% additional stuff\r
\usepackage{miller}\r
\r
+% roman numbers\r
+\newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}\r
+\r
\begin{document}\r
\r
%\title{Mobility of Carbon in Silicon -- a first principles study}\r
Ionic relaxation was realized by the conjugate gradient algorithm.\r
Spin polarization has been fully accounted for.\r
\r
-Migration and recombination pathways have been ivestigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.\r
+Migration and recombination pathways have been investigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.\r
The defect formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ is defined by chosing SiC as a particle reservoir for the C impurity, i.e. the chemical potentials are determined by the cohesive energies of a perfect Si and SiC supercell after ionic relaxation.\r
-The binding energy of a defect pair is given by the difference of the formation energy of the complex and the sum of the two separated defect configurations, i.e. energetically favorable configurations show binding energies below zero and non-interacting isolated defects would result in a binding energy of zero.\r
+The binding energy of a defect pair is given by the difference of the formation energy of the complex and the sum of the two separated defect configurations.\r
+Accordingly, energetically favorable configurations show binding energies below zero while non-interacting isolated defects result in a binding energy of zero.\r
\r
\section{Results}\r
\r
The investigations proceed with pairs of the ground state and, thus, most probable defect configurations that are believed to be fundamental in the Si to SiC transition.\r
\r
\subsection{Separated defects in silicon}\r
+\label{subsection:sep_def}\r
% we need both: Si self-int & C int ground state configuration (for combos)\r
\r
Several geometries have been calculated to be stable for individual intrinsic and C related defects in Si.\r
\begin{table*}\r
\begin{ruledtabular}\r
\begin{tabular}{l c c c c c c c c c}\r
- & Si$_{\text{i}}$ \hkl<1 1 0> DB & Si$_{\text{i}}$ H & Si$_{\text{i}}$ T & Si \hkl<1 0 0> DB & V & C$_{\text{s}}$ & C$_{\text{i}}$ \hkl<1 0 0> DB & C$_{\text{i}}$ \hkl<1 1 0> DB & C$_{\text{i}}$ BC \\\r
+ & Si$_{\text{i}}$ \hkl<1 1 0> DB & Si$_{\text{i}}$ H & Si$_{\text{i}}$ T & Si$_{\text{i}}$ \hkl<1 0 0> DB & V & C$_{\text{s}}$ & C$_{\text{i}}$ \hkl<1 0 0> DB & C$_{\text{i}}$ \hkl<1 1 0> DB & C$_{\text{i}}$ BC \\\r
\hline\r
Present study & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\\r
\multicolumn{10}{c}{Other ab initio studies} \\\r
On the other hand, if elevated temperatures enable migrations with huge activation energies, comparably small differences in configurational energy can be neglected resulting in an almost equal occupation of such configurations.\r
In both cases the configuration yielding a binding energy of \unit[-2.25]{eV} is promising.\r
First of all, it constitutes the second most energetically favorable structure.\r
-Secondly, a migration path with a barrier as low as \unit[?.?]{eV} exists starting from a configuration of largely separated defects exhibiting a low binding energy (\unit[-1.88]{eV}).\r
+Secondly, a migration path with a barrier as low as \unit[0.47]{eV} exists starting from a configuration of largely separated defects exhibiting a low binding energy (\unit[-1.88]{eV}).\r
The migration barrier and correpsonding structures are shown in Fig.~\ref{fig:188-225}.\r
% 188 - 225 transition in progress\r
\begin{figure}\r
\includegraphics[width=\columnwidth]{188-225.ps}\r
-\caption{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 5 (left) into a C$_{\text{i}}$ \hkl[1 0 0] DB at position 1 (right). An activation energy of \unit[?.?]{eV} is observed.}\r
+\caption{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 5 (left) into a C$_{\text{i}}$ \hkl[1 0 0] DB at position 1 (right). An activation energy of \unit[0.47]{eV} is observed.}\r
\label{fig:188-225}\r
\end{figure}\r
Finally, this type of defect pair is represented four times (two times more often than the ground state configuration) within the systematically investigated configuration space.\r
%./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\r
\begin{figure}\r
