X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=posic%2Fthesis%2Fsimulation.tex;fp=posic%2Fthesis%2Fsimulation.tex;h=850014ef857318e65ba58fd9ebec9a01d62ae8b9;hp=a32d343d9791a2070771b697a951a8667b9a5bbf;hb=308fce67d5c701822d5cb2d960a056644498cd03;hpb=4c26ce3943e912490d74ffa043fe0f297a02c04b diff --git a/posic/thesis/simulation.tex b/posic/thesis/simulation.tex index a32d343..850014e 100644 --- a/posic/thesis/simulation.tex +++ b/posic/thesis/simulation.tex @@ -24,7 +24,7 @@ Type 3 (Fig.~\ref{fig:simulation:sc3}) contains 4 primitive cells with 8 atoms a The basis is simple cubic. In the following, an overview of the different simulation procedures and respective parameters is presented. -These procedures and parameters differ depending on whether classical potentials or {\em ab initio} methods are used and on what is going to be investigated. +These procedures and parameters differ depending on whether classical potentials or {\em ab initio} methods are used as well as on the object of investigation. \section{DFT calculations} \label{section:simulation:dft_calc} @@ -47,7 +47,7 @@ In MD simulations the equations of motion are integrated by a fourth order predi % todo - point defects are calculated for the neutral charge state. Most of the parameter settings, as determined above, constitute a tradeoff regarding the tasks that need to be addressed. -These parameters include the size of the supercell, cut-off energy and $k$ point mesh. +These parameters include the size of the supercell, cut-off energy and $\vec{k}$-point mesh. The choice of these parameters is considered to reflect a reasonable treatment with respect to both, computational efficiency and accuracy, as will be shown in the next sections. Furthermore, criteria concerning the choice of the potential and the exchange-correlation (XC) functional are being outlined. Finally, the utilized parameter set is tested by comparing the calculated values of the cohesive energy and the lattice constant to experimental data. @@ -63,8 +63,8 @@ Obviously, the interaction reduces with increasing system size and will be negli \caption{Defect formation energies of several defects in c-Si with respect to the size of the supercell.} \label{fig:simulation:ef_ss} \end{figure} -To estimate a critical size the formation energies of several intrinsic defects in Si with respect to the system size are calculated. -An energy cut-off of \unit[250]{eV} and a $4\times4\times4$ Monkhorst-Pack $k$-point mesh~\cite{monkhorst76} is used. +To estimate a critical size, the formation energies of several intrinsic defects in Si with respect to the system size are calculated. +An energy cut-off of \unit[250]{eV} and a $4\times4\times4$ Monkhorst-Pack $\vec{k}$-point mesh~\cite{monkhorst76} is used. The results are displayed in Fig.~\ref{fig:simulation:ef_ss}. The formation energies converge fast with respect to the system size. Thus, investigating supercells containing more than 56 primitive cells or $112\pm1$ atoms should be reasonably accurate. @@ -75,11 +75,11 @@ Throughout this work sampling of the BZ is restricted to the $\Gamma$ point. The calculation is usually two times faster and half of the storage needed for the wave functions can be saved since $c_{i,q}=c_{i,-q}^*$, where the $c_{i,q}$ are the Fourier coefficients of the wave function. As discussed in section~\ref{subsection:basics:bzs}, this does not pose a severe limitation if the supercell is large enough. Indeed, it was shown~\cite{dal_pino93} that already for calculations involving only 32 atoms, energy values obtained by sampling the $\Gamma$ point differ by less than \unit[0.02]{eV} from calculations using the Baldereschi point~\cite{baldereschi73}, which constitutes a mean-value point in the BZ. -Thus, the calculations of the present study on supercells containing $108$ primitive cells can be considered sufficiently converged with respect to the $k$-point mesh. +Thus, the calculations of the present study on supercells containing $108$ primitive cells can be considered sufficiently converged with respect to the $\vec{k}$-point mesh. \subsection{Energy cut-off} -To determine an appropriate cut-off energy of the plane-wave basis set a $2\times2\times2$ supercell of type 3 containing $32$ Si and $32$ C atoms in the 3C-SiC structure is equilibrated for different cut-off energies in the LDA. +To determine an appropriate cut-off energy of the plane-wave basis set, a $2\times2\times2$ supercell of type 3 containing $32$ Si and $32$ C atoms in the 3C-SiC structure is equilibrated for different cut-off energies in the LDA. \begin{figure}[t] \begin{center} \includegraphics[width=0.7\textwidth]{sic_32pc_gamma_cutoff_lc.ps} @@ -93,9 +93,9 @@ Obviously, an energy cut-off of \unit[300]{eV}, although the minimum acceptable, \subsection{Potential and exchange-correlation functional} -To find the most suitable combination of potential and XC functional for the C/Si system a $2\times2\times2$ supercell of type 3 of Si and C, both in the diamond structure, as well as 3C-SiC is equilibrated for different combinations of the available potentials and XC functionals. -To exclude a possibly corrupting influence of the other parameters highly accurate calculations are performed, i.e.\ an energy cut-off of \unit[650]{eV} and a $6\times6\times6$ Monkhorst-Pack $k$-point mesh is used. -Next to the ultra-soft pseudopotentials~\cite{vanderbilt90} \textsc{vasp} offers the projector augmented-wave method (PAW)~\cite{bloechl94} to describe the ion-electron interaction. +To find the most suitable combination of potential and XC functional for the C/Si system, a $2\times2\times2$ supercell of type 3 of Si and C, both in the diamond structure, as well as 3C-SiC is equilibrated for different combinations of the available potentials and XC functionals. +To exclude a possibly corrupting influence of the other parameters, highly accurate calculations are performed, i.e.\ an energy cut-off of \unit[650]{eV} and a $6\times6\times6$ Monkhorst-Pack $\vec{k}$-point mesh is used. +Next to the ultra-soft pseudopotentials~\cite{vanderbilt90}, \textsc{vasp} offers the projector augmented-wave method (PAW)~\cite{bloechl94} to describe the ion-electron interaction. The two XC functionals included in the test are of the LDA~\cite{ceperley80,perdew81} and GGA~\cite{perdew86,perdew92} type as implemented in \textsc{vasp}. \begin{table}[t] @@ -213,7 +213,7 @@ Since, on the other hand, properties of the 3C-SiC precipitate, its surrounding To construct a spherical and topotactically aligned 3C-SiC precipitate in c-Si, the approach illustrated in the following is applied. A total simulation volume $V$ consisting of 21 unit cells of c-Si in each direction is created. -To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary according to experimental results as discussed in section~\ref{subsection:ibs} and~\ref{section:assumed_prec}. +To obtain a minimal and stable precipitate, 5500 carbon atoms are considered necessary according to experimental results as discussed in section~\ref{subsection:ibs} and~\ref{section:assumed_prec}. This corresponds to a spherical 3C-SiC precipitate with a radius of approximately \unit[3]{nm}. The initial precipitate configuration is constructed in two steps. In the first step the surrounding Si matrix is created.