In the following the simulation methods used within the scope of this study are introduced.
Enabling the investigation of the evolution of structure on the atomic scale, molecular dynamics (MD) simulations are chosen for modeling the behavior and precipitation of C introduced into an initially crystalline Si environment.
To be able to model systems with a large amount of atoms computational efficient classical potentials to describe the interaction of the atoms are most often used in MD studies.
-For reasons of flexibility in executing this non-standard task and in order to be able to use a novel interaction potential~\cite{albe_sic_pot} an appropriate MD code called \textsc{posic}\footnote{\textsc{posic} is an abbreviation for {\bf p}recipitation {\bf o}f {\bf SiC}} including a library collecting respective MD subroutines was developed from scratch\footnote{Source code: http://www.physik.uni-augsburg.de/\~{}zirkelfr/posic}.
+For reasons of flexibility in executing this non-standard task and in order to be able to use a novel interaction potential~\cite{albe_sic_pot}, an appropriate MD code called \textsc{posic}\footnote{\textsc{posic} is an abbreviation for {\bf p}recipitation {\bf o}f {\bf SiC}} including a library collecting respective MD subroutines was developed from scratch\footnote{Source code: http://www.physik.uni-augsburg.de/\~{}zirkelfr/posic}.
The basic ideas of MD in general and the adopted techniques as implemented in \textsc{posic} in particular are outlined in section~\ref{section:md}, while the functional form and derivative of the employed classical potential is presented in appendix~\ref{app:d_tersoff}.
An overview of the most important tools within the MD package is given in appendix~\ref{app:code}.
Although classical potentials are often most successful and at the same time computationally efficient in calculating some physical properties of a particular system, not all of its properties might be described correctly due to the lack of quantum-mechanical effects.
Although the Tersoff potential is one of the most widely used potentials, there are some shortcomings.
Describing the Si-Si interaction Tersoff was unable to find a single parameter set to describe well both, bulk and surface properties.
-Due to this and since the first approach labeled T1~\cite{tersoff_si1} turned out to be unstable~\cite{dodson87}, two further parametrizations exist, T2~\cite{tersoff_si2} and T3~\cite{tersoff_si3}.
+Due to this, and since the first approach labeled T1~\cite{tersoff_si1} turned out to be unstable~\cite{dodson87}, two further parametrizations exist, T2~\cite{tersoff_si2} and T3~\cite{tersoff_si3}.
While T2 describes well surface properties, T3 yields improved elastic constants and should be used for describing bulk properties.
However, T3, which is used in the Si/C potential, suffers from an underestimation of the dimer binding energy.
Similar behavior is found for the C-C interaction.
For this reason, Erhart and Albe provide a reparametrization of the Tersoff potential based on three independently fitted parameter sets for the Si-Si, C-C and Si-C interaction~\cite{albe_sic_pot}.
The functional form is similar to the one proposed by Tersoff.
Differences in the energy functional and the force evaluation routine are pointed out in appendix~\ref{app:d_tersoff}.
-Concerning Si the elastic properties of the diamond phase as well as the structure and energetics of the dimer are reproduced very well.
+Concerning Si, the elastic properties of the diamond phase as well as the structure and energetics of the dimer are reproduced very well.
The new parameter set for the C-C interaction yields improved dimer properties while at the same time delivers a description of the bulk phase similar to the Tersoff potential.
The potential succeeds in the description of the low as well as high coordinated structures.
The description of elastic properties of SiC is improved with respect to the potentials available in literature.
\subsection{Statistical ensembles}
\label{subsection:statistical_ensembles}
-Using the above mentioned algorithms the most basic type of MD is realized by simply integrating the equations of motion of a fixed number of particles ($N$) in a closed volume $V$ realized by periodic boundary conditions (PBC).
+Using the above mentioned algorithms, the most basic type of MD is realized by simply integrating the equations of motion of a fixed number of particles ($N$) in a closed volume $V$ realized by periodic boundary conditions (PBC).
