The most common empirical potentials for covalent systems are the Stillinger-Weber\cite{stillinger85}, Brenner\cite{brenner90}, Tersoff\cite{tersoff_si3} and environment-dependent interatomic potential\cite{bazant96,bazant97,justo98}.
These potentials are assumed to be reliable for large-scale simulations\cite{balamane92,huang95,godet03} on specific problems under investigation providing insight into phenomena that are otherwise not accessible by experimental or first-principles methods.
Until recently\cite{lucas10}, a parametrization to describe the C-Si multicomponent system within the mentioned interaction models did only exist for the Tersoff\cite{tersoff_m} and related potentials, e.g. the one by Gao and Weber\cite{gao02} as well as the one by Erhart and Albe\cite{albe_sic_pot}.
-All these potentials are short range potentials employing a cut-off function, which drops the atomic interaction to zero in between the first and second next neighbor distance.
+All these potentials are short range potentials employing a cut-off function, which drops the atomic interaction to zero in between the first and second nearest neighbor distance.
In a combined ab initio and empirical potential study it was shown that the Tersoff potential properly describes binding energies of combinations of C defects in Si\cite{mattoni2002}.
-However, investigations of brittleness in covalent materials\cite{mattoni2007} identified the short range character of these potentials to be responsible for overestimated forces necessary to snap the bond of two next neighbored atoms.
+However, investigations of brittleness in covalent materials\cite{mattoni2007} identified the short range character of these potentials to be responsible for overestimated forces necessary to snap the bond of two neighbored atoms.
In a previous study\cite{zirkelbach10a} we approved explicitly the influence on the migration barrier for C diffusion in Si.
Using the Erhart/Albe (EA) potential\cite{albe_sic_pot}, an overestimated barrier height compared to ab initio calculations and experiment is obtained.
A proper description of C diffusion, however, is crucial for the problem under study.
At constant temperature 10 atoms were inserted at a time.
Three different regions within the total simulation volume were considered for a statistically distributed insertion of the C atoms: $V_1$ corresponding to the total simulation volume, $V_2$ corresponding to the size of the precipitate and $V_3$, which holds the necessary amount of Si atoms of the precipitate.
After C insertion the simulation has been continued for \unit[100]{ps} and is cooled down to \unit[20]{$^{\circ}$C} afterwards.
-A Tersoff-like bond order potential by Erhart and Albe (EA)\cite{albe_sic_pot} has been utilized, which accounts for nearest neighbor interactions only realized by a cut-off function dropping the interaction to zero in between the first and second next neighbor distance.
+A Tersoff-like bond order potential by Erhart and Albe (EA)\cite{albe_sic_pot} has been utilized, which accounts for nearest neighbor interactions only realized by a cut-off function dropping the interaction to zero in between the first and second nearest neighbor distance.
The potential was used as is, i.e. without any repulsive potential extension at short interatomic distances.
Constant pressure simulations are realized by the Berendsen barostat\cite{berendsen84} using a time constant of \unit[100]{fs} and a bulk modulus of \unit[100]{GPa} for Si.
The temperature is kept constant by the Berendsen thermostat\cite{berendsen84} with a time constant of \unit[100]{fs}.
The C interstitial defect with the lowest energy of formation has been found to be the C-Si \hkl<1 0 0> interstitial dumbbell (C$_{\text{i}}$ \hkl<1 0 0> DB), which, thus, constitutes the ground state of an additional C impurity in otherwise perfect c-Si.
This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94} and experimental\cite{watkins76,song90} investigations.
Astonishingly EA and DFT predict almost equal formation energies.
-There are, however, geometric differences with regard to the DB position within the tetrahedron spanned by the four next neighbored Si atoms, as already reported in a previous study\cite{zirkelbach10a}.
+There are, however, geometric differences with regard to the DB position within the tetrahedron spanned by the four neighbored Si atoms, as already reported in a previous study\cite{zirkelbach10a}.
