQ is the density of defects with activation energy between Q and
Q + dQ.

The density of defects, N(Q), however, can vary from one sample to an-
other, depending on its history. Therefore, it is useful to develop another

concept, the density of defect states, D(Q), such that D(Q)dQ is the density

Chapter 1 6

of defect states that have activation energy between Q and Q + dQ. Physi-
cally, D(Q) is the theoretical upper limit of N(Q) that is intrinsic to a given
amorphous solid, such as could be obtained by fully unrelaxinga given sam-
ple at 0 K. As defects are being introduced, the actual defect density N (Q)
would evolve toward the density of defect states, D(Q).

The density of defect states D(Q) differs from other commonly used den-
sity of states functions, such as the electronic density of states in a solid.
It is not uniquely determined by a fundamental equation, and thus would
be very difficult to derive from an ab initio calculation. However, the ex-
istence of such an upper limit of defects as a function of activation energy
may still be understood as follows. Defects cannot accumulate indefinitely
in an amorphous solid while still maintaining the structural integrity of the
amorphous solid. For example, a continuous random network cannot con-
tain more defects than the percolation threshold for such a network and still
remain continuous. Therefore, there must exist an upper limit to the total
defect concentration, most likely much lower than the percolation threshold.
Furthermore, in a given amorphous solid, there probably exists a continuum
of possible defect configurations, each particular configuration being associ-

ated with a particular activation energy. We then may define the density of

Chapter 1 7

defect states, D(Q) by the relationship

D(Q) = p(Q) x Maximum total defect density, (1.1)

where p(Q) is the probability function, intrinsic to the material under study,
that a random defect will have activation energy between Q and Q + dQ.
In the case of metallic glasses, p(Q) may be the normalized distribution of
the free volume around an atom. In the case of amorphous semiconductors,
it may be the normalized degree of bond angle distortions around a point
defect.

In the case of the amorphous silicon, there are many experimental ev-
idences which support the idea that such a density of defect states exists,
and that it is intrinsic to amorphous silicon. For example, the maximum
defect density obtainable at room temperature seems to be of the order of 1
atomic percent, even when measured with many different methods [7, 10, 26].
Futhermore, experimental data show that the density of defect states does
not vary arbitrarily from sample to sample, but has a consistent form (7, 31].

Finally, we define the characteristic annealing function, f(Q, T(t), t), such

that

Chapter 1 8

N(Q,t) = f(Q,T(t),t)N(Q, 0) (1.2)

where T(t) is the thermal history of the sample, and ¢ is the time. It may
be possible that rather than annihilating, defects recombine to form other
defects, such that f(Q, T(t), ¢) is a function of initial defect distribution as
well. In this thesis, we shall neglect that possibility.

The characteristic annealing function f(Q,T(t),t) is determined by the
dynamics of structural relaxation. In order to accurately model the dynamics
of structural relaxation, the possibility of interaction between defects with
different activation energies must be taken into account. However, in the
activation energy spectrum theory that has been used so far, a simplifying
assumption is made that the defects with different activation energies are
independent of each other. That is, activation of relaxation processes within
an energy range between Q and Q+dQ is assumed not to affect the relaxation
processes in other activation energy ranges. With this simplification, we can

write down the dynamics of structural relaxation as

Chapter 1 9

where v, is the reaction constant, and n is the order of the reaction.
In the case of an isothermal anneal, f(Q,7,t) can easily be written down.

If structural relaxation occurs through a first-order reaction, then

f(Q,T,t) = exp |-vtexp Fal . (1.4)

If relaxation occurs through a second-order reaction, then

f(Q,T,t) = 1+vtexp[—Q/kT]’

(1.5)

For both Eq. (1.4) and Eq. (1.5), v is expected to be of the order of the
Debye frequency, or 10'* — 10'4sec~!. For a first order reaction, vy = v,, and
for second order reaction, vy = v,N(Q,0).

The density of defects, and thus the dynamics of structural relaxation,
usually cannot be measured directly. However, it can be inferred by measur-
ing a change of some property P which changes as the material relaxes. We

can write the property change AP as follows:

AP = [dp= [Mig = [ccqune(Qyan, (1.6)

Chapter 1 10

where dp(Q) is the change in property being measured due to activation of
relaxation processes with activation energy between Q and Q+dQ. N7“(Q)
is the density of defects that have contributed to relaxation at time t,, which
is simply [(1— f(Q, T, t))N(Q, 0)]. The term C(Q) is the function that relates
property change dp(Q) to the defect density N7*!(Q).

For reasonably high values of v (v > 10° sec~"), f(Q,7,t) changes
rapidly from 0.05 to 0.95 within a fairly narrow range of activation ener-
gies, of the order of a few kT’. Thus, it has been customary to approximate

f(Q,T,t) during an isothermal anneal by a step function such that

f(Q,T,t) = 0, Q< Qo

where

Qo = kT ln(vt). (1.8)

In other words, we make the approximation that all of the defects with ac-

tivation energy Q < Qo have been annihilated, while none of the defects

Chapter 1 il

with activation energy Q > Qe have been annihilated. This approximation
is valid for both first and second order reaction kinetics. Indeed, the main
conclusions of Gibbs’ activation energy spectrum theory do not depend sen-
sitively on the order of the reaction. This also implies, however, that it is not
possible to distinguish between different orders of reaction using this method.

Since Qo depends linearly on the anneal temperature but only logarith-
mically on the anneal time, during an isothermal anneal of a typical experi-
mental duration, it will sweep only a narrow activation energy range after a
brief transient. Therefore, if both N(Q) and C(Q) are broad with no sharp
features of the order of the energy range swept by Qe, we can greatly simplify

Kq. (1.6) as follows:

AP = / °° C(Q)Nte!(Q)dQ = TNT In(vt) « In(t), (1.9)

where C and N are the average values of C(Q) and N(Q), respectively, within
the activation energy range swept by Qo during an isothermal anneal (after
a brief transient). Thus, after a brief transient, a property of an amorphous
material will change logarithmically with time during an isothermal anneal

if there exists a density of relaxation states, and if the approximations used

Chapter 1

in deriving Eq. (1.9) are valid.

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Chapter 1 14

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[13] S. Roorda, R. A. Hakvoort, A. van Veen, P. A. Stolk, and F. W. Saris, in
Phase Formation and Modification by Beam-Solid Interactions, edited
by G. S. Was, D. M. Follstaedt, and L. E. Rehn, Mat. Res. Soc. Symp.

Proc. 235, 39 (1992).

[14] C. N. Waddell, W. G. Spitzer, J. E. Frederickson, G. K. Hubler, and T.

A. Kennedy, J. Appl. Phys. 55, 4361 (1984).

Chapter 1 15

[15] A. Witvrouw and F. Spaepen, in Kinetics of Phase Transformations,
edited by M. M. Thompson, M. J. Aziz, and G. B. Stephenson, Mat.

Res. Soc. Symp. Proc. 205, 21 (1991).

[16] S. Roorda, S$. Doorn, W. C. Sinke, P. M. L. O. Scholte, and E. van

Loenen, Phys. Rev. Lett. 62, 1880 (1989).

[17] E. P. Donovan, F. Spaepen, J. M. Poate, and D. C. Jacobson, Appl.

Phys. Lett. 55, 1516 (1989).

[18] J.S. Lannin, L. J. Pilione, S. T. Kshirsagar, R. Messier, and R. C. Ross,

Phys. Rev. B 26, 3506 (1982).

[19] W. C. Sinke, T. Wabarisako, M. Miyao, T. Tokuyama, S. Roorda, and

F. W. Saris, J. Non-Cryst. Solids, 99, 308 (1988).

[20] B. Park, F. Spaepen, J. M. Poate, and D. C. Jacobson, J. Appl. Phys.

69, 6430 (1991).
[21] C. A. Volkert, J. Appl. Phys. 70, 3521 (1991).

[22] W. Beyer and J. Stuke, In Amorphous and Liquid Semiconductors, Pro-

ceedings of the Fifth International Conference on Amorphous and Liquid

Chapter 1 16

Semiconductors, edited by J. Stuke and W. Brenig (Taylor & Francis,

London, 1974) p. 251.

[23] J. H. Shin, J. S. Im, and H. A. Atwater, in Phase Formation and Mod-
ification by Beam-Solid Interactions, edited by G. S. Was, D. M. Foll-

staedt, and L. E. Rehn, Mat. Res. Soc. Symp. Proc. 235, 21 (1992).

[24] A. L. Greer, in Rapidly Solidified Alloys: Processes, Structures and Prop-
erties, edited by H. H. Liebermann (Marcel Dekker, New York 1993) Ch.

10.
[25] K. Bothe and H. Neuhauser, Script. Metall. 16, 1053 (1982).

[26] P.A. Stolk, L. Calcagnile, 8. Roorda, W. C. Sinke, A. J. M. Berntsen,

Appl. Phys. Lett. 60 1688 (1992).
[27] C. A. Volkert, private communication.
[28] O.S. Narayanaswamy, J. Am. Ceram. Soc. 61 146 (1978).
[29] W. Primak, Phys. Rev. 100, 1677 (1955).

[30] M. R. J. Gibbs, J. E. Evetts, and T.A. Leake, J. Mater. Sci. 18, 278

(1983).

Chapter 1 17

[31] L. de Wit, S. Roorda, W. C. Sinke, F. W. Saris, A. J. M. Berntsen, and
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Proc. 205, 3 (1991).

Chapter 2

The Activation Energy
Spectrum of Amorphous
Silicon

2.1 Introduction

Although it is well established that the kinetics of structural relaxation of
amorphous silicon (a-Si) is described by a spectrum of activation energies, so
far no systematic investigation of the activation energy spectrum has been
made. For instance, Roorda et al. [1] have shown, using differential isother-
mal calorimetry measurements, that defects with many different activation
energies contribute to relaxation. However, due to the lack of an explicit
understanding of the enthalpy released upon defect annihilation in a-Si, no’

quantitative analysis of the activation energy spectrum has been attempted.

“Ss

Chapter 2 19

On the other hand, theories for electrical conductivity of a-Si are well
developed and supported by experimental data. It is also well established
that the electrical conductivity of a-Si is sensitively dependent on the degree
of relaxation [3, 4]. Furthermore, electrical conductivity also has the exper-
imental advantages of accuracy and in-situ capability, allowing us to probe
time and temperature regimes that have not been accessible to date. There-
fore, by combining existing models of electrical conductivity of a-Si with the
activation energy spectrum theory discussed in Chapter 1, we can for the
first time quantitatively analyze the activation energy spectrum for struc-
tural relaxation in a-Si. The results of the analysis enables us to comment
on the relationship between structural relaxation and the rate of solid phase
epitaxy (SPE) of c-Si into a-Si, and on irradiation-enhanced nucleation of
crystal silicon in an a-Si matrix.

Using electrical conductivity measurements, we only measure the struc-
tural relaxation kinetics of defects that are electrically active. Recently, how-
ever, Coffa and Poate [5] have shown that passivating ion-irradiated (i.e.,
unrelaxed) a-Si with hydrogen has the same effect on the diffusion of transi-
tion metals in a-Si as structural relaxation by annealing does. Hydrogen is

well known to passivate electrically active defects in a-Si, and decrease the

Chapter 2 20

electrical conductivity of a-Si by reducing the number of localized electronic
states in the energy gap [6]. The fact that passivating such electrically active
defects with hydrogen has the same effect on diffusion of transition metals as
structural relaxation by annealing suggests that it is reasonable to suppose
that the electrically active defects are representative of the defect population

as a whole.

2.2 Electron Transport in Amorphous Semi-
conductors |

Optical and electrical measurements have shown the Mott [7] theory of elec-
tron conduction in amorphous solids to be successful in explaining the ob-
served electrical properties of pure (i.e., unhydrogenated), undoped, and
tetrahedrally bonded amorphous semiconductors, such as a-Ge or a-Si [8,
9, 10}. Within this theory, the concepts of the electronic density of states
g(E) and the energy band gap are considered to be still conceptually valid
and useful. The lack of long range order is thought to create a large density
of localized states near the band edges, and electron mobilities in these lo-

calized band edge states are thought to be about three orders of magnitude

smaller than the mobility in the extended states. Furthermore, other de-

Chapter 2 21

fects inherent in amorphous solids are thought to create a band of localized
electronic states near the middle of the energy gap which pin the Fermi level.

This picture is supported by experimental data. A recently estimated
electronic density of states of amorphous silicon is shown in F ig. 2.1 [11].
The electronic density of states is estimated from optical measurements of
the absorption coefficient a(hv), shown in Fig. 2.la. Measurements in the
photon energy range of 1.2 to 2.5 eV represent absolute measurements, while
measurements in the photon energy range 0.6 to 1.0 eV were obtained by
scaling absorption data from Photothermal Deflection Spectroscopy (PDS).
Photothermal Deflection Spectroscopy measures the absorption coefficient by
measuring the gradient of the index of refraction modulated by the gradient
in the temperature of the medium, typically heated by a laser. More detailed
description of PDS can be found elsewhere [12]. It must be noted, however,
that the measured values for the density of electronic states is dependent on
assumptions used in deriving them. Thus, Fig. 2.1b gives a reliable indication
of the density of electronic states rather than a final determination. How-
ever, it is evident that the density of localized electron states near mid-gap
decreases upon relaxation, which is consistent with our assertion that elec-

trically active defects are representative of the defect population as a whole.

Chapter 2 22

The apparent shift in the Fermi level upon relaxation of a-Si is an artifact
of fixing the valence band edge to be zero, and the widening of the energy
band gap upon relaxation due to annihilation of localized states near the
band edges.

Conduction, then, is characterized by three different mechanisms, de-
pendent upon the temperature. Conduction will occur through (a) phonon-
assisted tunneling (hopping) of electrons near the Fermi level; (b) hopping
conduction of electrons in the localized band tails; or (c) extended state
(non-localized) conduction. At sufficiently high temperatures, enough elec-
trons are excited to extended states to dominate the conduction process. In

this case, the conductivity is given by

Text = of exp|—(E, — Es)/kT], (2.1)

where EF, and E; are the conduction band edge and the Fermi level, respec-
tively. At moderate temperatures, conductivity is thought to be dominated
by localized state transport in the band tail states, and is treated in a way

similar to the extended state conduction. That is,

Chapter 2 23

/.
‘U7 © Relaxed a-Si.
10°71. ( -
a) DOS

i } { { mI
2.0 —1.5 —-1.0 —-0.5 0.0 0.5
E (eV)
10° t { q ‘
(b) absorption

10*}- 4
i a
E ae
oO
"107 a 4

yY -—as—Implanted a—Si
/ —~— -
10° / colaxed a-Si 2
L { { { {
0.0 0.5 1.0 1.5 2.0 2.5
hy (eV)

Figure 2.1: (a) Experimentally derived electronic density of states of amor-
phous silicon. It is found by fitting the optical data shown in (b). A Gaussian
localized states distribution is assumed, and the Fermi level is assumed to be
fixed in the center of the Gaussian peak.

Chapter 2 24

Crit = of exp[—(E. — E's + W)/kT}, (2.2)

where W is the activation energy for hopping. Like 07, o% is difficult to
estimate. However, it is expected to be at least three order of magnitude
smaller than 0%, since the mobility in localized states near the band edges
is smaller by this factor. This supposed conduction due to hopping in the
localized band tail states, however, is not observed for a-Si or a-Ge [3, 9, 10].

At low temperatures, conduction is expected to be dominated by the
hopping (phonon-assisted tunneling) of electrons near the Fermi level from

one localized state to another. The probability that an electron can tunnel

from one localized state to another is given by

Ptunnel = wexp(—2aR — W/kT) (2.3)

where w is the attempt frequency of localized electrons, a-! is the decay
constant of the localized electron wavefunction, R is the jump distance, and
W is the energy difference between the two localized states. The conduction
will be dominated by a combination of the tunnel distance, R, and the energy

difference, W, which minimizes the exponent in Eq. (2.3).

Chapter 2 25

If the electronic density of states near the Fermi level is g(Es), the total
number of electronic states with a energy difference W within distance R

from a localized electronic state near the Fermi level is given by

> Rg Ey)W. (2.4)

In order for a tunneling to be possible, the number of accessible state must

be at least one, or

W = In Feg(Ey) (2.5)

Substituting Eq. (2.5) in Eq. (2.3), we find

Ptunnel = wexp(—2akR ~ 3/40 R°g(E,)kT). (2.6)

And the value of R which minimizes the exponent in Kq. (2.6) is

us scanner . 2-7)

Using above equations, and Einstein’s relation B= eD/kT, with D =

Chapter 2 26

(1/6) Ptunner R?, Mott derived an expression for the hopping conduction. It is

given by
Chop = oO exp[—A/T™4]. (2.8)
where
2 1/2
n _ © |g(Es)
%o = ae [ee — e8)
oe 1/4
A = 21 |=] , (2.10)

This exp[—A/T'/4] dependence of electrical conductivity on temperature
is widely accepted and observed for amorphous semiconductors. We note,
however, that the exact forms of o, and A are still somewhat controversial,
and deviations from the simplifying assumptions used in obtaining Eqs. (2.10)
and (2.10) will lead to slightly different results. We will assume Eqs. (2.10)
and (2.10) to be accurate. However, it will be shown later in this chapter
that for our purposes, it is sufficient to have the prefactor a” to be only some

weak function of g(/;), and the exponential factor A to be proportional to

gE).

