For the vacancy the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.
No other intrinsic defect configuration, within the ones that are mentioned, is affected by spin polarization.
-In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, i.e. if the distance of two atoms is within the cut-off radius $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
+In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, i.e.\ if the distance of two atoms is within the cut-off radius $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
For the tetrahedral and the slightly displaced configurations four bonds to the atoms located in the center of the planes of the unit cell exist in addition to the four tetrahedral bonds.
The length of these bonds are, however, close to the cut-off range and thus are weak interactions not constituting actual chemical bonds.
The same applies to the bonds between the interstitial and the upper two atoms in the \si{} \hkl<1 1 0> DB configuration.
The next energetically favorable defect configuration is the \hkl<1 1 0> C-Si DB interstitial.
Fig. \ref{fig:defects:110_mig_vasp} shows the migration barrier of the \hkl<1 1 0> C-Si DB to the BC, \hkl<0 0 -1> and \hkl<0 -1 0> (in place) transition.
Indeed less than \unit[0.7]{eV} are necessary to turn a \hkl<0 -1 0>- to a \hkl<1 1 0>-type C-Si DB interstitial.
-This transition is carried out in place, i.e. the Si DB pair is not changed and both, the Si and C atom share the initial lattice site.
+This transition is carried out in place, i.e.\ the Si DB pair is not changed and both, the Si and C atom share the initial lattice site.
Thus, this transition does not contribute to long-range diffusion.
Once the C atom resides in the \hkl<1 1 0> DB interstitial configuration it can migrate into the BC configuration requiring approximately \unit[0.95]{eV} of activation energy, which is only slightly higher than the activation energy needed for the \hkl<0 0 -1> to \hkl<0 -1 0> pathway as shown in Fig. \ref{fig:defects:00-1_0-10_mig}.
As already known from the migration of the \hkl<0 0 -1> to the BC configuration discussed in Fig. \ref{fig:defects:00-1_001_mig}, another \unit[0.25]{eV} are needed to turn back from the BC to a \hkl<1 0 0>-type interstitial.
Activation energies of roughly \unit[2.8]{eV} and \unit[2.7]{eV} are needed for migration.
The \ci{} \hkl[1 1 0] configuration seems to play a decisive role in all migration pathways in the classical potential calculations.
-As mentioned above, the starting configuration of the first migration path, i.e. the BC configuration, is fixed to be a transition point but in fact is unstable.
+As mentioned above, the starting configuration of the first migration path, i.e.\ the BC configuration, is fixed to be a transition point but in fact is unstable.
Further relaxation of the BC configuration results in the \ci{} \hkl[1 1 0] configuration.
Even the last two pathways show configurations almost identical to the \ci{} \hkl[1 1 0] configuration, which constitute local minima within the pathways.
Thus, migration pathways involving the \ci{} \hkl[1 1 0] DB configuration as a starting or final configuration are further investigated.
There is still a low interaction remaining, which is due to the equal orientation of the defects.
By changing the orientation of the second DB interstitial to the \hkl<0 -1 0>-type, the interaction is even more reduced resulting in an energy of \unit[-0.05]{eV} for a distance, which is the maximum that can be realized due to periodic boundary conditions.
Energetically favorable and unfavorable configurations can be explained by stress compensation and increase respectively based on the resulting net strain of the respective configuration of the defect combination.
-Antiparallel orientations of the second defect, i.e. \hkl[0 0 1] for positions located below the \hkl(0 0 1) plane with respect to the initial one (positions 1, 2 and 4) form the energetically most unfavorable configurations.
+Antiparallel orientations of the second defect, i.e.\ \hkl[0 0 1] for positions located below the \hkl(0 0 1) plane with respect to the initial one (positions 1, 2 and 4) form the energetically most unfavorable configurations.
In contrast, the parallel and particularly the twisted orientations constitute energetically favorable configurations, in which a vast reduction of strain is enabled by combination of these defects.
\begin{figure}[tp]
The two \ci{} atoms form a strong C-C bond, which is responsible for the large gain in energy resulting in a binding energy of \unit[-2.39]{eV}.
This bond has a length of \unit[1.38]{\AA} close to the next neighbor distance in diamond or graphite, which is approximately \unit[1.54]{\AA}.
