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}
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_0-10_vasp_s.ps}
-%\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[1.6cm]
-%\begin{picture}(0,0)(140,0)
-%\includegraphics[width=2.5cm]{vasp_mig/00-1_a.eps}
-%\end{picture}
-%\begin{picture}(0,0)(20,0)
-%\includegraphics[width=2.5cm]{vasp_mig/00-1_0-10_sp.eps}
-%\end{picture}
-%\begin{picture}(0,0)(-120,0)
-%\includegraphics[width=2.5cm]{vasp_mig/0-10.eps}
-%\end{picture}
-%\begin{picture}(0,0)(25,20)
-%\includegraphics[width=2.5cm]{100_arrow.eps}
-%\end{picture}
-%\begin{picture}(0,0)(200,0)
-%\includegraphics[height=2.2cm]{001_arrow.eps}
-%\end{picture}
\end{center}
\caption[Migration barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition.]{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition. Bonds of the C atom are illustrated by blue lines. {\color{red} Prototype design, adjust related figures!}}
% todo read above caption! enable [] hkls in short caption
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_110_0-10_mig_albe.ps}
\end{center}
-\caption[Migration barrier and structures of the \ci{} \hkl<0 0 -1> (left) to the \hkl<0 -1 0> DB (right) transition involving the \hkl<1 1 0> DB (center) configuration.]{Migration barrier and structures of the \ci{} \hkl[0 0 -1] (left) to the \hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration. Migration simulations were performed utilizing time constants of \unit[1]{fs} (solid line) and \unit[100]{fs} (dashed line) for the Berendsen thermostat.}
+\caption[Migration barrier and structures of the \ci{} \hkl<0 0 -1> (left) to the \hkl<0 -1 0> DB (right) transition involving the \hkl<1 1 0> DB (center) configuration.]{Migration barrier and structures of the \ci{} \hkl[0 0 -1] (left) to the \hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration. Migration simulations are performed utilizing time constants of \unit[1]{fs} (solid line) and \unit[100]{fs} (dashed line) for the Berendsen thermostat.}
\label{fig:defects:involve110}
\end{figure}
Approximately \unit[2.2]{eV} are needed to turn the \ci{} \hkl[0 0 -1] into the \hkl[1 1 0] DB located at the neighbored lattice site in \hkl[1 1 -1] direction.
The study proceeds with a structural and energetic investigation of pairs of the ground-state and, thus, most probable defect configurations that are believed to be fundamental in the Si to SiC conversion.
Investigations are restricted to quantum-mechanical calculations.
\begin{figure}[tp]
+% ./visualize_contcar -w 640 -h 480 -d results/.../CONTCAR -nll -0.20 -0.20 -0.6 -fur 1.2 1.2 0.6 -c 0.5 -1.5 0.3 -L 0.5 0 0 -r 0.6 -m 3.0 0.0 0.0 0.0 3.0 0.0 0.0 0.0 3.0 -A -1 2.465
\begin{center}
-\subfigure[]{\label{fig:defects:combos_ci}\includegraphics[width=0.3\textwidth]{combos_ci.eps}}
+\subfigure[]{\label{fig:defects:combos_ci}\includegraphics[width=0.3\textwidth]{combos_ci_col.eps}}
\hspace{0.5cm}
\subfigure[]{\label{fig:defects:combos_si}\includegraphics[width=0.3\textwidth]{combos.eps}}
\end{center}
-\caption{Position of the initial \ci{} \hkl[0 0 -1] DB (I) (a) and of the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (\si) (b). Lattice sites for the second defect used for investigating defect pairs are numbered from 1 to 5.}
+\caption{Position of the initial \ci{} \hkl[0 0 -1] DB (I) (a) and of the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (\si) (b). Lattice sites for the second defect used for investigating defect pairs are numbered from 1 to 5. For black/red/blue numbers, one/two/four possible atom(s) exist for the second defect to create equivalent defect combinations.}
\label{fig:defects:combos}
\end{figure}
Fig.~\ref{fig:defects:combos} schematically displays the initial \ci{} \hkl[0 0 -1] DB structure (Fig.~\ref{fig:defects:combos_ci}) as well as the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (Fig.~\ref{fig:defects:combos_si}) and various positions for the second defect (1-5) that are used for investigating defect pairs.
+The color of the number denotes the amount of possible atoms for the second defect resulting in equivalent configurations.
Binding energies of the defect pair are determined by equation \ref{eq:basics:e_bind}.
