\begin{document}\r
\r
%\title{Mobility of Carbon in Silicon -- a first principles study}\r
-\title{Extensive first principles study of carbon defects in silicon}\r
+\title{First principles study of defects in carbon implanted silicon}\r
\author{F. Zirkelbach}\r
\author{B. Stritzker}\r
\affiliation{Experimentalphysik IV, Universit\"at Augsburg, 86135 Augsburg, Germany}\r
\affiliation{Department Physik, Universit\"at Paderborn, 33095 Paderborn, Germany}\r
\r
\begin{abstract}\r
-A first principles investigation of the mobility of carbon interstitials in silicon is presented.\r
-The migration mechanism of a carbon \hkl<1 0 0> interstitial and a silicon \hkl<1 1 0> self-interstitial in otherwise defect-free silicon has been investigated using density functional theory calculations.\r
-Furthermore, the influence of nearby vacancies, another carbon interstitial and substitutional defects as well as silicon self-interstitials has been investigated systematically.\r
+A first principles investigation of the mobility of carbon and silicon interstitials in silicon is presented.\r
+The migration mechanism of a carbon \hkl<1 0 0> interstitial in otherwise defect-free silicon has been investigated using density functional theory calculations.\r
+Furthermore, the influence of a nearby vacancy, another carbon interstitial and a substitutional defect as well as a silicon self-interstitial has been investigated systematically.\r
Interactions of various combinations of defects have been characterized including a couple of selected migration pathways within these configurations.\r
Almost all of the investigated pairs of defects tend to agglomerate allowing for a reduction in strain.\r
The formation of structures involving strong carbon-carbon bonds was found to occur very unlikely.\r
Based on these results conclusions regarding the precipitation mechanism of silicon carbide in bulk silicon are derived and its conformability to experimental findings is discussed.\r
\end{abstract}\r
\r
-\keywords{point defects, migration, interstitials, first principles calculations }\r
-\pacs{ find out later... }\r
+\keywords{point defects, defect clusters, migration, interstitials, ion implantation, first principles calculations}\r
+\pacs{61.72.J-,61.72.Yx,61.72.uj,66.30.J-,79.20.Rf,31.15.A-}\r
\maketitle\r
\r
% --------------------------------------------------------------------------------\r
However, the process of the formation of SiC precipitates in Si during C implantation is not yet fully understood.\r
Based on experimental high resolution transmission electron microscopy (HREM) studies\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03} it is assumed that incorporated C atoms form C-Si dimers (dumbbells) on regular Si lattice sites.\r
The highly mobile C interstitials agglomerate into large clusters followed by the formation of incoherent 3C-SiC nanocrystallites once a critical size of the cluster is reached.\r
-In contrast, investigations of the precipitation in strained Si$_{1-y}$C$_y$/Si heterostructures formed by molecular beam epitaxy (MBE)\cite{strane94,guedj98} suggest an initial coherent clustering of substitutional instead of interstitial C followed by a loss of coherency once the increasing strain energy surpasses the interfacial energy of an incoherent 3C-SiC precipitate in c-Si.\r
+In contrast, investigations of the precipitation in strained Si$_{1-y}$C$_y$/Si heterostructures formed by molecular beam epitaxy (MBE)\cite{strane94,guedj98} suggest an initial coherent precipitation by an agglomeration of substitutional instead of interstitial C followed by a loss of coherency once the increasing strain energy surpasses the interfacial energy of the incoherent 3C-SiC precipitate and c-Si.\r
These two different mechanisms of precipitation might be attributed to the respective method of fabrication, i.e. whether it occurs inside the Si bulk or on a Si surface.\r
-However, in another IBS study Nejim et al. propose a topotactic transformation remaining structure and orientation likewise based on the formation of substitutional C and a concurrent reaction of the excess Si self-interstitials with further implanted C atoms in the initial state\cite{nejim95}.\r
+However, in another IBS study Nejim et al. propose a topotactic transformation remaining structure and orientation that is likewise based on the formation of substitutional C and a concurrent reaction of the excess Si self-interstitials with further implanted C atoms in the initial state\cite{nejim95}.\r
Solving this controversy and understanding the effective underlying processes will enable significant technological progress in 3C-SiC thin film formation driving the superior polytype for potential applications in high-performance electronic device production\cite{wesch96}.\r
\r
Atomistic simulations offer a powerful tool of investigation providing detailed insight not accessible by experiment.\r