\includegraphics[width=\columnwidth]{093-095.ps}\r
-\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 3 (left) into a configuration of a twofold coordinated Si$_{\text{i}}$ located inbetween two C$_{\text{s}}$ atoms occupying the lattice sites of the initial DB and position 3 (right). An activation energy of \unit[0.?]{eV} is observed.}\r
+\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 3 (left) into a configuration of a twofold coordinated Si$_{\text{i}}$ located inbetween two C$_{\text{s}}$ atoms occupying the lattice sites of the initial DB and position 3 (right). An activation energy of \unit[0.44]{eV} is observed.}\r
\label{fig:093-095}\r
\end{figure}\r
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 next neighbor.\r
% mattoni: A favored by 0.4 eV - NO, it is indeed B (reinforce Song and Capaz)!\r
%\r
% AB transition\r
-% ...\r
+The migration barrier was 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.\r
+Keeping in mind the formidable agreement of the energy difference with experiment, the overestimated activation energy is quite unexpected and disappointing.\r
+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.\r
+% not satisfactory!\r
\r
% a b\r
\begin{figure}\r
Configurations a, A and B are not affected by spin polarization and show zero magnetization.\r
Mattoni et~al.\cite{mattoni2002}, in contrast, find configuration b less favorable than configuration A by \unit[0.2]{eV}.\r
Next to differences in the XC-functional and plane-wave energy cut-off this discrepancy might be attributed to the missing accounting for spin polarization in their calculations, which -- as has been shown for the C$_{\text{i}}$ BC configuration -- results in an increase of configurational energy.\r
-\r
+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, was obtained with zero magnetization and an increase in configurational energy by \unit[0.2]{eV}.\r
+Obviously a different energy minimum of the electronic system is obatined indicating hysteresis behavior.\r
+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.\r
+%\r
+% a b transition\r
A low activation energy of \unit[0.1]{eV} is observed for the a$\rightarrow$b transition.\r
Thus, configuration a is very unlikely to occur in favor of configuration b.\r
-However, migration barriers yielding\r
-...\r
\r
-% mig a-b\r
+% repulsive along 110\r
+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.\r
+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.\r
+This finding agrees well with results by Mattoni et~al.\cite{mattoni2002}.\r
+% all other investigated results: attractive interaction. stress compensation.\r
+In contrast, all other investigated configurations show attractive interactions.\r
+The most favorable configuration is found for C$_{\text{s}}$ at position 3, which corresponds to the lattice site of one of the upper next neighbored Si atoms of the DB structure that is compressively strained along \hkl[1 -1 0] and \hkl[0 0 1] by the C-Si DB.\r
+The substitution with C allows for most effective compensation of strain.\r
+This structure is followed by C$_{\text{s}}$ located at position 2, the next neighbour atom below the two Si atoms bound to the C$_{\text{i}}$ DB atom.\r
+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.\r
+The latter is partially compensated by the C$_{\text{s}}$ atom.\r
+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].\r
+\r
+% c agglomeration vs c clustering ... migs to b conf\r
% 2 more migs: 051 -> 128 and 026! forgot why ... probably it's about probability of C clustering\r
+Obviously agglomeration of C$_{\text{i}}$ and C$_{\text{s}}$ is energetically favorable except for seprations along one of the \hkl<1 1 0> directions.\r
+The eneregtically 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.\r
+Again, conclusions concerning the probability of formation are drawn by investigating migration paths.\r
+Since C$_{\text{s}}$ is unlikely to exhibit a low activation energy for migration the focus is on C$_{\text{i}}$.\r
+Pathways starting from the two next most favored configurations were investigated, all of them showing activation energies above \unit[2.?-2.?]{eV}.\r