-Next to formation and binding energies, migration barriers are investigated, which allow to draw conclusions on the probability of the formation of such defect complexes by thermally activated diffusion processes.
+Next to formation and binding energies, migration barriers are investigated, which allow to draw conclusions on the probability of the formation of such defect complexes by thermally activated diffusion processes.
\subsection[Pairs of \ci{} \hkl<1 0 0>-type interstitials]{\boldmath Pairs of \ci{} \hkl<1 0 0>-type interstitials}
\label{subsection:defects:c-si_comb}
Based on the lowest energy migration path of a single \ci{} \hkl<1 0 0> DB, the configuration, in which the second \ci{} DB is oriented along \hkl[0 1 0] at position 2 is assumed to constitute an ideal starting point for a transition into the ground state.
In addition, the starting configuration exhibits a low binding energy (\unit[-1.90]{eV}) and is, thus, very likely to occur.
-However, a barrier height of more than \unit[4]{eV} is detected resulting in a low probability for the transition.
+However, a smooth transition path is not found.
+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}).
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}.
%\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.
%
\label{fig_defects:245csub}
\end{figure}
Fig.~\ref{fig_defects:245csub} lists the remaining configurations and binding energies of the relaxed structures obtained by creating a \cs{} at positions 2, 4 and 5 in the \ci{} \hkl[0 0 -1] DB configuration.
+% todo explain some configurations, source: old text some lines below
% 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.
-Again, conclusions concerning the probability of formation are drawn by investigating migration paths.
+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$_{\text{i}}$.
-Pathways starting from the two next most favored configurations were investigated, which show activation energies above \unit[2.2]{eV} and \unit[3.5]{eV} respectively.
+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<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.
For the same reasons as in the last subsection, structures other than the ground-state configuration are, thus, assumed to arise more likely due to much lower activation energies necessary for their formation and still comparatively low binding energies.
Si atoms 1 and 2, which form the initial DB, occupy Si lattice sites in the final configuration while Si atom 3 is transferred from a regular lattice site into the interstitial lattice.
These results support the above assumptions of an increased entropic contribution to structural formation involving C$_{\text{s}}$ to a greater extent.
-
-% todo migration of \si{}!
-
+% 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}
+\includegraphics[width=0.7\textwidth]{si_110_110_mig_02_conf.ps}
+\end{center}
+\caption[Migration barrier and structures of the \si{} \hkl<1 1 0> DB.]{Migration barrier and structures of the \si{} \hkl[0 -1 1] DB (left) to the \hkl[1 1 0] DB (right) transition. Bonds are illustrated by blue lines.}
+% todo read above caption! enable [] hkls in short caption
+\label{fig:defects:si_mig1}
+\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]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{si_mig_rest.ps}
+\end{center}
+\caption{Migration barrier of the \si{} \hkl[1 1 0] DB into the hexagonal (H) and tetrahedral (T) configuration as well as the hexagonal \si{} to tetrahedral \si{} transition.}
+% todo read above caption! enable [] hkls in short caption
+\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 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!
Comparatively high energies necessary for the reverse process reveal this configuration to be extremely stable.
Thus, C interstitials and vacancies located close together are assumed to end up in such a configuration, in which the C atom is tetrahedrally coordinated and bound to four Si atoms as expected in SiC.
-Investigating configurations of C$_{\text{s}}$ and Si$_{\text{i}}$, formation energies higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB were obtained keeping up previously derived assumptions concerning the ground state of C$_{\text{i}}$ in otherwise perfect Si.
+Investigating configurations of C$_{\text{s}}$ and Si$_{\text{i}}$, formation energies higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB are obtained keeping up previously derived assumptions concerning the ground state of C$_{\text{i}}$ in otherwise perfect Si.
However, a small capture radius is identified for the respective interaction that might prevent the recombination of defects exceeding a separation of \unit[0.6]{nm} into the ground state configuration.
In addition, a rather small activation energy of \unit[0.77]{eV} allows for the formation of a C$_{\text{s}}$-Si$_{\text{i}}$ pair originating from the C$_{\text{i}}$ \hkl<1 0 0> DB structure by thermally activated processes.
Thus, elevated temperatures might lead to thermodynamically unstable configurations of C$_{\text{s}}$ and a remaining Si atom in the near interstitial lattice, which is supported by the result of an {\em ab initio} molecular dynamics run.
projects / lectures / latex.git / blobdiff