\section{Silicon self-interstitials}
-For investigating the \si{} structures a Si atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section~\ref{section:basics:defects}.
+For investigating the \si{} structures, a Si atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section~\ref{section:basics:defects}.
The formation energies of \si{} configurations are listed in Table~\ref{tab:defects:si_self} for both methods used in this work as well as results obtained by other {\em ab initio} studies~\cite{al-mushadani03,leung99}.
\bibpunct{}{}{,}{n}{}{}
\begin{table}[tp]
\subsection{Defect structures in a nutshell}
-For investigating the \ci{} structures a C atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section~\ref{section:basics:defects}.
+For investigating the \ci{} structures, a C atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section~\ref{section:basics:defects}.
Formation energies of the most common C point defects in crystalline Si are summarized in Table~\ref{tab:defects:c_ints}.
The relaxed configurations are visualized in Fig.~\ref{fig:defects:c_conf}.
Again, the displayed structures are the results obtained by the classical potential calculations.
An experimental value of the formation energy of \cs{} was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\,\text{eV}/k_{\text{B}}T)} \text{ cm}^{-3}$~\cite{bean71}.
However, there is no particular reason for treating the prefactor as a free parameter in the fit to the experimental data.
It is simply given by the atomic density of pure silicon, which is $5\times 10^{22}\text{ cm}^{-3}$.
-Tersoff~\cite{tersoff90} and Dal Pino et al.~\cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6--1.89]{eV} an excellent agreement with the experimental solubility data within the entire temperature range of the experiment is obtained.
-This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal~Pino~et~al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.
+Tersoff~\cite{tersoff90} and Dal~Pino~et~al.~\cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6--1.89]{eV} an excellent agreement with the experimental solubility data within the entire temperature range of the experiment is obtained.
+This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal~Pino~et~al.\ is in good agreement with the results of the quantum-mechanical calculations performed in this work.
Unfortunately the EA potential undervalues the formation energy roughly by a factor of two, which is a definite drawback of the potential.
Except for Tersoff's results for the tetrahedral configuration, the \ci{} \hkl<1 0 0> DB is the energetically most favorable interstitial configuration.
An activation energy of \unit[2.2]{eV} is necessary to reorientate the \hkl[0 0 -1] into the \hkl[1 1 0] DB configuration, which is \unit[1.3]{eV} higher in energy.
Residing in this state another \unit[0.90]{eV} is enough to make the C atom form a \hkl[0 0 -1] DB configuration with the Si atom of the neighbored lattice site.
In contrast to quantum-mechanical calculations, in which the direct transition is the energetically most favorable transition and the transition composed of the intermediate migration steps is very unlikely to occur, the just presented pathway is much more conceivable in classical potential simulations, since the energetically most favorable transition found so far is likewise composed of two migration steps with activation energies of \unit[2.2]{eV} and \unit[0.5]{eV}, for which the intermediate state is the BC configuration, which is unstable.
-Thus the just proposed migration path, which involves the \hkl[1 1 0] interstitial configuration, becomes even more probable than the initially proposed path, which involves the BC configuration that is, in fact, unstable.
+Thus, the just proposed migration path, which involves the \hkl[1 1 0] interstitial configuration, becomes even more probable than the initially proposed path, which involves the BC configuration that is, in fact, unstable.
Due to these findings, the respective path is proposed to constitute the diffusion-describing path.
The evolution of structure and configurational energy is displayed again in Fig.~\ref{fig:defects:involve110}.
\begin{figure}[tp]
In both configurations, the far-off atom of the second DB resides in threefold coordination.
The interaction of \ci{} \hkl<1 0 0> DBs is investigated along the \hkl[1 1 0] bond chain assuming a possible reorientation of the DB atom at each position to minimize its configurational energy.
-Therefore, the binding energies of the energetically most favorable configurations with the second DB located along the \hkl[1 1 0] direction and resulting C-C distances of the relaxed structures are summarized in Table~\ref{tab:defects:comb_db110}.
+Therefor, the binding energies of the energetically most favorable configurations with the second DB located along the \hkl[1 1 0] direction and resulting C-C distances of the relaxed structures are summarized in Table~\ref{tab:defects:comb_db110}.
\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c c c c}
Intermediate configurations within the investigated turbulent pathway identify barrier heights of more than \unit[4]{eV} resulting in a low probability for the transition.
The high activation energy is attributed to the stability of such a low energy configuration, in which the C atom of the second DB is located close to the initial DB.
Due to an effective stress compensation realized in the respective low energy configuration, which will necessarily be lost during migration, a high energy configuration needs to get passed, which is responsible for the high barrier.
-Low barriers are only identified for transitions starting from energetically less favorable configurations, e.g. the configuration of a \hkl[-1 0 0] DB located at position 2 (\unit[-0.36]{eV}).
+Low barriers are only identified for transitions starting from energetically less favorable configurations, e.g.\ the configuration of a \hkl[-1 0 0] DB located at position 2 (\unit[-0.36]{eV}).
Starting from this configuration, an activation energy of only \unit[1.2]{eV} is necessary for the transition into the ground state configuration.
The corresponding migration energies and atomic configurations are displayed in Fig.~\ref{fig:036-239}.
\begin{figure}[tp]
Obviously, agglomeration of C$_{\text{i}}$ and C$_{\text{s}}$ is energetically favorable except for separations along one of the \hkl<1 1 0> directions.
