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\def\t.c.{thermal conductivity}
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\def\im{influence of mismatch}
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\def\taC{{\it ta}-C}
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\title{Molecular-dynamics simulation of thermal stress
at the (100) diamond/substrate interface: effect of film continuity}
\author{Irina Rosenblum}
\address{Departments of Physics and Chemical Engineering, Technion-IIT, 32000, Haifa, Israel}
\author{Joan Adler~\cite{footnote_corresponding author}}
\address{Department of Physics, Technion-IIT, 32000, Haifa, Israel}
\author{Simon Brandon}
\address{Department of Chemical Engineering, Technion-IIT, 32000, Haifa, Israel}
\author{Alon Hoffman}
\address{Department of Chemistry and the Solid State Institute,Technion-IIT, 32000, Haifa, Israel}
\date{\today}
\maketitle
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\begin{abstract} %DO NOT CHANGE THIS LINE
We propose an approach to modeling the mismatch-induced residual
thermal stress in microscopic film/substrate systems using an
atomistic simulation.
Criteria for choosing model parameters necessary for successful
prediction of macroscopic stress-induced phenomena (quantitatively
characterized by a reduction in binding energy) are discussed.
The model is implemented in a molecular dynamics simulation
of compressive thermal stress at the (100) diamond/substrate
interface.
The stress-induced binding energy reduction obtained in the
simulation is in good agreement with our
model.
The effect of sample size and local amorphization on obtained stress
values is considered
and
the maximum on the stress-strain dependence is explained in terms of
the ``thermal spike'' behavior.
Similarly to results from plasma deposition experiments,
the
dominant
stress-induced defect is found to be the tetrahedrally coordinated
amorphous carbon ({\it ta}-C). At higher film continuities these
defects are partially converted into
\splita s; at lower stresses transformation of a small fraction
of {\it ta}-C into the graphitic $sp^2$ configuration takes place.
The penetration depths and the distribution of the
stress-induced defects are determined.
The influence of residual stress on diamond thermal conductivity is
studied; defects formed due to stress are shown to reduce
the thermal conductivity, this effect being partially offset
by the counteracting influence of stress on the phonon density of
states.
\end{abstract}
\pacs{PACS numbers: 71.15.D, 81.05.Tp, 81.15.Gh, 68.55.L, 68.60.B}
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\normalsize
%\newpage
\section{ Introduction \label{INTRODUCTION}}
The remarkable properties of diamond which include its extreme
hardness, high \t.c., transparency, semiconductivity, low coefficient
of friction and chemical inertness make it attractive for a variety of
applications \cite{Seal}.
The potential of diamond as an engineering material has
considerably increased in the last two decades with the development of
advanced CVD techniques \cite{SatoKamo}
%(a review of these is given in \cite{SatoKamo})
producing polycrystalline diamond of quality approaching that of the
best single crystals \cite{Graebner_review}. Moreover, CVD
%technique
makes possible a wide range of products not accessible using bulk
diamond \cite{Klages,Zhu,Yarbrough}.
The quality of
CVD films is often limited by
residual stresses, arising during the growth process.
These stresses have been ascribed to the mismatch between the thermal
expansion coefficients of diamond and of the underlying substrate (the
so called {\it ``thermal stress''}) \cite{AlonOlga1,AlonOlga2} and
to several factors leading to {\it ``intrinsic
stress''}, such as the incorporation of non-diamond phases at grain
boundaries, the presence of hydrogen and the porosity arising during
the growth \cite{Windischmann1,Windischmann2,Chen}. In films obtained
by the CVD technique, thermal stress has been shown to be the main
component ($\sim90\%$) of the residual stress \cite{AlonOlga1}.
A detailed understanding of microscopic processes induced by residual
thermal stress, the mechanism for its relaxation, as well as its
effect on the structure and properties of diamond, is crucial for
predicting reliability and wear resistance of CVD diamond films.
In spite of a large number of experimental analyses of
residual stress in CVD diamond films
\cite{AlonOlga1,AlonOlga2,Windischmann1,Windischmann2,Chen,Stuart,Sails,Bernardez,Baglio,AlonOlga3},
this topic is still not well understood; many questions remain open.
For example, not much is known regarding
the evolution of defects arising in diamond during the CVD process
under conditions of residual stress. Though it is generally known
that the application of stress may lead to considerable structural
changes in the crystal, such as the formation of defects and other
local inhomogeneities \cite{Kittel}, experimental evidence concerning
the nature of stress-induced defects in diamond films is still lacking.
The problem of defects arising as a result of stress is
closely related to phenomena observed in Raman spectra during the film
deposition process.
Stress-induced shifts and splittings of the zone-center optical mode
of diamond \cite{Ramdas} render Raman spectroscopy a technique well
suited for the characterization of the developing stresses \cite{Ager}.
Several empirical
models relating either shifts \cite{Knight,Yoshikawa,Wanlu,Grimsditch}
or splittings \cite{Ager} to stress values have been
derived. However,
the behavior of Raman peaks associated with non-diamond phases and
defects
remains unexplained. In particular, it has been observed that the
intensity of the amorphous carbon peak, initially present in the
deposited film, first increases and then decreases with the deposition
time in correspondence with the evolution of film continuity
\cite{AlonOlga1,AlonOlga2,Olga_thesis}. This observation, apparently
connected to the nature and evolution of defects, has not yet been
explained.
The influence of residual stress on \t.c.\ of diamond is of great
technological importance. Many applications of CVD diamond
are based on its ability to effectively dissipate heat
\cite{Graebner_review,Li}.
The influence of stress on \t.c.\ can be inferred
from the known effect of stress, due both to the change of lattice
parameters and as a result of induced defects, on the
vibrational spectrum. The relative importance of these two mechanisms
can be investigated using the phonon spectrum (PS) method
for the calculation of \t.c., discussed in our earlier study
\cite{RAB1}, which establishes a direct connection between the
phonon spectrum of a dielectric and its thermal properties.
Atomistic simulation in general and molecular dynamics (MD) in
particular
is a powerful technique for investigating
thermal stress with very fine spatial and time resolution, thereby
providing information which is difficult or impossible to obtain
experimentally. To date, several authors have addressed the behavior
of diamond under stress using atomistic simulation techniques.
Two studies are of special relevance.
Uemura \cite{Uemura1,Uemura2} studied the behavior of the {\it
bulk} diamond under uniaxial tensile \cite{Uemura1} and compressive
\cite{Uemura2} stress using a modified tight-binding approximation; in
particular, the critical strength of perfect diamond was determined.
The papers of the group of Pailthorpe and McKenzie
\cite{Pailthorpe1,Pailthorpe2,Pailthorpe4,Pailthorpe6,Pailthorpe7,Pailthorpe9,Pailthorpe10,Pailthorpe3,Pailthorpe12}
are more relevant to the present study. They studied the process of
plasma deposition of diamond films using either a modified
Stillinger-Weber
\cite{Pailthorpe1,Pailthorpe2,Pailthorpe4,Pailthorpe6,Pailthorpe7,Pailthorpe9,Pailthorpe10}
or the Lennard-Jones\cite{Pailthorpe3,Pailthorpe12} potential, and
investigated phenomena that produce compressive stress and allow
it to be relieved. In particular, a ``thermal spike'', defined
as a molten zone or simply as the region in which the local structure
is significantly distorted \cite{Pailthorpe3}, was shown to form
following ion bombardment. Above some critical size this spike causes
the relaxation of stresses
\cite{Pailthorpe2,Pailthorpe4,Pailthorpe10,Pailthorpe3,Pailthorpe12}.
Another important result
is that the compressive stress generated by ion impact induces the
formation of tetrahedrally coordinated amorphous carbon ({\it
ta}-C) \cite{Pailthorpe2,Pailthorpe6,Pailthorpe9,Pailthorpe5}.
The above studies, however, are concerned with the intrinsic
component of residual stress prevailing when plasma deposition is used
\cite{Pailthorpe2}. Recall that when CVD is utilized, thermal
stress plays the major role \cite{AlonOlga1} and the aforementioned
phenomena,
which are likely to result from ion bombardment, may be of little
relevance.
Here, we present a study based on various experimental
results obtained for the CVD diamond films (in particular those of
Hoffman and coworkers
\cite{AlonOlga1,AlonOlga2,AlonOlga3,Olga_thesis}), thereby placing an
emphasis on the thermal component of stress.
We report on molecular-dynamics (MD) simulations of diamond
deposited on a substrate with a thermal expansion coefficient larger
than that of diamond (\eg\ silicon), hence producing a compressive
mismatch-induced thermal stress. Throughout our paper the Brenner
potential \cite{Brenner} is used.
The paper is structured as follows:
in Sec.~\ref{MODEL} we present the model used for the simulation of
thermal stress,
film continuity
and experimental conditions.
In Sec.~\ref{MOLECULAR DYNAMICS} we briefly describe our simulation
method and the details of its implementation.
In Sec.~\ref{RESULTS AND DISCUSSION} we discuss
the reduction of the binding energy due to thermal stress,
the effect of sample size and amorphous inclusions on the value of
obtained stresses,
the nature, formation mechanism and the penetration depth of
arising defects,
and the \t.c.\ of diamond obtained with residual stresses.
The conclusions are summarized in Sec.~\ref{CONCLUSIONS}.
The dependence of the mismatch influence on sample size and
mismatch factor is analytically derived in Appendix~\ref{APPENDIX 1},
and the Rayleigh coefficients for the defects considered here are
evaluated in Appendix~\ref{APPENDIX 2}.
\section{ Modeling of thermal stress in a microscopic system
\label{MODEL}}
There are two principal approaches to the MD simulation of
stress.
In the first (``formalistic'') approach the Hamiltonian of the
system is modified with either stress or strain being introduced into
the equations of motion
\cite{ParRahm1,ParRahm2,RayRahm1,RayRahm2,RibarskyLandman,WangLandman}.
In the second group of methods (referred to below as the
``physical'' approach) the particular physical processes that cause
the stress are simulated at the atomic level (\eg\ the application of
an external force \cite{Uemura1,Uemura2} or the ion bombardment
involved in plasma deposition
\cite{Pailthorpe1,Pailthorpe2,Pailthorpe4,Pailthorpe6,Pailthorpe7,Pailthorpe9,Pailthorpe10,Pailthorpe3,Pailthorpe12}).
This second way is preferable either when the particular
mechanism leading to the appearance of stress is expected to be
important, or when the produced stress is inhomogeneous and hence
cannot be well modeled by a single general variable. In the case of
thermal stress
both these factors are relevant.
We describe a model for the ``physical'' simulation of
mismatch-induced thermal stress, and discuss important modeling
issues related to the small size of the simulation system.
Note that this size problem is relevant for all atomistic simulations
thereby suggesting that our approach may be applicable to other
systems.
Our model also addresses the simulation
of the increase of film continuity that is observed during the
deposition process. The straightforward approach would be to consider
discrete grains with a realistic and continuously increasing size;
however, at present such a simulation is impossible for systems with
non-trivial potentials \cite{Uemura1}, in which sample sizes are
restricted to tens of angstroms, whereas the realistic grain size is
much larger \cite{AlonOlga1}. To avoid this problem, we propose an
alternative method for modeling the increase of film continuity.
A qualitative presentation of our model is given below;
the corresponding equations are derived in Appendix~\ref{APPENDIX 1}.
\subsection{ Modeling of stress \label{MODELING OF STRESS}}
The simulated diamond crystal consisted of 512 atoms
($4\times4\times4$ unit cells) interacting via the Brenner potential
\cite{Brenner} (see Sec.~\ref{MOLECULAR DYNAMICS} for more details).
%
Biaxial thermal substrate stress in the (100)
plane was simulated by compressing the two bottom (100) layers (the
``substrate'') by a ``mismatch factor'' $f_m$ in two directions, [010]
and [001], coinciding with the $x$ and $y$ axes, respectively
(Fig.~\ref{100_plane.fig}). Accordingly, the mismatch factor
\begin{equation}
f_m\equiv a_s/a_d
\label{fm_definition.eqn}
\end{equation}
is the ratio of the lattice parameters of substrate ($a_s$) and diamond
($a_d$),
and the strain tensor in the principal axes is
\vskip -10pt
\begin{equation}
E=\left( \begin{array}{ccc}
e_x & 0 & 0 \\
0 & e_y & 0 \\
0 & 0 & 0
\end{array} \right),
\label{deformation.eqn}
\end{equation}
\vskip -10pt
\noindent where $e_x=e_y\equiv -e=f_m-1$.