Since the energetic description is considered more important than the structural description, minor discrepancies of the latter are assumed non-problematic.
The second most favorable configuration is the C$_{\text{i}}$ \hkl<1 1 0> DB followed by the C$_{\text{i}}$ bond-centered (BC) configuration.
For both configurations EA overestimates the energy of formation by approximately \unit[1]{eV} compared to DFT.
\subsection{Formation energies of C$_{\text{i}}$ and C$_{\text{s}}$-Si$_{\text{i}}$}
-As has been shown in a previous study\cite{zirkelbach10b}, the energetically most favorable configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ is obtained for C$_{\text{s}}$ located at the next neighbored lattice site along the \hkl<1 1 0> bond chain of a Si$_{\text{i}}$ \hkl<1 1 0> DB.
+As has been shown in a previous study\cite{zirkelbach10b}, the energetically most favorable configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ is obtained for C$_{\text{s}}$ located at the neighbored lattice site along the \hkl<1 1 0> bond chain of a Si$_{\text{i}}$ \hkl<1 1 0> DB.
However the energy of formation is slightly higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB, which constitutes the ground state for a C impurity introduced into otherwise perfect c-Si.
For a possible clarification of the controversial views on the participation of C$_{\text{s}}$ in the precipitation mechanism by classical potential simulations, test calculations need to ensure the proper description of the relative formation energies of combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ compared to C$_{\text{i}}$.
Erhart/Albe & 3.88 & 4.93 & 5.25$^{\text{a}}$/5.08$^{\text{b}}$/4.43$^{\text{c}}$
\end{tabular}
\end{ruledtabular}
-\caption{Formation energies of defect configurations of a single C impurity in otherwise perfect c-Si determined by classical potential and ab initio methods. The formation energies are given in electron volt. T denotes the tetrahedral and the subscripts i and s indicate the interstitial and substitutional configuration. Superscripts a, b and c denote configurations of C$_{\text{s}}$ located at the first, second and third next neighbored lattice site with respect to the Si$_{\text{i}}$ atom.}
+\caption{Formation energies of defect configurations of a single C impurity in otherwise perfect c-Si determined by classical potential and ab initio methods. The formation energies are given in electron volt. T denotes the tetrahedral and the subscripts i and s indicate the interstitial and substitutional configuration. Superscripts a, b and c denote configurations of C$_{\text{s}}$ located at the first, second and third nearest neighbored lattice site with respect to the Si$_{\text{i}}$ atom.}
\label{tab:defect_combos}
\end{table}
Obviously the EA potential properly describes the relative energies of formation.
Combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ T are energetically less favorable than the ground state C$_{\text{i}}$ \hkl<1 0 0> DB configuration.
With increasing separation distance the energies of formation decrease.
However, even for non-interacting defects, the energy of formation, which is then given by the sum of the formation energies of the separated defects (\unit[4.15]{eV}) is still higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB.
-Unexpectedly, the structure of a Si$_{\text{i}}$ \hkl<1 1 0> DB and a next neighbored C$_{\text{s}}$, which is the most favored configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ according to quantum-mechanical calculations\cite{zirkelbach10b}, likewise constitutes an energetically favorable configuration within the EA description, which is even preferred over the two least separated configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ T.
+Unexpectedly, the structure of a Si$_{\text{i}}$ \hkl<1 1 0> DB and a neighbored C$_{\text{s}}$, which is the most favored configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ according to quantum-mechanical calculations\cite{zirkelbach10b}, likewise constitutes an energetically favorable configuration within the EA description, which is even preferred over the two least separated configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ T.
This is attributed to an effective reduction in strain enabled by the respective combination.
Thus, a proper description with respect to the relative energies of formation is assumed for the EA potential.
To accurately model the SiC precipitation, which involves the agglomeration of C, a proper description of the migration process of the C impurity is required.