Chapter 2 27

The importance of the electron transport in a-Si via hopping is twofold.
First, conductivity is overwhelmingly dominated by those electronic states
which lie very close to the Fermi level, of the order of few kT. This is true even
though the most favorable energy difference for hopping, W, may be of the
order of few tenths of an electron volt [8]. However, as we shall see later, the
activation energies necessary to anneal out these electronically active defects
near the Fermi level can vary by nearly 2 eV. This shows that there is very
little correlation between the electronic energy of an electrically active defect
and the activation energy necessary for its annihilation, and further supports
our assumption that electrically active defects are representative of the defect
population as a whole. Second, because the transport occurs via tunneling
of electrons from one localized electronic state to another, the conductivity
is fully controlled by localized states (presumably created by defects in a-Si),
and not by some kind of resonant scattering of charge carriers. Therefore, the
change in hopping conductivity accurately reflects the change in the localized

electronic states near the Fermi level, and thus the defects which create them.

Chapter 2 28

2.3. Experimental Procedure

Pure amorphous silicon films 450 nm thick were grown on thermally-oxidized
Si substrates by ultrahigh vacuum electron-beam deposition. The oxide layer
was 1.1 wm thick. Prior to loading into the deposition system, substrates were
cleaned by immersion in a 5:1:1 solution of HzO : H2O2 : NH4OH at 80°C,
followed by a rinse in ultrahigh purity deionized H,O. Prior to the deposi-
tion, the substrates were sputter cleaned by 100 eV Ar-ion bombardment,
and the silicon evaporation source material was outgassed. The system base
pressure was in the 107!° Torr range, but the pressure rose to 5 x 107° Torr
during the deposition. The substrate temperature during the deposition was
= 400 K, and the deposition rate was 2 Asec~}. Immediately following the
deposition, and prior to exposure to air, the films were heated in ultra high
vacuum at 620 K for 1h to densify the a-Si film and to remove macroscopic
voids which otherwise might lead to substantial trapping of oxygen and wa-
ter vapor [13]. Following deposition, a-Si resistors were defined as mesas
by photolithography and wet chemical etching. The photolithographic steps
necessary for defining resistors and contact formation required samples to

be annealed at 473 K. Aluminum contacts were evaporated for samples sub-

Chapter 2 29

ject to temperatures less than 573 K during the conductivity measurements.
Samples subject to temperatures higher than 573 K during the conductiv-
ity measurements employed sputter-deposited Ag contacts with amorphous
Ti/Si/N diffusion barriers between the Ag contacts and a-Si. Amorphous
Ti/Si/N diffusion barriers are known to be effective up to 900 K with crys-
tallization temperatures in excess of 1273 K [14]. -

Conductivity measurements were made in a two-point probe configura-
tion. The extremely high sample resistance made four-point probe measure-
ments unnecessary, since the contact resistance was negligible compared to
the sample resistance. For certainty, all contacts were verified to be Ohmic
prior to use.

All anneals were performed in a high-vacuum furnace with a base pressure
of 4 x 107° Torr. All irradiations were performed with 600 KeV Krtt+ ions
with total doses equal to or greater than one displacement per atom (dpa)
as calculated by using the TRIM [15, 16, 17] collision cascade simulation
program. A total dose of 1 dpa is known to be sufficient to completely
unrelax a-Si [1]. The beam spot size was approximately 1 cm?, and the beam
was rastered across the sample in excess of 1 KHz.

For in situ conductivity measurements at elevated temperatures, samples

Chapter 2 30

were clipped onto a resistively-heated metal hot stage. For in situ conduc-
tivity measurements at low temperatures, samples were glued onto a liquid
cooled cold stage using heat conducting paste. For high temperature mea-
surements, the sample temperature was measured by a thermocouple clipped
onto the sample surface. This step was necessary, since temperature calibra-
tion showed that at elevated temperatures, the sample temperature and the
stage temperature can differ by as much as 100 K. For low temperature ex-
periments, the sample temperature was assumed to be equal to the stage
temperature. A schematic of the experimental setup for in situ measure-
ments is shown in Fig. 2.2. No effort was made to separate the contribution
of ion beam current to the measured conductivity, since the ion beam current
was much smaller than the sample current.

Simultaneous measurement of the ion beam current, the sample current,
and the sample temperature proved to be difficult. Thus, for in situ ex-
periments, ion beam current was measured indirectly by first stabilizing the
beam at the desired dose rate, and then adjusting the accelerator controls
during the actual irradiation so as to maintain a constant accelerating cur-
rent and sample secondary electron current. The accuracy of dosimetry mea-

surements made in this fashion was calibrated by Rutherford backscattering

Chapter 2

31
Faraday Cage
Flap Sample Block
—_—__» Sample a Cc...
non ——_}> C Heating
cam Elements
—_———__> 1
O d
Tc [5
| Wires
tT
Power Ammeter y Temperature
Supply To Current Controller
Integrater

Figure 2.2: Schematic of the experimental setup used for in situ conductivity

measurements.

Chapter 2 32

spectroscopy (RBS) measurements to be within + 15 %. This level of accu-
racy in dosimetry is acceptable, since the total irradiation dose in all cases

greatly exceeded 1 dpa.

2.4 Experimental Conductivity Measurements

2.4.1 Temperature Dependent Conductivity of Re-
laxed Amorphous Silicon

The variation of electrical conductivity with measurement temperature for
amorphous silicon that was relaxed by annealing at 873 K for 15 min prior to
the conductivity measurement is illustrated in F ig. 2.3. The values obtained
are in good agreement with previous reports of the conductivity of a-Si [3).
There is a clear change in the temperature dependence of the conductivity
near 473K. For measurement temperatures greater than 473 K, the conduc-
tivity shows exp(—1/T) dependence. The solid line is a least-squares fit to
a functional form of o,.) = 0, exp[—(E. — E's)/kT)]. The values obtained
for o> and (£. — Ey) are 1.1 x 10*Q-1 cm™! and 0.8 eV, respectively. For
lower temperatures, Hause et al. have shown that the electrical conductiv-
ity of a-Si shows exp[-(1/T')'/4] dependence up to room temperature [18],

which was the highest temperature used in their experiments. We note that

Chapter 2 33

previous work has indicated that the electrical conductivity of a-Ge, which
has a band gap of about 0.6 eV and an activation energy for high tempera-
ture conductivity of 0.25 eV, also shows exp(—(1/T)*/4] behavior up to room
temperature [19]. Amorphous silicon has a band gap of about 1.4 eV and
an activation energy for high-temperature conductivity of 0.8 eV. Therefore,
it is reasonable to assume that hopping conduction must dominate to much
higher temperatures in a-Si than in a-Ge. Consequently, a functional form
of g = ot exp[—A/T™4] was fitted; and was found to fit the data well. The

values obtained for o* and A are 8.4 10°Q7! em=! and 110K'/*, respectively.

2.4.2 Variation of Room Temperature Conductivity
with Anneal Temperature

To discern the relationship between the extent of structural relaxation and
the conductivity, samples were first fully unrelaxed by irradiation at room
temperature to a total dose of 1 dpa and then partially relaxed by anneal-
ing at various temperatures for 15 min. Conductivities of the samples were
then measured at room temperature. Since samples were stored at room
temperature, the value of the conductivity after an anneal at room temper-

ature is taken to be that which is measured 15 min after termination of ion

Chapter 2 34

T (K)
» 800600 400 200
10 | i 1
's 10°? 4
uIP So~ |
— 10°F ~~ L =
b O~ ~
10° 4 ———!
1 2 3 4 5
1000/T(K)

Figure 2.3: Temperature dependence of the electrical conductivity of a-Si
relaxed by an 873 K anneal prior to measurement. The solid line is a fit to
o x exp[(—E/kT], and the dashed line is a fit to o x exp[—(A/kT)'/4]. The
fits yield E = 0.8 eV, and A= 110 K™4.

Chapter 2 35

irradiation at room temperature. For samples annealed at less than 473 K,
Al contacts were applied prior to irradiation and post-irradiation annealing.
For samples annealed at higher temperatures, contact evaporation was the
last step prior to the measurement of the conductivity. Figure 2.4 illustrates
the measured room temperature conductivity. The solid curve is a guide to
the eye. The values of conductivity of two samples that were re-irradiated
to ldpa are shown as solid circles. Contact evaporation again required a
prebake at 473 K. As indicated by Fig. 2.4, post-anneal irradiation causes
the conductivity to revert back to the value of the original sample annealed

at 473 K, represented by the broken line.

2.4.3 In Situ Measurements of Conductivity

Prior to the in situ conductivity measurements, samples were first relaxed
by annealing at 873 K for 15 min. Figure 2.5 shows in situ conductivity
measurements made before, during and after irradiation at various irradia-
tion temperatures. Throughout the measurements, samples were held at the
irradiation temperature. The dose rate was 7 x 10'! Kr ions cm~? sec™},

and the irradiation time was 30 min. Thus the total dose is calculated to

be approximately 3 dpa. The conductivity is normalized to the value prior

Chapter 2 36

Reirradiated and
annealed at 473K

“7 | | | | | |
800 400 500 600 700 800 900 1000

Toanneal (K)

Figure 2.4: Room temperature conductivity of a-Si as a function of the anneal
temperature. All samples were fully unrelaxed prior to anneals. The solid
circles are values of the conductivity, after re-irradiation, of samples initially
annealed at indicated temperatures. The solid line js a guide to the eye. The
dashed line is the value of the conductivity of a-Si after initial relaxation by
an anneal at 473K.

Chapter 2 37

to irradiation. No data are given for the sample irradiated at 77 K because
the conductivity was unmeasurably small prior to irradiation. However, the
observed changes in the conductivity were similar to observed changes of the
samples irradiated at higher temperatures. Note that the scale of the ab-
scissa of Fig. 2.5 changes abruptly in order to show the extremely rapid and
immediate increase in the conductivity upon the onset of irradiation. All
curves show a rapid and immediate increase in the conductivity upon the
onset of irradiation, and quick saturation to an apparent steady-state value.
It is evident that lower implantation temperatures correspond to greater in-
creases in the conductivity. The rapid conductivity decay in F ig. 2.5 marks
the moment when ion irradiation was terminated. The conductivity decay
transient after termination of irradiation is extremely rapid at first, but soon
exhibits a slow decay, which continues even 45 min after the termination of
irradiation.

For the sample irradiated at 773 K, however, the measured change in the
conductivity is different, and is shown separately in Fig. 2.6. The conductiv-
ity rises rapidly with the onset of irradiation, and saturates to an apparent,
steady state value, as at other irradiation temperatures. But after ~ 10 min,

the conductivity starts to rise rapidly and linearly with time. Irradiation of

Chapter 2 38

10° r—+ e&— | |
218 k
10° —-273 K@—
_ -——@o >> 1
co 10° tei
~ —-473 K
S02 pe Qa, 7
6 .
tee S73 Kee
10t pe Qe 7
: B75 Kr
per Gee —
0 | 2 | Tr ~~ = I ee, —-:
QO 50 1000 2000 5000 400
Time (sec)

Figure 2.5: In situ measurement of the conductivity of initially relaxed a-
Si during and after ion irradiation at indicated temperatures. The sharp
drop in the conductivity marks the end of irradiation. The conductivities
are normalized to the initial value at the irradiation temperature. Notice the
change in the scale of the x axis.

Chapter 2 39

this sample was stopped earlier than other samples, but the total dose was
still greater than ldpa. The decrease in the conductivity following termina-
tion of irradiation is small, but measurable, and will be shown in more detail
later in Fig. 2.7.

The most plausible explanation for the increase in conductivity during ir-
radiation at 773 K after reaching an apparent steady-state value is the onset
of crystal nucleation in a-Si. Since crystalline Si has much higher conduc-
tivity that a-Si, nucleation of crystalline silicon will irreversibly increase the
sample conductivity. Ion irradiation is known to greatly enhance the rate of
crystal nucleation in an a-Si matrix [20], and the irradiation parameters were
such that the onset of crystal nucleation is anticipated. Indeed, the presence
of crystal grains was later confirmed by x-ray diffraction. Furthermore, the
observed change in the conductivity is consistent with crystallization exhibit-
ing an incubation period followed by steady-state nucleation at a constant
rate.

Part of the fluctuations in the conductivity both during and after irradi-
ation, especially those observed for samples at high temperatures, is due to
fluctuations in temperature, which could be maintained only to within + 3 K

for elevated temperatures. Part of the fluctuation in the conductivity during

Chapter 2 40

10' re —

Termination of
irradiation

10° | || ! |
0 50~ 1000 2000 3000 4000

Time (sec)

Figure 2.6: In situ measurement of the conductivity of initially relaxed a-Si
during and after ion irradiation at 773 K. The scatter in data after termina-
tion of irradiation is due to fluctuations in temperature. The conductivity
is normalized to the initial value at the irradiation temperature. Notice the
change in the scale of the z axis.

Chapter 2 4]

ion irradiation, however, was due to fluctuation in the jon flux. Indeed, it
was observed that the conductivity increased when the ion flux increased.
Finally, some samples were re-annealed at 873 K for 15 min following irradi-
ation, and the room temperature conductivities of such samples were verified

to revert to the pre-implanation value.
2.5 Activation Energy Spectrum

Both in situ and ex situ measurements suggest clearly that structural relax-
ation of a-Si is associated with a decrease in the conductivity. Eqs. (2.10) and
(2.10) indicate that the change in the conductivity can arise either through
change in a~!, the decay constant for electron wavefunction, or g(Fs), the
electronic density of states near the Fermi level. As F ig. 2.1 shows, however,
structural relaxation of a-Si is associated with a decrease in g( Es). We will
therefore make the assumption that a-! remains relatively unchanged dur-
ing structural relaxation, and that changes in electrical conductivity which
accompany structural relaxation are due to changes in g(Es) alone. This
seems reasonable since electronic energy of an electrically active defect and
its activation energy for annihilation seem to be uncorrelated. Previous re-

ports of the effect of ion irradiation on the conductivity of a-Ge and a-Si are

Chapter 2 42

consistent with this interpretation [21].

Structural relaxation of a-Si, however, is associated with annihilation of
point defects, whose activation energies form a continuous spectrum [1]. Since
the localized electronic states near the Fermi level are presumed to have been
created by defects, we define a function E (Q), the density of defects with
activation energy Q associated with the localized electron states near the

Fermi level, such that

gE) = | E(Q)dQ. (2.11)

Since we are assuming that these electrically active defects are representative
of all defects (including those without any effect on g(Es)), we further assume
that E(Q) is proportional to N(Q), the overall density of defects in a-Si. In
addition to E(Q), we can also define G(Q), the density of electrically active
defect states. G(Q) is analogous to D(Q), the overall density of defect states.
It is the saturation value of E(Q), just as D(Q) is the saturation value of
N(Q).

Information about the existence and qualitative features of the activa-

tion energy spectrum of electrically active defects can be obtained from the

Chapter 2 43

insitu measurement of the electrical conductivity. It is important to note
that as samples are being irradiated, they also undergo an unavoidable ther-
mal anneal at the irradiation temperature T';. During irradiation, nearly all
defects with activation energy Q < kT;; In(vt;), where t; is the ion interar-
rival time, will recombine before the next ion arrives, while almost none of
the defects with activation energy Q > kT;;|n(vt;) will recombine. These
high activation energy defects will accumulate until they reach saturation
density, G(Q), and a steady-state defect concentration under irradiation will
be established, as is observed from in situ measurements of the conductivity
during ion irradiation. A quantitative analysis of conductivity measurements
during irradiation is difficult, however, since no clear information exists about
carriers generated by impinging ions.

Changes in the conductivity (after a short transient to ensure that both
the excess carriers generated by irradiation and defects with low activa-
tion energies created just prior to termination of irradiation have all re-
combined) measured after termination of irradiation, however, will reflect
the corresponding decrease in g(E;), the density of electronic states near
the Fermi level. Since defects undergo thermal anneal during irradiation, a

post-irradiation anneal is essentially a continuation of an anneal. Since Qo

Chapter 2 44

of Eq. (1.8) depends logarithmically on total anneal time, Qo will increase
only slightly over the value kT; In(vt;) during the 45 min anneal following ir-
radiation. Correspondingly, g(E,) will decrease only slightly, such that Ag/g
is small at all times.