The substrate atoms were immobilized to prevent expansion
by making their masses very
large (equal to $10^{11}~M_{\rm carbon}$). The interactions within the
substrate and between the substrate and the diamond film were
described by the same Brenner potential \cite{Brenner} used for the
bulk diamond (see Sec.~\ref{MOLECULAR DYNAMICS}), thus simulating
a diamond-like substrate, such as silicon \cite{Windischmann1}.
The lateral normal stresses ($\sigma_{xx}$ and $\sigma_{yy}$)
were calculated by summing forces across an imaginary vertical slice
\cite{Pailthorpe1} in the $yz$ and $xz$ planes and subtracting the
reference values of ideal non-stressed diamond.
The reported values of stress are the average of these quantities over
the $x$ and $y$ directions.
\subsection{ Choice of the mismatch factor: microscopic correction to
macroscopic mismatch factor \label{CHOICE OF MISMATCH FACTOR}}
At the deposition temperature (\Tdep=800\degree C) it is
assumed that there is no mismatch between the diamond film and the
substrate ($a_s^{\tdep}=a_d^{\tdep}$) \cite{AlonOlga1}.
At a lower temperature $T$, the mismatch factor $f_m$, which is a
result of the difference between linear thermal expansion coefficients
of diamond ($\alpha_d$) and substrate ($\alpha_s$), can be calculated
using the definition of $\alpha$ \cite{footnote_isotropy_of_alpha}
($\alpha_s\equiv \frac{1}{a_s} \frac{{\rm d}a_s}{{\rm
d}T}=\frac{{\rm d~ln}a_s}{{\rm d}T},
~~~~~\alpha_d\equiv \frac{1}{a_d} \frac{{\rm d}a_d}{{\rm
d}T}=\frac{{\rm d~ln}a_d}{{\rm d}T}$) to be
\begin{eqnarray}
f_m(T) \equiv \frac{a_s}{a_d}=
\frac{a_s^{\tdep}~\exp\left(\int\limits_{\tdep}^{T}\alpha_s
{\rm d}T\right)}
{a_d^{\tdep}~\exp\left(\int\limits_{\tdep}^{T}\alpha_d
{\rm d}T\right)}= &&\nonumber \\
= \exp\left(\int\limits_{\tdep}^{T}(\alpha_s - \alpha_d)
{\rm d}T\right)
\approx 1 + \int\limits_{\tdep}^{T}(\alpha_s - \alpha_d) {\rm d}T.&&
\label{fm_from_T_macroscopic.eqn}
\end{eqnarray}
The dependence of $f_m$ on $T$ for two substrates, one very similar to
diamond (silicon) and another very different (chromium carbide), as
calculated with Eq.~(\ref{fm_from_T_macroscopic.eqn}), is presented
in Fig.~\ref{fm_from_T_macroscopic.fig}. This Fig. shows that even
when $\alpha_s-\alpha_d$ is quite large, the mismatch factor is not
smaller than 0.99.
In order to reproduce the \im\ seen in experiments, it
would seem appropriate to accept the above value of $f_m$
for our simulations. However, if we quantitatively define the ``\im''
either as the average deviation from equilibrium of a bond between the
diamond and the nearest substrate atom, \Dr, or alternatively, as the
degree to which the binding energy is reduced due to a mismatch, \DU,
such a choice of $f_m$ turns out to be incorrect. This is a result of
the fact that our simulation system is microscopic in size;
contrary to macroscopic systems, where the \im\ is completely
determined by the mismatch factor $f_m$, for microscopic systems it
is also dependent on the sample size $L$.
The general analytical expressions for $\dr=f(L,f_m)$ and
$\dU=f(L,f_m)$ are derived in Appendix~\ref{APPENDIX 1} with a
graphical presentation of these dependencies given in
Fig.~\ref{effect_from_L_microscopic.fig}. The behavior of both
quantities is highly oscillatory at small $L$
and converges to size independent values as $L\rightarrow\infty$.
These large-sample values vary with the mismatch
factor in a non-monotonic manner (see
Appendix~\ref{APPENDIX 1}); however, in the range of $f_m=0.90 - 0.99$,
this variation is small and corresponds to approximately the same
maximal ($\sim 10\%$) weakening of the binding energy for all values
of $f_m$ (Fig.~\ref{effect_from_L_microscopic.fig}b and
Eq.~(\ref{limit_uav_fm_close_to_1_estimation})).
This ($10\%$) is the magnitude of the mismatch influence expected in
experiments (at $f_m \approx 0.99$), where
the samples are not smaller than hundreds of angstroms
(point A in Fig.~\ref{effect_from_L_microscopic.fig}).
However, applying the same ``experimental'' value of mismatch,
$f_m=0.99$, to our small sample size, $L\sim15\A$, we emerge at the
highly-oscillatory part of the curves (point B in
Fig.~\ref{effect_from_L_microscopic.fig}). At this point the
mismatch-induced strain of the bond and the corresponding weakening of
the interaction are close to zero - either initially
(Fig.~\ref{effect_from_L_scheme.fig}a), or after an easy shift of all
atoms which pushes the non-matching one out of the sample
(Fig.~\ref{effect_from_L_scheme.fig}b). This cannot happen in large
samples, where too many ``internal'' atoms have to be shifted, and the
freedom of their movement is too small to enable such ``free surface''
relaxation (Fig.~\ref{effect_from_L_scheme.fig}c).
Consequently, if we would apply the ``experimental'' mismatch
factor $f_m\approx 0.99$ to our small sample, we would obtain a
mismatch influence much smaller than that obtained in
experiment. As shown in Fig.~\ref{effect_from_L_microscopic.fig},
this ``small size effect'' can be compensated by decreasing the
mismatch factor
with an effect similar to that of
increasing the sample size. If, for example, $f_m=0.90$ is taken, even
the small system is located at the saturation limit (point C in
Fig.~\ref{effect_from_L_microscopic.fig}) and the interaction is
weakened to the same degree ($\sim 10\%$) as in experiment.
Thus, in order to obtain the same ``mismatch influence'' as
the factor $f_m\approx 0.99$ produces in experiment, in our simulation
a smaller mismatch factor such as $f_m\approx 0.90$ is applied.
\subsection{ Modeling the increase of film continuity
\label{MODELING CONTINUITY}}
SEM and micro Raman studies of the evolution of
deposited films have shown \cite{AlonOlga1,AlonOlga2} that
the density of diamond particles grows as time advances and reaches a
maximum value after 30 minutes of deposition. A continuous film is
obtained after 60 minutes and, for longer deposition times (120
minutes), secondary nucleation takes place. This increase of film
thickness and continuity leads to the build-up of residual stresses,
which has been explained in Ref.~\cite{AlonOlga1} as follows.
In the case of non-continuous films,
partial stress relaxation is enabled by the relatively large freedom
of movement of different parts of the film, presumably the free
crystallites surfaces, while in a continuous film the stress cannot
relax this way and
therefore builds up.
As explained above (Sec.~\ref{CHOICE OF MISMATCH FACTOR}),
this is similar to what happens (Fig.~\ref{effect_from_L_scheme.fig})
when the \Dr\ and \DU\ curves achieve the saturation limit, either by
increase in the sample size or by decrease of the mismatch
factor (insets in Fig.~\ref{effect_from_L_microscopic.fig}).
We used the latter method to approximate the effect of increasing film
continuity (leading to the build-up of stress) by varying the mismatch
factor $f_m$ from 1.0 to 0.9;
the latter value is expected to give results, qualitatively similar to
those of a macroscopic, continuous film with the maximal thermal stress.
Although this approach appears artificial (it is based on the {\it
mathematical}
equivalence
between the impact of decreasing mismatch factor and increasing sample
size), using the more realistic approach of varying sample size is
non-practical due to the
large increase in demands on
computational resources.
\subsection{ Modeling the experimental set-up (temperature regime and
initial amorphization) \label{TEMPERATURE REGIME}}
In order to
mimic the experimental conditions of diamond CVD, our ``computer
experiment'', unless stated otherwise, included
three
stages:
{\it 1. ``Deposition'':}
At this first stage the diamond/substrate system was equilibrated for
2.5 ps at the deposition temperature (a typical value
\cite{AlonOlga1,AlonOlga2,Windischmann1,Zuiker} of \Tdep=800\degree C
was used).
Some {\it initial} ``local'' amorphization of diamond near the
diamond/substrate interface, caused by thermal fluctuations and
facilitated by the attachment of the diamond to a rigidly fixed
substrate resulting in the ``growth strain'' \cite{Chen}, was already
present in our sample before any mismatch was introduced.
This reproduces experimental CVD conditions, where such non-diamond phases
were detected immediately after the deposition, in the ``as-grown''
film \cite{AlonOlga1,Ager}.
{\it 2. "Cooling" and equilibration:}
The system was then ``cooled'' to room temperature (25\degree C) at
which the
experimental measurements are made. At this stage the whole lattice
was compressed in accordance with the thermal expansion coefficient of
diamond \cite{thermal_expansion_coefficients} (cooling from 800\degree
C to 25\degree C results in the change of the lattice parameter from
3.576\A\ to 3.567\A). The substrate part was additionally compressed
by the mismatch factor $f_m$, thus simulating the difference in
thermal expansion coefficients of diamond and substrate. An
appropriate choice of $f_m$ is discussed above (Secs.~\ref{CHOICE
OF MISMATCH FACTOR} and \ref{MODELING CONTINUITY}).
The cooled system was then equilibrated during 2.5 ps.
{\it 3. Data collection:}
During this stage various properties of the equilibrated system (phonon
spectrum, radial distribution function, thermal conductivity {\it
etc.}) were ``measured'' during 5 ps (this time has been shown to be
appropriate for obtaining the phonon spectra with sufficient
resolution \cite{RAB1}).
\section{ Simulation method \label{MOLECULAR DYNAMICS}}
The evolution of the diamond/substrate system was simulated
using the molecular dynamics (MD) technique
efficiently implemented on a parallel high performance
computer as described in Ref.~\cite{RAB2}.
We used 16 processors of an SP2 for 20 hours for each run, making some
25 runs in the course of the project.
The equations of motion were integrated for 10 ps by the ``leap-frog''
algorithm \cite{Allen,Rapaport} with an integration time step of
$\Delta t=5\cdot10^{-5}$ ps.
The simulated system consisted of 512 carbon atoms
($4\times4\times4$ unit cells, 16 monoatomic layers in each of [100],
[010] and [001] directions) with minimum-image periodic boundary
conditions \cite{Allen,Rapaport}.
The periodic boundary conditions
allow us to overcome the effect of surface on the calculated
properties \cite{Allen}. They do not increase the size of the system
to make it ``macroscopic''
since the periodicity suppresses any density waves with a wavelength
greater than the simulation box \cite{Allen}. The fact that an atom
relaxing via a ``free surface'' (Fig.~\ref{effect_from_L_scheme.fig}a)
enters through the opposite face of the simulation box does not
influence relaxation; the simulation system remains ``small'', and the
above model holds.
Similarly, the ``grain size'' in the calculation of the \t.c.\ (see
Sec.~\ref{THERMAL CONDUCTIVITY WITH STRESSES}) should be considered to
be equal to the length of the simulation cell.
Interatomic interactions were described by the potential of
Brenner \cite{Brenner}, which is believed to accurately model diamond
and various carbon forms intermediate between diamond and graphite,
as well as defects with various types of hybridization; this is
important for the study of stress which produces a variety of defects.
The details of our implementation of this potential are given in
Ref.~\cite{RAB1}.
The required temperature regime (see Sec.~\ref{TEMPERATURE
REGIME}) was maintained by the periodic rescaling of all atomic
velocities followed by equilibration of the system
\cite{Allen} (the reasons for such a choice of temperature maintenance
method are explained in
Ref.~\cite{footnote_explanation_of_temperature_method_choice}).
\section{ Results and discussion \label{RESULTS AND DISCUSSION}}
\subsection{ Reduction of binding energy as a result of thermal stress
\label{BINDING ENERGY}}
As defined above,
the influence of thermal stress can be quantitatively characterized by
the reduction of binding energy caused by stress-induced mismatch.
