omitted LJ + more precise ef definition

[lectures/latex.git] / posic / thesis / const_sic.tex

\chapter{Investigation of self-constructed 3C-SiC precipitates}

\section{3C-SiC precipitate in crystalline silicon}
\label{section:const_sic:prec}

{\color{red}Todo: Phase stability as Kai Nordlund proposed}

A spherical 3C-SiC precipitate enclosed in a c-Si surrounding is constructed as it is expected from IBS experiments and from simulations that finally succeed in simulating the precipitation event.
On the one hand this sheds light on characteristic values like the radial distribution function or the total amount of free energy for such a configuration that is aimed to be reproduced by simulation.
On the other hand, assuming a correct alignment of the precipitate with the c-Si matrix, properties of such precipitates and the surrounding as well as the interface can be investiagted.
Furthermore these investigations might establish the prediction of conditions necessary for the simulation of the precipitation process.

To construct a spherical and topotactically aligned 3C-SiC precipitate in c-Si, the approach illustrated in the following is applied.
A total simulation volume $V$ consisting of 21 unit cells of c-Si in each direction is created.
To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary.
This corresponds to a spherical 3C-SiC precipitate with a radius of approximately 3 nm.
The initial precipitate configuration is constructed in two steps.
In the first step the surrounding silicon matrix is created.
This is realized by just skipping the generation of silicon atoms inside a sphere of radius $x$, which is the first unknown variable.
The silicon lattice constant $a_{\text{Si}}$ of the surrounding c-Si matrix is assumed to not alter dramatically and, thus, is used for the initial lattice creation.
In a second step 3C-SiC is created inside the empty sphere of radius $x$.
The lattice constant $y$, the second unknown variable, is chosen in such a way, that the necessary amount of carbon is generated and that the total amount of silicon atoms corresponds to the usual amount contained in the simulation volume.
This is entirely described by the system of equations \eqref{eq:md:constr_sic_01}
\begin{equation}
\frac{8}{a_{\text{Si}}^3}(
\underbrace{21^3 a_{\text{Si}}^3}_{=V}
-\frac{4}{3}\pi x^3)+
\underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
=21^3\cdot 8
\label{eq:md:constr_sic_01}
\text{ ,}
\end{equation}
which can be simplified to read
\begin{equation}
\frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
\Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
\label{eq:md:constr_sic_02}
\end{equation}
and
\begin{equation}
%x^3=\frac{16\pi}{5500 \cdot 3}y^3=
%\frac{16\pi}{5500 \cdot 3}\frac{5500 \cdot 3}{32 \pi}a_{\text{Si}}^3
%\Rightarrow
y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
\text{ .}
\label{eq:md:constr_sic_03}
\end{equation}
By this means values of 2.973 nm and 4.309 \AA{} are obtained for the initial precipitate radius and lattice constant of 3C-SiC.
Since the generation of atoms is a discrete process with regard to the size of the volume the expected amounts of atoms are not obtained.
However, by applying these values the final configuration varies only slightly from the expected one by five carbon and eleven silicon atoms, as can be seen in table \ref{table:md:sic_prec}.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{l c c c c}
\hline
\hline
 & C in 3C-SiC & Si in 3C-SiC & Si in c-Si surrounding & total amount of Si\\
\hline
Obtained & 5495 & 5486 & 68591 & 74077\\
Expected & 5500 & 5500 & 68588 & 74088\\
Difference & -5 & -14 & 3 & -11\\
Notation & $N^{\text{3C-SiC}}_{\text{C}}$ & $N^{\text{3C-SiC}}_{\text{Si}}$ 
         & $N^{\text{c-Si}}_{\text{Si}}$ & $N^{\text{total}}_{\text{Si}}$ \\
\hline
\hline
\end{tabular}
\caption{Comparison of the expected and obtained amounts of Si and C atoms by applying the values from equations \eqref{eq:md:constr_sic_02} and \eqref{eq:md:constr_sic_03} in the 3C-SiC precipitate construction approach.}
\label{table:md:sic_prec}
\end{center}
\end{table}

After the initial configuration is constructed some of the atoms located at the 3C-SiC/c-Si interface show small distances, which results in high repulsive forces acting on the atoms.
Thus, the system is equilibrated using strong coupling to the heat bath, which is set to be $20\,^{\circ}\mathrm{C}$.
Once the main part of the excess energy is carried out previous settings for the Berendsen thermostat are restored and the system is relaxed for another 10 ps.