In such cases, we can isolate changes in g( Ey). Letting g’(Es) = g(E;) —

Ag(Es), we can combine Eqs. (2.8) and (2.10) to obtain

Expanding (g/g’)'/4, and neglecting higher order terms,

(2.13)

o(g) o*(q) 4 \gkT) “g

, o(g') oh(g’) ox |-44 ( ae i Ag

If the prefactor o” is some weak function of g( Ey as in Eq. (2.10), then we
can neglect it in comparison with the exponential when taking the logarithm

of both sides. In such cases, we have

o 2.1 [ 8 \4 Ag
—)= Sf —, 2.14
n= (2) A (2.14)

Therefore, in the cases where the hopping conduction mechanism dominates,

the logarithm of the normalized conductivity will be proportional to the

Chapter 2 45

changes in g(Es), the density of electronic states near the Fermi level. Now
the assumptions made in deriving Eq. (1.9), which predicted a In(t) behavior,
hold. If there exists a G(Q), and if it is broad and slowly varying, then we
expect the logarithm of the normalized conductivity to decrease as In(t) after
termination of irradiation. Note that this conclusion does not depend on the
exact form of A in Eq. (2.10).

Figure 2.7 illustrates the measured decay of the conductivity starting 5
sec after termination of irradiation, plotted against In(t). For low tempera-
tures where we expect hopping conduction to be the dominant conduction
mechanism, In(o’/c) is plotted; for high temperatures where we expect the
extended state conduction to be dominant, (o’/c) is plotted instead. Hopping
conduction was assumed to be the dominant carrier transport mechanism for
the sample irradiated at 573 K because previous reports suggested that un-
relaxation tends to increase the temperature at which the transition from
hopping conduction to extended state conduction occurs [3]. Indeed, the
conductivity decrease after termination of irradiation at 573 K more closely
resembles the one at 473 K than the one at 673 K. Thus it is deemed appro-
priate to assume hopping conduction at 573 K. Most curves display excellent

linearity, and the rates of decay for the low-temperature relaxation (< 573

Chapter 2
0.0
-0.5
“=
~~
a -1.0
Nar
e)
NS
—1.5
—2.0

Figure 2.7: Conductivity transients for in situ measurements

Extended—state |
Conduction

4 6
In(t)

1.0
0.5
0.0 OS
a,
4 —0.5
~—1.0

, plotted against

In(t). The left vertical axis is the logarithm of the conductivity normalized
to the value of conductivity during irradiation. The right vertical axis is the
value of the conductivity normalized to the value during irradiation.

Chapter 2 47

K) are all similar. The two measurements made at below room temperatures
deviate somewhat from the linear behavior, but the behavior can still be
approximated by a straight line.

The observed In(t) decay of the conductivity indicates clearly that the
density of electronic states near the Fermi level g(E;) decreases as In(t).
This In(¢) decay and the similar rates of decay for different anneal temper-
atures indicate clearly that there indeed exists a G(Q), the density of de-
fect states with activation energy Q, associated with the localized electronic
states near the Fermi level. Furthermore, the In(t) behavior also indicates
that G(Q) is a slowly varying function that is nearly constant on a scale
of a few tenths of an electron volt, the energy range spanned by the post
irradiation anneal. The fact that the two samples irradiated and annealed
at below room temperature deviate somewhat from the linear decay might
be an indication of some features in the activation energy spectrum at low
activation energies. However, the fact that they can be approximated by a
straight line means that G(Q), and also presumably D(Q), the total density
of defect states with activation energy Q, have non-zero values in the low
Q range, in this case below 0.23 eV. The high temperature (> 573 K) data

suggest that G(Q) probably extends into high activation energy ranges as

Chapter 2 48

well, although in such cases Eq. (2.14) is clearly invalid, since conduction
at these high temperatures is dominated by the extended state conduction.
It is possible that the onset of crystal nucleation has rendered the sample
irradiated at 773 K a composite of amorphous and crystalline silicon whose
conductivity is not amenable to a simple interpretation. Nonetheless, the
fact that the conductivity does decrease during the post irradiation anneal is
evidence that some post-irradiation structural relaxation occurs in a-Si even
when irradiated at high temperatures, and that G(Q) probably extends into
high activation energy ranges.

A more quantitative insight into G(Q) is possible from changes in the
room temperature conductivity of a-Si that was initially fully unrelaxed by
room temperature irradiation to doses greater than 1 dpa, then partially

unrelaxed by 15 min isochronal anneals at different temperatures. We rewrite

Eq. (2.11) as follows:

By) = 40+ [ o(QvaQ, (2.15)

where g, is the electronic density of states at the Fermi level after 873 K

anneal. The upper limit of integration is the value of Qo after 873 K anneal,

Chapter 2 49

and the lower limit is Qo(T’) after anneal for 15 min at temperature T. Note
that we are integrating over the density of defect states, G(Q), rather than
the density of defects, E(Q). This is because the samples were fully unrelaxed

prior to 15 min anneals. Now we rewrite Kq. (2.12) as follows:

’ h(t 3 1/4 . ua
o . 79) 5 —2.1 ( e 2 —1 .
a of(g) gokT Go + Seer AQ)dQ

(2.16)

From Fig. 2.3, we know the value of the factor 2.1(a°/g.kT')*/4 to be 26.43 at
room temperature. If we assume a reasonable value for aq', we can calculate
the value for g,. Assuming a7! to be = 10 A, we find that Go © 1.45 x 108
eV~* cm7°. Taking the logarithm of both sides of Eq. (2.12), and again

neglecting the prefactor in comparison to the exponent, we can isolate G(Q):

n.d 7%
G@) = “IO Lowen (OD
d o'\)~"
= 907% }:~o.038m ()] : (2.17)

If a functional form of A that is different from Eq. (2.10) is used, a slightly
different form of Eq. (2.17) will result. However, the main conclusion of the

analysis, again, is not affected.

Chapter 2 50

] l
o 1019L 4107"?
& &
O oe) Co O
S rare) fe) >
© 108L ane) 4 107 ©
~— O
“= OOO o~
'S) ‘Si
nd NZ
ro) - QO
4917 l l l +1919
0.5 1.0 1.5 2.0 2.5: 3.0
Q (eV)

Figure 2.8: The density of the defect states of amorphous silicon. The left
axis gives the density of electrically active defect states G(Q) as derived from
Fig. 2.4. The right axis gives the total density of defect states D(Q), inferred
from G(Q).

Chapter 2 51

In order to perform this differentiation, it is necessary to convert the
abscissa of Fig. 2.4 from the anneal temperature to the activation energy Q.
This requires an estimate of the reaction constant v. We assume v to be of
the order of the Debye frequency, or 10'% sec~! for silicon. Similar values
have been used previously for recombination of ion induced defects in a-Si
{1, 27]. The result of the differentiation, using 10'* sec™! for v, is illustrated
in Fig. 2.8. This is the central result of this chapter, an experimentally-
measured density of defect states of amorphous silicon. The left vertical axis
shows G(Q), the density of electrically active defects. The right vertical
axis shows D(Q), the overall density of defect states in a-Si. It is estimated
by comparing the saturation value of g(Es), the density of electronic states
near the Fermi level, with the estimated saturation density of defects in a-Si,
which is found to be © 1 at. % [1, 22]. Assuming that one defect creates one
electronic state, we find D(Q) ~ 90 eV xG(Q).

As expected, G(Q) is broad, and extends from 0.9 eV to greater than
2.7 eV. It also seems to extend to lower values of the activation energy Q,
corroborating the results of in situ measurements made below room temper-
ature. It must be pointed out, however, that the activation energy range

probed by the below-room-temperature irradiation and anneal lies outside

Chapter 2 52

the region covered by the deduced activation energy spectrum in Fig. 2.8,
and thus no quantitative conclusion can be drawn about the features of the
density of defect states at low (< 0.9 eV) activation energies. The density of
defect states decreases slowly and monotonically with increasing activation
energy, with no sharp features, thus justifying the approximation that we
employed in deriving it. The lack of any discernible features in the density
of relaxation states does not contradict the in situ low temperature measure-
ments (which indicated the possibility of some features) since the range of

activation energies covered in Fig. 2.8 does not extend to such low values.

2.6 Discussion

Care must be taken in interpreting Fig. 2.8, since rigorous conversion of the
annealing temperature to the activation energy requires knowledge of the
reaction constant v. But because Qe depends only logarithmically on v, the
conversion is fairly insensitive to errors in estimation. Indeed, Av would have
to be much greater than one order of magnitude for there to be a significant
shift in the activation energy scale. Even with such uncertainty in the abso-
lute conversion of annealing temperature to activation energy, it is evident

that the vast majority of defects in a-Si have low activation energies, and

Chapter 2 53

that D(Q) reaches small values at high (Q > 2.5 eV) activation energies.
Such a preponderance of defects with low activation energies is consistent
with previous experimental results. For instance, using Raman spectroscopy,
de Wit et al.. [23] have shown that the rate of increase in the degree of
structural relaxation decreases with increasing anneal temperature; and Ro-
orda et al., [1] have shown using differential isothermal calorimetry (DIC)
that more stored enthalpy was released during annihilation of low activation
energy defects than during annihilation of high activation energy defects.
The relative dearth of relaxation processes with activation energies greater
than 2.5 eV is tantalizing, since 2.7 eV is the activation energy of the rate
of the solid phase epitaxy (SPE) of crystalline silicon into a-Si. Unlike other
processes involving a-Si, the rate of SPE has been shown to be conspicuously
independent of the degree of structural relaxation (24], and is characterized
by a single activation energy (Q = 2.7 eV) over nearly ten orders of mag-
nitude in the SPE rate [25]. The observed dearth of relaxation processes
with activation energies greater than 2.5 eV is consistent with the notion
that structural relaxation and solid phase epitaxy are controlled by different
processes. Structural relaxation is by nature a bulk phenomenon, whereas

there is strong evidence that solid phase epitaxy is limited by interface re-

Chapter 2 54

arrangement kinetics [24, 26, 27]. Furthermore, the likeliest candidate for
the controlling event in solid phase epitaxy is dangling bond rearrangement
[24, 28], while recent reports suggest that structural relaxation is associated
with the annihilation of vacancy-like defects in a-Si [29, 30].

One other remarkable consequence of this analysis, as evidenced by the
presence of defects with high activation energies and the in situ measurements
of relaxation at high irradiation temperatures, is the ability of irradiation to
maintain an excess concentration of structural defects in a-Si during irradi-
ation even when the irradiation temperature is high (573 - 773K). In other
words, irradiation creates a substantial number of defects with relatively high
activation energies and long lifetimes to measurably increase the steady-state
defect concentration during irradiation at 773 K relative to the defect con-
centration following a 773 K anneal. This increased defect concentration has
been demonstrated to substantially increase the defect enthalpy and thus the
overall free energy of a-Si. In later chapters, we will discuss in more detail the
implication of such an increase in the free energy for the possible mechanism
responsible for the observed enhancement of nucleation of crystal silicon in
an amorphous silicon matrix under irradiation at elevated temperatures.

Recently, Coffa et al., using identical methods, have also measured the

Chapter 2 55

density of defect states D(Q), of a-Si [31]. Unlike the work presented in this
chapter, they probed D(Q) at low (< 0.5 eV) activation energies as well. The
values of D(Q) measured by Coffa et al. within the activation energy range
covered in this thesis (0.9 eV < Q < 2.8 eV) differ from the values measured
in this thesis, since he takes the decay length of electron wavefunction a7}
to be 3 A. When corrected for this different value of a, his values of D(Q)
and the values of D(Q) derived in this thesis agree reasonably well, within
+ ~ 3x10" eV-! cm=*. A pronounced disagreement occurs near 1.5 eV,
however, where they observe a pronounced dip in the density of defect states
near 1.5 eV, something not observed in Fig. 2.8. However, the least square fit
to his values of D(Q) within the activation energy range covered in this thesis
agrees very well with the least square fit to the values of D(Q) measured in
this thesis.

Such a dip in the density of defect states may be an artifact of the ex-
perimental procedure. It is interesting to note that the supposed dip in the
density of defect states measured by Coffa et al. occurs right after the acti-
vation energy corresponding to a room temperature anneal. Since structural
relaxation takes place indefinitely (albeit as In(¢)), a prolonged storage at

room temperature will sufficiently relax the samples that temperatures much

Chapter 2 56

greater than room temperature are needed to further relax the material. In
such cases, the electrical conductivity will change only a little with an intial
increase in anneal temperature beyond room temperature, and the density
of defect states will indeed display a dip after the activation energy corre-
sponding to a room temperature anneal. On the other hand, the possibility
that the fine details of the density of defect states of a-Si are dependent on
the mode of production cannot be completely discounted. Coffa et al. have
produced their a-Si film by low temperature chemical vapor depostion, unlike
the a-Si films used in this chapter, which were produced by ultrahigh vacuum °

electron-beam deposition.
2.7 Conclusion

Through measurement of irradiation-induced conductivity changes, we have
probed G(Q), the density of relaxation states with localized electron states
near the Fermi level, and thus D(Q), the total density of relaxation states in
a-Si. It is a function that decreases slowly with activation energy Q, ranging
from below 0.9 eV (and probably below 0.23 eV) to beyond 2.6 eV. The dearth
of defect states with activation energies greater than the activation energy for

solid phase epitaxy is consistent with the notion that structural relaxation

Chapter 2 57

and solid phase epitaxy are controlled by different processes, and may explain
the observed independence of the rate of solid phase epitaxy on the extent
of structural relaxation. Jn situ measurements of the conductivity show that
D(Q) decreases linearly with In(t) following irradiation, consistent with the
predictions of an activation energy spectrum model and further supports the
conclusion that D(Q) varies slowly over the energy range investigated. High
temperature irradiation shows that even at elevated irradiation temperatures
(5734 < T < 773K) ion irradiation unrelaxes a-Si. Furthermore, under ion
irradiation, the steady-state defect concentration (and thus the free energy
of a-Si) is maintained at a level higher than that following an anneal at the

irradiation temperature.

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and L. E. Rehn, Mat. Res. Soc. Symp. Proc. 235, 21 (1992).

Chapter 2 59

[5] S. Coffa and J. M. Poate, Appl. Phys. Lett. 59, 2296 (1991).

[6] M. H. Brodsky, in Amorphous Semiconductors, edited by M. H. Brodsky,

2nd. ed. (Springer-Verlag, New York, 1985) p. 3.

[7] N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline

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[8] M. Pollak, M. L. Knotek, H. Kurtzman, and H. Glick, Phys. Rev. Lett.

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(1973).
[10] P. Nagels, in Amorphous Semiconductors (Ref. [6]) p. 113.

fll] Ph. D. thesis, FOM Instituut voor Atoom- en Mulecuulfysica, The
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[13] J. C. Bean and J. M. Poate, Appl. Phys. Lett. 36 59 (1980).

Chapter 2 60

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[15] J. F. Ziegler, J. P. Biersack, and U. Littmark, The Stopping and Range

of Ions in Solids (Pergammon, New York, 1985).

[16] J. P. Biersack and L. J. Haggmark, Nucl. Instrum. Methods 174, 257

(1980).
{17} Copyright J. F. Ziegler, 1990.
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[20) J. S. Im and Harry A. Atwater, Appl. Phys. Lett. 57, 1766 (1990).

[21] N. Apsley, E. A. Davis, A. P. Troup, and A. D. Yoffe, in Amorphous
and Liquid Semiconductors: Proceedings of the Seventh International
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E. Spear (Centre for Industrial Consultancy & Liaison, University of

Edinburgh, Edinburgh, 1977) p. 447.

Chapter 2 61

[22] S. Coffa, J. M. Poate, D. C. Jacobson, W. Frank, and W. Gustin, Phys.

Rev. B 45, 15 8355 (1992)

[23] L. de Wit, S. Roorda, W. C. Sinke, F. W. Saris, A. J. M. Berntsen, and
W. F. van der Weg, in Kinetics of Phase Transformations, edited by M.
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Proc. 205, 3 (1991).
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[26] J. S. Custer, A. Battaglia, M. Saggio, and F. Priolo, Phys. Rev. Lett.

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Chapter 2 62

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Chapter 3

Generalized Activation Energy
Spectrum Theory

3.1 Introduction

Using the activation energy spectrum theory as formulated by Gibbs et al.,
{1] we were able to quantitatively measure D(Q), the density of defect states
responsible for structural relaxation in a-Si. However, as mentioned often in
Chapter 2, the simple theory relies on several approximations and implicit as-
sumptions that render the theory inappropriate for an accurate, quantitative
description of defect kinetics under many circumstances. One of these is the
approximation of the annealing function f(Q,T,t), by a step function. Since
this approximation is valid for both first and second order defect recombina-

tion kinetics, we cannot distinguish between them using the simple theory.