This quantity, measured immediately after the mismatch was
introduced (``as-grown'' film) and again after the relaxation, is
plotted in Fig.~\ref{binding_energy_reduction_per.fig}.
Initially (circles in
Fig.~\ref{binding_energy_reduction_per.fig}), the reduction of the
binding energy is rather large and reaches approximately 0.6 eV/atom
for the largest considered mismatch, $f_m=0.90$.
This reduction is due both to mismatch and to a small number of
defects, present already after the deposition stage.
When the sample is allowed to relax (triangles in
Fig.~\ref{binding_energy_reduction_per.fig}), the accounted reduction
of the binding energy becomes smaller since the system
has time to adjust itself to the optimal structure. The agreement of
these
simulation values of the binding energy reduction with those predicted
by our theoretical model (crosses; calculated using Eq.~(\ref{uav}))
is very good considering the roughness of the model and the fact that
the modeled values were averaged over the sample thickness under the
assumption that the mismatch influence distributes uniformly over the
entire sample, while in reality this is not so (see
Sec.~\ref{PENETRATION DEPTH}).
%
For higher mismatches ($f_m<0.94$) the reduction of binding energy is
due not only to the mismatch itself, but also to the stress-induced
formation of defects (Sec.~\ref{MECHANISM OF DEFECTS
FORMATION}) not taken into account in the model but accounted by the
simulation; hence a larger and a more monotonous reduction of energy
in the simulation compared to that given by the model.
\subsection{ Effect of sample size and local amorphization on
obtained stresses \label{STRESSES}}
To single out the
``pure''
dependence of
measured
stress on the value of
substrate strain, we considered the samples subjected {\it only} to a
mismatch (without preliminary deposition stage heating resulting in
initial amorphization).
The initial substrate layer stress in these samples is
presented in Fig.~\ref{initial_stress.fig} (solid line).
As expected, the mismatch-induced thermal stress is
compressive. The slope of a linear fit to the curve allows to
evaluate the biaxial Young's modulus
\cite{AlonOlga1,Ager,handbook_physics}:
\begin{equation}
\frac{E}{1-\nu}=\sigma/e \approx 469 {\rm~GPa},
\end{equation}
where $E$ is the Young's modulus and $\nu$ is the Poisson ratio.
This result is almost three times smaller than
the value from the literature (1345~GPa \cite{AlonOlga1,Ager}),
corresponding to $E=1050$~GPa \cite{Grimsditch1} and $\nu=0.219$.
This discrepancy is most probably the result of the decrease
in the influence of mismatch in the case of small samples, discussed
in Sec.~\ref{CHOICE OF MISMATCH FACTOR}
\cite{footnote_greater_strength_for_small_crystals}.
The literature value of the elastic modulus, suitable for the
{\it bulk}, was determined experimentally for large samples located in
the ``saturation'' region (point A in
Fig.~\ref{effect_from_L_microscopic.fig}), while the stress-strain
relationship in a {\it thin} film (or a {\it monoatomic} substrate
layer for which our estimations were made) is not as straightforward
\cite{Chen,Binder_new}; the \im\ produced by the same strain is expected to be
much smaller (point B in Fig.~\ref{effect_from_L_microscopic.fig}).
The maximum in the stress-strain dependence
can be explained in terms of a ``thermal spike'' behavior, observed in
ion bombardment (plasma deposition) experiments \cite{Pailthorpe2} and
substantiated theoretically
\cite{Pailthorpe4,Pailthorpe10,Pailthorpe3,Pailthorpe12,David2}. The
thermal spike is defined as a zone in which a {\it local} melting
followed by rapid chilling had occurred due to a highly energetical
ion impact \cite{Pailthorpe2,Pailthorpe3}; roughly speaking, it
is
the region in which local structure is significantly distorted
\cite{Pailthorpe3}. With increasing impact energy the thermal spike
region grows; starting from some critical value of energy, it becomes
large enough to allow the relief of stresses, thus giving rise to a
maximum in the dependence of stress on impact energy
\cite{Pailthorpe1,Pailthorpe2}.
A similar phenomenon can be expected also for thermal stress
resulting from the mismatch. The larger the mismatch, the more defects
are formed (see Sec.~\ref{MECHANISM OF DEFECTS FORMATION}); when
some critical concentration of defects is achieved, a partial
relaxation of stress
on
these defects becomes possible. According to
Fig.~\ref{initial_stress.fig}, this happens when the strain $e$
achieves a value of 0.07 ($f_m=0.93$).
In a realistic system produced by the deposition at
800\degree C and then cooled (dotted line in
Fig.~\ref{initial_stress.fig}), some initial amorphization is present
in addition to mismatch (see Sec.~\ref{TEMPERATURE REGIME}). Such
non-diamond inclusions are known to cause an intrinsic stress
\cite{Windischmann1,Chen,Pailthorpe2} which can be either
compressive \cite{Pailthorpe2} or tensile \cite{Windischmann1}. At
small strains (before the critical concentration of defects is
obtained) the difference between the stress value with and
without this amorphization is positive; hence in our case the intrinsic
stress is compressive. However, due to the presence of some defects
already at zero mismatch, the apparent critical concentration of
defects necessary for a stress to relax is achieved much earlier (at
$e=0.04,~f_m=0.96$) than in the perfect crystal, and the reduction of
stress is considerably larger due to its more efficient
relaxation. As seen in Fig.~\ref{concentration_of_defects_per.fig},
the apparent critical concentration of defects is equal to $\sim 3.3$at.\%
(the value corresponding to $f_m=0.96$).
After passing the maximum, at $f_m\approx 0.92$
($e \approx 0.08$), the concentration of defects
reaches saturation and stops growing with mismatch
(Fig.~\ref{concentration_of_defects_per.fig}). For a further increase of
mismatch, the relief of stress due to this concentration of defects
becomes insufficient to overcome the build-up of stress due to a
larger strain; hence, in this case,
stress again increases with increasing strain
(Fig.~\ref{initial_stress.fig}).
Finally, we consider an alternative approach to the estimation
of residual stress. According to
Ager and Drory \cite{Ager},
the value of stress in the (100) plane is related to the splitting of the
zone-center optical phonon (ZCOP) frequency by the following relation:
\begin{equation}
\sigma_{(100)}=0.384 {\rm~GPa/cm^{-1}}~(\nu_d - \nu_0).
\end{equation}
Here $\nu_0$ is the ZCOP frequency in diamond;
stress splits it into singlet (lower value, $\nu_s$) and
doublet (higher value, $\nu_d$) frequencies
\cite{Ramdas,Grimsditch,Anastassakis}. These frequencies can be
determined from the simulated phonon spectra (see
Sec.~\ref{MECHANISM OF DEFECTS FORMATION}); the resultant
$\sigma_{(100)}$ is plotted in Fig.~\ref{initial_stress.fig} (dashed
line). Note that, though the model of Ager and Drory is not quite
suitable for our system (it was derived for a polycrystalline film
different from ours, and the coefficient is known to be
structure-specific \cite{Ramdas,Ager}), the presented dependence
shows qualitative and even semi-quantitative agreement with that
obtained directly from the simulation.
\subsection{ Formation of defects as a result of thermal stress
\label{MECHANISM OF DEFECTS FORMATION}}
Three types of defects were found to form as a result of
the thermal stress:
(1) isolated atoms with approximately 3-fold coordination,
intermediate between the diamond $sp^3$ and the graphite $sp^2$
configurations (referred to as ``tetrahedrally coordinated amorphous
carbons'', {\it ta}-C \cite{Pailthorpe2}, or as ``intermediate carbons''
\cite{Gheeraert1}),
(2) isolated 3-fold atoms possessing the graphitic $sp^2$ configuration,
and (3) split interstitials, which can be defined as dumbbell pairs of
3-fold atoms \cite{point_defects_structures}.
The formation and the evolution of these defects with
increase in mismatch (corresponding to the build-up of stress and to
increase in film continuity) were
followed
using several quantities:
the concentration of each type of defect
(Fig.~\ref{concentration_of_defects_per.fig}),
the radial distribution functions (RDF) (Fig.~\ref{rdf_per.fig}),
and the phonon spectra (density of states) calculated by Fourier
transformation of the velocity-velocity autocorrelation functions
\cite{Wang} (Fig.~\ref{phonon_spectra_per.fig}).
The concentration curves were calculated based on atomic
coordination numbers; these numbers are integer
\cite{footnote_integer_coordination_numbers}
and hence do not allow us to exactly
distinguish
the tetrahedral amorphous carbon
(whose coordination number is 3.7 \cite{Pailthorpe2,Pailthorpe5}) from
the perfect 4-fold diamond, on one hand, and from the 3-fold atoms
in $sp^2$ configuration on the other hand. For the following, we will
refer all atoms having less than 4 neighbors inside the cutoff radius
($r_c=2$\AA\ \cite{Brenner}) and hence considered ``3-fold'' to the
class of intermediate carbons. However, we will bear in mind that
their coordination is not {\it exactly} equal to 3 and their
configuration is not {\it necessarily} $sp^2$; in order to decide
whether this is the case or not, additional evidence will be used.
The radial distribution function was used to extract
information both on the bond length (given by the nearest-neighbor
distance $r_1$) and on the bond angle $\theta$ obtained from the ratio
of the first- and second-neighbor distances \cite{Pailthorpe2}:
$\theta=2~{\rm arcsin}(r_2/2 r_1)$ .
In summary, the results presented in
Figs.~\ref{concentration_of_defects_per.fig}-\ref{phonon_spectra_per.fig}
allow us to suggest the following mechanism of defect evolution.
{\it 1. Breaking of diamond bonds - formation of ``intermediate
tetrahedral amorphous carbons'', {\it ta}-C:}\\
%
First, when the mismatch is small (loosely corresponding to a
low film continuity),
the dominant defects are atoms with an approximately 3-fold
coordination (Fig.~\ref{concentration_of_defects_per.fig}), resulting
from the breaking of one of the diamond bonds. For these defects the
formation energy is relatively low: according to Ref.~\cite{Kittel},
the energy of a single C-C bond (to be broken) is equal to 3.6eV.
As noted above, in order to decide whether this defect is
the tetrahedral amorphous carbon ({\it ta}-C) whose coordination
number is 3.7 \cite{Pailthorpe2,Pailthorpe5} or the ``true'' 3-fold
atom with the graphitic $sp^2$ configuration, additional evidence is
obtained from the radial distribution functions and phonon spectra.
Based on the RDF (peak {\it B} in Fig.~\ref{rdf_per.fig}) analysis,
the defect in question corresponds to the bond length $r_1=1.525$\AA\
and the bond angle $\theta=111$\degree; these values are almost
identical to those reported for the tetrahedral amorphous carbon,
$r_1=1.53$\AA\ and $\theta=110$\degree\ \cite{Pailthorpe2}.
%
In the phonon spectrum, this defect
gives rise to the local mode vibration $\sim 1450$ \cm-1 (peak {\it B}
in Fig.~\ref{phonon_spectra_per.fig}); similar peaks in the 1400-1500
\cm-1\ region were observed and attributed to various C-C vibrations
in amorphous diamond
\cite{Lin-Chung}, in particular, to the stretching vibrations between
the 4-fold and 3-fold atoms \cite{Wang} accounted \eg\ in case of a
vacancy \cite{my_Steve,Steve's_thesis}.
The formation of {\it ta}-C defect as a result of compressive
stress was experimentally and theoretically proven for plasma
deposition experiments \cite{Pailthorpe2,Pailthorpe5} in which the
dominating component of stress is intrinsic
\cite{Pailthorpe2}. The above data, obtained from our simulation,
serve as an unequivocal evidence of the compressive-stress-induced
formation of this defect also under CVD conditions promoting the
development of thermal rather than intrinsic stress
\cite{AlonOlga1}.
The presence of a small concentration of this defect even in
samples with no mismatch ($f_m=1.0$) is accounted for by an initial
``local amorphization'' resulting from the deposition at high
temperature.
{\it 2a. Conversion of a part of the $sp^3$ {\it ta}-C to the
graphitic $sp^2$ configuration:}\\
%
Stabilization of the diamond-like {\it ta}-C defect
compared to the
$sp^2$ graphitic defect under the conditions of compressive stress is
consistent with the phase diagram of carbon \cite{Dresselhaus_Kalish},
showing that at higher pressures the diamond state is favorable.
