The formation energy of \unit[4.48]{eV} is determined by this low kinetic energy configuration shortly before the relaxation process starts.
The \si{} atom then begins to slowly move towards an energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes.
The formation energy of \unit[3.96]{eV} for this type of interstitial is equal to the result for the hexagonal one in the original work \cite{albe_sic_pot}.
-Obviously the authors did not carefully check the relaxed results assuming a hexagonal configuration.
+Obviously, the authors did not carefully check the relaxed results assuming a hexagonal configuration.
In Fig. \ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
\begin{figure}[tp]
\begin{center}
%\end{figure}
%
Table~\ref{tab:defects:c-s} lists the energetic results of \cs{} combinations with the initial \ci{} \hkl[0 0 -1] DB.
-For \cs{} located at position 1 and 3, the configurations a and A correspond to the naive relaxation of the structure by substituting the Si atom by a C atom in the initial \ci{} \hkl[0 0 -1] DB structure at positions 1 and 3 respectively.
-However, small displacements of the involved atoms near the defect result in different stable structures labeled b and B respectively.
-Fig.~\ref{fig:093-095} and \ref{fig:026-128} show structures A, B and a, b together with the barrier of migration for the A to B and a to b transition respectively.
+For \cs{} located at position 1 and 3, the configurations $\alpha$ and A correspond to the naive relaxation of the structure by substituting the Si atom by a C atom in the initial \ci{} \hkl[0 0 -1] DB structure at positions 1 and 3 respectively.
+However, small displacements of the involved atoms near the defect result in different stable structures labeled $\beta$ and B respectively.
+Fig.~\ref{fig:093-095} and \ref{fig:026-128} show structures A, B and $\alpha$, $\beta$ together with the barrier of migration for the A to B and $\alpha$ to $\beta$ transition respectively.
% A B
%./visualize_contcar -w 640 -h 480 -d results/c_00-1_c3_csub_B -nll -0.20 -0.4 -0.1 -fur 0.9 0.6 0.9 -c 0.5 -1.5 0.375 -L 0.5 0 0.3 -r 0.6 -A -1 2.465
% mattoni: A favored by 0.4 eV - NO, it is indeed B (reinforce Song and Capaz)!
%
% AB transition
-The migration barrier ss identified to be \unit[0.44]{eV}, almost three times higher than the experimental value of \unit[0.16]{eV} \cite{song90_2} estimated for the neutral charge state transition in p- and n-type Si.
+The migration barrier is identified to be \unit[0.44]{eV}, almost three times higher than the experimental value of \unit[0.16]{eV} \cite{song90_2} estimated for the neutral charge state transition in p- and n-type Si.
Keeping in mind the formidable agreement of the energy difference with experiment, the overestimated activation energy is quite unexpected.
Obviously, either the CRT algorithm fails to seize the actual saddle point structure or the influence of dopants has exceptional effect in the experimentally covered diffusion process being responsible for the low migration barrier.
% not satisfactory!
% a b
\begin{figure}[tp]
\begin{center}
-\includegraphics[width=0.7\textwidth]{026-128.ps}
+\includegraphics[width=0.7\textwidth]{comb_mig_026-128_vasp.ps}
\end{center}
\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 1 (left) into a C-C \hkl[1 0 0] DB occupying the lattice site at position 1 (right). An activation energy of \unit[0.1]{eV} is observed.}
\label{fig:026-128}
\end{figure}
-Configuration a is similar to configuration A, except that the C$_{\text{s}}$ atom at position 1 is facing the C DB atom as a neighbor resulting in the formation of a strong C-C bond and a much more noticeable perturbation of the DB structure.
+Configuration $\alpha$ is similar to configuration A, except that the C$_{\text{s}}$ atom at position 1 is facing the C DB atom as a neighbor resulting in the formation of a strong C-C bond and a much more noticeable perturbation of the DB structure.
Nevertheless, the C and Si DB atoms remain threefold coordinated.
Although the C-C bond exhibiting a distance of \unit[0.15]{nm} close to the distance expected in diamond or graphite should lead to a huge gain in energy, a repulsive interaction with a binding energy of \unit[0.26]{eV} is observed due to compressive strain of the Si DB atom and its top neighbors (\unit[0.230]{nm}/\unit[0.236]{nm}) along with additional tensile strain of the C$_{\text{s}}$ and its three neighboring Si atoms (\unit[0.198-0.209]{nm}/\unit[0.189]{nm}).
Again a single bond switch, i.e. the breaking of the bond of the Si atom bound to the fourfold coordinated C$_{\text{s}}$ atom and the formation of a double bond between the two C atoms, results in configuration b.
% mattoni: A favored by 0.2 eV - NO! (again, missing spin polarization?)
A net magnetization of two spin up electrons, which are equally localized as in the Si$_{\text{i}}$ \hkl<1 0 0> DB structure is observed.
In fact, these two configurations are very similar and are qualitatively different from the C$_{\text{i}}$ \hkl<1 0 0> DB that does not show magnetization but a nearly collinear bond of the C DB atom to its two neighbored Si atoms while the Si DB atom approximates \unit[120]{$^{\circ}$} angles in between its bonds.