The minimum of the binding energy observed for this configuration suggests preferred C clustering as a competing mechanism to the \ci{} DB interstitial agglomeration inevitable for the SiC precipitation.
-However, the second most favorable configuration ($E_{\text{f}}=-2.25\,\text{eV}$) is represented four times, i.e. two times more often than the ground-state configuration, within the systematically investigated configuration space.
+However, the second most favorable configuration ($E_{\text{f}}=-2.25\,\text{eV}$) is represented four times, i.e.\ two times more often than the ground-state configuration, within the systematically investigated configuration space.
Thus, particularly at high temperatures that cause an increase of the entropic contribution, this structure constitutes a serious opponent to the ground state.
In fact, following results on migration simulations will reinforce the assumption of a low probability for C clustering by thermally activated processes.
The energetically most unfavorable configuration ($E_{\text{b}}=0.26\,\text{eV}$) is obtained for the \ci{} \hkl[0 0 1] DB, which is oppositely orientated with respect to the initial one.
A DB taking the same orientation as the initial one is less unfavorable ($E_{\text{b}}=0.04\,\text{eV}$).
Both configurations are unfavorable compared to far-off, isolated DBs.
-Nonparallel orientations, i.e. the \hkl[0 1 0], \hkl[0 -1 0] and its equivalents, result in binding energies of \unit[-0.12]{eV} and \unit[-0.27]{eV}, thus, constituting energetically favorable configurations.
+Nonparallel orientations, i.e.\ the \hkl[0 1 0], \hkl[0 -1 0] and its equivalents, result in binding energies of \unit[-0.12]{eV} and \unit[-0.27]{eV}, thus, constituting energetically favorable configurations.
The reduction of strain energy is higher in the second case, where the C atom of the second DB is placed in the direction pointing away from the initial C atom.
\begin{figure}[tp]
\label{fig:defects:comb_db110}
\end{figure}
The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:defects:comb_db110}.
-The interaction is found to be proportional to the reciprocal cube of the C-C distance for extended separations of the \ci{} DBs and saturates for the smallest possible separation, i.e. the ground-state configuration.
+The interaction is found to be proportional to the reciprocal cube of the C-C distance for extended separations of the \ci{} DBs and saturates for the smallest possible separation, i.e.\ the ground-state configuration.
The ground-state configuration was ignored in the fitting process.
Not considering the previously mentioned elevated barriers for migration, an attractive interaction between the \ci{} \hkl<1 0 0> DB defects indeed is detected with a capture radius that clearly exceeds \unit[1]{nm}.
The interpolated graph suggests the disappearance of attractive interaction forces, which are proportional to the slope of the graph, in between the two lowest separation distances of the defects.
\label{fig:093-095}
\end{figure}
Configuration A consists of a C$_{\text{i}}$ \hkl[0 0 -1] DB with threefold coordinated Si and C DB atoms slightly disturbed by the C$_{\text{s}}$ at position 3, facing the Si DB atom as a neighbor.
-By a single bond switch, i.e. the breaking of a Si-Si in favor of a Si-C bond, configuration B is obtained, which shows a twofold coordinated Si atom located in between two substitutional C atoms residing on regular Si lattice sites.
+By a single bond switch, i.e.\ the breaking of a Si-Si in favor of a Si-C bond, configuration B is obtained, which shows a twofold coordinated Si atom located in between two substitutional C atoms residing on regular Si lattice sites.
This configuration has been identified and described by spectroscopic experimental techniques \cite{song90_2} as well as theoretical studies \cite{leary97,capaz98}.
Configuration B is found to constitute the energetically slightly more favorable configuration.
However, the gain in energy due to the significantly lower energy of a Si-C compared to a Si-Si bond turns out to be smaller than expected due to a large compensation by introduced strain as a result of the Si interstitial 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.
+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.
The two C atoms form a \hkl[1 0 0] DB sharing the initial C$_{\text{s}}$ lattice site while the initial Si DB atom occupies its previously regular lattice site.
The transition is accompanied by a large gain in energy as can be seen in Fig.~\ref{fig:026-128}, making it the ground-state configuration of a C$_{\text{s}}$ and C$_{\text{i}}$ DB in Si yet \unit[0.33]{eV} lower in energy than configuration B.