The energetically most favorable configuration (configuration $\beta$) forms a strong but compressively strained C-C bond with a separation distance of \unit[0.142]{nm} sharing a Si lattice site.
Again, conclusions concerning the probability of formation are drawn by investigating respective migration paths.
-Since C$_{\text{s}}$ is unlikely to exhibit a low activation energy for migration the focus is on C$_{\text{i}}$.
+Since C$_{\text{s}}$ is unlikely to exhibit a low activation energy for migration, the focus is on C$_{\text{i}}$.
Pathways starting from the next most favored configuration, i.e.\ \cs{} located at position 2, into configuration $\alpha$ and $\beta$ are investigated, which show activation energies above \unit[2.2]{eV} and \unit[2.5]{eV}.
The respective barriers and structures are displayed in Fig.~\ref{fig:051-xxx}.
For the transition into configuration $\beta$, as before, the non-magnetic configuration is obtained.
The C atom is slightly displaced in \hkl[0 1 -1] direction.
A binding energy of \unit[-0.59]{eV} indicates the occurrence of much less strain reduction compared to that in the latter configuration.
Evidently this is due to a smaller displacement of Si atom 1, which would be directly bound to the replaced Si atom at position 2.
-In the case of a vacancy created at position 4, even a slightly higher binding energy of \unit[-0.54]{eV} is observed, while the Si atom at the bottom left, which is bound to the \ci{} DB atom, is vastly displaced along \hkl[1 0 -1].
+In the case of a vacancy created at position 4, even a slightly higher binding energy of \unit[-0.54]{eV} is observed while the Si atom at the bottom left, which is bound to the \ci{} DB atom, is vastly displaced along \hkl[1 0 -1].
However the displacement of the C atom along \hkl[0 0 -1] is less compared to the one in the previous configuration.
Although expected due to the symmetric initial configuration, Si atom number 1 is not displaced correspondingly and also the \si DB atom is displaced to a greater extent in \hkl[-1 0 0] than in \hkl[0 -1 0] direction.
The symmetric configuration is, thus, assumed to constitute a local maximum, which is driven into the present state by the conjugate gradient method used for relaxation.
The migration path is best described by the reverse process.
Starting at \unit[100]{\%}, energy is needed to break the bonds of Si atom 1 to its neighbored Si atoms as well as the bond of the C atom to Si atom number 5.
At \unit[50]{\%} displacement, these bonds are broken.
-Due to this and due to the formation of new bonds, e.g. the bond of Si atom number 1 to Si atom number 5, a less steep increase of configurational energy is observed.
+Due to this, and due to the formation of new bonds, e.g.\ the bond of Si atom number 1 to Si atom number 5, a less steep increase of configurational energy is observed.
In a last step, the just recently formed bond of Si atom number 1 to Si atom number 5 is broken up again as well as the bond of the initial Si DB atom and its Si neighbor in \hkl[-1 -1 -1] direction, which explains the repeated boost in energy.
Finally, the system gains some configurational energy by relaxation into the configuration corresponding to \unit[0]{\%} displacement.
%
In summary, pairs of C$_{\text{i}}$ DBs and vacancies, like no other before, show highly attractive interactions for all investigated combinations independent of orientation and separation direction of the defects.
Furthermore, small activation energies, even for transitions into the ground state exist.
-If the vacancy is created at position 1 the system will end up in a configuration of C$_{\text{s}}$ anyways.
+If the vacancy is created at position 1, the system will end up in a configuration of C$_{\text{s}}$ anyways.
Based on these results, a high probability for the formation of C$_{\text{s}}$ must be concluded.
\subsection{Combinations of \si{} and \cs}
Obtained results for separated point defects in Si are in good agreement to previous theoretical work on this subject, both for intrinsic defects~\cite{leung99,al-mushadani03} as well as for C point defects~\cite{dal_pino93,capaz94}.
The ground-state configurations of these defects, i.e.\ the Si$_{\text{i}}$ \hkl<1 1 0> and C$_{\text{i}}$ \hkl<1 0 0> DB, are reproduced and compare well to previous findings of theoretical investigations on Si$_{\text{i}}$~\cite{leung99,al-mushadani03} as well as theoretical~\cite{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental~\cite{watkins76,song90} studies on C$_{\text{i}}$.
-A quantitatively improved activation energy of \unit[0.9]{eV} for a qualitatively equal migration path based on studies by Capaz et.~al.~\cite{capaz94} to experimental values~\cite{song90,lindner06,tipping87} ranging from \unit[0.70--0.87]{eV} reinforce their derived mechanism of diffusion for C$_{\text{i}}$ in Si
+A quantitatively improved activation energy of \unit[0.9]{eV} for a qualitatively equal migration path based on studies by Capaz et~al.~\cite{capaz94} to experimental values~\cite{song90,lindner06,tipping87} ranging from \unit[0.70--0.87]{eV} reinforce their derived mechanism of diffusion for C$_{\text{i}}$ in Si
However, it turns out that the BC configuration is not a saddle point configuration as proposed by Capaz et~al.~\cite{capaz94} but constitutes a real local minimum if the electron spin is properly accounted for.
A net magnetization of two electrons, which is already clear by simple molecular orbital theory considerations on the bonding of the $sp$ hybridized C atom, is settled.
By investigating the charge density isosurface it turns out that the two resulting spin up electrons are localized in a torus around the C atom.
projects / lectures / latex.git / blobdiff