However, when the stress is not too large, the conversion of
some $sp^3$ {\it ta}-C defects to the graphitic $sp^2$ configuration
becomes possible \cite{footnote_activation_energy_for_sp3_to_sp2}.
The presence of the small 1.41\AA\ peak and of the
bond angle $\theta=126$\degree\
in the radial distribution function, as well as of the 1533 \cm-1\
feature in the phonon spectra (peaks {\it C} in
Fig.~\ref{rdf_per.fig}, \ref{phonon_spectra_per.fig}) indicate that
the graphitic $sp^2$ configuration exists
approximately
for $f_m \geq 0.975$ and for $f_m \leq 0.930$ (compare to the
corresponding values 1.45\AA, 120\degree\ \cite{Berman} and 1582
\cm-1\ ~ \cite{Wang} for graphite). This can be explained based on the
non-monotonic stress-strain relationship, discussed in
Sec.~\ref{STRESSES}. When the strain is small ($f_m \geq 0.975$, $e
\leq 0.025$) the stress is rather low, and the $sp^2$ configuration
can form. When the strain is large enough, the concentration of
defects approaches the critical value enabling a partial relaxation of
stress, and at $f_m \leq 0.930$ ($e \geq 0.070$) the stress again
becomes low enough to make the existence of the $sp^2$ configuration
possible (see Fig.~\ref{initial_stress.fig} where the stress below
which the graphitic defect exists is shown).
{\it 2b. Additional stabilization of above
configurations at higher stresses - formation of split interstitials:}\\
%
The predominant
{\it ta}-C defect is stable enough
\cite{Pailthorpe5} and present up to the highest considered
mismatches. However, this defect, being intermediate between the
diamond and the (less dense) graphite configuration, requires more
volume than the perfect diamond atom would require; indeed, its
fractional volume difference relative to diamond is positive,
$p_{ta-{\rm C}} \approx 0.166$ (see Appendix~\ref{APPENDIX
2}). Yet more considerable is this volume difference for the second
defect, the $sp^2$ graphitic carbon ($p_{sp^2} \approx 0.552$).
Therefore, accumulation of both {\it ta}-C and $sp^2$ graphitic carbon
defects (which can be commonly referred to as {\it isolated} 3-fold
atoms) with the increase of mismatch
requires
a volume expansion of the crystal, experimentally known as
``swelling'' \cite{footnote_swelling_origin,Maby,Prins}.
As the continuity of the film increases,
it becomes more difficult to gain the required additional volume. This
causes combining of some of the isolated 3-fold atoms
into pairs with
a stabilization corresponding to the formation of split interstitial
defects (Fig.~\ref{concentration_of_defects_per.fig}). These are
presumably \splita s, known to be the most stable point defects in
diamond \cite{David1,Breuer}. As shown in Appendix~\ref{APPENDIX 2},
this new defect is more compact than both
types of isolated 3-fold coordinated
defects ($p_{\rm si}
\approx 0.157 < p_{ta-{\rm C}} \ll p_{sp^2}$) and is hence consistent
with the more
continuous
system with less degrees of freedom.
The formation energy of \splita, 16.5-16.6eV
\cite{David1,Bernholc}, is higher than that of the isolated 3-fold atom,
explaining the absence of these defects at low stresses, \ie\ at small
mismatches ($f_m > 0.960$) (Fig.~\ref{concentration_of_defects_per.fig}).
%
When the split interstitial defect is formed, the radial
distribution function starts to exhibit new, non-diamond, peaks, such
as that located at about 2.425\A (peak {\it C} in
Fig.~\ref{rdf_per.fig}). As the compressive stress builds up, the
defect stabilizes, and this peak evolves towards smaller distances;
its value 2.268\A\ at $f_m=0.90$ is close to the distance 2.129\A\
between one of the central atoms of the \splita\ and the terminating
dumbbell atom farthest from it for the \split\ configuration predicted
in Ref.~\cite{RAB1}.
Conversion of a part of the isolated 3-fold
atoms into split interstitials
becomes apparent also when looking at the
phonon spectra
(Fig.~\ref{phonon_spectra_per.fig}): as the film continuity increases,
a new local peak ({\it D}), evolving towards 1607 \cm-1\ and most
likely attributed to the \splita\ \cite{my_Steve}, appears.
The proposed mechanism of defect evolution provides an
explanation for the experimental behavior of the amorphous carbon peak
in Raman spectra \cite{AlonOlga1,AlonOlga2}. In our case,
as the film continuity increases, the intensity of the broad
``amorphous carbon'' feature, located at $\sim 1420-1560$ \cm-1\ and
attributed to two kinds of isolated 3-fold defects,
(Fig.~\ref{relative_intensity.fig}), first increases with the growth
of the total concentration of these defects ($fm \geq 0.960$,
Fig.\ref{concentration_of_defects_per.fig}), and then decreases, when
some of these defects convert to split interstitials and their
concentration reduces ($fm < 0.960$,
Fig.\ref{concentration_of_defects_per.fig})
\cite{footnote_increase_of_amorphous_peak_intensity_at_highest_mismatches};
a similar behavior was observed in experimental Raman spectra
\cite{AlonOlga1,AlonOlga2}.
Finally, two additional features of the evolution of the RDF
and phonon spectrum with the increase of mismatch should be
mentioned.
Firstly, the higher the compressive stress, the smaller
the stable lengths of all bonds (note, for example, the downward
shifting of the aforementioned split interstitial line, initially
located at 2.425\A\ (Fig.~\ref{rdf_per.fig})). Increase of
the compressive stress also causes the upward shift and splitting of
the diamond (``A'') peak in phonon spectrum
(Fig.~\ref{phonon_spectra_per.fig}); quantitative estimation of the
stress value from the degree of this splitting was discussed in
Sec.~\ref{STRESSES}.
Secondly, as the stress builds up, stress-induced defects
develop, and the order of the structure decreases (it becomes more
amorphous); as a result, the RDF peaks become broader and more
``grassy'' (Fig.~\ref{rdf_per.fig}).
\subsection{ Depth of the defects penetration \label{PENETRATION DEPTH}}
The penetration depth of various defects can be obtained by
mapping their concentrations along the direction normal to the
diamond/substrate interface
(Fig.~\ref{coord_numbers_profiling_per.fig}).
This Fig. demonstrates that the effect of substrate stress on
the crystal structure is very deep: when allowed to relax, the defects,
initially located near the substrate/diamond interface, expand over
the whole crystal, though their distribution remains non-uniform. At
small mismatches the defects are the ``intermediate carbons''
(isolated 3-fold carbon atoms); the further from substrate, the
smaller is their concentration. At larger mismatches, a considerable
proportion
of these defects turn into split interstitials (see
Sec.~\ref{MECHANISM OF DEFECTS FORMATION}). In this case, the
split interstitials form predominantly near the substrate/diamond
interface where the stress is maximal, while the remaining
``intermediate carbons'' (requiring more volume) concentrate in
deeper, less compressed, layers.
In experiment the presence of defects can be analyzed using
the full width of the half-maximum (FWHM) of the band-center phonon
peak in Raman spectra: a decrease in the degree of crystallinity
(increasing defects or amorphicity) is known to increase the value of
FWHM \cite{Sails}.
Unfortunately, the penetration depths deduced experimentally based on
FWHM cannot be quantitatively compared to our results, since the
former are extremely sensitive to the deposition conditions and to the
film thickness and orientation. However, the measurements of Sails
\etal\ for the $\langle100\rangle$ oriented film \cite{Sails}
qualitatively confirm our conclusion about the defects propagating
deep enough and affecting the whole film.
\subsection{ Thermal conductivity of diamond with residual stresses
\label{THERMAL CONDUCTIVITY WITH STRESSES}}
For reasons discussed in Ref.~\cite{RAB1}, the molecular dynamics
thermal conductivity is best calculated using the phonon spectrum (PS)
method.
This method \cite{RAB1} is based on the following expression for the
\t.c.\ of dielectrics \cite{Kittel,Belay,Olson}:
\begin{equation}
\kappa=\frac{\hbar^2v^2}{3kT^2V_{mol}}\int\limits_
{0}^{\omega_R}\tau(\omega)D(\omega)\frac{\omega^2e^{\hbar\omega/kT}}{(e^{\hbar\omega/kT}-1
)^2}d\omega,
\label{kphonon}
\end{equation}
where $T$ is the temperature,
$v$ is the average velocity of sound in the crystal,
$V_{mol}$ is the molar volume,
$\omega_R$ is the maximal frequency up to which the integration is
accomplished (it corresponds to the wave vector ${\bf k}=0$),
$D(\omega)$ is the normalized mode density at frequency $\omega$,
% the term involving exponential functions comes from the Planck distribution,
and $\tau(\omega)$ is the relaxation time comprising all mechanisms
participating in the phonon scattering
\cite{Belay,Graebner50,BermanfromBelay}.
Under the assumption that the scattering mechanisms are
independent one from another, the scattering rates of individual
mechanisms are additive \cite{Graebner_review,Van13}, and include
(in our case) relaxation due to Umklapp processes ($\tau_{\rm U}^{-1}$),
grain boundaries ($\tau_{\rm gb}^{-1}$) and point defects, earlier
shown to include the tetrahedrally coordinated amorphous carbon atoms,
{\it ta}-C ($\tau_{ta{\rm-C}}^{-1}$), the split interstitials
($\tau_{\rm si}^{-1}$) and the isolated $sp^2$ graphitic
carbons ($\tau_{sp^2}^{-1}$):
\begin{equation}
\tau^{-1}(\omega)=\tau_{\rm U}^{-1}(\omega)+
\tau_{\rm gb}^{-1}(\omega)+
\tau_{ta{\rm-C}}^{-1}(\omega)+
\tau_{\rm si}^{-1}(\omega)+
\tau_{sp^2}^{-1}(\omega).
\label{tau_complete}
\end{equation}
Based on the intensity of those RDF and PS peaks which
correspond to the latter defect (Figs.~\ref{rdf_per.fig},
\ref{phonon_spectra_per.fig}), its concentration is rather low, and
the concentration of the ``isolated 3-fold'' atoms can be attributed
to the {\it ta}-C defect alone; hence
\begin{equation}
\tau^{-1}(\omega) \approx \tau_{\rm U}^{-1}(\omega)+
\tau_{\rm gb}^{-1}(\omega)+
\tau_{ta{\rm-C}}^{-1}(\omega)+
\tau_{\rm si}^{-1}(\omega).
\label{tau}
\end{equation}
The phonon density $D(\omega)$ was extracted from the phonon
spectra (Fig.~\ref{phonon_spectra_per.fig}). The expressions for each of the
relaxation mechanisms present in Eq.~(\ref{tau}), as well as the
parameters necessary for Eqs.~(\ref{kphonon})-(\ref{tau}), were taken
from Ref.~\cite{RAB1}. The Rayleigh coefficients $I$ for the three
types of defects
not considered in Ref.~\cite{RAB1} are evaluated in
Appendix~\ref{APPENDIX 2}.
The above expressions show that, in principle, the influence
of residual stress on \t.c.\ can be inferred in three possible
mechanisms:
(1) effect of stress on the phonon density of states, $D(\omega)$,
due to the change of the lattice parameters,
(2) appearance of the stress-induced defects serving as an additional
mechanism of phonon scattering (appearance or increase of the point
defects terms $\tau_{\rm 3i}$ and $\tau_{\rm si}$ in Eq.~(\ref{tau}))
and (3) modification of the density of states $D(\omega)$ by defects.
Of these mechanisms, the third does not occur: as shown in
Fig.~\ref{phonon_spectra_per.fig}, the defects in question do not
noticeably affect the spectrum in the quasilocal frequency region
($\omega < \omega_R$), while the local vibrations with frequencies
lying above the maximum possible frequency of pure diamond are unable
to propagate through the crystal \cite{Houghton} and hence have no
effect on its thermal properties.
Influence of two other mechanisms is analyzed in
Fig.~\ref{thermoconductivity_per.fig}.
Change of the \t.c.\ due to the ``pure'' first mechanism
(effect of stress on $D(\omega)$) can be isolated by omitting the
terms resulting from the relaxation on point defects, \ie\ accepting
$\tau^{-1}=\tau_{\rm U}^{-1}+\tau_{\rm gb}^{-1}$ (dashed line).