This finding is in good agreement with a combined {\em ab initio} and experimental study of Liu et~al.~\cite{liu02}, who first proposed this structure as the ground state identifying an energy difference compared to configuration B of \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 $\alpha$ 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}.
+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.
+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.
%
% a b transition
A low activation energy of \unit[0.1]{eV} is observed for the a$\rightarrow$b transition.
Thus, configuration a is very unlikely to occur in favor of configuration b.
% repulsive along 110
-A repulsive interaction is observed for C$_{\text{s}}$ at lattice sites along \hkl[1 1 0], i.e. positions 1 (configuration a) and 5.
+A repulsive interaction is observed for C$_{\text{s}}$ at lattice sites along \hkl[1 1 0], i.e.\ positions 1 (configuration a) and 5.
This is due to tensile strain originating from both, the C$_{\text{i}}$ DB and the C$_{\text{s}}$ atom residing within the \hkl[1 1 0] bond chain.
This finding agrees well with results by Mattoni et~al.~\cite{mattoni2002}.
% all other investigated results: attractive interaction. stress compensation.
This structure is followed by C$_{\text{s}}$ located at position 2, the lattice site of one of the neighbor atoms below the two Si atoms that are bound to the C$_{\text{i}}$ DB atom.
As mentioned earlier, these two lower Si atoms indeed experience tensile strain along the \hkl[1 1 0] bond chain, however, additional compressive strain along \hkl[0 0 1] exists.
The latter is partially compensated by the C$_{\text{s}}$ atom.
-Yet less of compensation is realized if C$_{\text{s}}$ is located at position 4 due to a larger separation although both bottom Si atoms of the DB structure are indirectly affected, i.e. each of them is connected by another Si atom to the C atom enabling the reduction of strain along \hkl[0 0 1].
+Yet less of compensation is realized if C$_{\text{s}}$ is located at position 4 due to a larger separation although both bottom Si atoms of the DB structure are indirectly affected, i.e.\ each of them is connected by another Si atom to the C atom enabling the reduction of strain along \hkl[0 0 1].
\begin{figure}[tp]
\begin{center}
\subfigure[\underline{$E_{\text{b}}=-0.51\,\text{eV}$}]{\label{fig:defects:051}\includegraphics[width=0.25\textwidth]{00-1dc/0-51.eps}}
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 $\alpha$ and $\beta$ are investigated, which show activation energies above \unit[2.2]{eV} and \unit[2.5]{eV}.
+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 $\beta$.
Strain reduced by this huge displacement is partially absorbed by tensile strain on Si atom number 1 originating from attractive forces of the C atom and the vacancy.
A binding energy of \unit[-0.50]{eV} is observed.
-The migration pathways of configuration \ref{fig:defects:314} and \ref{fig:defects:059} into the ground-state configuration, i.e. the \cs{} configuration, are shown in Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively.
+The migration pathways of configuration \ref{fig:defects:314} and \ref{fig:defects:059} into the ground-state configuration, i.e.\ the \cs{} configuration, are shown in Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{314-539.ps}
\end{figure}
An activation energy as low as \unit[0.12]{eV} is necessary for the migration into the ground-state configuration.
Accordingly, the C$_{\text{i}}$ \hkl<1 0 0> DB configuration is assumed to occur more likely.
-However, only \unit[0.77]{eV} are needed for the reverse process, i.e. the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state.
+However, only \unit[0.77]{eV} are needed for the reverse process, i.e.\ the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state.
Due to the low activation energy this process must be considered to be activated without much effort either thermally or by introduced energy of the implantation process.
\begin{figure}[tp]
\end{figure}
Fig.~\ref{fig:dc_si-s} shows the binding energies of pairs of C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB with respect to the separation distance.
The interaction of the defects is well approximated by a Lennard-Jones (LJ) 6-12 potential, which is used for curve fitting.
-Unable to model possible positive values of the binding energy, i.e. unfavorable configurations, located to the right of the minimum, the LJ fit should rather be thought of as a guide for the eye describing the decrease of the interaction strength, i.e. the absolute value of the binding energy, with increasing separation distance.