Chapter 3 64

Also, no mechanism is provided for unrelaxation, thus preventing the theory
from being applicable to both creation and annihilation of defects. The abil-
ity to address both defect creation and annihilation, however, is essential for
a quantitative description of defect kinetics under ion irradiation.

A more serious shortcoming of the simple theory is the ad hoc assumption
that defects with different activation energies are independent of each other.
Such an assumption might be justified in the case of unimolecular recombi-
nation of defects with fixed sink sites, especially if defects are isolated point
defects. In the case of bimolecular recombination, it is unphysical to assume
that defects in amorphous solids somehow do not interact with each other if
their activation energies differ by an amount dQ.

It is therefore the aim of this chapter to develop a new, more physically
reasonable theory, hereafter referred to as the generalized activation energy
spectrum theory, which takes into account the possibility that defect dynam-
ics may depend on other defects with different activation energies. To test
and compare the generalized theory with the simple theory, models of defect
creation and annihilation of a-Si based on the simple and on the generalized

theory will be developed, and compared against experimental data.

Chapter 3 65

3.2. Generalized Activation Energy Spectrum
Theory

As written in Chapter 1, the kinetics of defect annihilation is described by

the simple theory as

Mt) 2 —v,exp (Z) may’ (3.1)

where vy, is the reaction constant, and n is the order of the reaction. In
essence, we have a continuum of independent equations for all possible values
of Q.

The main difference between the simple and the generalized theory is the
interaction between defects with different activation energies. This can be

achieved by rewriting Eq. (3.1) as

avg) _ 2M) lexp (=2) [n(a,aym ayaa’ A

[eo (FF) xa. ormarna. (32)
The function o;(Q,Q’) and o2(Q, Q’) are generalized reaction cross sections

appropriate to the relaxation dynamics considered. For bimolecular recom-

bination kinetics, they represent reaction cross sections between defects with

Chapter 3 66

activation energies @ and Q’. The physical meaning of Eq. (3.2) becomes

more clear if the terms are considered separately. The first term on the right,

voN(Q) [exe (Fr) [ ol@,@N(Q'dQ (3.3)
kT
is the rate at which a defect with activation energy Q annihilates other defects

with activation energy Q’. The second term on the right,

voW(Q)| [exe (FF) (2, Qvm(@yaa, (3.4)
is the rate at which a defect with activation energy Q is annihilated by other
defects with activation energy Q’. The factor of 1/2 is needed because we are
counting each defect recombination event twice. Although Eq. (3.2) is cast
in a form of bimolecular recombination kinetics, by choosing suitable forms

for o1(Q,Q’) and o2(Q,Q’), we can transform Eq. (3.2) to represent other

recombination kinetics. For example, if

01(Q,Q’) = 02(Q, Q') = 6(Q, 2"), (3.5)

we recover the simple theory. If we let

Chapter 3 67

01(Q,Q') = 25,6(Q")/N(Q'), 2(Q, Q') = 0, (3.6)

we recover the unimolecular recombination kinetics, provided we have S, =
density of fixed sinks. The generalized theory can also be employed to model
defect recombination kinetics in phases other than the amorphous phase.
For example, the defect dynamics of crystalline solids may be modelled by
having the defect density D(@Q) consist of one or more sharp peaks. It is
also possible that a form of D(Q) intermediate between the crystalline and
amorphous forms may be used to describe the defect dynamics of heavily

damaged crystals.

3.3. Defect Dynamics of Amorphous Silicon:
Modelling and Experiments

3.3.1 Modelling Defect Dynamics

In order to test the generalized theory, the defect dynamics of ion-irradiated
a-Si is modelled, and the results are compared with experimental data and
with the simple activation energy spectrum theory. To model the evolu-
tion of defect population in a-Si during ion irradiation and anneal, we make

the following assumptions. First, we assume that an impinging ion creates

Chapter 3 68

defects with different activation energies with equal probabilities. Second,
we assume that annihilation occurs via bimolecular recombination, which is
consistent with previous experimental observations [2, 3]. Finally, we as-
sume that o1(Q,Q’) = o2(Q.Q’) = constant; that is, defects with different

activation energies recombine with equal cross section. We then write

pe = 98 (1 LD) — NO) Foxe (22) [avcanaa’ +

[exp (4) veqiag' (3.7)
where g and ¢ are defects/flux and ion flux, respectively. The first term on
the right side of Eq. (3.7) is the defect generation term. It ensures that the
defect population saturates at D(Q). A similar form for generation of defects

in a-Si by irradiation has been used successfully in the past [8]. If we use the

simple theory [1], we can write

ag) _ g¢ (: _ non — Vo exp (=) N(Q) (3.8)

With Eq. (3.8) and (3.7), the time evolution of defects in a-Si during both
irradiation and the subsequent anneal can be solved. In both cases, we have

g and vy, as fitting parameters to be determined.

Chapter 3 69

The assumption underlying the interaction of defects with different ac-
tivation energies is that the defects are of similar kinds (presumably point
defects), and differ only in their activation energies. The activation energies
are unlikely to be that of self-diffusion, since the activation energy for self-
diffusion in a-Si is between 0.13 and 0.22 eV [9]. Most likely, the activation
energies are that of some rate-limiting step to defect migration, possibly that
of the transition states necessary for the defects to become mobile.

For Eq. (3.8) for the simple theory, an analytical solution can be found. |

It is given by [4]

| Otenh E + 2D(Q) tanh-1(9¢/Q) - i os

| 2D(Q)
N@,T,Q,¢) = 2v, exp(—Q/kT) D(Q) ®)

where

2 = 4/98)’ + dvoexp(—Q/kT)96D(Q)?. (3.10)

For Eq. (3.7), however, no analytical solution could be found. Therefore,
it was solved by computer simulation. To ensure an accurate comparison
between the results of the two theories, Eq. (3.8) was also solved by computer

simulation. The simulation of the defect dynamics during irradiation and

Chapter 3 70

anneal consisted of calculating dN/dt using either Eq. (3.7) or Eq. (3.8), and
then altering N(Q) such that N(Q,t + 6t) = N(Q,t) + 6t(dN/dt). This
step was repeated for the desired length of time. One implicit result of
this simulation is the approximation of the defect generation as a steady
stream. Under real conditions, defects are more likely to be generated as
pulses corresponding to the collision cascades produced by impinging ions.
For both theories, the range of activation energy Q covered in the simulations
was from 0.6 eV to 2.8 eV. A minimum value of 0.6 eV was chosen for
two reasons. First, D(Q), the density of relaxation states, from Chapter: 2
extends only down to 0.9 eV. Therefore, we introduce more uncertainty as
the lower limit is lowered, especially since D(Q) increases exponentially with
decreasing (). Second, the time step required for accurate simulation becomes
unpractically short as the lower limit is lowered. The value of 0.6 eV was a
compromise between the practicability of simulation and the need to consider
as much of N(Q) as possible. For the simulation of irradiation, the value of
6t was 2 x 107° sec. This value was chosen because it is much less than the
expected lifetime of the defects with activation energy 0.6 eV, as evidenced
by the fact that further decrease in é¢ did not result in any changes in the

final results. For simulations of defect dynamics during an anneal, the value

Chapter 3 71

of 6¢ was adapted during calculation to be 1/10 of the lifetime of most mobile
defects that were still present in significant numbers, defined to be 1 x 1072°
cm~*. At this concentration, the recombination rates of even the most mobile
defects are negligible compared with others. Finally, the numerical solutions
to Eq. (3.8) were compared with the analytical results, and found to agree
very well, thus confirming the validity of the simulation.

By explicitly solving the equations for the defect dynamics, we remove
the step function approximation of the annealing function. Therefore, the
solutions of Eq. (3.8), whether numerical or analytical, are already a great
improvement over the results derived using the existing simple theory. Hence-
forth, the simple theory without the step function approximation will be re-
ferred to as the modified simple theory. The three different theories discussed

in this chapter, and the assumptions associated with them are summarized

in table 3.1.

3.3.2 Experiments

In all experiments described in this chapter, same a-Si resistors as described
in Chapter 2 were used. The anneal protocol was identical to that described

in Chapter 2. All irradiation was performed at room temperature. However,

Chapter 3 72

Theories Assumptions

Simple theory Constant reaction cross section
Bimolecular recombination
Step function approximation

No interaction between defects with different Q

Modified simple theory | Constant reaction cross section
Bimolecular recombination

No interaction between defects with different Q

Generalized theory Constant reaction cross section

Bimolecular recombination

Table 3.1: Theories discussed in this paper, along with significant assump-
tions associated with them.

since accurate measurement of dose and dose rate was critical, no in situ
measurements of the conductivity during irradiations were attempted. In-
stead, the conductivity was measured in situ immediately after irradiation.

A schematic of the experimental setup for the in situ measurements is shown

in Fig. 3.1.

Chapter 3 73

Faraday Cage
Flap Sample Block
——_>> Sample
Ion >
Beam
————__ >
| _, To Current
Integrater
Power Ammeter
Supply

Figure 3.1: Schematic of the experimental setup used for the second sets of in

situ conductivity measurements. Note that, ion current is measured directly
from the sample.

Chapter 3 74

3.4 Results and Discussion

In all experiments, the quantity actually measured was the electrical conduc-
tivity of a-Si. However, using values obtained in Chapter 2, we can obtain
the total defect density. Therefore, we will hereafter discuss structural relax-
ation of a-Si in terms of an experimentally determined defect density without

explicit reference to the measured conductivity.

3.4.1 Evolution of Defect Density vs. Irradiation Dose

Figure 3.2 shows the defect density of a-Si, measured 15 minutes after termi-
nation of irradiation, as a function of the dose. Prior to irradiation, samples
were relaxed by a 15 min anneal at 900 K. The symbols are experimental
data. The ions used were 600 KeV Krt+, 350 KeV Artt, 250 KeV Sit, and
190 KeV Net. These energies were chosen to confine the damage distribution
within the a-5i film. As expected, the evolution of defect density with irradia-
tion dose depends strongly on the irradiating ion, with heavy ions being much
more efficient in unrelaxing a-Si. However, the evolution of the defect density
follows the same sigmoidal curve regardless of the ion species. Furthermore,
the saturation level of the defect density is independent of irradiating ion.

Figure 3.3 shows the defect population vs. irradiating dose, but this time

Chapter 3 75

Figure 3.2: Experimentally measured defect population, plotted vs. irradiat-
ing dose. The ions used were 600 KeV Krt*, 350 KeV Ar++, 250 KeV Sit,
and 190 KeV Net.

Chapter 3 , 76

the dose is normalized to the displacement per atom (dpa) as calculated by
the TRIM simulation program (5, 6, 7]. Again, the symbols are experimental
data. The solid curve in Fig. 3.3 is the result of fitting the generalized the-
ory, and the dashed curve is the result of fitting the modified simple theory.
Both theories give an excellent fit to the data, and they are nearly indis-
tinguishable from each other. The values of g and v, used to fit the data
are 4,1x10??cm73dpa7! and 1.7x107!°cm°sec7! for the generalized activa-
tion energy theory, and 4.1x10?? cm7°dpa7! and 5.1x107® cm*sec™! for the
modified simple theory. The same values of g and vy, are used in all of the
following figures. Since there are no more adjustable parameters, the results
which follow are the results of calculation, not a fit to the data. It needs to
be remarked that the value of the reaction constant for the generalized ac-
tivation energy theory depends on the lower limit of activation energy used.
However, the conclusions that can be drawn from the analysis are not signif-
icantly affected, provided that the same values of y, and the lower limit of
activation energy are used consistently.

It is clear that once scaled to dpa, the evolution of defect population
shows a universal behavior. Therefore, we will discuss unrelaxation in terms

of dpa without making specific reference to the irradiating ion. The defect

Chapter 3 77

6.x107° | | I
m 4.6 _
= fx O Krypton _
O 3 A Argon
Neon
Oob- * Si _
—— generalized theory
7 > 4 — -modified simple theory |
O. | | |
107° 1o* 10°? 10°

dpa

Figure 3.3: Experimentally measured defect population, plotted vs. dpa,
as calculated from TRIM. Symbols are experimental data. The solid line
is the result of fitting the results of calculations based on the generalized
activation energy theory. The dashed line is the result of fitting the results
of calculations based on the modified simple theory.

Chapter 3 78

population starts to saturate near the dose of 0.02 dpa, in agreement with
results obtained by others using different methods [2, 8]. However, the defect
density continues to increase with increasing dose, albeit very slowly and
slightly. One important implication of Fig. 3.3 is the irrelevance of cascade
structure to structural unrelaxation. A 600 KeV Kr*t ion is expected to
create a dense, compact cascade, while a 190 KeV Net ion is expected to
create widely scattered regions of high defect density. TRIM simulations
of cascade created by an Ne ion and a Xe ion are shown in Figs. 3.4 and
3.5, respectively. The fact that it is possible to scale such disparate ion
damages by such a simple quantity as dpa indicates that a heavy ion does
not completely unrelax a-Si within its cascade. Rather, unrelaxation of a-Si
seems to be controlled by point defects that have lifetimes that are much
longer than that of a typical cascade, and thus diffuse out from cascade
regions to the surrounding matrix. This is in contrast to ion damage in
crystal silicon, in which a heavy ion such as Kr creates small regions that are
very heavily defective or even amorphous.

With the values of g and v, fixed, we can follow the time evolution of
N(Q), the defect population as a function of the activation energy, during

irradiation. Figure 3.6 a) shows the results of calculations using the gener-

Chapter 3

T 7 y ‘
ooh, wigs gt ae eee ee ar a be
‘ 4,
m Me
Baw
5 : 4
a]
fn
Le “
t—
H 1.

Figure 3.4: Cascade generated by a 190 KeV Ne ion.

Chapter 3

q i
Cc
ce
= 4

= :

Doth

Figure 3.5: Cascade generated by a 600 KeV Xe ion.

“568A

Chapter 3 81

alized activation energy theory, and Fig. 3.6 b) shows the results of calcula-
tions using the modified simple theory. The doses were 0.001, 0.002, 0.005,
0.01, 0.02, and 0.05 dpa. The symbols are D(Q) as calculated in Chapter
2. The results are qualitatively very similar. However, Fig. 3.6 highlights
a crucial difference between the two theories. As shown in Fig. 3.6 b), the
modified simple theory predicts that the defect population increases mono-
tonically with increasing dose for all activation energies. The generalized
theory predicts a quite different and unexpected result that the population
of low-activation energy defects will first increase with increasing dose, then
actually decrease with further increases in the dose. Figure 3.7 shows this
difference more clearly. This counterintuitive prediction is due to the ability
of low activation energy defects to recombine with defects across the entire
activation energy spectrum. As the total defect density increases, so does
the recombination rate of low activation energy defects. After a critical dose,
the recombination rates of low activation energy defects exceed the genera-
tion rate, which is assumed to be independent of the activation energy. The
defect population of low activation energy defects then decreases with further

increases in the dose, although the total defect population increases.

Chapter 3

82
1071 | | !
—— Generalized theory
te oO D(Q)
> 107° -
Oo
com
‘S}
oa’ ° ,
Zz | Increasing dose
1019 4 | | | |
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Q (eV)
1071 | | | | 7
— —Modified simple theory
fe 08-9 0 GQ)
> 107° -
(cD)
‘S)
NH
< I Increasing dose
10919 | | | |
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Q (eV)

Figure 3.6: Time evolution of the defect population. Figure a) shows the
results of generalized theory, and b) shows the results of the modified simple

theory. The symbols are D(Q) from Chapter 2. The doses are 0.001, 0.002,
0.005, 0.01, 0.02, and 0.05 dpa.

Chapter 3

83
21
10 - | | |!
—— Generalized theory /
m9 —-Modified simple theory ‘ oO .
is O D(Q) Via |
| Zo ~
~ ee
(S) a
Ww CL
= ye Increasing dose
10'9 v | |
0.5 0.6 0.7 0.8 0.9 1.0

Q (eV)

Figure 3.7: Time evolution of the low activation energy defect population.
Solid curves are the results of the generalized theory, and dashed curves are

the results of the modified simple theory. The doses are 0.001, 0.002, 0.005,
0.01, 0.02, and 0.05 dpa.

Chapter 3 84
3.4.2 Evolution of Defect Density During Anneal

The ability of low activation energy defects to annihilate defects across the
entire activation energy spectrum is much more readily apparent when the
defect dynamics during an anneal are calculated, and the resulting change
in the defect population compared. Figure 3.8 shows the N (Q), the defect
density as function of the activation energy Q, for a-Si irradiated to satura-
tion and then annealed for 15 min at different temperatures. The symbol is
D(Q) from Chapter 2. The solid curves are the results of calculations using
the generalized theory, and the dashed curves are the results of calculations
using the modified simple theory. Also included are the results of the simple
theory, with its step function approximation.