As shown in a previous study\cite{zirkelbach10a}, quantum-mechanical results properly describe the C$_{\text{i}}$ \hkl<1 0 0> DB diffusion resulting in a migration barrier height of \unit[0.90]{eV}, excellently matching experimental values of \unit[0.70-0.87]{eV}\cite{lindner06,tipping87,song90} and, for this reason, reinforcing the respective migration path as already proposed by Capaz et~al.\cite{capaz94}.
-During transition a C$_{\text{i}}$ \hkl[0 0 -1] DB migrates towards a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the next neighbored lattice site in \hkl[1 1 -1] direction.
+During transition a C$_{\text{i}}$ \hkl[0 0 -1] DB migrates towards a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the neighbored lattice site in \hkl[1 1 -1] direction.
However, it turned out that the description fails if the EA potential is used, which overestimates the migration barrier (\unit[2.2]{eV}) by a factor of 2.4.
In addition a different diffusion path is found to exhibit the lowest migration barrier.
-A C$_{\text{i}}$ \hkl[0 0 -1] DB turns into the \hkl[0 0 1] configuration at the next neighbored lattice site.
+A C$_{\text{i}}$ \hkl[0 0 -1] DB turns into the \hkl[0 0 1] configuration at the neighbored lattice site.
The transition involves the C$_{\text{i}}$ BC configuration, which, however, was found to be unstable relaxing into the C$_{\text{i}}$ \hkl<1 1 0> DB configuration.
If the migration is considered to occur within a single step, the kinetic energy of \unit[2.2]{eV} is enough to turn the \hkl<1 0 0> DB into the BC and back into a \hkl<1 0 0> DB configuration.
If, on the other hand, a two step process is assumed, the BC configuration will most probably relax into the C$_{\text{i}}$ \hkl<1 1 0> DB configuration resulting in different relative energies of the intermediate state and the saddle point.
\caption{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration.}
\label{fig:mig}
\end{figure}
-Approximately \unit[2.24]{eV} are needed to turn the C$_{\text{i}}$ \hkl[0 0 -1] DB into the C$_{\text{i}}$ \hkl[1 1 0] DB located at the next neighbored lattice site in \hkl[1 1 -1] direction.
+Approximately \unit[2.24]{eV} are needed to turn the C$_{\text{i}}$ \hkl[0 0 -1] DB into the C$_{\text{i}}$ \hkl[1 1 0] DB located at the neighbored lattice site in \hkl[1 1 -1] direction.
Another barrier of \unit[0.90]{eV} exists for the rotation into the C$_{\text{i}}$ \hkl[0 -1 0] DB configuration.
Roughly the same amount would be necessary to excite the C$_{\text{i}}$ \hkl[1 1 0] DB to the BC configuration (\unit[0.40]{eV}) and a successive migration into the \hkl[0 0 1] DB configuration (\unit[0.50]{eV}) as displayed in our previous study\cite{zirkelbach10a}.
The former diffusion process, however, would more nicely agree with the ab initio path, since the migration is accompanied by a rotation of the DB orientation.
By comparing the Si-C peaks of the low concentration simulation with the resulting Si-C distances of a C$_{\text{i}}$ \hkl<1 0 0> DB it becomes evident that the structure is clearly dominated by this kind of defect.
One exceptional peak exists, which is due to the Si-C cut-off, at which the interaction is pushed to zero.
Investigating the C-C peak at \unit[0.31]{nm}, which is also available for low C concentrations as can be seen in the inset, reveals a structure of two concatenated, differently oriented C$_{\text{i}}$ \hkl<1 0 0> DBs to be responsible for this distance.
-Additionally, the Si-Si radial distribution shows non-zero values at distances around \unit[0.3]{nm}, which, again, is due to the DB structure stretching two next neighbored Si atoms.
+Additionally, the Si-Si radial distribution shows non-zero values at distances around \unit[0.3]{nm}, which, again, is due to the DB structure stretching two neighbored Si atoms.