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=12cm]{pc_0.ps}
\end{center}
\caption[Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$.]{Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$. The Si-Si radial distribution of plain c-Si is plotted for comparison. Green arrows mark bumps in the Si-Si distribution of the precipitate configuration, which do not exist in plain c-Si.}
\label{fig:md:pc_sic-prec}
\end{figure}
Figure \ref{fig:md:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
The Si-Si radial distribution for both, plain c-Si and the precipitate configuration show a maximum at a distance of 0.235 nm, which is the distance of next neighboured Si atoms in c-Si.
Although no significant change of the lattice constant of the surrounding c-Si matrix was assumed, surprisingly there is no change at all within observational accuracy.
Looking closer at higher order Si-Si peaks might even allow the guess of a slight increase of the lattice constant compared to the plain c-Si structure.
A new Si-Si peak arises at 0.307 nm, which is identical to the peak of the C-C distribution around that value.
It corresponds to second next neighbours in 3C-SiC, which applies for Si as well as C pairs.
The bumps of the Si-Si distribution at higher distances marked by the green arrows can be explained in the same manner.
They correspond to the fourth and sixth next neighbour distance in 3C-SiC.
It is easily identifiable how these C-C peaks, which imply Si pairs at same distances inside the precipitate, contribute to the bumps observed in the Si-Si distribution.
The Si-Si and C-C peak at 0.307 nm enables the determination of the lattic constant of the embedded 3C-SiC precipitate.
A lattice constant of 4.34 \AA{} compared to 4.36 \AA{} for bulk 3C-SiC is obtained.
This is in accordance with the peak of Si-C pairs at a distance of 0.188 nm.
Thus, the precipitate structure is slightly compressed compared to the bulk phase.
This is a quite surprising result since due to the finite size of the c-Si surrounding a non-negligible impact of the precipitate on the materializing c-Si lattice constant especially near the precipitate could be assumed.
However, it seems that the size of the c-Si host matrix is chosen large enough to even find the precipitate in a compressed state.

The absence of a compression of the c-Si surrounding is due to the possibility of the system to change its volume.
Otherwise the increase of the lattice constant of the precipitate of roughly 4.31 \AA{} in the beginning up to 4.34 \AA{} in the relaxed precipitate configuration could not take place without an accompanying reduction of the lattice constant of the c-Si surrounding.
If the total volume is assumed to be the sum of the volumes that are composed of Si atoms forming the c-Si surrounding and Si atoms involved forming the precipitate the expected increase can be calculated by
\begin{equation}
 \frac{V}{V_0}=
 \frac{\frac{N^{\text{c-Si}}_{\text{Si}}}{8/a_{\text{c-Si of precipitate configuration}}}+
 \frac{N^{\text{3C-SiC}}_{\text{Si}}}{4/a_{\text{3C-SiC of precipitate configuration}}}}
 {\frac{N^{\text{total}}_{\text{Si}}}{8/a_{\text{plain c-Si}}}}
\end{equation}
with the notation used in table \ref{table:md:sic_prec}.
The lattice constant of plain c-Si at $20\,^{\circ}\mathrm{C}$ can be determined more accurately by the side lengthes of the simulation box of an equlibrated structure instead of using the radial distribution data.
By this a value of $a_{\text{plain c-Si}}=5.439\text{ \AA}$ is obtained.
The same lattice constant is assumed for the c-Si surrounding in the precipitate configuration $a_{\text{c-Si of precipitate configuration}}$ since peaks in the radial distribution match the ones of plain c-Si.
Using $a_{\text{3C-SiC of precipitate configuration}}=4.34\text{ \AA}$ as observed from the radial distribution finally results in an increase of the initial volume by 0.12 \%.
However, each side length and the total volume of the simulation box is increased by 0.20 \% and 0.61 \% respectively compared to plain c-Si at $20\,^{\circ}\mathrm{C}$.
Since the c-Si surrounding resides in an uncompressed state the excess increase must be attributed to relaxation of strain with the strain resulting from either the compressed precipitate or the 3C-SiC/c-Si interface region.
This also explains the possibly identified slight increase of the c-Si lattice constant in the surrounding as mentioned earlier.
As the pressure is set to zero the free energy is minimized with respect to the volume enabled by the Berendsen barostat algorithm.
Apparently the minimized structure with respect to the volume is a configuration of a small compressively stressed precipitate and a large amount of slightly stretched c-Si in the surrounding.

In the following the 3C-SiC/c-Si interface is described in further detail.
One important size analyzing the interface is the interfacial energy.
It is determined exactly in the same way than the formation energy as described in equation \eqref{eq:defects:ef2}.
Using the notation of table \ref{table:md:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
\begin{equation}
 E_{\text{f}}=E-
 N^{\text{3C-SiC}}_{\text{C}} \mu_{\text{SiC}}-
 \left(N^{\text{total}}_{\text{Si}}-N^{\text{3C-SiC}}_{\text{C}}\right)
 \mu_{\text{Si}} \text{ ,}
\label{eq:md:ife}
\end{equation}
with $E$ being the free energy of the precipitate configuration at zero temperature.
An interfacial energy of 2267.28 eV is obtained.
The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of 29.93 \AA.
Thus, the interface tension, given by the energy of the interface devided by the surface area of the precipitate is $20.15\,\frac{\text{eV}}{\text{nm}^2}$ or $3.23\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$.
This is located inside the eperimentally estimated range of $2-8\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$ \cite{taylor93}.

