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.
+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.
\subsection{Energy cut-off}
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 controlled by the Berendsen thermostat~\cite{berendsen84} with a time constant of \unit[100]{fs}.
Integration of the equations of motion is realized by the velocity Verlet algorithm~\cite{verlet67} using a fixed time step of \unit[1]{fs}.
-For structural relaxation of defect structures the same algorithm is utilized with the temperature set to zero Kelvin.
+For structural relaxation of defect structures, the same algorithm is utilized with the temperature set to zero Kelvin.
This also applies for the relaxation of structures within the CRT calculations to find migration pathways.
In the latter case the time constant of the Berendsen thermostat is set to \unit[1]{fs} in order to achieve direct velocity scaling, which corresponds to a steepest descent minimization driving the system into a local minimum, if the temperature is set to zero Kelvin.
However, in some cases a time constant of \unit[100]{fs} turned out to result in lower barriers.
\subsection{Time step}
The quality of the integration algorithm and the occupied time step is determined by the ability to conserve the total energy.
-Therefore, simulations of a $9\times9\times9$ 3C-SiC unit cell containing 5832 atoms in total are carried out in the $NVE$ ensemble.
+Therefor, simulations of a $9\times9\times9$ 3C-SiC unit cell containing 5832 atoms in total are carried out in the $NVE$ ensemble.
The calculations are performed for \unit[100]{ps} corresponding to $10^5$ integration steps and two different initial temperatures are considered, i.e.\ \unit[0]{$^{\circ}$C} and \unit[1000]{$^{\circ}$C}.
\begin{figure}[t]
\begin{center}
However, it seems that the size of the c-Si host matrix is chosen large enough to even find the precipitate in a compressed state.
The absence of a compression of the c-Si surrounding is due to the possibility of the system to change its volume.
-Otherwise the increase of the lattice constant of the precipitate of roughly \unit[4.31]{\AA} in the beginning up to \unit[4.34]{\AA} in the relaxed precipitate configuration could not take place without an accompanying reduction of the lattice constant of the c-Si surrounding.
+Otherwise, the increase of the lattice constant of the precipitate of roughly \unit[4.31]{\AA} in the beginning up to \unit[4.34]{\AA} in the relaxed precipitate configuration could not take place without an accompanying reduction of the lattice constant of the c-Si surrounding.
If the total volume is assumed to be the sum of the volumes that are composed of Si atoms forming the c-Si surrounding and Si atoms involved forming the precipitate, the expected increase can be calculated by
\begin{equation}
\frac{V}{V_0}=
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