+Although lower than the barriers for obtaining the ground state of two C$_{\text{i}}$ defects the activation energy is yet considered too high.\r
+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.\r
\r
\subsection{C$_{\text{i}}$ next to V}\r
\r
+In the last subsection configurations of a C$_{\text{i}}$ DB with C$_{\text{s}}$ occupying a vacant site created by the implantation process have been investigated.\r
+Additionally, configurations might arise in IBS, in which the impinging C atom creates a vacant site near a C$_{\text{i}}$ DB but does not occupy it.\r
+Resulting binding energies of a C$_{\text{i}}$ DB with a nearby vacancy are listed in the second row of Table~\ref{table:dc_c-sv}.\r
+All investigated structures are prefered compared to isolated largely separated defects.\r
+In contrast to C$_{\text{s}}$ this is also valid for positions along \hkl[1 1 0] resulting in an entirely attractive interaction between defects of these types.\r
+Even for the largest possible distance (R) achieved in the calculations of the periodic supercell a binding energy as low as \unit[-0.31]{eV} is observed.\r
+The ground state configuration is obtained for a V at position 1.\r
+The C atom of the DB moves towards the vacant site forming a stable C$_{\text{s}}$ configuration resulting in the release of a huge amount of energy.\r
+The second most favored configuration is accomplished for a V located at position 3 due to the reduction of compressive strain of the Si DB atom and its two upper Si neighbours present in the C$_{\text{i}}$ DB configuration.\r
+This configuration is follwed by the structure, in which a vacant site is created at position 2.\r
+Similar to the observations for C$_{\text{s}}$ in the last subsection a reduction of strain along \hkl[0 0 1] is enabled by this configuration.\r
+Relaxed structures of the latter two defect combinations are shown in the bottom left of Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively together with their energetics during transition into the ground state.\r
+\begin{figure}\r
+\includegraphics[width=\columnwidth]{314-539.ps}\r
+\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created at position 3 (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.1]{eV} is observed.}\r
+\label{fig:314-539}\r
+\end{figure}\r
+\begin{figure}\r
+\includegraphics[width=\columnwidth]{059-539.ps}\r
+\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created at position 2 (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.6]{eV} is observed.}\r
+\label{fig:059-539}\r
+\end{figure}\r
+Activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} are observed.\r
+In the first case the Si and C atom of the DB move towards the vacant and initial DB lattice site respectively.\r
+In total three Si-Si and one more Si-C bond is formed during the transition.\r
+In the second case the lowest barrier is found for the migration of Si number 1 , which is substituted by the C$_{\text{i}}$ atom, towards the vacant site.\r
+A net amount of five Si-Si and one Si-C bond are additionally formed during the transition.\r
+The direct migration of the C$_{\text{i}}$ atom onto the vacant lattice site results in a somewhat higher barrier of \unit[1.0]{eV}.\r
+In both cases, the formation of additional bonds is responsible for the vast gain in energy rendering almost impossible the reverse processes.\r
+\r
+In summary, pairs of C$_{\text{i}}$ DBs and Vs, like no other before, show highly attractive interactions for all investigated combinations indpendent of orientation and separation direction of the defects.\r
+Furthermore, small activation energies, even for transitions into the ground state exist.\r
+Based on these results, a high probability for the formation of C$_{\text{s}}$ must be concluded.\r
+\r
\subsection{C$_{\text{s}}$ next to Si$_{\text{i}}$}\r
\r
+As shown in section~\ref{subsection:sep_def} C$_{\text{s}}$ exhibits the lowest energy of formation.\r
+Considering a perfect Si crystal and conservation of particles, however, the occupation of a Si lattice site by a slowed down implanted C atom is necessarily accompanied by the formation of a Si self-interstitial.\r
+There are good reasons for the existence of regions exhibiting such configurations with regard to the IBS process.\r