This is accompanied by a reduction of the number of bonds at regular Si distances of c-Si.
A more detailed description of the resulting C-Si distances in the C$_{\text{i}}$ \hkl<1 0 0> DB configuration and the influence of the defect on the structure is available in a previous study\cite{zirkelbach09}.
For high C concentrations, the defect concentration is likewise increased and a considerable amount of damage is introduced in the insertion volume.
A subsequent superposition of defects generates new displacement arrangements for the C-C as well as Si-C pair distances, which become hard to categorize and trace and obviously lead to a broader distribution.
-Short range order indeed is observed, i.e. the large amount of strong next neighbored C-C bonds at \unit[0.15]{nm} as expected in graphite or diamond and Si-C bonds at \unit[0.19]{nm} as expected in SiC, but hardly visible is the long range order.
+Short range order indeed is observed, i.e. the large amount of strong neighbored C-C bonds at \unit[0.15]{nm} as expected in graphite or diamond and Si-C bonds at \unit[0.19]{nm} as expected in SiC, but hardly visible is the long range order.
This indicates the formation of an amorphous SiC-like phase.
In fact, resulting Si-C and C-C radial distribution functions compare quite well with these obtained by cascade amorphized and melt-quenched amorphous SiC using a modified Tersoff potential\cite{gao02}.
New accelerated methods have been developed to bypass the time scale problem retaining proper thermodynamic sampling\cite{voter97,voter97_2,voter98,sorensen2000,wu99}.
However, the applied potential comes up with an additional limitation, as previously mentioned in the introduction.
-The cut-off function of the short range potential limits the interaction to next neighbors, which results in overestimated and unphysical high forces between next neighbor atoms.
+The cut-off function of the short range potential limits the interaction to nearest neighbors, which results in overestimated and unphysical high forces between neighbored atoms.
This behavior, as observed and discussed for the Tersoff potential\cite{tang95,mattoni2007}, is supported by the overestimated activation energies necessary for C diffusion as investigated in section \ref{subsection:cmob}.
Indeed, it is not only the strong, hard to break C-C bond inhibiting C diffusion and further rearrangements in the case of the high C concentration simulations.
This is also true for the low concentration simulations dominated by the occurrence of C$_{\text{i}}$ \hkl<1 0 0> DBs spread over the whole simulation volume, which are unable to agglomerate due to the high migration barrier.
Obviously, the structure obtained at \unit[450]{$^{\circ}$C}, which was found to be dominated by C$_{\text{i}}$, transforms into a C$_{\text{s}}$ dominated structure with increasing temperature.
Comparing the radial distribution at \unit[2050]{$^{\circ}$C} to the resulting bonds of C$_{\text{s}}$ in c-Si excludes any possibility of doubt.
-The phase transformation is accompanied by an arising Si-Si peak at \unit[0.325]{nm}, which corresponds to the distance of second next neighbored Si atoms along \hkl<1 1 0> bond chain with C$_{\text{s}}$ in between.
+The phase transformation is accompanied by an arising Si-Si peak at \unit[0.325]{nm}, which corresponds to the distance of next neighbored Si atoms along \hkl<1 1 0> bond chain with C$_{\text{s}}$ in between.
Since the expected distance of these Si pairs in 3C-SiC is \unit[0.308]{nm} the existing SiC structures embedded in the c-Si host are stretched.
According to the C-C radial distribution, agglomeration of C fails to appear even for elevated temperatures, as can be seen on the total amount of C pairs within the investigated separation range, which does not change significantly.
-However, a small decrease in the amount of next neighbored C pairs can be observed with increasing temperature.
+However, a small decrease in the amount of neighbored C pairs can be observed with increasing temperature.
This high temperature behavior is promising since breaking of these diamond- and graphite-like bonds is mandatory for the formation of 3C-SiC.
Obviously, acceleration of the dynamics occurred by supplying additional kinetic energy.