-Configurations a, A and B are not affected by spin polarization and show zero magnetization.
-Mattoni et~al.~\cite{mattoni2002}, in contrast, find configuration b less favorable than configuration A by \unit[0.2]{eV}.
+Configurations $\alpha$, A and B are not affected by spin polarization and show zero magnetization.
+Mattoni et~al.~\cite{mattoni2002}, in contrast, find configuration $\beta$ less favorable than configuration A by \unit[0.2]{eV}.
Next to differences in the XC functional and plane-wave energy cut-off, this discrepancy might be attributed to the neglect of spin polarization in their calculations, which -- as has been shown for the C$_{\text{i}}$ BC configuration -- results in an increase of configurational energy.
-Indeed, investigating the migration path from configurations a to b and, in doing so, reusing the wave functions of the previous migration step the final structure, i.e. configuration b, is obtained with zero magnetization and an increase in configurational energy by \unit[0.2]{eV}.
+Indeed, investigating the migration path from configurations $\alpha$ to $\beta$ and, in doing so, reusing the wave functions of the previous migration step the final structure, i.e. configuration $\beta$, is obtained with zero magnetization and an increase in configurational energy by \unit[0.2]{eV}.
Obviously a different energy minimum of the electronic system is obtained indicating hysteresis behavior.
However, since the total energy is lower for the magnetic result it is believed to constitute the real, i.e. global, minimum with respect to electronic minimization.
%
% c agglomeration vs c clustering ... migs to b conf
% 2 more migs: 051 -> 128 and 026! forgot why ... probably it's about probability of C clustering
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 b) forms a strong but compressively strained C-C bond with a separation distance of \unit[0.142]{nm} sharing a Si lattice site.
+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$_{\text{i}}$.
-Pathways starting from the next most favored configuration, i.e. \cs{} located at position 2, into configuration a and b are investigated, which show activation energies above \unit[2.2]{eV} and \unit[2.5]{eV}.
-The respective barriers and structures are shown in Fig.~\ref{fig:051-xxx}.
-Again, the non-magnetic configuration is obtained.
-If not forced by the CRT algorithm, the structures beyond \perc{50} displacement of the transition into configuration a would likewise settle into configuration b.
+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.
+If not forced by the CRT algorithm, the structures beyond \perc{50} and below \perc{90} displacement of the transition approaching configuration $\alpha$ would settle into configuration $\beta$.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{comb_mig_051-xxx_conf.ps}
\end{center}
-\caption{Migration barrier and structures of the transition of a configuration equivalent to the one of the initial \hkl<1 0 0> \ci{} DB with \cs{} located at position 2 into the a and b configurations.}
+\caption{Migration barrier and structures of the transition of a configuration equivalent to the one of the initial \hkl<0 0 -1> \ci{} DB with \cs{} located at position 2 into the $\alpha$ and $\beta$ configurations.}
\label{fig:051-xxx}
\end{figure}
Although lower than the barriers for obtaining the ground state of two C$_{\text{i}}$ defects, the activation energies are yet considered too high.
% link to migration of \si{}!
The possibility for separated configurations of \cs{} and \si{} becomes even more likely if one of the constituents exhibits a low barrier of migration.
In this case, the \si{} is assumed to constitute the mobile defect compared to the stable \cs{} atom.
+Thus, migration paths of \si{} are investigated in the following excursus.
Acoording to Fig.~\ref{fig:defects:si_mig1}, an activation energy of \unit[0.67]{eV} is necessary for the transition of the \si{} \hkl[0 -1 1] to \hkl[1 1 0] DB located at the neighbored Si lattice site in \hkl[1 1 -1] direction.
\begin{figure}[tp]
\begin{center}
\end{figure}
The barrier, which is even lower than the one for \ci{}, indeed indicates highly mobile \si.
In fact, a similar transition is expected if the \si{} atom, which does not change the lattice site during transition, is located next to a \cs{} atom.
+Due to the low barrier the initial separation of the \cs{} and \si{} atom are very likely to occur.
Further investigations revealed transition barriers of \unit[0.94]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to the hexagonal Si$_{\text{i}}$, \unit[0.53]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to the tetrahedral Si$_{\text{i}}$ and \unit[0.35]{eV} for the hexagonal Si$_{\text{i}}$ to the tetrahedral Si$_{\text{i}}$ configuration.
The respective configurational energies are shown in Fig.~\ref{fig:defects:si_mig2}.
\begin{figure}[tp]
\label{fig:defects:si_mig2}
\end{figure}
The obtained activation energies are of the same order of magnitude than values derived from other ab initio studies \cite{bloechl93,sahli05}.
-The low barriers indeed enable configurations of separated \cs{} and \si{}.
+The low barriers indeed enable configurations of further separated \cs{} and \si{} atoms by the highly mobile \si{} atom departing from the \cs{} defect as observed in the previously discussed MD simulation.
% kept for nostalgical reason!
projects / lectures / latex.git / blobdiff