+Highly energetic C atoms are able to kick out a Si atom from its lattice site, resulting in a Si self-interstitial accompanied by a vacant site, which might get occupied by another C atom, which lost almost all of its kinetic energy.\r
+Thus, configurations of C$_{\text{s}}$ and Si self-interstitials are investigated in the following.\r
+The Si$_{\text{i}}$ \hkl<1 1 0> DB, which was found to exhibit the lowest energy of formation within the investigated self-interstitial configurations, is assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$.\r
+\r
+\begin{table}\r
+\begin{ruledtabular}\r
+\begin{tabular}{l c c c c c c}\r
+ & \hkl[1 1 0] & \hkl[-1 1 0] & \hkl[0 1 1] & \hkl[0 -1 1] &\r
+ \hkl[1 0 1] & \hkl[-1 0 1] \\\r
+\hline\r
+1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\\r
+2 & \RM{2} & \RM{6} & \RM{6} & \RM{2} & \RM{8} & \RM{5} \\\r
+3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\\r
+4 & \RM{4} & \RM{7} & \RM{9} & \RM{10} & \RM{10} & \RM{9} \\\r
+5 & \RM{5} & \RM{8} & \RM{6} & \RM{2} & \RM{6} & \RM{2} \\\r
+\end{tabular}\r
+\caption{Equivalent configurations labeld \RM{1}-\RM{10} of \hkl<1 1 0>-type Si$_{\text{i}}$ DBs created at position I and C$_{\text{s}}$ created at positions 1 to 5 according to Fig.~\ref{fig:combos_si}. The respective orientation of the Si$_{\text{i}}$ DB is given in the first row.}\r
+\label{table:dc_si-s}\r
+\end{ruledtabular}\r
+\end{table}\r
+\begin{table*}\r
+\begin{ruledtabular}\r
+\begin{tabular}{l c c c c c c c c c c}\r
+ & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & \RM{6} & \RM{7} & \RM{8} & \RM{9} & \RM{10} \\\r
+\hline\r
+$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\\r
+$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\\r
+$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\\r
+\end{tabular}\r
+\caption{Formation energies $E_{\text{f}}$, binding energies $E_{\text{b}}$ and C$_{\text{s}}$-Si$_{\text{i}}$ separation distances of the combinational C$_{\text{s}}$ and Si$_{\text{i}}$ configurations as defined in table \ref{table:dc_si-s}. Energies are given in eV while the separation is given in nm.}\r
+\label{table:dc_si-s_e}\r
+\end{ruledtabular}\r
+\end{table*}\r
+Table~\ref{table:dc_si-s} classifies equivalent configurations of \hkl<1 1 0>-type Si$_{\text{i}}$ DBs created at position I and C$_{\text{s}}$ created at positions 1 to 5 according to Fig.~\ref{fig:combos_si}.\r
+Corresponding formation as well as binding energies and the C$_{\text{s}}$-Si$_{\text{i}}$ distances are listed in Table~\ref{table:dc_si-s_e}.\r
+\r
+\begin{figure}\r
+\includegraphics[width=\columnwidth]{c_sub_si110.ps}\r
+\caption{Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance. The binding energies of the defect pairs are well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}\r
+\label{fig:dc_si-s}\r
+\end{figure}\r
+\r
Non-zero temperature, entropy, spatial separation of these defects possible, indeed observed in ab initio MD run.\r
\r
\section{Discussion}\r
\bibliography{../../bibdb/bibdb}{}\r
\bibliographystyle{h-physrev3}\r
\r
-%\begin{thebibliography}{99}\r
-%\bibitem{kresse96} G. Kresse and J. Furthm\"uller, \r
-% Comput. Mater. Sci. {\bf 6}, 15 (1996). \r
-%\bibitem{perdew92} J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais, \r
-% Phys. Rev. B {\bf 46}, 6671 (1992). \r
-%\bibitem{ceperley80} D. M. Ceperley and B. J. Alder, \r
-% Phys. Rev. Lett. {\bf 45}, 556 (1980). \r
-%\bibitem{perdew81} J. P. Perdew and A. Zunger, \r
-% Phys. Rev. B {\bf 23}, 5048 (1981). \r
-%\bibitem{bloechel94} P. E. Bl\"ochl, \r
-% Phys. Rev. B {\bf 50}, 17953 (1994).\r
-%\bibitem{kresse99} G. Kresse and D. Joubert, \r
-% Phys. Rev. B {\bf 59}, 1758 (1999).\r
-%\bibitem{monk76} H. J. Monkhorst and J. D. Pack, \r
-% Phys. Rev. B {\bf 13}, 5188 (1976). \r
-%\bibitem{rauls03a} E. Rauls, A. Gali, P. De´ak, and Th. Frauenheim, Phys. Rev. B, 68, 155208 (2003).\r
-%\bibitem{rauls03b} E. Rauls, U. Gerstmann, H. Overhof, and Th. Frauenheim, Physica B, Vols. 340-342, p. 184-189 (2003). \r
-%\bibitem{gerstmann03} U. Gerstmann, E. Rauls, Th. Frauenheim, and H. Overhof, Phys. Rev. B, 67, 205202, (2003).\r
-%\end{thebibliography}\r
-\r
\end{document}\r
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