Within the modified simple theory, it is evident that there really does
exist an activation energy Qe such that most of the defects with Q< Qe
have recombined after an anneal at a temperature T,,, for an anneal time tgni,
while most of the defects with Q ® Qe have not. However, Fig. 3.8 shows
clearly that the step function approximation of f(Q,T, t) made in the simple
theory is inaccurate, especially for high temperature anneals. Therefore, the

simple theory will not be discussed further. Within the generalized theory,

Chapter 3 85

—— Generalized theory
— —Modified simple theory —
trtees Simple theory
O D(Q), expt.

Increasing To

0.5 . 2.0 2.9 5.0

Figure 3.8: N(Q) after isochronal anneals, for samples irradiated to satura-
tion prior to anneals. Symbols are D(Q) from Chapter 2. Solid curves are
the results of calculations using the generalized theory, and dashed curves
are results of calculations using the modified simple theory. Dotted curves
are the results of calculations using thesimple theory.

Chapter 3 86

however, Qe does not have a real meaning. The generalized theory predicts
that after an anneal, even some of the defects with highest activation energies
will have been annihilated by the low activation energy defects.

When the total integrated defect density is compared with the experimen-
tal data, this annihilation of the high activation energy defects even after an
anneal at low to moderate temperatures is found to be essential to achieve
good agreement between the experimental data and the theoretical predic-
tions, as shown in Fig. 3.9. Symbols are the experimental data. For accuracy
in dosimetry, low-, medium-, and saturation dose irradiations were performed
with 250 KeV Sit, 190 KeV Net, and 600 KeV Krt*, respectively. Solid
curves are the results of calculation using the generalized theory, and the
dashed curves are the results of the modified simple theory. For calculations
using the generalized theory, it was necessary to take into account the fact
that the samples were stored at room temperature after irradiation and prior
to the anneals. No such precaution was necessary for the modified simple
theory.

Neither the generalized theory nor the modified simple theory results in
exact agreement with the experimental data, nor is such exact agreement

expected, since the results shown in Fig. 3.9 are direct predictions based on

Chapter 3 87

x Saturation 2.0x10% BRENT TT
4 0.0042 _dpa ‘\ Cross—over
5. |. XO 0.00087 dpa
7 4b
c | a
a eP
1.2 !
oS 600 700 800
2.7 = =~ _lanneait 7
i =
| L

Figure 3.9: Integrated defect density after isochronal anneals. Symbols are
experimental data. Solid curves are the results of calculations using the
generalized theory, and dashed curves are the results of calculations using

modified simple theory. The inset shows the “cross-over” effect in more
detail.

Chapter 3 88

parameters derived by fitting the experimental data of Fig. 3.3. It is pos-
sible that a better fit could be achieved, should an attempt be made to do
so. Yet it is clear that the values predicted by the generalized theory are in
better agreement with the experimental data than the values predicted by
the modified simple theory, especially for low- and medium dose irradiated
samples. For such samples, the modified simple theory predicts a decay in
the defect density that is nearly linear with annealing temperature, which is
not observed. Even for low dose irradiated samples, experimental data show
that the decrease in defect concentration exhibits a pronounced concave cur-
vature. Such curvature, however, is correctly predicted by the generalized
theory. This disparity between the two theories comes about because Qo
still plays an important role in the modified simple theory. At doses below
saturation, N(Q) is nearly constant and is independent of the activation en-
ergy Q, as can be seen in Fig. 3.6. For constant N(Q), the decrease in the
defect concentration is solely determined by Qe. Since Qe « Tani, the de-
fect concentration in such cases will indeed decrease linearly with increasing
anneal temperature. Within the generalized theory, however, even some of
the defects with activation energies so high that they are immobile at the

given anneal temperature are annihilated during an anneal by the low acti-

Chapter 3 89

vation energy defects. Therefore, the defect density decreases more rapidly
than linearly with an initial increase in the anneal temperature even when
N(Q) is constant, and thus predicts the concave curvature that is observed
experimentally.

Furthermore, as the inset in Fig. 3.9 shows, the generalized theory pre-
dicts an experimentally observed “cross-over” effect. That is, for a certain
combination of irradiation dose and anneal temperature, samples irradiated
to a higher dose have a lower defect density after an anneal, since they
initially contain more mobile (low-activation energy) defects to annihilate
defects that are immobile (i.e., have high activation energies). This “cross-
over” effect cannot occur in the simple and modified simple theories. The
prediction of the experimentally observed “cross-over” effect by the gener-
alized theory thus represents a substantial step in understanding the defect

dynamics of a-Si, more important than a simple improvement in accuracy.
3.4.3 In Situ Measurements of Conductivity
Figure 3.10 shows the defect density immediately following termination of

irradiation to same doses but at different fluxes, plotted against In(t). The

symbols are the experimental data measured in situ, and lines are results of

Chapter 3 90

calculations using either the modified simple or the generalized theory. The
ion used for irradiations was 600 KeV Kr**, and the total irradiation dose for
all samples was 0.23 dpa, well past the saturation dose. Again, no attempt
was made to fit the data in Fig. 3.10. The results are predictions using the
parameters derived from Fig. 3.3.

It is clear from Fig. 3.10 that the results of the generalized theory and the
experimental data converge quickly to the same values within experimental
resolution. Furthermore, the generalized theory again predicts the observed
“cross-over” effect, shown more clearly in Fig. 3.11. The reason for this
“cross-over” effect is this: with higher flux, more mobile defects are available
immediately after termination of irradiation to annihilate the defects that
are immobile. This “cross-over” effect, however, need not always be present.
Other investigations into the defect dynamics of a-Si failed to observe this
effect [10]. However, it was verified by calculation that if the defect generation
rate is small enough, as was the case in Ref. [10], there will not be enough
low activation energy defects to produce the “cross-over” effect. The “cross-
over” effect will never occur, however, if defects of different activation energies
are independent of each other, as is assumed in the modified simple theory.

Therefore, the observed “cross-over” effect in the decay transient of defect

Chapter 3 91

I ! I i

.00093 dpa/sec
.00023 dpa/sec
.00012 dpa/sec

6.2x107° F-

+bO
200

6.0
Experiments

Generalized Theory

5.8
Modified Simple Theory

5.6 b ~ S —

3.25 NN an 7

Figure 3.10: Jn situ measurement of decay transient following irradiations
at different ion fluxes. The symbols are the experimental data. The lines
are results of calculation using either the modified simple or the generalized
theory.

Chapter 3

5.8x107° F

5.7

5.6

3.9

5.4

Generalized
Theory

i ! I

—,0O 0.00093 dpa/sec
—-,A 0.00023 dpa/sec
—-~,+ 0.00012 dpa/sec

4.0

4.5 5.0

In(t)

Figure 3.11: The in situ measurements of “cross-over” effect is shown in
detail, marked by arrows. The symbols are the experimental data. The lines
are results of calculation using either the modified simple or the generalized

theory.

Chapter 3 93

concentration is strong evidence supporting generalized activation energy
spectrum theory.

Such independence of defects with different activation energies might be
reasonable, however, if defect recombination were to occur via unimolecular
recombination of locally isolated defects (such as isolated point defects) at
fixed sinks. Therefore, the existence of a “cross-over” effect may be inter-
preted as a strong evidence for the bimolecular recombination of defects with

different activation energies in a-Si.
3.5 Conclusion

In conclusion, a generalized theory is presented to describe the structural re-
laxation dynamics of amorphous solids. It is specifically designed to allow for
interactions between defects with different activation energies. Furthermore,
other forms of defect recombination dynamics may by modelled by suitable
choices of parameters, although such modelling is not attempted here. Ap-
plied to structural relaxation dynamics of a-Si, the generalized theory is
found to be needed for an accurate description of the relaxation dynamics.
The results also suggest that defects in a-Si annihilate via bimolecular recom-

bination. The success of the generalized theory does not obviate the need for

Chapter 3 94

the simple theory, however. After all, D(Q), the density of defect states in
a-Si, was derived in Chapter 2 using the simple theory. The simple theory
is still valid for a quick, qualitative description of relaxation dynamics. The
generalized theory is necessary when an accurate, quantitative description of

the relaxation dynamics is needed.

Bibliography

[1] M. R. J. Gibbs, J. E. Evetts, and T.A. Leake, J. Mater. Sci. 18, 278

(1983),

[2] S. Roorda, W. C. Sinke, J. M. Poate, D. C. Jacobson, S. Dierker, B. S.
Dennis, D. J. Eaglesham, F. Spaepen, and P. Fuoss, Phys. Rev. B 44

3702 (1991).
[3] C. A. Volkert, J. Appl. Phys. 70, 3521 (1991).

[4] J. H. Shin and H. A. Atwater, in Thermodynamics and Kinetics of
Phase Transformation in Thin Films, edited by M. Atzmon, A. Greer,

J. Harper, and M. Libera, Mat. Res. Soc. Symp. Proc. 311.

[5] J. F. Ziegler, J. P. Biersack, and U. Littmark, The Stopping and Range

of Ions in Solids (Pergammon, New York, 1985).

Chapter 3 96

[6] J. P. Biersack and L. J. Haggmark, Nucl. Instrum. Methods 174, 257

(1980).
[7] Copyright J. F. Ziegler, (1990).

[8] P.A. Stolk, L. Calcagnile, S. Roorda, W. C. Sinke, A. J. M. Berntsen,

Appl. Phys. Lett. 60 1688 (1992).
[9] B. Park and F. Spaepen, J. Appl. Phys. 68 4556 (1990)

[10] S. Coffa, F. Priolo, and A. Battaglia, Phys. Rev. Lett. 70 3756 (1993).

Chapter 4

Modification of Crystallization
Dynamics of Amorphous
Silicon by Irradiation

4.1 Introduction

Amorphous silicon is thermodynamically unstable with respect to crystal sil-
icon, as is evidenced by the highly non-equilibrium methods necessary to
produce it. At elevated temperatures, a-Si will inevitably crystallize. If a
single crystal template is present, single crystal silicon will grow epitaxially
into a-Si. This process is known as solid phase epitaxy (SPE), and begins
to proceed at measurable rates from 700 K. SPE is a thermally activated
process, with a single activation energy of 2.7 eV over 10 orders of magnitude

in SPE rate, and temperature range in the excess of 1300 K [1]. If no crys-

Chapter 4 98

tal template is present, a-Si crystalizes via random nucleation and growth
(RNG). That is, small crystal grains nucleate randomly in the a-Si matrix
and grow in size, until the a-Si is transformed into polycrystalline silicon [2].
If steps are taken to ensure a clean environment, nucleation of crystal sili-
con occurs homogenously within the a-Si matrix rather than heterogeneously
[3, 4]. In explaining the observed nucleatin dynamics of c-Si, the classical
theory of nucleation is often invoked (2, 3, 5, 6].

This random nucleation and growth of crystal silicon in an amorphous
matrix is of a great technological interest, since large grained polycrystalline
silicon films for thin film transistor applications with high ( > 100 cm? /V s)
channel mobility can be produced this way. [7]. Random nucleation of crystal
grains ahead of the advancing epitaxial crystal growth interface is also one
of the fundamental limiting steps in the production of single crystal silicon
on insulator films [8]. Therefore, a thorough understanding of nucleation
kinetics is needed to obtain more control over these technologically important
processes.

Furthermore, since the crystallization of amorphous silicon occurs in a
homogeneous, single component system, and no significant volumetric or

morphological change occurs during nucleation, crystallization of amorphous

Chapter 4 99

silicon is an ideal candidate to study the phenomenon of nucleation and
growth in general. Nucleation and growth is the path taken by many phase
transformations, and has been the subject of intense theoretical studies for
more than 50 years since first treated by Volmer and Weber [10-17]. However,
the theory of nucleation is by no means settled, and arbitrary control of
nucleation kinetics is still out of reach.

One of the reasons for the inability to control the evolution of microstruc-
ture (e.g., grain size and grain size distribution) during nucleation is the fact
that the relative rates of nucleation and growth are usually coupled via ther-
mally activated interface kinetic processes, usually making it impossible to
increase (or decrease) one without affecting the other. Another fundamental
limit in most nucleation processes is the inability to observe the basic events
in nucleation, since the critically sized clusters, which control nucleation, are
usually very small. In the crystallization of amorphous silicon, for example,
the critical size is comparable to or smaller than the minimum observable size.
This inability to directly observe the clusters that control nucleation has to
date prevented the direct experimental characterization of the evolution of
the subcritical and critical cluster size distribution during nucleation. As

a consequence, many critical parameters that characterize nucleation (such

Chapter 4 100

as the time lag for steady state nucleation to be established) can only be
inferred by observing the much larger visible clusters. Unfortunately, it has
been shown [15] that the values obtained by such inferences depend on ex-
perimental and observational conditions, making quantitative comparison of
different experiments very difficult.

Ion irradiation is well known to greatly increase the rate of SPE at ele-
vated temperatures [18, 19]. At low temperatures, ion irradiation can reverse
the motion of an a-Si/c-Si interface, and amorphize crystal silicon in a layer
by layer fashion [20]. Ion irradiation is also known to greatly enhance the
crystal nucleation kinetics [22, 4, 5]. No clear understanding yet exists for
the dependence of these phenomena on ion irradiation. A widely accepted
model for irradiation enhanced SPE exists [23, 24], but it is only phenomeno-
logical. Similarly, the mechanism behind this enhancement remains neither
well defined nor understood, although the classical theory of nucleation is
often invoked to explain the ion enhanced nucleation of crystal silicon.

With ion irradiation, it is possible to allow the ratio of the nucleation
rate to the growth rate to be varied much more widely than is possible under
purely thermal conditions, and thus gain a greater control over nucleation.

Jackson’s model, [23], for example, predicts that at high temperatures and

Chapter 4 . 101

low irradiation rates, the growth rate will be purely thermal. Nucleation,
on the other hand, is known to be affected even under such conditions [25].
With in situ observation of crystal nucleation under irradiation, it is pos-
sible to eliminate many experimental difficulties associated with the study
of nucleation, and thus isolate the effect of ion irradiation. By continuously
observing the entire transformation of one area, many variables that affect
the final analysis can be eliminated, and the observed changes correlated to
the changes in irradiation conditions. In this chapter, we will use zn situ
observation of nucleation under irradiation to obtain information on the evo-
lution of the cluster size distribution, and on the role the irradiation plays in

the modification of the nucleation rate.

4.2 Classical Theory of Nucleation

4.2.1 Formulation of the Theory

The classical theory of nucleation envisions the nucleation and growth process
as the result of a series of small, bimolecular reactions between the growing
cluster and the atoms of the metastable phase [26, 27]. The driving force
for the transformation is the free energy difference between the metastable

phase and the nucleating phase. The free energy of formation, AG, of a

Chapter 4 102

cluster of size n, however, initially rises with increasing size due to the large
number of atoms that are on the surface. Only for large clusters does the
free energy of formation decrease with increasing size. The maximum value
of the free energy of formation is the free energy barrier to nucleation, and
is denoted by AG,. The value of n at which AG, occurs is the critical size,
and is denoted by n,. As we shall see later, AG, and n, play determining
roles in controlling the nucleation kinetics.

Two of the important quantities in the characterization of the nucleation
dynamics are the nucleation rate J, and the time lag for establishment of
steady state nucleation 7, also known as the incubation time. These are
determined by the time evolution of clusters and by the time evolution of
the cluster size distribution, C,,, respectively. Within the classical theory of
nucleation, the net rate of flux between clusters of size n and (n + 1) is given

Lit = Crekt — n+ tknyas (4.1)

where k* and k> are the rate of addition and deletion of one atom for a

cluster of size n, respectively. The subscript ¢ is a reminder that the flux

Chapter 4 103

is, in general, time dependent. For solid phase nucleation which does not
involve long range diffusion or compositional changes, Turnbull and Fisher

[12] have shown that &+ can be written as

ky = One

Ye = yexpl-(AGrii — AG) /2kT], (4.2)
and

Kogr = OntiYntis

Vasey = expl(AGryi — AG, )/2kT). (4.3)

O,, is the number of possible attachment sites of a cluster of size n. For most
atomic configurations, it can be approximated by 4n?/8 [15]. + is the term
that governs the rearrangement kinetics at the interface of growing nuclei.
For the crystallization of a-Si, it is simply related to the rate of movement of

a-Si/c-Si interface, as shown more explicitly in Appendix A. Notice that

kt < ky for n

Chapter 4 104
kt >kz for n> ne. (4.4)

A cluster of a size less than the critical size will tend to shrink, while a cluster
with a size greater than the critical size will tend to grow.
In the case of thermal equilibrium, the cluster size distribution would

simply be

Ce = C,exp(—AG,/kT), (4.5)

where C, is the number of atoms in the initial phase. At such thermal
equilibrium, however, the net rate of nucleation, J, would be zero. In other

words,

I, = Ciky - Croiknys = 0. (4.6)

The “equilibrium” cluster size distribution C¢, which yields a zero nucle-
ation rate, is very useful in the classical theory of nucleation. In fact, some
authors [16, 17] have changed the order of definition, and defined C% to be
the cluster distribution which satisfies Eq. 4.6. AG, is then defined as the
free energy which yields C£, with no physical connection to the free energy

ot formation.