-A lot of theoretical work has been done on intrinsic point defects in Si\cite{bar-yam84,bar-yam84_2,car84,batra87,bloechl93,tang97,leung99,colombo02,goedecker02,al-mushadani03,hobler05,posselt08,ma10}, threshold displacement energies in Si\cite{mazzarolo01,holmstroem08} important in ion implantation, C defects and defect reactions in Si\cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,zhu98,mattoni2002,park02,jones04}, the SiC/Si interface\cite{chirita97,kitabatake93,cicero02,pizzagalli03} and defects in SiC\cite{bockstedte03,rauls03a,gao04,posselt06,gao07}.\r
+A lot of theoretical work has been done on intrinsic point defects in Si\cite{bar-yam84,bar-yam84_2,car84,batra87,bloechl93,tang97,leung99,colombo02,goedecker02,al-mushadani03,hobler05,sahli05,posselt08,ma10}, threshold displacement energies in Si\cite{mazzarolo01,holmstroem08} important in ion implantation, C defects and defect reactions in Si\cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,zhu98,mattoni2002,park02,jones04}, the SiC/Si interface\cite{chirita97,kitabatake93,cicero02,pizzagalli03} and defects in SiC\cite{bockstedte03,rauls03a,gao04,posselt06,gao07}.\r
However, none of the mentioned studies consistently investigates entirely the relevant defect structures and reactions concentrated on the specific problem of 3C-SiC formation in C implanted Si.\r
% but mattoni2002 actually did a lot. maybe this should be mentioned!\r
In fact, in a combined analytical potential molecular dynamics and ab initio study\cite{mattoni2002} the interaction of substitutional C with Si self-interstitials and C interstitials is evaluated.\r
-However, investigations are, first of all, restricted to interaction chains along the \hkl[1 1 0] and \hkl[-1 1 0] direction, secondly lacking combinations of C interstitials and, finally, not considering migration barriers giving further information about the probability of defect agglomeration.\r
+However, investigations are, first of all, restricted to interaction chains along the \hkl[1 1 0] and \hkl[-1 1 0] direction, secondly lacking combinations of C interstitials and, finally, not considering migration barriers providing further information on the probability of defect agglomeration.\r
\r
By first principles atomistic simulations this work aims to shed light on basic processes involved in the precipitation mechanism of SiC in Si.\r
During implantation defects such as vacancies (V), substitutional C (C$_{\text{s}}$), interstitial C (C$_{\text{i}}$) and Si self-interstitials (Si$_{\text{i}}$) are created, which play a decisive role in the precipitation process.\r
Instead, Capaz et al.\cite{capaz94}, investigating migration pathways of the C$_{\text{i}}$ \hkl<1 0 0> DB, find this defect to be \unit[2.1]{eV} lower in energy than the bond-centered (BC) configuration, which is claimed to constitute a saddle point configuration in the migration path within the \hkl(1 1 0) plane and, thus, interpreted as the barrier of migration for the respective path.\r
However, the present study indicates a local minimum state for the BC defect if spin polarized calculations are performed resulting in a net magnetization of two electrons localized in a torus around the C atom.\r
Another DFT calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerate Kohn-Sham states and an increase of the total energy by \unit[0.3]{eV} for the BC configuration.\r
-Regardless of the rather small correction due to the spin, the difference we found is much smaller (\unit[0.9]{eV}), which would nicely compare to experimental findings $(\unit[0.70-0.87]{eV})$\cite{lindner06,tipping87,song90} for the migration barrier.\r
+Regardless of the rather small correction of \unit[0.3]{eV} due to the spin, the difference we found is much smaller (\unit[0.9]{eV}), which would nicely compare to experimental findings $(\unit[0.70-0.87]{eV})$\cite{lindner06,tipping87,song90} for the migration barrier.\r
However, since the BC configuration constitutes a real local minimum another barrier exists which is about \unit[1.2]{eV} in height.\r
Indeed Capaz et al. propose another path and find it to be the lowest in energy\cite{capaz94}, in which a C$_{\text{i}}$ \hkl[0 0 -1] DB migrates into a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.\r
Calculations in this work reinforce this path by an additional improvement of the quantitative conformance of the barrier height (\unit[0.9]{eV}) to experimental values.\r
For the latter two the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site and in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms respectively.\r
No other configuration, within the ones that are mentioned, is affected.\r
\r
-Concerning the mobility of the ground state Si$_{\text{i}}$, an activation energy shortly below \unit[0.7]{eV} was found for the migration of a Si$_{\text{i}}$ \hkl[0 1 -1] into a \hkl[1 1 0] DB configuration located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.\r
+Concerning the mobility of the ground state Si$_{\text{i}}$, an activation energy of \unit[0.67]{eV} was found for the Si$_{\text{i}}$ \hkl[0 1 -1] to \hkl[1 1 0] DB configuration located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.\r