Since the precipitate configuration is artificially constructed the resulting interface does not necessarily correspond to the energetically most favorable configuration or to the configuration that is expected for an actually grown precipitate.
Thus annealing steps are appended to the gained structure in order to allow for a rearrangement of the atoms of the interface.
The precipitate structure is rapidly heated up to $2050\,^{\circ}\mathrm{C}$ with a heating rate of approximately $75\,^{\circ}\mathrm{C}/\text{ps}$.
From that point on the heating rate is reduced to $1\,^{\circ}\mathrm{C}/\text{ps}$ and heating is continued to 120 \% of the Si melting temperature, that is 2940 K.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=12cm]{fe_and_t_sic.ps}
\end{center}
\caption{Free energy and temperature evolution of a constructed 3C-SiC precipitate embedded in c-Si at temperatures above the Si melting point.}
\label{fig:md:fe_and_t_sic}
\end{figure}
Figure \ref{fig:md:fe_and_t_sic} shows the free energy and temperature evolution.
The sudden increase of the free energy indicates possible melting occuring around 2840 K.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=12cm]{pc_500-fin.ps}
\end{center}
\caption{Radial distribution of the constructed 3C-SiC precipitate embedded in c-Si at temperatures below and above the Si melting transition point.}
\label{fig:md:pc_500-fin}
\end{figure}
Investigating the radial distribution function shown in figure \ref{fig:md:pc_500-fin}, which shows configurations below and above the temperature of the estimated transition, indeed supports the assumption of melting gained by the free energy plot.
However the precipitate itself is not involved, as can be seen from the Si-C and C-C distribution, which essentially stays the same for both temperatures.
Thus, it is only the c-Si surrounding undergoing a structural phase transition, which is very well reflected by the difference observed for the two Si-Si distributions.
This is surprising since the melting transition of plain c-Si is expected at temperatures around 3125 K, as discussed in section \ref{subsection:md:tval}.
Obviously the precipitate lowers the transition point of the surrounding c-Si matrix.
This is indeed verified by visualizing the atomic data.
\begin{figure}[!ht]
\begin{center}
\begin{minipage}{7cm}
\includegraphics[width=7cm,draft=false]{sic_prec/melt_01.eps}
\end{minipage}
\begin{minipage}{7cm}
\includegraphics[width=7cm,draft=false]{sic_prec/melt_02.eps}
\end{minipage}
\begin{minipage}{7cm}
\includegraphics[width=7cm,draft=false]{sic_prec/melt_03.eps}
\end{minipage}
\end{center}
\caption{Cross section image of atomic data gained by annealing simulations of the constructed 3C-SiC precipitate in c-Si at 200 ps (top left), 520 ps (top right) and 720 ps (bottom).}
\label{fig:md:sic_melt}
\end{figure}
Figure \ref{fig:md:sic_melt} shows cross section images of the atomic structures at different times and temperatures.
As can be seen from the image at 520 ps melting of the Si surrounding in fact starts in the defective interface region of the 3C-SiC precipitate and the c-Si surrounding propagating outwards until the whole Si matrix is affected at 720 ps.
As predicted from the radial distribution data the precipitate itself remains stable.

For the rearrangement simulations temperatures well below the transition point should be used since it is very unlikely to recrystallize the molten Si surrounding properly when cooling down.
To play safe the precipitate configuration at 100 \% of the Si melting temperature is chosen and cooled down to $20\,^{\circ}\mathrm{C}$ with a cooling rate of $1\,^{\circ}\mathrm{C}/\text{ps}$.
However, an energetically more favorable interface is not obtained by quenching this structure to zero Kelvin.
Obviously the increased temperature run enables structural changes that are energetically less favorable but can not be exploited to form more favorable configurations by an apparently yet too fast cooling down process.

\section{Coherent to incoherent transition of 3C-SiC precipitates in crystalline silicon}

As already pointed out, some of the previous results indicate the very likely possibility of another precipitation mechanism.
This mechanism is based on the successive formation of substitutional C sites, which might result in coherent 3C-SiC structures within the c-Si matrix assuming that Si self-interstitials might diffuse out of the affected region easily.
Reaching a critical size these coherent SiC structures release the alignement on the c-Si lattice spacing by contracting to an incoherent SiC precipitate with lower lattice constant.

Precipitation -> contraction ... free 'space' might be compensated by volume changes due to the barostat ...

In contrary to the last constructed precipitates 

{\color{red}Todo: TEM simulations to check whether coherent SiC in c-Si would also lead to dark contrasts on an undisturbed Si lattice structure.}