Fig.~\ref{thermoconductivity_per.fig} demonstrates that in this case
(if there were no scattering on point defects), the \t.c.\ of diamond
under residual stress would be higher than that of ideal diamond (thin
solid line). This would-be increase is accounted for by the fact that
compressive stress shifts the maximum frequency upwards, thus
enhancing the frequency region in which the phonons can propagate.
However, the true \t.c.\ of diamond with residual stress
(thick solid line) is lower that that of ideal diamond.
This reduction is due to the second route of the stress influence,
namely to the appearance of defects serving as an additional mechanism
of the phonon scattering.
The extent to which these defects reduce the \t.c.\ results
from the interplay of two factors: the type of prevailing defects and
their concentration. On one hand, as the mismatch increases and the
stress builds up, the concentration of defects grows
(Fig.~\ref{concentration_of_defects_per.fig}), and the defect-induced
reduction of \t.c.\ must become stronger. On the other hand, the
build-up of stress was shown to cause the conversion of some of the
``intermediate carbon'' defects to more compact split interstitials,
possessing the lower Rayleigh coefficient per single defect (9.31
instead of 9.84, see Appendix~\ref{APPENDIX 2}),
%; this means that the phonon scattering on split interstitials is
%less intensive.
and hence scattering the phonons less intensively.
However, the difference in Rayleigh parameters is
rather small, and the latter effect is negligible, so that the resulting
degree of the \t.c.\ reduction by point defects grows with their
concentration, \ie\ with the increase of mismatch (dashed line in
Fig.~\ref{thermoconductivity_per.fig}).
\section{ Conclusions \label{CONCLUSIONS}}
(1) A model for the atomistic simulation of macroscopic thermal
residual stress in microscopic film/substrate systems was presented.
In this model the mismatch between the film and underlying substrate,
resulting from the difference in their thermal expansion coefficients,
was simulated using substrate compression by a factor $f_m$.
The dependence of the mismatch influence (quantitatively defined as
the binding energy reduction due to mismatch) on the mismatch factor
$f_m$ and sample size $L$ was derived analytically.
Using the dependence of the mismatch influence on $L$, it
was demonstrated that in small (simulated) systems the influence of
a given level of mismatch is considerably less than that produced in
large (experimental) samples by the same value of mismatch. The reason
for this lies in a larger freedom of movement in the small sample,
which enables free-surface relaxation of stress. This freedom can be
reduced either by increasing the sample size (this happens, \eg, when
the film continuity grows with the deposition time) or by introducing
a decrease in the mismatch factor
thereby producing effects similar to those obtained when increasing
the sample size.
Two conclusions can be drawn from
the similar effect of increasing the sample size and that of
decreasing the mismatch factor.
First, in order to
produce the same \im\ as is obtained in a macroscopic experiment, in a
microscopic system one should use mismatch factors considerably
smaller than the realistic ones. Second, the increase of the film
continuity,
%observed during the film deposition,
corresponding to reduction in free-surface relaxation and increase in
compressive stress, can be
semi-quantitatively
modeled using the continuous decrease of the mismatch factor. This
approach is favorable compared to the direct increase in size of a
simulated system;
such an increase is limited by computer resources which currently
render realistic sample sizes unachievable.
(2) The model was applied in the simulation of compressive thermal
stress at a (100) diamond surface using the molecular dynamics
technique;
the resultant binding energy reduction was calculated. Initially, when
mismatch was introduced, the energy reduction amounted to
0.6eV/atom for the largest mismatch studied, and became almost 5
times smaller when the system was allowed to relax. The value of the
binding energy reduction in the relaxed system agrees well with that
predicted by our theoretical model except
when applying large mismatches which lead to the formation of
mismatch-induced defects, not accounted in the model; these noticeably
contribute to the binding energy reduction.
(3) The biaxial Young's modulus, determined from the obtained stress
dependence on strain, was found to be smaller than the corresponding
literature value. This was explained in terms of the smaller size of the
simulation system, which causes a reduction in the mismatch influence
thereby favoring stress relaxation and making the macroscopic literature
value
inappropriate
for microscopic simulation. This conclusion is consistent both with
our theoretical model and with experimental data
\cite{footnote_greater_strength_for_small_crystals}.
A maximum was observed in the stress-strain relationship.
This phenomenon was explained by the fact that for large mismatches
the concentration of defects achieves some critical value allowing a
partial relaxation of stresses, similarly to the ``thermal spike''
behavior observed in ion bombardment experiments in which a maximum
of the stress-impact energy dependence was obtained
\cite{Pailthorpe2,Pailthorpe4,Pailthorpe10,Pailthorpe3,Pailthorpe12}.
This suggests that, although the origin of stress is different in two
cases (the mismatch-induced thermal stress in our ``CVD experiments''
and the impact-induced intrinsic stress in plasma deposition
\cite{Pailthorpe2}), the mechanism of stress relaxation
is the same.
Amorphous inclusions and internal defects, arising as a result
of thermal stress, were shown to cause a compressive intrinsic stress.
(4) The evolution of crystal structure with increasing film continuity
was investigated.
Three types of defects were found to form as a result of
stress.
At low film continuities the prevailing defect was shown to
be an isolated atom with an approximately 3-fold coordination. In
agreement with plasma deposition experiments
\cite{Pailthorpe2,Pailthorpe5} in which defects originated from an
{\it intrinsic} compressive stress, in our simulated CVD experiment,
resulting mainly in {\it thermal} stress, this defect was shown to be
the tetrahedrally coordinated amorphous carbon {\it ta}-C whose
configuration is intermediate between diamond and graphite,
rather than the graphite-like $sp^2$ carbon.
Interestingly, for low enough stresses, an insignificant concentration
of the graphite-like $sp^2$ carbon defects was also found.
For large film continuity values, both of these defects were
shown to be partially converted into \splita s.
These transformations were interpreted in terms of phase
diagrams and relative volumes and were found to be consistent with
experiment. In particular, the proposed mechanism explains the time
evolution of the ``amorphous carbon'' peak in Raman spectra, observed
during the CVD of diamond \cite{AlonOlga1}.
(5) The penetration depth of the the stress-induced defects was
studied. It was demonstrated that the effect of stress on the crystal
structure is long ranged: when allowed to relax, the stress-induced
defects, initially located near the substrate/diamond interface,
propagate over the whole crystal. When both split interstitials and
``intermediate carbons'' are present, the former concentrate near the
substrate/diamond interface where the stress is maximal, and the
latter remain predominantly in deeper layers.
(6) The influence of residual stress on \t.c.\
%, so far uninvestigated,
was studied by the PS method \cite{RAB1}. The residual thermal stress
was shown to reduce the diamond \t.c., this reduction being due to the
formation of stress-induced defects,
leading to additional
phonon scattering. This effect, growing with
the concentration of defects, counteracts the effect of stress on
the phonon density of states, namely the enhancement of the region of
phonon propagation with the stress build-up. As a result, the degree
to which compressive stress reduces the \t.c.\ is approximately
independent of the stress value for the entire range of mismatches
considered here.
(7)
Two directions in which this study could be extended are as follows.
The investigation of thermal stress on other diamond surfaces
should enhance understanding of the dependence of the above phenomena
on the surface orientation.
In addition, modeling more realistic films with a columnar structure
and a grain size varying
in the direction of film growth
\cite{Graebner17,Graebner32} is of interest. However,
the latter option could lead to
considerable computational difficulties, since the typical
thicknesses of such films range between 50-400 $\mu$m
\cite{Graebner32}, while the maximum sample size
that we can reach with
the MD simulation of diamond using the Brenner potential on a parallel
16-processor supercomputer is about 40\A\ \cite{RAB2}.
\addcontentsline{toc}{section}{\hskip 0.65cm Acknowledgments}
\section*{ Acknowledgments \label{ACKNOWLEDGEMENTS}}
The authors would like to thank O. Glozman, A. Reznik and
S. Prawer for valuable discussions concerning experimental
measurements of residual stress and swelling.
The support of Israel Ministry of Science (research grant
No. 9608-2-96) in the early stages of this calculation and the support
of the German Israel Foundation in the later stages are gratefully
acknowledged.
The calculations were carried out on the SP2 of the
Inter-University Computer Center and on several computers of the
Computational Physics Group and the Minerva Non-Linear Center at the
Technion.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FOR TWO COLUMN ...
% If you REALLY want "twocolumn", DEACTIVATE THE LINE BELOW.
% However, I had to introduce it even if everything else is in twocolumn
% format, because appendices do not fit into two columns and have
% to be one column anyway; after that I can renew the "twocolumn" command:
\onecolumn
\appendix
%\newpage
%\addcontentsline{toc}{section}{\hskip 0.65cm Appendix~\protect{\ref{APPENDIX 1}}}
\section{ Influence of mismatch as the function of the
sample size $L$ and the mismatch factor $f_m$ \label{APPENDIX 1}}
%%%%%%%%%%%%%%%%%%%
%\vskip 0.5cm
%\subsection*{1. Definition of the ``influence of mismatch''}
In this Appendix we calculate the ``\im'' as a function of the
mismatch factor $f_m$ and the sample size $L$.
%
The ``\im'' can be
defined quantitatively as the average deviation from equilibrium of a
bond between a diamond atom and the nearest substrate atom, \Dr, or,
alternatively, as the degree to which the binding energy is reduced
due to a mismatch, \DU.
%
%%%%%%%%%%%%%%%%%%%
%\vskip 0.5cm
%\subsection*{2. Notation}
%
For the sake of clarity, in this Appendix we deal with a
simple cubic lattice, and use the Morse potential \cite{Morse_form}
for carbon, with parameters $D=4.006$eV and
$\beta=1.5$\A$^{-1}$~~\cite{Morse_parameters}.
Let us consider
the diamond/substrate interface with the mismatch factor $f_m$,
previously defined (Eq.~\ref{fm_definition.eqn}) as the ratio of the
lattice parameters of substrate ($a_s$) and diamond ($a_d$), and,
obviously, equal to the ratio of the corresponding interatomic
distances, $l_s$ and $l_d$, in the plane parallel to the compression
(the substrate is compressed, hence $l_s \leq l_d$ and $f_m \leq
1$). To illustrate our notations geometrically, in
Fig.~\ref{mismatch_notation.fig} we present a sample with the mismatch
factor equal to $f_m=l_s/l_d=4/5$. For the simple cubic lattice
considered here the interatomic distances in the layer are equivalent
to interatomic bond lengths; we use the value of the bond length in
diamond, $l_d=1.54$\AA.
We define the absolute mismatch $\Delta l \equiv l_d-l_s$ and
the least common multiple of $l_d$ and $l_s$, ~~$l_{ds} \equiv n_d
l_d=n_s l_s$. This is the length at which for the first time a
complete substrate cell fits into a multiple of $l_d$ (in
Fig.~\ref{mismatch_notation.fig} $l_{ds}=4 l_d=5 l_s$). The numbers of
diamond (substrate) bonds in the length $l_{ds}$ are denoted by
$n_d=l_{ds}/l_d$ and $n_s=l_{ds}/l_s$, respectively; according to the
definition of $l_{ds}$, $n_s\equiv n_d+1$ (in
Fig.~\ref{mismatch_notation.fig} $n_d=4, ~ n_s=5$). A combination that
will be used frequently below is
\begin{equation}
k_s \equiv n_s+1-{\rm mod}(n_s,2)=\left\{ \begin{array}{ll}
n_s & {\rm if~}n_s{\rm ~ is~odd} \\
n_s + 1 & {\rm if~}n_s{\rm ~ is~even} \\
\end{array} \right..
\end{equation}
The length of the whole sample (not shown in
Fig.~\ref{mismatch_notation.fig}) is denoted by $L$.
Next we establish notation for differences between equilibrium
and stressed configurations. If the equilibrium length of the bond
between atoms in different layers (equal to the interlayer distance,
0.89\A\ in case of diamond) is denoted by $r_e$, and the real
(strained) length of this bond by $r$, then the deviation of the bond
from equilibrium is given by $\Delta r \equiv r-r_e$.
We will also refer to one-dimensional deviation of the atom starting
this bond from its position in a non-compressed ideal sample, $\Delta d$.
Another quantity characterizing the ``\im'' (see above), is the
per atom reduction of the binding energy of the real stressed sample
compared to the non-stressed case; we denote it by $\Delta U$.