+Further investigations revealed a barrier of \unit[0.94]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ H, \unit[0.53]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ T and \unit[0.35]{eV} for the Si$_{\text{i}}$ H to Si$_{\text{i}}$ T transition.\r
+The obtained values are within the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.\r
% look for values in literature for neutraly charged Si_i diffusion\r
+% T seems to constitute a saddle point according to migration calculations\r
\r
\subsection{Pairs of C$_{\text{i}}$}\r
\r
-C$_{\text{i}}$ pairs of the \hkl<1 0 0>-type have been considered in the first part.\r
+First of all C$_{\text{i}}$ pairs of the \hkl<1 0 0>-type have been investigated.\r
Fig.~\ref{fig:combos_ci} schematically displays the position of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and the various positions for the second defect (1-5) used for investigating the defect pairs.\r
-Table~\ref{table:dc_c-c} summarizes the binding energies for the combination with a second C-Si \hkl<1 0 0> DB obtained for different orientations.\r
+Table~\ref{table:dc_c-c} summarizes the binding energies for the combination with a second C-Si \hkl<1 0 0> DB obtained for different orientations at positions 1 to 5.\r
\begin{figure}\r
%\begin{minipage}{0.49\columnwidth}\r
\subfigure[]{\label{fig:combos_ci}\includegraphics[width=0.45\columnwidth]{combos_ci.eps}}\r
\caption{Binding energies $E_{\text{b}}$ and C-C distance of energetically most favorable C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs separated along bonds in the \hkl[1 1 0] direction.}\r
\label{table:dc_110}\r
\end{table}\r
-The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:dc_110}\r
+The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:dc_110}.\r
\begin{figure}\r
\includegraphics[width=\columnwidth]{db_along_110_cc_n.ps}\r
\caption{Minimum binding energy of dumbbell combinations separated along \hkl[1 1 0] with respect to the C-C distance. The blue line is a guide for the eye and the green curve corresponds to the most suitable fit function consisting of all but the first data point.}\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[2.2-3.5]{eV}.\r
+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.\r
Although lower than the barriers for obtaining the ground state of two C$_{\text{i}}$ defects the activation energies are 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
\r
As mentioned above, configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ DBs might be especially important at higher temperatures due to the low activation energy necessary for its formation.\r
At higher temperatures the contribution of entropy to structural formation increases, which might result in a spatial separation even for defects located within the capture radius.\r
-Indeed an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} starting from configuration \RM{1}, which -- based on the above findings -- is assumed to recombine into the ground state configuration, results in a separation the C$_{\text{s}}$ and Si$_{\text{i}}$ DB by more than 4 next neighbor distances realized in a repeated migration mechnism of annihilating and arising Si DBs.\r
+Indeed an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} starting from configuration \RM{1}, which -- based on the above findings -- is assumed to recombine into the ground state configuration, results in a separation of the C$_{\text{s}}$ and Si$_{\text{i}}$ DB by more than 4 next neighbor distances realized in a repeated migration mechnism of annihilating and arising Si DBs.\r
The atomic configurations for two different points in time are shown in Fig.~\ref{fig:md}.\r
Si atoms 1 and 2, which form the initial DB, occupy usual Si lattice sites in the final configuration while atom 3 occupies an interstitial site.\r
\begin{figure}\r
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.\r
\r
The investigation of defect pairs indicates a general trend of defect agglomeration mainly driven by the potential of strain reduction.\r
-Obtained results for the most part compare well with results gained in previous studies\cite{leary97,capaz98,mattoni2002,liu02} and show an astnishingly good agreement with experiment\cite{song90}.\r
+Obtained results for the most part compare well with results gained in previous studies\cite{leary97,capaz98,mattoni2002,liu02} and show an astonishingly good agreement with experiment\cite{song90}.\r
Configurations involving two C impurities indeed exhibit the ground state for structures consisting of C-C bonds, which are responsible for the vast gain in energy.\r
However, based on investigations of possible migration pathways, these structures are less likely to arise than structures, in which both C atoms are interconnected by another Si atom, which is due to high activation energies of the respective pathways or alternative pathways with less high activation energies, which, however, involve intermediate unfavorable configurations.\r