Providing a stable integration algorithm the total energy $E$, i.e.\ the kinetic and configurational energy of the particles, is conserved.
This is known as the $NVE$, or microcanonical ensemble, describing an isolated system composed of microstates, among which the number of particles, volume and energy are held constant.
\end{equation}
where $\beta$ is the isothermal compressibility and $p$ corresponds to the current pressure, which is determined by equation \eqref{eq:basics:ps}.
-Using this method the system does not behave like a true $NpT$ ensemble.
+Using this method, the system does not behave like a true $NpT$ ensemble.
On average $T$ and $p$ correspond to the expected values.
For large enough time constants, i.e.\ $\tau > 100 \delta t$, the method shows realistic fluctuations in $T$ and $p$.
The advantage of the approach is that the coupling can be decreased to minimize the disturbance of the system and likewise be adjusted to suit the needs of a given application.
The challenging problem of determining the exact ground-state is now formally reduced to the determination of the $3$-dimensional function $n(\vec{r})$, which minimizes the energy functional.
However, the complexity associated with the many-electron problem is now relocated in the task of finding the well-defined but, in contrast to the potential energy, not explicitly known functional $F[n(\vec{r})]$.
-It is worth to note, that this minimal principle may be regarded as exactification of the TF theory, which is rederived by the approximations
+It is worth to note that this minimal principle may be regarded as exactification of the TF theory, which is rederived by the approximations
\begin{equation}
T=\int n(\vec{r})\frac{3}{10}k_{\text{F}}^2(n(\vec{r}))d\vec{r}
\text{ ,}
\subsection{Pseudopotentials}
As discussed in the last part of the previous section, an extremely large basis set of plane waves would be required to perform an all-electron calculation and a vast amount of computational time would be required to calculate the electronic wave functions.
-It is worth to stress out one more time, that this is mainly due to the orthogonalization wiggles of the wave functions of valence electrons near the nuclei.
+It is worth to stress out one more time that this is mainly due to the orthogonalization wiggles of the wave functions of valence electrons near the nuclei.
Thus, existing core states practically prevent the use of a PW basis set.
However, the core electrons, which are tightly bound to the nuclei, do not contribute significantly to chemical bonding or other physical properties of the solid.
This fact is exploited in the pseudopotential (PP) approach~\cite{cohen70} by removing the core electrons and replacing the atom and the associated strong ionic potential by a pseudoatom and a weaker PP that acts on a set of pseudo wave functions rather than the true valance wave functions.
The standard generation procedure of pseudopotentials proceeds by varying its parameters until the pseudo eigenvalues are equal to the all-electron valence eigenvalues and the pseudo wave functions match the all-electron valence wave functions beyond a certain cut-off radius determining the core region.
Modified methods to generate ultra-soft pseudopotentials were proposed, which address the rapid convergence with respect to the size of the plane wave basis set~\cite{vanderbilt90,troullier91}.
-Using PPs the rapid oscillations of the wave functions near the core of the atoms are removed considerably reducing the number of plane waves necessary to appropriately expand the wave functions.
+Using PPs, the rapid oscillations of the wave functions near the core of the atoms are removed considerably reducing the number of plane waves necessary to appropriately expand the wave functions.
More importantly, less accuracy is required compared to all-electron calculations to determine energy differences among ionic configurations, which almost totally appear in the energy of the valence electrons that are typically a factor $10^3$ smaller than the energy of the core electrons.
\subsection{Brillouin zone sampling}
Thus, $E_{\text{b}}$ indeed can be best thought of a binding energy, which is required to bring the defects to infinite separation.
The methods presented in the last two chapters can be used to investigate defect structures and energetics.
-Therefore, a supercell containing the perfect crystal is generated in an initial process.
+Therefor, a supercell containing the perfect crystal is generated in an initial process.
If not by construction, the system should be fully relaxed.
The substitutional or vacancy defect is realized by replacing or removing one atom contained in the supercell.
Interstitial defects are created by adding an atom at positions located in the space between regular lattice sites.
projects / lectures / latex.git / blobdiff