%%%%%%%%%%%%%%%%%%%
%\vskip 0.5cm
%\subsection*{3. Useful relationships}
The following additional relationships can be derived using
the above definitions, in particular, that of $l_{ds}=n_d l_d=n_s
l_s=(n_d+1) l_s$:\\
\begin{equation}
n_d=\frac{l_s}{l_d-l_s}=\frac{f_m l_d}{l_d- f_m l_s}=\frac{f_m}{1-f_m};
\label{n_d}
\end{equation}
\begin{equation}
n_s=n_d+1=\frac{f_m}{1-f_m}+1=\frac{1}{1-f_m};
\label{n_s}
\end{equation}
\begin{equation}
k_s=\frac{2-f_m}{1-f_m}-{\rm mod}\left(\frac{1}{1-f_m},2\right);
\label{k_s}
\end{equation}
\begin{equation}
\Delta l \equiv l_d-l_s=l_d~(1-f_m)=l_s~(1-f_m)/f_m.
\label{Delta l}
\end{equation}
\def\L{l_{ds}} % My notation of l_ds for the further convenient use
%%%%%%%%%%%%%%%%%%%
%\vskip 0.5cm
%\subsection*{4. Displacement of an atom starting the $i$-th
%bond in the first period $\L$ from its location without mismatch,
%$\Delta d_i^I$}
We start by deriving an expression for the displacement of atom
starting the $i$-th bond in the {\it first} period $\L$ (in
Fig.~\ref{mismatch_notation.fig} only this period is shown) from its
location in the absence of mismatch, $\Delta d_i^I$.
For the case of $n_s$ odd, illustrated in
Fig.~\ref{mismatch_notation.fig}, these displacements are:
$$
\Delta d_1^I=0,~
\Delta d_2^I=\Delta l,~
\Delta d_3^I=2\Delta l,~
\Delta d_4^I=2\Delta l,~
\Delta d_5^I=\Delta l,~
\Delta d_6^I=\Delta d_1^I=0.
$$
%
The general expression for $n_s$ odd reads:
\begin{equation}
\Delta d_i^I=\Delta l~\left(\frac{n_s}{2}-\Bigg | i-\frac{n_s+2}{2}\Bigg |\right)=
\left\{ \begin{array}{ll}
(i-1)\Delta l & {\rm for~} i < (n_s+2)/2\\
(n_s + 1 - i)\Delta l & {\rm for~} i > (n_s+2)/2 \\
\end{array} \right.
\end{equation}
%
Similarly, for $n_s$ even
\begin{equation}
\Delta d_i^I=\Delta l~\left(\frac{n_s+1}{2}-\Bigg | i-\frac{n_s+3}{2}\Bigg |\right)=
\left\{ \begin{array}{ll}
(i-1)\Delta l & {\rm for~} i < (n_s+3)/2\\
(n_s + 2 - i)\Delta l & {\rm for~} i > (n_s+3)/2 \\
\end{array} \right.
\end{equation}
%
Applying the notation $k_s$, we obtain the general expression for
arbitrary $n_s$:
\begin{equation}
\Delta d_i^I=\Delta l~\left(\frac{k_s}{2}-\Bigg | i-\frac{k_s+2}{2}\Bigg |\right)=
\left\{ \begin{array}{ll}
(i-1)\Delta l & {\rm for~} i < (k_s+2)/2\\
(k_s + 1 - i)\Delta l & {\rm for~} i > (k_s+2)/2 \\
\end{array} \right.
\label{d_i^I}
\end{equation}
%%%%%%%%%%%%%%%%%%%
%\vskip 0.5cm
%\subsection*{5. Displacement of atom starting an arbitrary
%$i$-th bond from its location without mismatch, $\Delta d_i$}
For an {\it arbitrary} period $\L$, the displacement of an atom
starting the $i$-th bond is equal to that of its image in the {\it
first} period:
\begin{equation}
\Delta d_i=
\left\{ \begin{array}{ll}
\Delta d^I_{{\rm mod}(i-1,n_s)} & {\rm for~} n_s {\rm~odd}\\
\Delta d^I_{{\rm mod}(i-1,n_s+1)} & {\rm for~} n_s {\rm~even}\\
\end{array} \right.,
\end{equation}
Using Eq.~(\ref{d_i^I}), we get the general expression for
$\Delta d_i$:
\begin{equation}
\Delta d_i=\Delta d^I_{{\rm mod}(i-1,k_s)}=\Delta l~\left(\frac{k_s}{2}-\Bigg | {\rm mod}(i-1,k_s)-\frac{k_s+2}{2}\Bigg |\right)
\end{equation}
This function is periodic in $i$, with the period equal to $k_s$.
%%%%%%%%%%%%%%%%%%%
%\vskip 0.5cm
%\subsection*{6. Sum of all displacements for a sample
%containing $N$ bonds, $s(N)$}
For a sample containing $N$ bonds the sum of all displacements, $s(N)$,
is given by the series
\begin{equation}
s(N)=\sum_{i=1}^{N} \Delta d_i=\Delta l~\sum_{i=1}^{N}\left(\frac{k_s}{2}-\Bigg | {\rm mod}(i-1,k_s)-\frac{k_s+2}{2}\Bigg |\right)
\end{equation}
%
To calculate the sum of this series, let us divide it into two
parts. Assuming that the $N$-th bond lies in the $J$-th period $\L$, we
get
\begin{equation}
s(N)=\sum_{j=1}^{J-1} s^j + \Delta s_N,
\end{equation}
where $s^j$ is the sum over a complete period $\L$ number $j$, and
$\Delta s_N$ is the sum over the last, incomplete, period ending by
the bond number $N$.
Considering the aforementioned periodicity, the sum over {\it any}
complete period is equal to that over the first one, $s^j = s^I$.
Hence,
\begin{equation}
s(N)=(J-1)s^I + \Delta s_N.
\label{s(N)}
\end{equation}
%
It is easy to see that
\begin{eqnarray}
s^I \equiv
\sum_{i=1}^{k_s}\Delta l~\left(\frac{k_s}{2}-\Bigg | {\rm
mod}(i-1,k_s)-\frac{k_s+2}{2}\Bigg |\right) =
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ & & \nonumber \\
=\sum_{i=1}^{\frac{k_s+1}{2}}\Delta l~(i-1) +
\sum_{i=\frac{k_s+3}{2}}^{k_s}\Delta l~(k_s+1-i) =
\frac{\Delta l (k_s^2-1)}{8}+\frac{\Delta l (k_s^2-1)}{8}=
\frac{\Delta l (k_s^2-1)}{4} & &
\label{s^I}
\end{eqnarray}
%
The number $J$ of the period to which the last bond number $N$ belongs
is given by
\begin{eqnarray}
J=[(N-1)/k_s]+1,
\label{J}
\end{eqnarray}
%
\def\iend{i_{\rm last}} % My notation of iend for the further convenient use
\def\iendprime{i'_{\rm last}} % My notation of iend' for the convenient use
%
\noindent and the sum over the last, $J$-th, incomplete period is
\begin{equation}
\Delta s_N=\frac{\Delta l}{8}~\left(
4 (\iendprime-1)\iendprime+
\left[\frac{\iend \cdot 2}{k_s+2}\right]~
\left(4\iend(2k_s+1)-4\iend ^2 -1-4k_s-3k_s^2\right)
\right),
\label{Delta s}
\end{equation}
where $\iend={\rm mod}(N-1,k_s)+1$
and $\iendprime={\rm min}(\iend,(k_s+1)/2)$.
%
The final expression for $s(N)$ is obtained by substituting
Eqs.~(\ref{s^I})-(\ref{Delta s}) into Eq.~(\ref{s(N)}).
%%%%%%%%%%%%%%%%%%%
%\vskip 0.5cm
%\subsection*{7. Sum of all displacements for a sample of length $L$, $S(L)$}
The next step is to convert the dependence on the number of bonds,
$s(N)$, to that on the sample size, $S(L)$.
For an odd $n_s$ the number of bonds in a sample of length $L$ is
equal to the number of atoms $L/l_s+1$ in the substrate layer (\eg\ in
Fig.~\ref{mismatch_notation.fig} $n_s$=5, and the sample of length
$L=\L=5 l_s$ contains $\L/l_s+1=6$ bonds).
%
If $n_s$ is even, one additional bond builds in into each period $\L$,
\ie\ for the whole sample $L/\L$ additional bonds build in.
%
Therefore, the number of bonds in the sample of length $L$ is
\begin{equation}
N=
\left\{ \begin{array}{ll}
L/l_s +1 & {\rm for~} n_s {\rm~odd}\\
L/l_s+1+L/\L=L/l_s+1+L/(n_s l_s)=L~(n_s+1)/(l_s n_s)+1& {\rm for~} n_s {\rm~even}\\
\end{array} \right.,
\end{equation}
and the general expression for arbitrary $n_s$ reads
\begin{equation}
N=L k_s / l_s n_s +1.
\end{equation}
%
The sum of all displacements for a sample of length $L$ is then equal to
\begin{equation}
S(L)=s(N)=s(L k_s / l_s n_s +1),
\end{equation}
where the expression for $s(L k_s / l_s n_s +1)$ must be taken from
Eqs.~(\ref{s(N)})-(\ref{Delta s}).
%\def\Nbonds{\frac{L k_s}{l_s n_s}+1} % My fractional notation for Nbonds
\def\Nbonds{L k_s/l_s n_s+1} % My notation for Nbonds without fraction
%%%%%%%%%%%%%%%%%%%
%\vskip 0.5cm
%\subsection*{8. Average displacement of a bond in the sample
%of length $L$, $\langle \Delta d(L,f_m) \rangle$}
Dividing by the number of bonds in the sample, we obtain the average
displacement of a bond in the sample of length $L$:
\begin{equation}
\langle \Delta d(L,f_m) \rangle=\frac{S(L)}{N}=
\frac{\Delta l (k_s^2-1)l_s n_s\left[L/l_s n_s\right]}{4(L k_s + l_s n_s)}+
\frac{\Delta s_{\Nbonds}}{\Nbonds}
\label{dav}
\end{equation}
(the non-explicit dependence on the mismatch factor $f_m$ is inferred
from the dependencies of $n_s$, $k_s$ and $\Delta l$ on this parameter,
Eqs.~(\ref{n_s})-(\ref{Delta l})).
The complete expression for $\Delta s_{\Nbonds}$ must be substituted
from Eq.~(\ref{Delta s}).
%%%%%%%%%%%%%%%%%%%
%\vskip 0.5cm
%\subsection*{9. Average strain of a bond in the sample of
%length $L$, $\langle \Delta r(L,f_m) \rangle$}
Since the mismatch we are considering occurs in two directions
(Fig.~\ref{mismatch_notation.fig}, lower panel), the bond length $r$ is
related to the deviations $\Delta d$ by
\begin{equation}
r=\sqrt{r_e^2+2 \Delta d^2}\approx r_e+ \Delta d^2/r_e.
\end{equation}
%
The bond strain $\Delta r$ is then
\begin{equation}
\Delta r = r-r_e \approx \Delta d^2/r_e.
\label{r}
\end{equation}
%
Using Eq.~(\ref{dav}), we obtain the average strain for the sample of
length $L$:
\begin{equation}
\langle \Delta r(L,f_m) \rangle \approx \langle \Delta d(L,f_m) \rangle^2/r_e=
\frac{\Delta l^2 (k_s^2-1)^2 l_s^2 n_s^2\left[L/l_s n_s\right]^2}{16 r_e(L k_s + l_s n_s)^2}+
\frac{\Delta s_{\Nbonds}^2}{r_e(\Nbonds)^2}
\label{rav}
\end{equation}
Again, the dependence on the mismatch factor $f_m$ is inferred from the
dependencies of $n_s$, $k_s$ and $\Delta l$ on this parameter
(Eqs.~(\ref{n_s},\ref{k_s},\ref{Delta l})), and
the complete expression for $\Delta s_{\Nbonds}$ can be substituted
from Eq.~(\ref{Delta s}).
The
%so
derived dependence of \Dr\ on sample size $L$ and on mismatch
factor $f_m$, Eq.~(\ref{rav}), is plotted in
Fig.~\ref{effect_from_L_microscopic.fig}a. The behavior of this
quantity appears to be highly oscillatory at small $L$ ($\sim
l_{ds}=l_d f_m/(1-f_m)$)
%($\sim l_{ds}=\frac{l_d f_m}{1-f_m}$)
and
saturates to a
limit at $L\rightarrow\infty$ (see
below). A qualitative explanation of such dependence is given in the
main text (Sec.~\ref{CHOICE OF MISMATCH FACTOR}).