Thus, agglomeration of C$_{\text{i}}$ is expected while the formation of C-C bonds is assumed to fail to appear by thermally activated diffusion processes.\r
\r
% add somewhere: nearly same energies of C_i -> Si_i + C_s, Si_i mig and C_i mig\r
\r
+% add somewhere: controversy c_i vs c_s agglomeration, we suggest both!\r
+\r
These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si.\r
Agglomeration of C$_{\text{i}}$ is energetically favored and enabled by a low activation energy for migration.\r
Although ion implantation is a process far from thermodynamic equlibrium, 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 clusters.\r
\r
-Unrolling these findings on the initially stated controversy present in the precipitation model, an increased participation of C$_{\text{s}}$ already in the initial stage must be assumed.\r
-Thermally activated C$_{\text{i}}$ might turn into C$_{\text{s}}$.\r
+Unrolling these findings on the initially stated controversy present in the precipitation model, an increased participation of C$_{\text{s}}$ already in the initial stage must be assumed due to its high probability of incidence.\r
+In addition, thermally activated, C$_{\text{i}}$ might turn into C$_{\text{s}}$.\r
The associated emission of Si$_{\text{i}}$ serves two needs: as a vehicle for other C$_{\text{s}}$ and as a supply of Si atoms needed elsewhere to form the SiC structure.\r
As for the vehicle, Si$_{\text{i}}$ is believed to react with C$_{\text{s}}$ turning it into a highly mobile C$_{\text{i}}$ again, allowing for the rearrangement of the C atom.\r
The rearrangement is crucial to end up in a configuration of C atoms only occupying substitutionally the lattice sites of one of the fcc lattices that build up the diamond lattice as expected in 3C-SiC.\r
However, there is no particular reason for the C species to reside in the interstitial lattice.\r
Contrasts are also assumed for Si$_{\text{i}}$.\r
Once precipitation occurs regions of dark contrasts disappear in favor of Moir\'e patterns indicating 3C-SiC in c-Si due to the mismatch in the lattice constant.\r
-Until then, however, these regions are either composed of stretched coherent SiC and interstitials or of yet contracted incoherent SiC surrounded by Si and interstitials too small to be detected in HREM.\r
+Until then, however, these regions are either composed of stretched coherent SiC and interstitials or of already contracted incoherent SiC surrounded by Si and interstitials too small to be detected in HREM.\r
In both cases Si$_{\text{i}}$ might be attributed a third role, which is the partial compensation of tensile strain either in the stretched SiC or at the interface of the contracted SiC and the Si host.\r
\r
In addition, the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate is statisfied by the mechanism of successive positioning of C$_{\text{s}}$.\r
In summary, C and Si point defects in Si, combinations of these defects and diffusion processes within such configurations have been investigated.\r
It is shown that C interstitials in Si tend to agglomerate, which is mainly driven by a reduction of strain.\r
Investigations of migration pathways, however, allow to conclude that C clustering fails to appear by thermally activated processes due to high activation energies of the the respective diffusion processes.\r
-A highly attractive interaction and a large capture radius has been identified for the C$_{\text{i}}$ \hkl<1 0 0> DB and the vacancy indicating a high probability for the formation of C$_{\text{s}}$\r
-In contrast, a rapidly decreasing interaction with respect to the separation distance has been identified for C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB resulting in a low probability of defects exhibiting respective separations ...\r
-These findings suggest an increased ... in prec model ....\r
+A highly attractive interaction and a large capture radius has been identified for the C$_{\text{i}}$ \hkl<1 0 0> DB and the vacancy indicating a high probability for the formation of C$_{\text{s}}$.\r
+In contrast, a rapidly decreasing interaction with respect to the separation distance has been identified for C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB resulting in a low probability of defects exhibiting respective separations to transform into the C$_{\text{i}}$ \hkl<1 0 0> DB, which constitutes the ground state configuration for a C atom introduced into otherwise perfect Si. \r
+Based on these findings conclusions on basic processes involved in the SiC precipitation in bulk Si are drawn.\r
+It is concluded that the precipitation process is governed by the formation of C$_{\text{s}}$ already in the initial stages.\r
+Agglomeration and rearrangement of C$_{\text{s}}$, however, is only possible by mobile C$_{\text{i}}$, which, thus, needs to be present at the same time.\r
+Si$_{\text{i}}$ constitutes the vehicle for the rearrangement of C$_{\text{s}}$.\r
+\r
\r
% ----------------------------------------------------\r
\section*{Acknowledgment}\r
projects / lectures / latex.git / blobdiff