A slight shift towards higher distances can be observed for the maximum located shortly above \unit[0.3]{nm}.
Arrows with dashed lines mark C-C distances resulting from C$_{\text{i}}$ \hkl<1 0 0> DB combinations while arrows with solid lines mark distances arising from combinations of C$_{\text{s}}$.
-The continuous dashed line corresponds to the distance of C$_{\text{s}}$ and a next neighbored C$_{\text{i}}$ DB.
+The continuous dashed line corresponds to the distance of C$_{\text{s}}$ and a neighbored C$_{\text{i}}$ DB.
Obviously, the shift of the peak is caused by the advancing transformation of the C$_{\text{i}}$ DB into the C$_{\text{s}}$ defect.
Quite high g(r) values are obtained for distances in between the continuous dashed line and the first arrow with a solid line.
For the most part these structures can be identified as configurations of C$_{\text{s}}$ with either another C atom that basically occupies a Si lattice site but is displaced by a Si interstitial residing in the very next surrounding or a C atom that nearly occupies a Si lattice site forming a defect other than the \hkl<1 0 0>-type with the Si atom.
No significant change in structure is observed.
However, the decrease of the cut-off artifact and slightly sharper peaks observed with increasing temperature, in turn, indicate a slight acceleration of the dynamics realized by the supply of kinetic energy.
However, it is not sufficient to enable the amorphous to crystalline transition.
-In contrast, even though next neighbored C bonds could be partially dissolved in the system exhibiting low C concentrations, the amount of next neighbored C pairs even increased in the latter case.
-Moreover, the C-C peak at \unit[0.252]{nm}, which gets slightly more distinct, equals the second next neighbor distance in diamond and indeed is made up by a structure of two C atoms interconnected by a third C atom.
+In contrast, even though bonds of neighbored C atoms could be partially dissolved in the system exhibiting low C concentrations, the amount of neighbored C pairs even increased in the latter case.
+Moreover, the C-C peak at \unit[0.252]{nm}, which gets slightly more distinct, equals the second nearest neighbor distance in diamond and indeed is made up by a structure of two C atoms interconnected by a third C atom.
Obviously, processes that appear to be non-conducive are likewise accelerated in a system, in which high amounts of C are incorporated within a short period of time, which is accompanied by a concurrent introduction of accumulating, for the reason of time non-degradable damage.
% non-degradable, non-regenerative, non-recoverable
Thus, for these systems even larger time scales, which are not accessible within traditional MD, must be assumed for an amorphous to crystalline transition or structural evolution in general.
For the agglomeration and rearrangement of C, Si$_{\text{i}}$ is needed to turn C$_{\text{s}}$ into highly mobile C$_{\text{i}}$ again.
Since the conversion of a coherent SiC structure, i.e. C$_{\text{s}}$ occupying the Si lattice sites of one of the two fcc lattices that build up the c-Si diamond lattice, into incoherent SiC is accompanied by a reduction in volume, large amounts of strain are assumed to reside in the coherent as well as at the surface of the incoherent structure.
Si$_{\text{i}}$ serves either as a supply of Si atoms needed in the surrounding of the contracted precipitates or as an interstitial defect minimizing the emerging strain energy of a coherent precipitate.
-The latter has been directly identified in the present simulation study, i.e. structures of two C$_{\text{s}}$ atoms with one being slightly displaced by a next neighbored Si$_{\text{i}}$ atom.
+The latter has been directly identified in the present simulation study, i.e. structures of two C$_{\text{s}}$ atoms with one being slightly displaced by a neighbored Si$_{\text{i}}$ atom.
It is, thus, concluded that precipitation occurs by successive agglomeration of C$_{\text{s}}$ as already proposed by Nejim et~al.\cite{nejim95}.
This agrees well with a previous ab initio study on defects in C implanted Si\cite{zirkelbach10b}, which showed C$_{\text{s}}$ to occur in all probability.
projects / lectures / latex.git / commitdiff