X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=posic%2Fthesis%2Fmd.tex;h=fab465627a4752ee2b2f74b25ae48e4d9672d463;hb=fbf04a7729cd69c3416caa748f53453d860f165f;hp=bd6d979b58f1a86bb3b6dc9a5d64823e08fba3ed;hpb=77317edfd6988a221dd90fe9a75fe6cecce2b6a7;p=lectures%2Flatex.git diff --git a/posic/thesis/md.tex b/posic/thesis/md.tex index bd6d979..fab4656 100644 --- a/posic/thesis/md.tex +++ b/posic/thesis/md.tex @@ -457,6 +457,8 @@ However, by applying these values the final configuration varies only slightly f Obtained & 5495 & 5486 & 68591 & 74077\\ Expected & 5500 & 5500 & 68588 & 74088\\ Difference & -5 & -14 & 3 & -11\\ +Notation & $N^{\text{3C-SiC}}_{\text{C}}$ & $N^{\text{3C-SiC}}_{\text{Si}}$ + & $N^{\text{c-Si}}_{\text{Si}}$ & $N^{\text{total}}_{\text{Si}}$ \\ \hline \hline \end{tabular} @@ -479,33 +481,59 @@ Once the main part of the excess energy is carried out previous settings for the Figure \ref{fig:md:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration. The Si-Si radial distribution for both, plain c-Si and the precipitate configuration show a maximum at a distance of 0.235 nm, which is the distance of next neighboured Si atoms in c-Si. Although no significant change of the lattice constant of the surrounding c-Si matrix was assumed, surprisingly there is no change at all within observational accuracy. -Each side length and the total volume of the simulation box is increased by 0.4 \% and 1.2 \% respectively of the initial state. -Indeed an increase of the total volume is expected due to the slightly lower Si density of 3C-SiC compared to c-Si. -The expected increase in volume can be calculated by +Looking closer at higher order Si-Si peaks might even allow the guess of a slight increase of the lattice constant compared to the plain c-Si structure. +A new Si-Si peak arises at 0.307 nm, which is identical to the peak of the C-C distribution around that value. +It corresponds to second next neighbours in 3C-SiC, which applies for Si as well as C pairs. +The bumps of the Si-Si distribution at higher distances marked by the green arrows can be explained in the same manner. +They correspond to the fourth and sixth next neighbour distance in 3C-SiC. +It is easily identifiable how these C-C peaks, which imply Si pairs at same distances inside the precipitate, contribute to the bumps observed in the Si-Si distribution. +The Si-Si and C-C peak at 0.307 nm enables the determination of the lattic constant of the embedded 3C-SiC precipitate. +A lattice constant of 4.34 \AA{} compared to 4.36 \AA{} for bulk 3C-SiC is obtained. +This is in accordance with the peak of Si-C pairs at a distance of 0.188 nm. +Thus, the precipitate structure is slightly compressed compared to the bulk phase. +This is a quite surprising result since due to the finite size of the c-Si surrounding a non-negligible impact of the precipitate on the materializing c-Si lattice constant especially near the precipitate could be assumed. +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 4.31 \AA{} in the beginning up to 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} -I_V=\frac{N^{\text{c-Si}}_{\text{Si}}/n_{\text{Si}}^{\text{c-Si}}+ - N^{\text{3C-SiC}}_{\text{Si}}/n_{\text{Si}}^{\text{3C-SiC}}} - {N^{\text{c-Si and 3C-SiC}}_{\text{Si}}/n_{\text{Si}}^{\text{c-Si}}} + \frac{V}{V_0}= + \frac{\frac{N^{\text{c-Si}}_{\text{Si}}}{8/a_{\text{c-Si of precipitate configuration}}}+ + \frac{N^{\text{3C-SiC}}_{\text{Si}}}{4/a_{\text{3C-SiC of precipitate configuration}}}} + {\frac{N^{\text{total}}_{\text{Si}}}{8/a_{\text{plain c-Si}}}} \end{equation} -with $N_{\text{Si}}$ and $n_{\text{Si}}$ being the number of Si atoms and the Si density respectively of the corresponding material. -Due to a slightly lower Si density of 3C-SiC compared to c-Si an increase of x \% of the total volume would be expected for precipitate with a radius of 3 nm embedded in - -Calc expected increase due to Si density mismatch ... -Obviously the surrounding matrix is chosen big enough to exclude size effects ... -Nice, since obviously matrix is big enough to exclude size effects in the system in which pbc are applied, we can consider it single precipitate in a infinite Si matrix. -A new peak