+Unable to model possible positive values of the binding energy, i.e.\ unfavorable configurations, located to the right of the minimum, the LJ fit should rather be thought of as a guide for the eye describing the decrease of the interaction strength, i.e.\ the absolute value of the binding energy, with increasing separation distance.
The binding energy quickly drops to zero.
The LJ fit estimates almost zero interaction already at \unit[0.6]{nm}.
indicating a low interaction capture radius of the defect pair.
-%As can be seen, the interaction strength, i.e. the absolute value of the binding energy, quickly drops to zero with increasing separation distance.
+%As can be seen, the interaction strength, i.e.\ the absolute value of the binding energy, quickly drops to zero with increasing separation distance.
%Almost zero interaction may be assumed already at distances about \unit[0.5-0.6]{nm}, indicating a low interaction capture radius of the defect pair.
In IBS, highly energetic collisions are assumed to easily produce configurations of defects exhibiting separation distances exceeding the capture radius.
For this reason C$_{\text{s}}$ without a Si$_{\text{i}}$ DB located within the immediate proximity, which is, thus, unable to form the thermodynamically stable C$_{\text{i}}$ \hkl<1 0 0> DB, constitutes a most likely configuration to be found in IBS.
However, this configuration is unstable involving a structural transition into the C$_{\text{i}}$ \hkl<1 1 0> DB interstitial, thus, not maintaining the tetrahedral Si nor the \cs{} defect.
Thus, the underestimated energy of formation of C$_{\text{s}}$ within the EA calculation does not pose a serious limitation in the present context.
-Since C is introduced into a perfect Si crystal and the number of particles is conserved in simulation, the creation of C$_{\text{s}}$ is accompanied by the creation of Si$_{\text{i}}$, which is energetically less favorable than the ground state, i.e. the C$_{\text{i}}$ \hkl<1 0 0> DB configuration, for both, the EA and {\em ab initio} treatment.
+Since C is introduced into a perfect Si crystal and the number of particles is conserved in simulation, the creation of C$_{\text{s}}$ is accompanied by the creation of Si$_{\text{i}}$, which is energetically less favorable than the ground state, i.e.\ the C$_{\text{i}}$ \hkl<1 0 0> DB configuration, for both, the EA and {\em ab initio} treatment.
In either case, no configuration more favorable than the C$_{\text{i}}$ \hkl<1 0 0> DB has been found.
Thus, a proper description with respect to the relative energies of formation is assumed for the EA potential.
\ifnum1=0
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$_{\text{i}}$.
+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
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.
-With an activation energy of \unit[0.9]{eV} the C$_{\text{i}}$ carbon interstitial can be expected to be highly mobile at prevailing temperatures in the process under investigation, i.e. IBS.
+With an activation energy of \unit[0.9]{eV} the C$_{\text{i}}$ carbon interstitial can be expected to be highly mobile at prevailing temperatures in the process under investigation, i.e.\ IBS.
Since the \ci{} \hkl<1 0 0> DB is the ground-state configuration and highly mobile, possible migration of these DBs to form defect agglomerates, as demanded by the model introduced in section \ref{section:assumed_prec}, is considered possible.
Unfortunately the description of the same processes fails if classical potential methods are used.
These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si.
% which is elaborated in more detail within the comprehensive description in chapter~\ref{chapter:summary}.
Agglomeration of C$_{\text{i}}$ is energetically favored and enabled by a low activation energy for migration.
-Although ion implantation is a process far from thermodynamic equilibrium, which might result in phases not described by the Si/C phase diagram, i.e. a C phase in Si, high activation energies are believed to be responsible for a low probability of the formation of C-C clusters.
+Although ion implantation is a process far from thermodynamic equilibrium, which might result in phases not described by the Si/C phase diagram, i.e.\ a C phase in Si, high activation energies are believed to be responsible for a low probability of the formation of C-C clusters.
In the context of the initially stated controversy present in the precipitation model, these findings suggest an increased participation of C$_{\text{s}}$ already in the initial stage due to its high probability of incidence.
In addition, thermally activated, C$_{\text{i}}$ might turn into C$_{\text{s}}$.
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