%%%%%%%%%%%%%%%%%%%
%\vskip 0.5cm
%\subsection*{10. Average reduction of the binding energy for
%the sample of length $L$, $\langle \Delta U(L,f_m) \rangle$}
The obtained mismatch-induced deviation from equilibrium of
the bond length can be used to estimate the binding energy reduction
caused by mismatch. For a rough estimation some simple pairwise
interaction potential, like that of Morse \cite{Morse_form},
$U_{\rm bond}(\Delta r)=D(1-\exp(-\beta \Delta r))^2$,
can be assumed.
%
Considering that in diamond there are 2 bonds per atom (each
atom is connected by 4 bonds, and each bond is shared by 2 atoms), the
reduction of the binding energy per atom can be estimated to be
\begin{equation}
\Delta U(\Delta r)=2~\left(U_{\rm bond}(0)-U_{\rm bond}(\Delta r)\right)=
2 D(-1-\exp(-2\beta \Delta r)+2\exp(-\beta\Delta r))
\end{equation}
%
Averaging over bonds with various strains (\ie\ substituting
$\langle \Delta r \rangle$ instead of $\Delta r$), we get the
average weakening of the binding energy of the sample of size $L$ due
to mismatch $f_m$:
\begin{equation}
\langle \Delta U(L,f_m) \rangle= 2 D\left(
-1-\exp({-2\beta \langle \Delta r(L,f_m)\rangle})
+2 \exp({-\beta \langle \Delta r(L,f_m)\rangle })
\right),
\label{uav}
\end{equation}
where the expression for $\langle \Delta r(L,f_m) \rangle$ must be
taken from Eq.~(\ref{rav}).
The function Eq.~(\ref{uav}) is plotted in
Fig.~\ref{effect_from_L_microscopic.fig}b. Again, the behavior
is oscillatory at small $L$ and
saturates to a
limit at
$L\rightarrow\infty$. This means that for large enough samples the
reduction of the binding energy is independent of $L$, and only a
(rather weak and non-monotonous) dependence on $f_m$ remains.
A qualitative explanation of such behavior is given in the main text
(Sec.~\ref{CHOICE OF MISMATCH FACTOR}).
%%%%%%%%%%%%%%%%%%%
%\vskip 0.5cm
%\subsection*{11. Limits of
%$\langle \Delta r(L,f_m) \rangle$, $\langle \Delta U(L,f_m) \rangle$
%at $L\rightarrow\infty$,
%$\langle \Delta r_{\infty} \rangle$ and
%$\langle \Delta U_{\infty} \rangle$}
As shown above, for large sample sizes $L$ the quantities
$\langle \Delta r(L,f_m) \rangle$
and $\langle \Delta U(L,f_m)\rangle$ converge to some asymptotic
values. Let us find these ``saturation limits''.
%
\def\liminf{L\rightarrow\infty} % My notation of l->infinity
%
Using Eq.~(\ref{dav}), we find the limit of $\langle \Delta d(L,f_m)
\rangle$ to be
\begin{equation}
\langle \Delta d_{\infty}(f_m)\rangle=
\lim_{\liminf}\langle \Delta d(L,f_m)\rangle=
\lim_{\liminf}
\frac{\Delta l (k_s^2-1)l_s n_s\left[L/l_s n_s\right]}{4(L k_s + l_s n_s)},
\end{equation}
since
$\frac{\Delta s_{\Nbonds}}{\Nbonds} \rightarrow 0$ at $\liminf$.
%
Considering that
\begin{equation}
\lim_{\liminf}\frac{l_s n_s \left[L/l_s n_s\right]}{L k_s + l_s n_s}=\frac{1}{k_s}
\end{equation}
and substituting expressions Eqs.~(\ref{n_s})-(\ref{Delta l}) for
$n_s$, $k_s$ and $\Delta l$, we get
\begin{equation}
\langle \Delta d_{\infty}(f_m)\rangle=
\left\{ \begin{array}{ll}
\frac{l_d f_m(2-f_m)}{4} & {\rm for~} n_s=1/(1-f_m) {\rm~odd}\\
\frac{l_d(3-2 f_m)}{4(2-f_m)}& {\rm for~} n_s=1/(1-f_m) {\rm~even}\\
\end{array} \right.,
\label{limit_dav}
\end{equation}
or, expressing this by one equation,
\begin{equation}
\langle \Delta d_{\infty}(f_m)\rangle=
{\rm mod}(1/(1-f_m),2) \cdot l_d~\frac{f_m(2-f_m)}{4}+
{\rm mod}(1/(1-f_m)+1,2) \cdot l_d~\frac{3-2 f_m}{4(2-f_m)}.
\end{equation}
Two features of this result should be noted. Firstly, the
dependence of the saturation limit on $f_m$ is non-monotonic: for
an odd $n_s$ it grows with $f_m$, while for an even $n_s$ it
decreases. In reality, $n_s$ does not have to be an integer, and,
although the general tendency for $\langle \Delta d_{\infty}\rangle$
is to decrease with $f_m$, the dependence still remains non-monotonic.
%
Secondly, for $f_m$ close to 1 (as it is in reality), for both
$n_s$ either odd or even
\begin{equation}
\langle \Delta d_{\infty,~f_m \sim 1}\rangle \approx l_d/4
\label{limit_dav_fm_close_to_1}
\end{equation}
Using the relationships (\ref{rav}) and (\ref{uav}), we finally obtain
\begin{equation}
\langle \Delta r_{\infty}(f_m)\rangle \approx
\langle \Delta d_{\infty}(f_m)\rangle ^2/r_e,
\label{limit_rav}
\end{equation}
\begin{equation}
\langle \Delta U_{\infty}(f_m)\rangle \approx
2 D\left(
-1-\exp({-2\beta \langle \Delta r_{\infty} (f_m)\rangle})
+2 \exp({-\beta \langle \Delta r_{\infty} (f_m) \rangle })
\right)
\label{limit_uav}
\end{equation}
These functions are plotted in Fig.~\ref{limit_at_large_L.fig}, where
we observe that, in accordance with
Eq.~(\ref{limit_dav_fm_close_to_1}), for a realistic value of $f_m$
close to unity, these limits are:
\begin{equation}
\langle \Delta r_{\infty,~f_m \sim 1}\rangle
\approx l_d^2/16 r_e,
\label{limit_rav_fm_close_to_1}
\end{equation}
\begin{equation}
\langle \Delta U_{\infty,~f_m \sim 1}\rangle \approx
2 D\left(
-1-\exp({-2\beta l_d^2/16 r_e}) + 2 \exp({-\beta l_d^2/16 r_e}) \right),
\label{limit_uav_fm_close_to_1}
\end{equation}
or, using the numeric values of parameters given above,
\begin{equation}
\langle \Delta r_{\infty,~f_m \sim 1}\rangle
\approx 0.108~l_d,
\label{limit_rav_fm_close_to_1_estimation}
\end{equation}
\begin{equation}
\langle \Delta U_{\infty,~f_m \sim 1}\rangle \approx -0.1 D
~~~(10\%{\rm ~weakening~of~the~interaction})
\label{limit_uav_fm_close_to_1_estimation}
\end{equation}
%%%%%%%%%%%%%%%%%%%
%\vskip 0.5cm
%\subsection*{12. Conclusion: character of the calculated dependence}
Summarizing, we can note that, though all above
estimations are rough (containing many approximations and
not considering the specific structure of the lattice),
%
they still allow to understand how the ``\im'' depends on
sample size $L$ and on mismatch factor $f_m$
(Fig.~\ref{effect_from_L_microscopic.fig}): for small samples
($L \sim l_{ds}=l_d f_m/(1-f_m)$) it is
highly sensitive to both sample size and mismatch factor, while for
large samples only the dependence on $f_m$ remains
(Eqs.~(\ref{limit_rav})-(\ref{limit_uav}).
Moreover, for large samples, if the mismatch factor is
close to 1 (as it is in reality), also this last dependence is almost
eliminated: for all values of $f_m$ the \im\ is nearly the same and
corresponds to approximately a $10\%$ weakening of the interaction
(Eqs.~(\ref{limit_rav_fm_close_to_1})-(\ref{limit_uav_fm_close_to_1_estimation})).
Finally, the
period necessary to achieve the ``saturation limit'' at which the
\im\ becomes insensitive to the sample size depends on the value of
mismatch factor: the smaller $f_m$ (\ie\ the larger the mismatch),
the sooner (for smaller $L$) this ``saturation limit'' is achieved
(Fig.~\ref{effect_from_L_microscopic.fig}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
%\addcontentsline{toc}{section}{\hskip 0.65cm Appendix~\protect{\ref{APPENDIX 2}}}
\section{ Evaluation of Rayleigh coefficients for three defects -
graphite-like carbon atom in $sp^2$ configuration,
tetrahedral amorphous carbon $ta$-C,
and $\langle 100 \rangle$ split interstitial \label{APPENDIX 2}}
%\vskip 0.5cm
%\subsection*{1. General expression for the Rayleigh coefficient $I$}
According to Ref.~\cite{TurkKlemens}, the Rayleigh coefficient, $I$, for the
scattering on point defects is given by
\begin{equation}
I=\frac{ca^3k^4}{4\pi v^3\hbar^4}\left(\frac{\Delta M}{M}+2\gamma p\right)^2,
\label{Rayleigh_general.eqn}
\end{equation}
where $c$ is the concentration of defects,
$a^3$ is the volume per atom (for diamond equal to $5.67\cdot 10^{-24}$
cm$^3$/atom at r.t.),
$v$ is the velocity of sound in the material, $1.195\cdot10^6$ cm/s
\cite{RAB1},
$\Delta M$ is the mass change due to replacement of the parent atom of
mass $M$ by the defect atom of mass $M+\Delta M$,
$\gamma$ is the Gr\"uneisen parameter (equal to 1.1 for diamond
\cite{BerMart}),
and $p$ is the fractional volume difference accounted due to the
defect.
In our simulation all defects are
local inhomogeneities and have
the same mass as the parent atoms. Hence, the point defect scattering
is due only to the difference in the volume occupied by an ideal
diamond atom and that of a defect. Therefore,
Eq.~(\ref{Rayleigh_general.eqn}) simplifies to
\begin{equation}
I=\frac{ca^3k^4}{4\pi v^3\hbar^4}~\left(2\gamma p\right)^2,
\label{Rayleigh_here.eqn}
\end{equation}
and the only defect-specific quantity which remains to be calculated
is the fractional volume difference $p$ of each particular defect:
\begin{equation}
p \equiv \frac{v_{\rm p.d.}-v_{\rm d}}{v_{\rm d}},
\label{p_general.eqn}
\end{equation}
where $v_{\rm p.d.}$ and $v_{\rm d}$ are the atomic volume of point
defect and of ideal diamond, correspondingly.
Let us now estimate the values of $p$ and $I$ for the three defects
relevant to the present study, namely
for the graphite-like carbon atom in \sp2\ configuration,
for the tetrahedral amorphous carbon \ta -C,
and for the $\langle 100 \rangle$ split interstitial.
%\vskip 0.7cm
%\subsection*{2. Isolated graphite-like 3-fold carbon atom in the $sp^2$
%configuration}
First we consider the isolated graphite-like 3-fold carbon
atom in the $sp^2$ configuration.
%
The configuration of this defect is equivalent to that of a
carbon atom in graphite. Therefore, its atomic volume $v_{sp^2}$
is equal to the atomic volume $v_{\rm g}$ of graphite, and the
fractional volume difference for this defect, $p_{sp^2}$, is given by
\begin{equation}
p_{sp^2} \equiv \frac{v_{sp^2}-v_{\rm d}}{v_{\rm d}}\approx
\frac{v_{\rm g}-v_{\rm d}}{v_{\rm d}},
\label{p_sp^2.eqn}
\end{equation}
where the subscript ``$sp^2$'' stands for the ``isolated graphite-like
3-fold carbon defect in the $sp^2$ configuration''.