for the silicon pairs arises at 0.307 nm. -It is identical to the peak of the C-C distribution around that value. -It corresponds to second next neighbours in 3C-SiC, which applies for Si as well as C pairs. -The bumps of the Si-Si distribution at higher distances, which are marked by green arrows and do not exist in plain c-Si, can be explained in the same manner. -They correspond to the fourth and sixth next neighbour in 3C-SiC. -Again, these peaks apply to Si and C pairs and indeed it is easily identifiale how the C-C peaks at contribute to the bumps observed in the Si-Si distribution. - -4.34 \AA{} compared to 4.36 \AA{}. +with the notation used in table \ref{table:md:sic_prec}. +The lattice constant of plain c-Si at $20\,^{\circ}\mathrm{C}$ can be determined more accurately by the side lengthes of the simulation box of an equlibrated structure instead of using the radial distribution data. +By this a value of $a_{\text{plain c-Si}}=5.439\text{ \AA}$ is obtained. +The same lattice constant is assumed for the c-Si surrounding in the precipitate configuration $a_{\text{c-Si of precipitate configuration}}$ since peaks in the radial distribution match the ones of plain c-Si. +Using $a_{\text{3C-SiC of precipitate configuration}}=4.34\text{ \AA}$ as observed from the radial distribution finally results in an increase of the initial volume by 0.12 \%. +However, each side length and the total volume of the simulation box is increased by 0.20 \% and 0.61 \% respectively compared to plain c-Si at $20\,^{\circ}\mathrm{C}$. +Since the c-Si surrounding resides in an uncompressed state the excess increase must be attributed to relaxation of strain with the strain resulting from either the compressed precipitate or the 3C-SiC/c-Si interface region. +This also explains the possibly identified slight increase of the c-Si lattice constant in the surrounding as mentioned earlier. +As the pressure is set to zero the free energy is minimized with respect to the volume enabled by the Berendsen barostat algorithm. +Apparently the minimized structure with respect to the volume is a configuration of a small compressively stressed precipitate and a large amount of slightly stretched c-Si in the surrounding. + +In the following the 3C-SiC/c-Si interface is described in further detail. +One important size analyzing the interface is the interfacial energy. +It is determined exactly in the same way than the formation energy as described in equation \eqref{eq:defects:ef2}. +Using the notation of table \ref{table:md:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by +\begin{equation} + E_{\text{f}}=E- + N^{\text{3C-SiC}}_{\text{C}} \mu_{\text{SiC}}- + \left(N^{\text{total}}_{\text{Si}}-N^{\text{3C-SiC}}_{\text{C}}\right) + \mu_{\text{Si}} \text{ ,} +\label{eq:md:ife} +\end{equation} +with $E$ being the free energy of the precipitate configuration at zero temperature. +An interfacial energy of 2267.28 eV is obtained. +The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of 29.93 \AA. +Thus, the interface tension, given by the energy of the interface devided by the surface area of the precipitate is $20.15\,\frac{\text{eV}}{\text{nm}^2}$ or $3.23\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$. +This is located inside the eperimentally estimated range of $2-8\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$ \cite{taylor93}. -New lattice constant -Surface energy -Now let's see, whether annealing will lead to some energetically more favorable configurations. +Since interface region is constructed and not neccesarily corresponds to the energetically most favorable layout we will now try hard to improve this ... +Let's see, whether annealing will lead to some energetically more favorable configurations. \subsection{Simulations at temperatures exceeding the silicon melting point} @@ -514,5 +542,11 @@ LL Cool J is hot as hell! A different simulation volume and refined amount as well as shape of insertion volume for the C atoms, to stay compareable to the results gained in the latter section, is used throughout all following simulations. +\subsection{Todo} + +{\color{red}TODO: self-guided MD!} + +{\color{red}TODO: other approaches!} + {\color{red}TODO: ART MD?}