Chapter 4 105

Combining Eqs. (4.6) and (4.1), we can express the time-dependent nu-

cleation rate as

C t C 1,t
Ine = C&kt | — Se). 4.7
yt n'’n Ce Ce 4, ( )
The time evolution of the cluster size distribution is given by
OCr

The entire problem of the classical theory of nucleation, then, is reduced to
solving the above set of coupled difference equations, subject to the appro-
priate boundary conditions.

Finally, the incubation time is defined in terms of the numbers of nucle-
ated grains. At long times, the steady state nucleation rate, J°, is reached,

and the number of nucleated grains is given by

Number of grains = J*(t—7T) jt>7. (4.9)

Equation (4.9) defines the incubation time, 7. An equivalent definition that

allows for the calculation of 7 is given by [15, 17]

Chapter 4 106

°° Tne,t
r= [ [1 — B24] a, (4.10)

Equation (4.11), however, gives the time lag it takes for the nucleation rate
to reach the steady state value at the critical size, which may not be visible.
A more general definition of the incubation time as a function of the cluster

size is given by [17]

°° Tit
Tn = f [1-24] at (4.11)

4.2.2 Solutions to the Problem

Equations (4.7) and (4.8) can be solved in its discrete form [16]. The results,
however, are given in forms of summations and integrals, and are unwieldy.
More compact forms can be obtained by replacing Eq. (4.8) by its continuum
approximation, and solving the resulting partial differential equation. The

differential form of Eq. (4.8) is

One Of yd (Cut
oon we (ed ( at)). (4.12)

Chapter 4 107

This is also known as the Zeldovich-Frenkel [14, 28] equation, and is
formally equivalent to the diffusion of clusters in the size space, with k* and
C,,,t as the diffusion coefficient and the concentration, respectively.

The accuracy of above continuum approximation has been estimated by
Wu [16]. He has found that it is accurate to a few percent, and that a
more accurate approximation can be obtained by using a slightly different
form of k*C'%. A more significant problem is the assumption that nucleation
proceeds by addition or deletion of single atoms of the metastable phase. In
reality, nucleation may proceed by bursts; that is, more than one atom may
be added to or deleted from a cluster at one time. In the case of a-Si, there
is evidence which suggests that solid phase epitaxy proceeds by the addition
of more than one atom at a time [29]. Such possibility has been examined
before. Wu, for example, has obtained expressions for Eq. (4.12) and the
incubation time in such cases [16, 17]. However, no closed expressions for
the nucleation rate and incubation time are given. On the other hand, no
complications would arise if nucleation proceeds by the addition of a fixed
number of atoms, since we could simply scale everything in terms of the
number of atoms added (or deleted) at one time.

Much more significant problem lies with AG,,. It is the most important

Chapter 4 108

factor in determining the dynamics of nucleation, since it controls both the
nucleation rate and the time evolution of the cluster size distribution. Tradi-
tionally, it has been identified with the the free energy of formation[26]. Yet
as Wu has pointed out, such identification is neither necessary nor wholly
justified. Even if we do identify AG, with the free energy of formation of a
cluster of size n, the exact functional form of AG,, remains controversial. In
the case of solid state nucleation, where the entropy due to movement and
rotation of clusters can be neglected, many authors [11, 12, 15, 26] have used

the form

AG, = —AGn+Aon?’, (4.13)

where AG is the difference in Gibbs free energy per atom between the phases,
n the number of atoms in a cluster, A a geometrical factor for the surface area,
and o is the free energy associated with the interface between the crystal and
the surrounding matrix.

Equation (4.13), however, is hard to justfy physically. It is not clear
whether it is physical to assign a mathematical surface to a cluster that

contains only a few atoms, especially when the width of the transition region

Chapter 4 109

between the cluster and the surrounding matrix may be of the same order as
the diameter of the cluster. Similarly, it is also not clear whether it is physical
to assign a surface free energy to such a “surface”. In the case of solid state
nucleation, it may also be possible that the possible strain contribution to the
volume free energy difference may be dependent on the cluster size, making
AG be a non-trivial function of n.

In the classical theory of nucleation, however, such difficulties are ignored,
and bulk values of AG and o are used. In this thesis, the classical theory
of nucleation will be assumed to be accurate, and Eq. (4.13) will be used
for all of the analysis. Furthermore, we will assume that nucleation occurs
as the result of the addition and deletion of single atoms. Doing so allows
us to use compact, closed expressions for the nucleation rate, the incubation
time, and the cluster size distribution. To its justification, we note that
despite its lack of firm physical justification, the classical theory of nucleation
with these simplifications has been used successfully many times to describe
the observed kinetics of nucleation. [2, 5, 30]. Assuming Eq. (4.13) to be

accurate, the free energy barrier to nucleation, AG,, and the critical cluster

Ne, may now be explictly calculated. They are

Chapter 4 110

AMF 3

2Ac 7°

In the case of amorphous silicon, assuming spherical clusters, AG and o are
0.1 eV and 0.32 J m~?, respectively [21]. The critical size, n,, is then 110
atoms. Figure 4.1 shows the calculated values of AG, for silicon.

Boundary conditions are required in order to solve Eq. (4.7). For the
steady state, it is usually assumed that as n — 0,C% — C®, and asn >
oo, C3 — 0. In such case, we can sum Eq. (4.1) to obtain

set a
DD Geke = Ge Ge =} (4.16)

where the limits of summation u and v are such that u< n,. < v. The sum is
evaluated by approximating it by an integral, expanding the integrand with
a Taylor expansion around n,, and then dropping the higher order terms.
Such an approximation is possible because near n., AG, is at its maximum,
and is slowly varying. Given these assumptions, most authors {11, 13, 12, 16]

have obtained a steady state nucleation rate of the form

Chapter 4 111

10 | I | |

—5- .

_l | | |
0 100 200 300 400 500

n (# of atoms in a cluster)

Figure 4.1: Free energy of formation for silicon.

Chapter 4 112

P= Zktce

—AG,
= C.On,yZ exp ( iT ) , (4.17)
where Z is the Zeldovich factor, given by
AG
Zo= (arm): (4.18)

Some controversy exists on the steady state cluster size distribution, and
the incubation time however. Depending on the approximation used, many
authors have derived somewhat different forms of C%. Solving Eq. (4.12)
numerically, Kelton et al. [15] have proposed that the expression derived by
Kashchiev to be the most accurate. Recently, however, Wu [17] has shown
that the agreement to be a fortuitous one, and proposed a new set of expres-
sions. In this thesis, the expressions derived by Wu will be used exclusively.

According to Wu, the steady state cluster size distribution near n, is given

Ce = sertel VR Z(n — n.)]C%. (4.19)

Chapter 4 113

The incubation time, assuming no prior cluster size distribution, is given by

DO L
Qn Z2kt. Sanz sy Zr

L = \n(VrZn,) + 0.3.

(4.20)

Notice that the incubation time is inversely proportional to 4, which we
stated to be simply related to the growth velocity. Although the Zeldovich
factor and the prefactor, ZL, both depend on temperature, they do so only
weakly. In the case of amorphous silicon, the deviation of incubation time
from Arrhenius behavior in the temperature range of 700K to 1000K is very
small. Therefore, the incubation time has nearly Arrhenius behaviour within
this limited temperature range. Therefore, the absolute value of the apparent
activation energy of incubation time within the temperature range should be
equal to the absolute value of activation energy of growth velocity. This is
indeed found to be the case, for both thermal and irradiation enhanced cases

(2, 6], and shows the validity of identifying y with the growth velocity.

Chapter 4 114

4.3 Experiments

The experiments consisted of two parts. In both parts, the samples used
were 100 nm thick a-Si films on S102 free of supercritical clusters created
by implanting chemical vapor deposited Si films with 70 KeV Si to a dose
of 1 x 101° ions cm? at 77 K. In the first part, performed at Caltech, the
’ nucleation and the growth rates of crystal silicon were investigated at high
temperatures and low dose rates to confirm that the growth rate is unaffected
by the ion beam under such conditions, but the nucleation rate is. The
samples were irradiated with 600 KeV Xet+ ions at a dose rate of 2 x 10!°

ions cm7~?

sec”! at an irradiation temperature range of 873K < T;,, < 903K.
Since the accurate control of the temperature is critical, a specially designed
heating stage was used. Figure 4.2 shows a schematic of the experimental
setup used for the nucleation experiments performed at Caltech. At these
high temperatures, it was impossible to measure the beam current directly
off the sample due to electrons emitted from the heating elements. Therefore,
the dose rate was maintained by first stabilizing the beam at the desired dose

rate, and then maintaining a constant accelerating current and secondary

electron current. The ion beam was also periodically interrupted briefly

Chapter 4 115

to ensure that the dose rate was constant within + 10 %. The samples
were then analyzed ez situ by transmission electron microscope (TEM). The
growth rate was determined by measuring the average of the square roots of
the areas of the ten largest grains in each micrograph, and following the time
evolution. Presumably, the largest grains were the first ones to nucleate in
each samples, all around the same time into irradiation.

In the second part, nucleation of crystal silicon under irradiation was mea-
sured in situ at the High Voltage Electron Microscope/Tandem Accelerator
facility at Argonne National Laboratory [31]. The HVEM/Tandem facility
consists of a 2 MeV tandem accelerator interfaced to a high voltage electron
microscope, with continuous video data-image acquisition during irradiation.
Free standing a-Si films created by back etching the substrate were irradiated
with 600 KeV Xe* ions at a constant dose rate of 4 x 10'° ions~? sec? at
Tirp = 903.4. In the second part, only the total irradiation dose was varied
between different samples. In both parts, TEM images were digitized using
a frame grabber, and analyzed with the aid of an image analysis program
[32]. For time-resolved analysis of nucleation rates, the time evolution of

each individual grain was followed to ensure accuracy of counting.

Chapter 4 116
Faraday Cage
Fl Heating
ap Sample Block Elements
> | Cc
Ion > SSS
Beam S
7 C
N __
O e
CWC Set Screw TC pp
33 Sample Wires
Vv |
Current Integrater panel

Figure 4.2: Experimental setup for the irradiation enhanced nucleation in
the high temperature, low ion flux regime, perfomed at Caltech.

Chapter 4 117

4.4 Results and Discussion

Results from the first part of experiments are shown in Figs. 4.4 and 4.5.
Table 4.4 shows the measured nucleation rate and incubation time under
irradiation, and the expected values for purely thermal transformations from
Iverson et al. [2]. The incubation time was obtained by subtracting the
estimated time needed for nucleated clusters to grow to observable size from
the intercept of plots of variation of grain density with time. Quantitative
comparison of thermal and irradiation-enhanced values is inappropriate, since
the measurement conditions were not identical. Nonetheless, it is evident that
both the nucleation rate and the incubation time are significantly affected by
irradiation. However, as Fig. 4.5 shows, the growth rate of nucleated clusters
is not affected by irradiation, as predicted by Jackson [23]. The solid line
in Fig. 4.5 is not a fit to the data, but the expected SPE rate for the [111]
orientation [1, 33]. The morphologies of the nucleated grains were irregular,
heavily twinned, and in many cases elongated. This is typical of thermally
grown grains, but is in contrast to round, isotropic shapes characteristic of
grains whose growth is enhanced by irradiation, as shown in Fig. 4.3. These

results are in agreement with previous reports {2, 34] which showed that

Chapter 4 118

the growth rate of nucleated grains seems to be limited by movement of the
[111] planes, and demonstrate convincingly that the growth rate, and thus
the interfacial rearrangement rate of nucleated grains, were only thermally

activated, and had no irradiation-enhanced component.

T(K) Nucleation rate Incubation time
(grains cm7> sec™!) (sec)
Under Thermal Under Thermal
irradiation | anneal irradiation anneal
873 1.1 x 10° 3.8 x 10° 1.4 x 10° 3.6 x 104
888 2.1 x 103° 1.2 x 108 1.1 x 10° 2.0 x 104
903 2.1 x 108 3.6 x 10° 7.2 x 10? 1.1 x 104

Table 4.1: Comparison of the irradiation enhanced and the thermal values
of the nucleation rate and the incubation time. The irradiation enhanced
values were measured from experiments performed at Caltech.

The nucleation rate is, as Eq. (4.17) shows, linearly dependent on the in-
terfacial rearrangement rate and exponentially dependent on the free energy
barrier to nucleation. The incubation time is also dependent on the free en-
ergy barrier, although not as strongly as is the nucleation rate. The observed
enhancement in the nucleation rate and the incubation time in the absence
of enhancement in the interfacial rearrangement rate, as evidenced by the
thermal rate of growth for the nucleated grains, strongly suggests that ion

irradiation modifies the thermodynamic component of the nucleation. The

Chapter 4 119

a) Irradiation-enhanced growth regime

Figure 4.3: Comparison of grain morphology of grains grown under thermal
and irradiation-enhanced growth conditions

Chapter 4 120
1.5x10'® I T T
Ve © 630° C
Oo 1.0 4 615° C
N + 600° C
6p)
5 0.54 4
‘an
i) Oo
0.0 | I l
0 500 1000 1500 2000 2500 3000

Time (sec)

Figure 4.4: Measured grain density, corrected for the crystal fraction, of
samples irradiated at Caltech with 600 KeV Xe** ions at a flux of 2 x 10'°

ions cm~?

square fits to the data.

sec! at the indicated temperatures. The solid lines are least

Chapter 4 121

T(K)
0 910 900 890 880 870 860
10 | |
® 107"
” Expected rate of movement for
oct, [111] planes
1072 | | |
1.10 1.12 1.14 1.16
1000/T (Kk)

Figure 4.5: Measured growth rate of the nucleated grains under irradiation.
The solid line is the expected SPE rate of the [111] planes from refs. [1, 33].

Chapter 4 122

relatively minor effect of irradiation on the incubation time compared with
the nucleation rate is also consistent with this conclusion.

Results of in situ measurements of the nucleation rate during irradiation
are shown in Figs. 4.6 and 4.7. Under continuous irradiation, the number
of grains rises linearly with time, indicating a steady state nucleation, as
shown in Fig. 4.6. The steady state nucleation rate under irradiation is
found to be 2.9 x 10!? grains~? sec-!. A control sample which was annealed
at 903 K under identical conditions but without ion irradiation did not show
any nucleation of crystal grains until after approximately 2 hrs of annealing.
Interestingly, when the irradiation is terminated after steady state nucleation
is observed, we do not observe any drastic changes in nucleation kinetics
following termination of irradiation. The arrow marks the moment when the
irradiation is terminated, at the dose of 8 x 10!° ions cm~?. Indeed, after
termination of irradiation, nucleation seems to continue for a few minutes at
the same rate as under irradiation, followed by a gradual decrease at later
times.

Figure 4.7 shows the nucleation kinetics of samples for which irradia-
tion was terminated prior to establishment of steady state nucleation under

irradiation. The arrows mark the times at which the ion irradiation was ter-

Chapter 4 123

| | | | | | _
1.5x10° F “0 Continuous irradiation
A A 8x10'" jons cm

Oo
Ol

Grains/cm
fo)

m9)
a9)
9)

| | | |
1000 2000 3000 4000 5000 6000 7000

Time (sec)

Figure 4.6: In situ measurements of the nucleation rate for samples for which
steady state nucleation under irradiation is observed.

Chapter 4 124

5 LI | | | i | _
1.5x10 O 3.7x10'8 ions cm72
- A 6.7x10'° jons cm72
c 10 + 7x10'% jons em72
OT |
NN
eC +
“— OOF + 7
‘o)
O yt A O
oO att dda
vy, + A
0.0 + O O ~
| l | ‘| I |

1000 2000 3000 4000 5000 6000 7000
Time (sec)

Figure 4.7: In situ measurements of the nucleation rate for samples for which
irradiation is terminated before steady state nucleation under irradiation is
observed.

Chapter 4 125

minated. The doses were, in the increasing order, 3.7, 6.7, and 7 x101° ions
cm”. For the smallest dose,the crystallization kinetics were similar to that
of purely thermally induced crystallization, except for one very important
difference. A few (<1x 10" cm™~*) grains nucleated early, and grew to large
(> 1000A) sizes with no further nucleation of crystal grains. Only later, at a
time comparable to thermal incubation time, did nucleation of crystal silicon
occur again. Similar transformation kinetics characterize other samples for
which irradiation was terminated before steady state nucleation under irra-
diation is observed. After an incubation time, a few grains nucleate initially,
followed by a characteristic “quiescent” time period during which negligi-
ble nucleation occurs. Subsequently, nucleation begins again and reaches a
steady-state rate of 2+ 0.5 x 101’ grains cm~? sec™!, which is one order of
magnitude slower than the steady state nucleation rate under irradiation.
With increasing dose, we find that more grains nucleate initially, and that
the “quiescent” period is shortened. However, the nucleation rate following
the quiescent period remains unchanged.