With the values $v_{\rm d}=5.65\cdot10^{-24}~{\rm cm}^3/{\rm
atom}$ and $v_{\rm g}=8.77\cdot10^{-24}~{\rm cm}^3/{\rm atom}$
\cite{Dresselhaus_Kalish}, $p_{sp^2}$ and, consequently, $I_{sp^2}$
take the values
\begin{equation}
p_{sp^2} \approx 0.552, ~~~~~ I_{sp^2} \approx 114.48 c~{\rm
K}^{-4}{\rm s}^{-1},
\label{p_Rayleigh_isolated_sp^2_value.eqn}
\end{equation}
$c$ being the dimensionless atomic concentration of defects in question.
%\vskip 0.7cm
%\subsection*{3. Intermediate tetrahedral amorphous carbon defect, {\it ta}-C}
Next, we estimate the values of $p$ and $I$ for the
intermediate tetrahedral amorphous carbon defect, {\it ta}-C.
%
The configuration of this defect is intermediate between
graphite and diamond; according to
Refs.~\cite{Pailthorpe2,Pailthorpe5}, its coordination number is equal
to 3.7. This means that this defect can be considered to be 70\%
diamond-like and 30\% graphite-like in character, and its atomic
volume to be approximately equal to
\begin{equation}
v_{ta{\rm-C}} \approx 0.7 v_{\rm d} + 0.3 v_{\rm g}.
\end{equation}
This gives the fractional volume difference of
\begin{equation}
p_{ta{\rm-C}} \equiv \frac{v_{ta{\rm-C}}-v_{\rm d}}{v_{\rm d}}\approx
\frac{0.3 (v_{\rm g}-v_{\rm d})}{v_{\rm d}}.
\label{p_ta-C.eqn}
\end{equation}
For the above values of $v_{\rm d}$ and $v_{\rm g}$ this volume
difference and the corresponding Rayleigh coefficient $I_{ta{\rm-C}}$
take the values
\begin{equation}
p_{ta{\rm-C}} \approx 0.166, ~~~~~ I_{ta{\rm-C}} \approx 9.84 c~{\rm
K}^{-4}{\rm s}^{-1},
\label{p_Rayleigh_ta-C_value.eqn}
\end{equation}
where $c$ is the atomic concentration of {\it ta}-C defects.
%\vskip 0.7cm
%\subsection*{4. \splita}
Finally, we consider the \splita .
%(marked by the subscript ``$si$'').
%
It would seem reasonable to determine the effective volume of
a split interstitial, $v_{\rm si}$, from the experiments on diamond
implantation, provided that (1) it is known that the produced defects
are split interstitials; (2) their concentration is known; (3) the
swelling of the diamond lattice due to these defects is known (\eg,
from the step height measurements \cite{PHK}).
This approach, however, is
subject to two main difficulties.
Firstly, the standard Monte Carlo computer code TRIM used for
the calculation of defects concentration in implantation experiments
completely ignores dynamic annealing which repairs the lattice almost
instantaneously \cite{Steve_private}; hence, the concentration of
defects given by TRIM is highly overestimated (sometimes, by several
orders) \cite{Steve_PRB}.
Secondly, in experiment it is impossible to obtain only one sort
of defects: the implantation always results in the formation of both
vacancies and split interstitials, as well as of the partially
graphitized regions \cite{Alla_NIMB,Steve_PRB}.
To overcome these difficulties, high-energy implantation
experiments in which a single vacancy-interstitial pair will be
created in one ion hit \cite{Steve_private} are desirable; however, to
the best of our knowledge, such experiments on diamond have not been
carried yet.
Thus, since appropriate experimental data are lacking, we
estimate the volume expansion of diamond due to a \splita\ from
simulation results. In Ref.~\cite{RAB1} a dumbbell was created at a
separation of 1.27\A, \ie\ one \splita\ was artificially inserted
into the {\it non-expanded} diamond lattice. This defect was annealed,
and the dumbbell separation relaxed to 1.47\A. Considering that the
dumbbell defect is linear, the ratio of these quantities, $1.47/1.27$,
may serve as a rough estimation of the extent to which a single \splita\
expands the surrounding diamond lattice:
\begin{equation}
v_{\rm si}/v_{\rm d} \approx 1.47/1.27
\label{ratio_splita.eqn}
\end{equation}
With this estimation the fractional volume difference due to \splita,
$p_{\rm si}$, and the corresponding Rayleigh coefficient, $I_{\rm
si}$, take the values:
\begin{equation}
p_{\rm si} \equiv \frac{v_{\rm si}-v_{\rm d}}{v_{\rm d}}=\frac{v_{\rm
si}}{v_{\rm d}}-1\approx 1.47/1.27-1=0.157, ~~~~~ I_{\rm si} \approx
9.31 c~{\rm K}^{-4}{\rm s}^{-1},
\label{p_Rayleigh_splita_value.eqn}
\end{equation}
%\vskip 0.7cm
%\subsection*{5. Comparison of volume expansion due to the three defects}
Comparison between the fractional volume expansion due
to an isolated $sp^2$-configurated 3-fold atom, $p_{sp^2} \approx 0.552$,
to an ``intermediate'' {\it ta}-C defect, $p_{ta{\rm-C}} \approx 0.166$,
and to a \splita, $p_{\rm 3i} \approx 0.157$,
demonstrates that the former defect is much less compact than the
others and hence is expected to be consistent with lower values of
pressure (stress). The most compact defect is the \splita; it is
likely to form when the volume is most restricted, \eg, for a high
degree of the film continuity.
%FOR TWO COLUMN ACTIVATE THE LINE BELOW (Irka's)
%\twocolumn
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\addcontentsline{toc}{section}{\hskip 0.65cm References}
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phone:~972-4-8293937,
fax:~972-4-8221514,
e-mail:~phr76ja@phjoan.technion.ac.il
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%predominates? exceeds? is prevalent?
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\onecolumn
\addcontentsline{toc}{section}{\hskip 0.65cm Figures}
%\section*{ Figures \label{FIGURES}} %"revtex" format makes it automatically
\pagestyle{empty}
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\begin{figure}[h]
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\caption{The (100) diamond surface at which the biaxial compressive
stress is applied (unit cell).
Numbers indicate the coordinate along the [100] ($z$) direction
in units of the lattice parameter of diamond, $a_d$
(positive $z$ direction is into the page).
Bonds with atoms of the deepest plane $z=a_d$ (underlying those at $z=0$)
are shown by dashed lines.
}
\label{100_plane.fig}
\end{figure}
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\begin{figure}[h]
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\caption{Mismatch factor $f_m$ as the function of temperature
for two substrates
calculated by
Eq.~(\protect{\ref{fm_from_T_macroscopic.eqn}}). Dependence of diamond
and silicon thermal expansion coefficients on temperature, used for
the calculation, was taken from
Ref.~\protect{\cite{thermal_expansion_coefficients}}; the coefficient
of chromium carbide, which is weakly dependent on temperature, was
taken equal to $11.7 \times 10^{-6} {\rm K}^{-1}$ for the entire
temperature range \protect{\cite{AlonOlga1}}.}
\label{fm_from_T_macroscopic.fig}
\end{figure}
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\begin{figure}[h]
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\caption{
Dependence of ({\it a}) the average bond strain (in units of the
diamond bond length, $l_d=1.54\A$), calculated using
Eq.~(\protect{\ref{rav}}),
and of ({\it b}) the reduction of the binding energy per atom (in
units of the well depth $D$), calculated using
Eq.~(\protect{\ref{uav}}), on sample size $L$,
for different values of the mismatch factor:
$f_m=0.90$ (solid line), $f_m=0.96$ (dotted line) and $f_m=0.99$
(dashed line).
Insets show magnification of the small-sample part of the
curves. Arrows on insets indicate the increase of the ``film
continuity'', corresponding to the decrease of the mismatch factor $f_m$.
See Section~\protect{\ref{CHOICE OF MISMATCH FACTOR}} for the
explanation of the A, B and C notations.
}
\label{effect_from_L_microscopic.fig}
\end{figure}
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\begin{figure}[h]
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\caption{
Scheme of the average bond strain and its relaxation
abilities for a small sample ("isolated particle" - {\it a,b}) and
a large one ("continuous film" - {\it c}). Diamond atoms are shown
by light grey, substrate atoms in general by dark grey, and substrate
atoms with the maximal mismatch are black. For clarity, in this Fig.
the close packing is shown.
}
\label{effect_from_L_scheme.fig}
\end{figure}
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\begin{figure}[h]
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\caption{Reduction of the binding energy due to mismatch: immediately
after the deposition stage, ``as-grown'' film (circles), and after
the relaxation (triangles).
Model values
(calculated using Eq.~(\protect{\ref{uav}}) and divided by 8 to
consider that the mismatch located in 2 layers acts over 16 and not 2
layers, as was assumed in the model)
are also shown for comparison (crosses).}
\label{binding_energy_reduction_per.fig}
\end{figure}
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\begin{figure}[h]
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\caption{Initial lateral stress $\sigma$ in the substrate layer as a
result of the strain $e$ for an ideal crystal (solid line) and for
defect-containing samples obtained by the simulated CVD (dotted line);
the thin line with arrows indicates the stress below which the
graphitic $sp^2$ defects can be formed. Stress estimated from phonon
spectra using Ager and Drory model \protect{\cite{Ager}} is shown by
the dashed line.}
\label{initial_stress.fig}
\end{figure}
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\begin{figure}[h]
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\caption{Evolution of defects concentration
with the increase of mismatch (corresponding to the increase of the
film continuity) calculated based on coordination
numbers. This Fig. does not reflect the difference between the
``intermediate tetrahedrally coordinated amorphous carbons'', {\it
ta}-C, and the $sp^2$ atoms in graphitic configuration (see the text);
both these defects are classified as ``isolated 3-fold atoms''.}
\label{concentration_of_defects_per.fig}
\end{figure}
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\begin{figure}[h]
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\caption{Evolution of radial distribution function
with the increase of mismatch (corresponding to the increase of the
film continuity).
Notation of peaks: {\it A} - ideal diamond; {\it B} - tetrahedral
amorphous carbon, {\it ta}-C; {\it C} - 3-fold graphitic $sp^2$ atoms;
{\it D} - split interstitials.}
\label{rdf_per.fig}
\end{figure}
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\begin{figure}[h]
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\caption{Evolution of phonon spectrum with the increase of mismatch
(corresponding to the increase of the film continuity).
Notation of peaks: {\it A} - ideal diamond; {\it B} - tetrahedral
amorphous carbon, {\it ta}-C; {\it C} - 3-fold graphitic $sp^2$ atoms;
{\it D} - split interstitials; a broad feature attributed to both
kinds of isolated 3-fold defects, {\it B} and {\it C}, and referred to
as an ``amorphous carbon peak'' is marked by ``a.c.''.
}
\label{phonon_spectra_per.fig}
\end{figure}
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\begin{figure}[h]
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\caption{Intensity of the ``amorphous carbon'' peak with respect to
the main diamond peak ({\it A} in
Fig.~\protect{\ref{phonon_spectra_per.fig}}),
calculated based on Fig.~\protect{\ref{phonon_spectra_per.fig}}.}
\label{relative_intensity.fig}
\end{figure}
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\begin{figure}[h]
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\caption{
Penetration depth of stress-induced defects: number of isolated 3-fold
atoms - ``intermediate carbons'' (solid line) and split interstitials
(dotted line) - as a function of the layer number.
The numbering of layers starts from substrate.
}
\label{coord_numbers_profiling_per.fig}
\end{figure}
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\begin{figure}[h]
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\caption{
Thermal conductivity of diamond with residual stress at
room temperature (solid line), and its components resulting from point
defects (``p.d.'') (dashed line) and from the rest scattering
mechanisms (dotted line). Thermal conductivity of ideal diamond is
shown for comparison (thin solid line).
}
\label{thermoconductivity_per.fig}
\end{figure}
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\begin{figure}[h]
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\caption{
Notation for Appendix~\protect{\ref{APPENDIX 1}}: ({\it a}) shows part
of the diamond/substrate interface and ({\it b}) illustrates details
of one bond.
Color code: light grey - diamond atoms, black - substrate atoms.
}
\label{mismatch_notation.fig}
\end{figure}
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\begin{figure}[h]
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\caption{
Large-sample limits ({\it a}) of the average bond strain (in
units of the diamond bond length, $l_d=1.54\A$)
and ({\it b}) of the reduction of the binding energy per atom (in
units of the well depth $D$) ({\it b}) drawn according to
~Eqs.~(\protect{\ref{limit_rav}})-(\protect{\ref{limit_uav}}).
}
\label{limit_at_large_L.fig}
\end{figure}
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