The observed dynamics of nucleation rate upon termination of irradiation
at, various doses can be explained in the context of classical nucleation theory,

if we assume that during the quiescent period, the free energy of formation

Chapter 4 126

and the cluster distribution evolves from values modified by irradiation to
thermal values. Defect injection into a-Si is known to significantly increase its
stored enthalpy [9]. Moreover, as we have shown in Chapter 2, ion irradiation
can maintain excess defect population in a-Si, even when the irradiation
temperature is high. It is not clear whether irradiation injected defects affect
AG, the free energy difference between a-Si and c-Si, or a, the interfacial
free energy of a crystalline cluster, or possibly both. However, as will be
shown later, the modification of either parameter by irradiation affects the
nucleation rate in a similar manner.

Assuming that the second nucleation regime following the quiescent pe-
riod corresponds to the onset of thermal nucleation, and using Eq. (4.17),
one can readily estimate the changes in: i) the free energy barrier to nu-
cleation, AG,; ii) the critical cluster size; iii) the change in the free energy
difference between a-Si and c-Si AG, assuming that change in AG, occurs
through change in AG only; iv) the change in the interfacial free energy o,
assuming that change in AG, occurs through change in o only. The results
are summarized in table 4.4. From the values given in table 4.4, we can also
calculate the free energy of formation, AG, the steady state cluster distri-

bution, C%, and its gradient, dC/dn. The results of these calculations are

Chapter 4 127

illustrated in Figs. 4.8, 4.9, and 4.10.

Under irradiation Under thermal conditions

AG. 5.31 eV 5.5 eV
Ne 104 atoms 111 atoms
AG 0.102 atom~! 0.1 atom7!
o 0.316 J m=? 0.32 J m~?

Table 4.2: Calculated changes in thermodynamic parameters controlling nu-
cleation.

It is clear from Figs. 4.9 and 4.10 that the changes in the cluster size
distribution and its gradient under irradiation are generally similar, whether
change in AG, occurs through change in free energy difference AG or a
change in the interfacial free energy o. In both cases, the critical cluster
size is smaller during irradiation than during a purely thermal treatment.
The cluster size distribution is also much larger, and much steeper during
irradiation than during a purely thermal treatment. This similarity means
that we cannot tell which effect is more dominant, however. In either case,
when irradiation is terminated, the cluster size distribution becomes too large
and steep to be able to be supported by new, thermal values of the free

energy of formation, AG,,. This now unstable cluster size distribution has to

Chapter 4 128

NX.
—S | S
O 100 200 500 400 500
n (# of atoms in a cluster)

Figure 4.8: Comparison of the free energies of formation, for thermal and
irradiation-enhanced cases. The solid line is the free energy of formation
curing thermal anneal. The free energy of formation under irradiation is cal-
culated assuming assuming i) modification of the free energy difference only
(dashed curve), and ii) assuming modification of amorphous-crystal interface
free energy only (dash-dot curve).

Chapter 4 129

—2
” 10
; aN
SO 1077 . -
)) —-Change in Ag only
D —--Change in o only
+ 107§L —— Thermal |
” i/i
-8 ~Lme
— 1078b ==
o pthermal
Cc
+ 10719 | | | |

60 80 _ 100 120 140
n (# of atoms in a cluster)

Figure 4.9: Comparison of the calculated cluster size distribution, C’ ac-
cording to Eq. (4.21). The solid curve is the cluster size distribution under
purely thermal conditions. Estimates of C% under irradiation were made
assuming i) modification of the free energy difference only (dashed curve),
and li) assuming modification of amorphous-crystal interface free energy only
(dash-dot curve). The critical cluster sizes under irradiation and under ther-
mal conditions are indicated.

Chapter 4 130

~4
10 "RK | |
_ —-Change in Ag only
1078 L Sy —--Change ino only
SS, —— Thermal

40712 | J ! | |
60 80 100 120 140

# of atoms

Figure 4.10: Comparison of the gradient of calculated cluster size distri-
bution, dC'g/dn according to Eq. (4.21). The solid curve is the expected
values under purely thermal conditions. Estimates of dC%/dn under irradi-
ation were made assuming i) modification of the free energy difference only
(dashed curve), and ii) assuming modification of the amorphous-crystal in-
terface free energy only (dash-dot curve). The critical cluster sizes under
irradiation and under thermal conditions are indicated.

Chapter 4 131

evolve toward the thermal steady state distribution. One possible scenario
is a burst of nucleation, caused by the growth of all the excess clusters near
the critical region. Such a burst, however, is not observed. Furthermore,
the thermal critical cluster size is larger than the critical cluster size under
irradiation, which implies that some clusters that would have grown under
irradiation now tend to shrink. Taken together, it is much more likely that
the cluster size distribution “crashes” toward thermal values. The net result
of such changes in cluster size distribution is a net backward flow of clusters
tn size space. Thus when irradiation is terminated, nucleation will cease
until the thermal cluster distribution is established, which is consistent with
observations. When the steady state under irradiation is already observed,
however, it implies that there are already many clusters that are supercritical
but still unobservable. Therefore, termination of irradiation in this case will
yield a smaller and much more gradual change in nucleation rate, since the
supercritical yet unobservable cluster population will continue to contribute
to the observed nucleation rate. These evolution of clusters size distribution

upon termination of irradiation is shown schematically in Figs. 4.11 and 4.12.

Chapter 4 132

Fon
n fen}
= n
5 . Ne
Q I
& {
iS
A \
f Time \
c| ~——
= O
'@) —a
, | n ! Ne
[~ Ih,

Figure 4.11: Schematic description of evolution of the cluster size distribution
function and the nucleation rate for samples for which steady state nucleation
under irradiation is not observed.

Chapter 4 133

ow}
oO
Ne

3)
a | JS
fot ————
‘dD ——<——
i I n
0 I Ny

f Time \
SG ————
rs
oOo Uv _
na n
| n Ne
I In C

Figure 4.12: Schematic description of evolution of the cluster size distribution
function and the nucleation rate for samples for which steady state nucleation
under irradiation is observed.

Chapter 4 134

4.5 Conclusion

We have demonstrated that ion irradiation greatly enhances the crystal nucle-
ation rate in an amorphous silicon matrix, even when irradiation conditions
were such that the crystal growth rate is unchanged by irradiation. In situ
observation of crystal nucleation has indicated that termination of irradja-
tion during the transient stage of nucleation produces changes in nucleation
rate that is best explained by assuming that irradiation affects the thermody-
namic quantities that control nucleation process, presumably through defect
injection into a-Si matrix. The results suggest clearly that irradiation can
be used to independently control thermodynamic and kinetic quantities that

control irradiation induced phase transformations.

Bibliography

[1] G. L. Olson and J. A. Roth, Mater. Sci. Reports, 3 1 (1988).
[2] R. B. Iverson and R. Reif, J. Appl. Phys., 62 1675 (1987).
[3] S. Roorda, Ph.D. dissertation, FOM, Netherlands (1990).

[4] Tomonori Yamaoka, Keiji Oyoshi, Takashi Tagami, Yasunori Arima,

Ken Yamashita, and Shuhei Tanaka, Appl. Phys. Lett., 57 1970 (1990).

[5] C. Spinella, A. Battaglia, F. Priolo and S. U. Campisano, Europhys.

Lett., 16 313 (1991)
[6] J. S. Im and Harry A. Atwater, Nucl. Inst. Meth. B, 59 422 (1992)

[7] T. Ohshima, T. Noguchi, and H. Hayashi, Jpn. J. Appl. Phys., 25 L291

(1986).

135

Chapter 4 136

[8] H. Ishiwara, A. Tamba, and S. Furukawa, Appl. Phys. Lett., 48, 773

(1986).

[9] S. Roorda, W. C. Sinke, J. M. Poate, D. C. Jacobson, S. Dierker, B. S.

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3702 (1991).
[10] M. Volmer and A. Weber, Z. Phys. Chem. 119, 227 (1926).
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[15] K. F. Kelton, A. L. Greer, and C. V. Thompson, J. Chem. Phys. 79

6162 (1983).
[16] David T. Wu, J. Chem. Phys. 97, 1922 (1992).
[17] David T. Wu, J. Chem. Phys. 97, 2644 (1992).

[18] J. Linnross, G. Holmen and B. Svennson, Phys. Rev. B 32 227 (1985).

Chapter 4 137

[19] J. Linnross, R. G. Ellliman and W. L. Brown, J. Mater. Res. 3 1209

(1988)

[20] W. L. Brown, R. G. Elliman, R. V. Knoel, A. Lieberich, J. Linnross,
and J. S. Williams, in Microscopy of Semiconductor Materials,edited by

A. G. Cullis (Institute of Physics, London, 1987) p. 61.
[21] K. N. Tu, Appl. Phys. A 53, 32 (1991).
[22] J. S. Im and Harry A. Atwater, Appl. Phys. Lett. 57, 1766 (1990)
[23] K. A. Jackson, J. Mater. Sci. 51 1218 (1988)

[24] J. S. Custer, A. Battaglia, M. Saggio, and F. Priolo, Phys. Rev. Lett.

69 780 (1992)

[25] J. S. Im, Jung H. Shin and Harry A. Atwater, presented at Fall ’91

Materials Research Society Conference.

[26] A good introduction to the classical theory of nucleation can be found
in J. W. Christian, The Theory of Transformation in Metals and Alloys,

2nd. ed. (Pergammon, Oxford, 1975) pp. 442-448.

[27] K. Binder and D. Stauffer, Adv. in Physics, 25 343 (1976).

Chapter 4 138

[28] J. Frenkel, Kinetic Theory of Liquids (Oxford University, Oxford 1946),
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Chapter 5

Estimate of Change in the
Free Energy of Crystallization
Using Generalized Activation
Energy Theory

5.1 Introduction

In Chapter 4, we explained the observed enhancement in kinetics of nucle-
ation under irradiation in the absence of the enhancement in the interfacial
rearrangement rate by postulating that defects generated by irradiation af-
fect the thermodynamic quantities that control nucleation. With the gener-
alized activation energy theory developed in Chapter 3, it is possible to follow
the time evolution of the defect population under irradiation conditions em-

ployed in the nucleation experiments. From previous work that relates the

139

Chapter 5 140

degree of structural unrelaxation (defect density) with excess enthalpy [1, 2],
this allows us to estimate the change in the free energy difference between
amorphous and crystal silicon during irradiation, and to compare with ex-
perimental data obtained in Chapter 4. It is true that by doing so, we are
neglecting the effect of irradiation on the surface free energy. However, there
are no good estimates of the effect of defect generation on the interfacial free

energy. Thus, no attempt will be made to include this effect.

5.2 Estimates of Change in the Free Energy

5.2.1 Estimating the Defect Population

Although it would be straightforward to directly simulate the defect popula-
tion during irradiation at high (903 K) temperature, it is not very practical.
At such high temperatures, the defects are very mobile; i.e. they have a very
short lifetime. At 903 K, the lifetime of a defect with activation energy of 0.6
eV is expected to be less than 1 x 10~® sec. For an accurate simulation, the
timestep taken during simulation would have to be of the order of 1 x 1077
sec. This would require computation time in the excess of 4000 hrs on IBM
RISC System/6000™. Computation time could be shortened by using an

activation energy greater than 0.6 eV as the lower limit for the activation

Chapter 5 141

energy. Doing so, however, renders the results inaccurate, since the reaction
constant vy, is dependent on the lower limit of activation energy used.

Such dependence of results on the time step taken and the lower limit for
activation energy is actually beneficial, since they can be used to put upper
and lower limits on the defect concentration during irradiation without hav-
ing to resort to excessively long computations. The lower limit on the defect
concentration was determined by simulating the evolution of defect popula-
tion with timesteps that are longer than the estimated defect lifetime. This is
equivalent to letting defects continue to annihilate even after they are already
annihilated, and results in an underestimate of the defect population. By do-
ing several simulations with 0.6 eV as the lower limit for activation energy,
but employing successively shorter (but still practical) timesteps, a good es-
timate for lower limit for defect population can be made. The upper limit
on defect concentration during irradiation was determined by simulating the
evolution of defect population with a lower limit of activation energy that is
greater than 0.6 eV. The defects with low activation energies are very mobile,
and exert controlling influence on the total defect density by annihilating all
defects, regardless of their activation energies. Therefore, using a lower limit

of activation energy that is higher than 0.6 eV is equivalent to neglecting

Chapter 5 142

their dominant contribution to defect annihilation, and results in an over-
estimate of the defect population. To ensure that the timesteps were long
enough, two simulations were performed, with two different timesteps but
with the same lower limit on activation energy. Simulations were performed
with a successively lower limit on the activation energy until the results of

the two simulation diverged by more than 0.1 %.

Qmin (eV) | Defect Concentration (cm~*)
1.7 5.65 x 10*°
1.4 4.23 x 10"
1.1 2.95 x 1019
0.85 2.05 x 10'°

Table 5.1: Effect of changing Qmin on steady state defect concentration under
irradiation.

Timestep (sec) | Defect Concentration (cm~*)
0.0002 3.17 x 1078
0.00005 7.28 x 10'®
0.00002 1.06 x 10'°

Table 5.2: Effect of changing timestep on steady state defect concentration
under irradiation. Qmin = 0.6 eV.

Chapter 5

143
Effect of changing Q,.,
107° 7
” 10'FL —~
° 108L ~
16 l | |
"10 1.5 2.0 2.5 3.0

Q (eV)

Figure 5.1: Effect of changing Qmin on steady state defect concentration
under irradiation.

Chapter 5

Q (eV)

Figure 5.2: Effect of changing the time step on steady state defect concen-
tration under irradiation. Qmin = 0.6 eV.

Chapter 5 145

The results are shown in Figs. 5.1 and 5.2, and summarized in tables 5.2.1
and 5.2.1 As the timestep is decreased, the total defect population increases,
as expected. As the lower limit of activation energy is decreased, the defect
density decreases, as expected. Both calculations seem to converge to a
value near 1.5 x 10° defects cm~*. The lower and upper limits to steady
state defect concentrations are 1.06 x 10’ defects cm~? and 2.05 x 101° defects
cm, respectively. These limits on increase in the defect concentration under
irradiation can be scaled to limits in the increase in the free energy difference,
if we assume that the each defect contributes equally to stored enthalpy of
unrelaxed a-Si. From previous Chapters, we know that increase in defect

density by 3.55 x 107° defects cm~® results in increase in enthalpy of a-Si by

5.28 x 10-7 eV atom™!. Scaling, we find the limits on increase in free energy

to be

0.00158 eV/atom < A(AG) < 0.00305 eV/atom. (5.1)

These limits are in excellent agreement with the experimentally derived
change in the free energy from Chapter 4, which showed change in free energy

of 0.002 eV/atom. Such an excellent agreement is not to be taken literally,

Chapter 5 146

for it is likely to be fortuitous. Rather, it should be considered only to show
the validity of the idea that excess defects generated by irradiation affect the

thermodynamic quantities which control nucleation.

5.3 Conclusion

Using the generalized activation energy spectrum theory developed for struc-
tural relaxation of a-Si, we were for the first time able to quantitatively esti-
mate the role that irradiation plays in affecting the thermodynamic factors
which control nucleation. The estimated values are in excellent agreement
with experimentally derived values. This structural unrelaxation approach
towards the effects of irradiation on nucleation may be extended to inves-
tigate other effects. For example, Spinella et al. have shown that the nu-
cleation rate under irradiation depends non-linearly with irradiating ion flux
[3]. Such a non-linear effects would be a natural consequence of generalized
activation energy theory, since with increasing dose rate, the dynamic an-
nealing effects will be more pronounced, resulting in a non-linear increase in

the defect density with increasing dose rate.

Bibliography

{1] E. P. Donovan, F. Spaepen, J. M. Poate, and D. C. Jacobson, Appl.

Phys. Lett. 55 1516 (1989).

[2] S. Roorda, W. C. Sinke, J. M. Poate, D. C. Jacobson, S. Dierker, B.S.
Dennis, D. J. Eaglesham, F. Spaepen, and P. Fuoss, Phys. Rev. B 44

3702 (1991).

[3] C. Spinella, A. Battaglia, F. Priolo and S. U. Campisano, Europhys.

Lett. 16 313 (1991).

147

Conclusion 148

Conclusion

This thesis consisted of two parts. The first part investigated the dynam-
ics of defect creation (structural unrelaxation) and annihilation (structural
relaxation) in amorphous silicon, while the second part investigated the en-
hancement of nucleation of crystal silicon in amorphous matrix by irradiation.

In the first part, using electrical conductivity as the probe, We have shown
that the density of defect states, D(Q), exists, and that the standard sim-
ple activation energy spectrum theory for structural relaxation is applicable
to structural relaxation of amorphous silicon. Using the standard theory of
structural relaxation, we have measured the value of the density of defect
states, D(Q), for amorphous silicon. We also developed the generalized ac-
tivation energy spectrum theory, which was shown to be a more accurate,
and more physically reasonable description of defect reaction kinetics during
structural relaxation of amorphous silicon than the standard, simple theory
of structural relaxation.

In the second part, we have shown that ion irradiation greatly enhances
the crystal nucleation rate in an amorphous silicon matrix, even when irra-

diation conditions were such that the crystal growth rate is unchanged by

Conclusion 149

irradiation. Jn situ observation of crystal nucleation has indicated that de-
fect injected by irradiation affect the thermodynamic quantities that control
nucleation process, presumably through defect injection into a-Si matrix.
Finally, the two parts are brought together by using the generalized ac-
tivation energy spectrum theory developed for structural relaxation of a-Si
to estimate the modification of the thermodynamic factors which control nu-
cleation by irradiation. The estimated and experimentally measured values
are in excellent agreement with each other, and confirms the validity of the

this thesis.

Appendix A

Growth Rate and Interface
Rearrangement

In Chapter 4, we stated that the interfacial rearrangement frequency, y, of
Eq. 4.2 is simply related to the growth rate of large clusters. Here we show
this explicitly, starting with Eqs. 4.2 and 4.3. These equations, respectively,
give the rate at which an atom attaches to or detaches from a cluster of size
n. Now, imagine a large cluster (such as the substrate), with n > n.. In

such a case, the rate at which the size of that particular cluster changes is

given by
dn
hn ee Po
n ~ n in —A n
= Ony (exp |S ST AG | — exp fee oT a a.)

150

Appendix A 151

For a large cluster, AG41 — AG, is simply -AG. We can therefore simplify

the above equation to

kt — ky

= Onyexp (Se) f — exp (-25)| , (A.2)

To convert dn/dt to a growth velocity, we note that On « n?/°. Since the

radius of a cluster, r, is proportional to n!/°, we can find the growth velocit
prop & y

= dr/dt.

dr AG AG
aX TP (se) f — exp (-F)| (A.3)

If we assume that 7 is thermally activated, we find

< O Yo exp (-E= 82) f — exp (-25)| (A.4)

where F is the activation energy for an interfacial jump. We immediately
recognize this equation as the equation for velocity of planar crystallization
front {1]. The exp(—AG/2kT) term arises because of the way F is defined in

Turnbull-Fisher theory, as shown in fig. A.1. A more conventional definition

Appendix A 152
of the activation energy for an interfacial jump would define the activation

energy as Ei’ in fig. A.1, which is simply EF — AG/2, in agreement with Eq.

A.4. This shows that 7 is related simply to the rate of SPE.

Appendix A 153

AGn

AGn4+1

Figure A.1: Schematic of activation energy for interfacial jump

Bibliography

[1] F. Spaepen and D. Turnbull, Materials, in Laser Annealing of Semicon-
ductors, edited by J. M. Poate, J. W. Mayer (Academic Press, New York

1982) pp. 15-42.

[2] G. L. Olson and J. A. Roth, Mater. Sci. Reports, 3 1 (1988).

154

Appendix B

Code for Simulation of
Thermal Anneal

The following is the computer code for simulating defect dynamics during
an anneal, using the generalized activation energy theory of Chapter 3. The
code is written in C. The code assumes presence of a file named “list.anl”,
which contains the names of data files. The data file is assumed to contain
a header line, then the values of anneal temperature, anneal time, and the
reaction constant, all in one line. The next 200 lines of the data files are
assumed to be the values of activation energies and the values of N(Q) prior
to anneal.

The time evolution of the defects is simulated by letting

155

Appendix B 156

N(Qt+81) = N(Q,t) + 6x MOO.

where ét is the time step. The value of time step is not fixed, but is changed
throughout the simulation to be 1/10th of the lifetime of the most mobile
(1.e., with lowest activation energy) defect that is still in siginificant number.
Integration is performed by simply trapezoidal approximation. Since the
functions to be integrated are smoothly varying, the error in integration is
estimated to be small. The simulation stops when the total anneal time
exceeds the anneal time read in from the data file. The results are then

written to a file named “name of data file.an]”.

/* Program for calculating time evolution of N(Q) */

/* upon annealing. It reads in data from files listed */
/* in ‘list.anl’. It uses adaptive time */

/* scaling to facilitate calculation */

#include
#include

/* Minimum population for defects */
#define min_population 1e-30

/* Bin size of activation energies */
#define Q_bin 200

/* Boltzmann constant */

Appendix B 157

#define k .0000862

#define max_list 20
#define name_length 15

double N_integrate(double NQ[]);

double N_eQ_integrate(double NQ[]);

double find_t (double NQ[] ,double integral_1,double integral_2);
void change_name(char f_list[]);

double v,g,T;
double Q[Q_bin];

main()

*/ Variables. v = reaction constant, g=generation term */
*/ T = anneal temperature * NQ[] are arrays of data */

extern double v,g,T;

extern double Q[Q_bin] ;

double integral_1,integral_2,annl_time,newN,
tot_time,t_interval;

double NQ_i[Q_bin], NQ_2[Q_bin];

char comments[80], name[name_length] ;

char f_list[max_list] [name_length] ;
register unsigned long int i;

int j,1,str_length;

FILE *fp;

/* initialize f_list so that they are all EOF’s */
for (1=0;1

f_list[1] [o]=’\0’;

/* Read the the list of files to read */

Appendix B 158

fp=fopen("list.anl","r") ;
fgets(comments, 80, fp);
1=0;

fgets (name,name_length, fp) ;
str_length = strlen(name)-1;
name[str_length]=’\0’;
strcpy(f_list[1] ,name) ;

l++;

while ((l

fclose(fp);

/* Read in the appropriate values for variables */

for (1=0 ; (1if ((fp=fopen(f_list [1] ,"r'")) !=NULL)

/* Read in sequence of temperature,anneal time,v, g */
fgets(comments, 80, fp);
fscanf (fp, "Alf ,A1f,%1£ , 41f",&T,&annl_time,&v,&g) ;

/* Read in the data, and put it into N(Q)_2 and NQ_1*/

for (j=0;j

fscanf(fp,"%1f%1£",&QCj] ,&NQ_20j]);
NQ_1[j]=NQ_2[j];

fclose (fp);

Appendix B

tot_time=0.0;
i=0;

159

*/ The main part of program. Finds time evolution of N(Q)’s */

while (tot_time

if ((i%2) !=0)

integral_1=N_integrate(NQ_2) ;
integral_2=N_eQ_integrate(NQ_2) ;
t_interval=find_t(NQ_2,integral_1,integral_2) ;

for (j=0;}

if (NQ_1[j]>0.0)

newN=NQ_2[j]-t_interval*(v/2.0)
*NQ_2[j]*(integral_1*exp(-Q[j]/(k*T) )+integral_2) ;
if (newN

NQ_iLjJ=0.0;

else

NQ_1Cj]=newN;

else

integral_1=N_integrate(NQ_1) ;

integral _2=N_eQ_integrate(NQ_1) ;
t_interval=find_t(NQ_1,integral_1,integral_2) ;

for (j=0;j

if (NQ_2[j]>0.0)

newN=NQ_1[j]-t_interval*(v/2.0)
*NQ_1[j]*(integral_1*exp(-Q(j]/(k*T) )+integral_2) ;
if (newN

Appendix B 160

NQ_2[j]=0.0;
else
NQ_2[j]=newN;

tot_time += t_interval;
it+;

/* change the file name to *.anl */
change_name(f_list[1]);
/* Write the values to file */

fp=fopen(f_list[1],"w");

fprintf(fp,"/* %s\n", comments) ;

fprintf(fp,"/* anneal time:\t/15.2f\n",annl_time) ;
fprintf(fp,"/* anneal temperature: \t%15.2f\n",T);
fprintf(fp,"/* anneal time:\t%15.2f\n",tot_time) ;

if ((i%2) !=0)
integral_1=N_integrate(NQ_2);

for (j=0;jfprintf(fp,"%15.8e\t%15.8e\n" ,Q[j] ,NQ_2[j]);

fprintf(fp,"Integral of N(Q) is:\n");
fprintf(fp,"%15.8e\n", integral_1);

else

integral_1=N_integrate(NQ_1) ;

for (j=0;j

Appendix B

fprintf(fp,"%15.8e\t%15.8e\n" Qj] ,NQ_1[j]);

fprintf(fp,"Integral of N(Q) is:\n");
fprintf(fp,"%15.8e\n", integral_1);

fclose(fp) ;

return 0;

/* function to integrate N(Q) */

double N_integrate(double NQ[])

int i;

double integral;

extern double Q[Q_bin] ;

integral=0.0;

for (i=1;iintegral+=((NQCiJ+NQ[i-1])/2.0)*(QLi]-Q[i-1]);

return integral;

/* function to integrate exp(-Q)+*NQ(Q) */

double N_eQ_integrate(double NQ[])

int i;

extern double T;

extern double Q[Q_bin];

double integral;

integral=0.0;

161

Appendix B 162

for (i=1;i

integral+=((NQ(i]+NQLi-1])/2.0)*(Q[iJ-QLi-1])*exp(-Q[i] /(k#T)) ;

return integral;

/* function to change the name to the output file name.*/
void change_name(char f_list[])

int i;

1=0;

while (f_list[i]!=’\0’)
itt;
f_list[i]=’.’;
f_list[it+1]=’a’;
f_list[i+2]=’n’;
f_list[i+3]=’3’;
f_list[i+4]=’\o’;

/* function to find time interval. Returns the time */
/* needed for the lowest non-zero bin to decay by 10 percent */

double find_t (double NQ[] ,double integral_1,double integral_2)

extern double Q[Q_bin] ,v,T;

int i;

double time_found;

for (i1=0; (NQLiJ==0.0)&&(i

time_found=0.1/((v/2.0)*(integral_1*exp(-Q[i]/(k#T))
+integral_2));

Appendix B

if (time_found>5.0)
return 5.0;

else
return time_found;

163

Appendix C

Code for Simulation of
Irradiation

The following is the computer code for simulating defect dynamics during an
irradiation, using the generalized activation energy theory of Chapter 3. The
code is written in C. The code assumes presence of a file named “list.dpa”,
which contains the names of data files. The data file is assumed to contain
a header line, then the values of irradiation temperature, irradiation dose,
the dose rate, the minimum value of the activation energy to be considered,
the maximum value of the activation energy to be considered, the reaction
constant, and the generation rate, all in one line. The density of defect
states, G(Q), from Chapter 2 is approximated by 4 segments of exponential
functions.

The time evolution of the defects is simulated by letting

164

Appendix C 165

N(Q,t+ 81) = N(Q,t) + 6x M29,

where ét is the time step. The value of time step is fixed to be 0.0002 sec.
Integration is performed by simply trapezoidal approximation. Since the
functions to be integrated are smoothly varying, the error in integration is
estimated to be small. The simulation stops when the desired dose is reached.
The results are then appended to data files, and written to the disk under

the same name.

/* Program for calculating time evolution of N(Q) */
/* during irradation It reads in data from files */
/* listed in ‘list.dpa’. */

#include
#include

/* Time step used in simulation */
#define t_interval 0.00005

/* Bin size of activation energies */
#tdefine Q_bin 200

/* Boltzmann constant */
#define k .0000862

#define max_list 20
#define name_length 15

Appendix C 166

double N_integrate(double NQ[]);
double N_eQ_integrate(double NQ[]);
void init_GQ(double GQ[]);

void init_NQ(double NQ[]);

/* Variables. v = reaction constant g = generation term */
/* T = temperature low_Q = minimum activation energy used */
/* high_Q = maximum activation energy used */

double v,g;
double low_Q,high_Q,dose_rate,dose,T,Q_step;

main()

extern double v,g;

extern double low_Q,high_Q,dose_rate,dose,T,Q_step;
double newN,integral_1,integral_2;

double NQ_1[Q_bin], NQ_2{[Q_bin], GQ[Q_bin];
char fname([name_length], comments[80],
f_list [max_list] [mame_length] ;

char name[name_length] ;

int j,1,str_length;

FILE *£p;

/* initialize f_list so that they are all EOF’s +*/
for (1=0;1l

f_list[1] [0]=’\0o’;

/* Read in the list of files*/
fp=fopen("list.dpa","r") ;

fgets(comments, 80, fp);

1=0;
do

Appendix C 167

fgets (name,name_length,fp);

str_length = strlen(name)-1;
name[str_length]=’\0’;
strcpy(f_list{[1] ,name) ;

l++;

while ((1(f_list[1-1] [0] '='’\n’));

fclose(fp);
/* Read in temperature,dose,dose rate, low_Q,high_Q,v,g */

for (1=0;(1

if ((fp=fopen(f_list [1] ,"r")) !=NULL)

fgets (comments ,80,fp) ;

fscanf (fp, "Alf Alf, “1f,“1f, 41f ,W1f,%1f" ,&T,&dose,&dose_rate,
&low_Q,&high_Q,&v,&g) ;

fclose (fp);

/* Initialze N(Q)’s to all zero, and initialize G(Q) */
init_NQ(NQ_1);

init_NQ(NQ_2);

init_GQ(GQ);

Q_step=(high_Q-low_Q)/Q_bin;

/* Main part of the program. Finds time evolution of N(Q) */
for (i=1; (ixdose_rate*t_interval)

if ((i%2) !=0)

integral_1=N_integrate(NQ_2);
integral_2=N_eQ_integrate(NQ_2) ;

for (j=0;j

Appendix C 168

newN=NQ_2[j]+t_interval*
(g+dose_rate*(1.0-NQ_2[j]/GQ[j])-(v/2.0)*NQ_2[j]*
(integral_i*exp(-(low_Q+j*Q_step) /(k*T))+integral_2));
if (newN<0)

NQ_1[j]=0.0;

else

NQ_1[jl]=newN;

else

integral_1=N_integrate(NQ_1) ;
integral_2=N_eQ_integrate(NQ_1) ;

for (j=0;j

newN=NQ_1[j]+t_interval*
(g*dose_rate*(1.0-NQ_1[j]/GQ[j])-(v/2.0)*NQ_1[j]*
(integral_1*exp(-(low_Q+j*Q_step) /(k*T))+integral_2));
if (newN<0o)

NQ_2[j]=0.0;

else

NQ_2[jJ=newN;

/* Write the values to file by appending the input file */
fp=fopen(f_list[1],"a");

if ((i%2) !=0)

integral_1=N_integrate(NQ_2) ;

for (j=0;jfprintf(fp,"415.8e\t%15.8e\n", j*Q_steptlow_Q,NQ_2[j]);

Appendix C 169

fprintf(fp,"Integral of N(Q) is:\n");
fprintf(fp,"415.8e\n", integral _1) ;

else

integral_1=N_integrate(NQ_1);

for (j=0;jfprintf(fp,"%15.8e\t%15.8e\n", j*Q_step+low_Q,NQ_1[j]);

fprintf(fp,"Integral of N(Q) is:\n");
fprintf(fp,"%15.8e\n", integral_1);

fclose(fp);

return 0;

/* function to initialize the NQ’s */

void init_NQ(double NQ[])

int i;

for (i=0;iNQ[i]=0.00;

/* function to initialize GQ, approximated by exponentials */

void init_GQ(double GQ[])

int i;

double Q_step;

extern double low_Q,high_Q;

Appendix C 170

Q_step = (high_Q-low_Q)/Q_bin;
1=0;

while ((i*Q_stept+low_Q)<1.29)

GQ ({i]=7 .808e18*exp(-0.459*(low_Q+i*Q_step));
it+;

while ((i*Q_steptlow_Q)<2.26)

GQ[i]=2.53e19*exp(-1.38*(low_Q+i*Q_step)) ;
itt;

while ((i*Q_step+low_Q)<2.55)

GQ[i]=2.3e20*exp (-2.3* (low_Q+i*Q_step) ) ;
it+;

while (i

GQ[i]=1.604e18*exp(-0.409*(low_Q+i*Q_step));
it+;

/* function to integrate N(Q) */

double N_integrate(double NQ[])

int 1;

extern double low_Q, high_Q;
double Q_step, integral;

Appendix C 171

integral=0.0;
Q_step = (high_Q-low_Q)/Q_bin;

for (i=1;iintegral+=((NQ[i]+NQ[i-1])/2.0)*Q_step;

return integral;

/* function to integrate exp(-Q)*NQ(Q) */

double N_eQ_integrate(double NQ[])

int i;

extern double low_Q, high_Q,T;
double Q_step, integral;

integral=0.0;
Q_step = (high_Q-low_Q)/Q_bin;

for (i=1;iintegral+=((NQCiJ+NQ[i-1])/2.0)*Q_step*
exp(-(low_Q+i*Q_step)/(k*T)) ;

return integral;