X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=posic%2Fthesis%2Fdefects.tex;h=756c45ba778f9c794cde539695bf842051a637ab;hb=886e55594bdd8a1a17ba824f3c1371e5b9709088;hp=b5d42f54c514ffd952d18cca53d1ae790b977127;hpb=68da9886cd4a86071560b5de2d16376bb1f17b37;p=lectures%2Flatex.git

diff --git a/posic/thesis/defects.tex b/posic/thesis/defects.tex
index b5d42f5..756c45b 100644
--- a/posic/thesis/defects.tex
+++ b/posic/thesis/defects.tex
@@ -655,6 +655,7 @@ The focus is on combinations of the \hkl<0 0 -1> dumbbell interstitial with a se
 The second defect is either another \hkl<1 0 0>-type interstitial occupying different orientations, a vacany or a substitutional carbon atom.
 Several distances of the two defects are examined.
 Investigations are restricted to quantum-mechanical calculations.
+
 \begin{figure}[th]
 \begin{center}
 \begin{minipage}{7.5cm}
@@ -722,7 +723,7 @@ For defects far away from each other the formation energy of the defect combinat
 Thus, $E_{\text{b}}$ can be best thought of a binding energy, which is required to bring the defects to infinite separation.
 In fact, a \hkl<0 0 -1> dumbbell interstitial created at position R with a distance of $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ ($\approx 12.8$ \AA) from the initial one results in an energy as low as -0.19 eV.
 There is still a low interaction which is due to the equal orientation of the defects.
-By changing the orientation of the second dumbbell interstitial to the \hkl<0 -1 0>-type the interaction is even mor reduced resulting in an energy of $E_{\text{b}}=-0.05\text{ eV}$ for a distance, which is the maximum that can be realized due to periodic boundary conditions.
+By changing the orientation of the second dumbbell interstitial to the \hkl<0 -1 0>-type the interaction is even more reduced resulting in an energy of $E_{\text{b}}=-0.05\text{ eV}$ for a distance, which is the maximum that can be realized due to periodic boundary conditions.
 The energies obtained in the R column of table \ref{eq:defects:e_of_comb} are used as a reference to identify, whether less distanced defects of the same type are favorable or unfavorable compared to the far-off located defect.
 Configurations wih energies greater than zero or the reference value are energetically unfavorable and expose a repulsive interaction.
 These configurations are unlikely to arise or to persist for non-zero temperatures.
@@ -801,7 +802,7 @@ Figure \ref{fig:defects:comb_db_02} c) displays the results of another \hkl<0 0
 The binding energy is -2.05 eV.
 Both dumbbells are tilted along the same direction remaining parallely aligned and the second dumbbell is pushed downwards in such a way, that the four dumbbell atoms form a rhomboid.
 Both carbon atoms form tetrahedral bonds to four silicon atoms.
-However, silicon atom 1 and 3, which are bond to the second carbon dumbbell interstitial are also bond to the initial carbon atom.
+However, silicon atom 1 and 3, which are bound to the second carbon dumbbell interstitial are also bound to the initial carbon atom.
 These four atoms of the rhomboid reside in a plane and, thus, do not match the situation in silicon carbide.
 The carbon atoms have a distance of 2.75 \AA.
 In figure \ref{fig:defects:comb_db_02} b) a second \hkl<0 1 0> dumbbell is constructed at position 2.
@@ -899,7 +900,6 @@ Figure \ref{fig:defects:comb_db110} shows the corresponding plot of the data inc
 The funtion found most suitable for curve fitting is $f(x)=a/x^3$ comprising the single fit parameter $a$.
 Thus, far-off located dumbbells show an interaction proportional to the reciprocal cube of the distance and the amount of bonds along \hkl<1 1 0> respectively.
 This behavior is no longer valid for the immediate vicinity revealed by the saturating binding energy of a second dumbbell at position 1, which is ignored in the fitting procedure.
-{\color{red}Todo: DB mig along 110?}
 
 \begin{figure}[t!h!]
 \begin{center}
@@ -968,62 +968,110 @@ On the other hand stretched silicon carbide is obtained by the transition of the
 \begin{figure}[t!h!]
 \begin{center}
 \begin{minipage}[t]{7cm}
-a) \underline{$E_{\text{b}}=-1.53\text{ eV}$}
+a) \underline{Pos: 2, $E_{\text{b}}=-0.59\text{ eV}$}
 \begin{center}
-\includegraphics[width=6.0cm]{00-1dc/1-53.eps}
+\includegraphics[width=6.0cm]{00-1dc/0-59.eps}
 \end{center}
 \end{minipage}
 \begin{minipage}[t]{7cm}
-b) \underline{$E_{\text{b}}=-1.66\text{ eV}$}
+b) \underline{Pos: 3, $E_{\text{b}}=-3.14\text{ eV}$}
 \begin{center}
-\includegraphics[width=6.0cm]{00-1dc/1-66.eps}
+\includegraphics[width=6.0cm]{00-1dc/3-14.eps}
 \end{center}
 \end{minipage}\\[0.2cm]
 \begin{minipage}[t]{7cm}
-c) \underline{$E_{\text{b}}=-1.88\text{ eV}$}
+c) \underline{Pos: 4, $E_{\text{b}}=-0.54\text{ eV}$}
 \begin{center}
-\includegraphics[width=6.0cm]{00-1dc/1-88.eps}
+\includegraphics[width=6.0cm]{00-1dc/0-54.eps}
 \end{center}
 \end{minipage}
 \begin{minipage}[t]{7cm}
-d) \underline{$E_{\text{b}}=-1.38\text{ eV}$}
+d) \underline{Pos: 5, $E_{\text{b}}=-0.50\text{ eV}$}
 \begin{center}
-\includegraphics[width=6.0cm]{00-1dc/1-38.eps}
+\includegraphics[width=6.0cm]{00-1dc/0-50.eps}
 \end{center}
 \end{minipage}
 \end{center}
 \caption{Relaxed structures of defect complexes obtained by creating vacancies at positions 2 (a)), 3 (b)), 4 (c)) and 5 (d)).}
-\label{fig:defects:comb_db_03}
+\label{fig:defects:comb_db_06}
+\end{figure}
+Figure \ref{fig:defects:comb_db_06} displays relaxed structures of vacancies in combination with the \hkl<0 0 -1> dumbbell interstital.
+The creation of a vacancy at position 1 results in a configuration of substitutional carbon on a silicon lattice site and no other remaining defects.
+The carbon dumbbell atom moves to position 1 where the vacancy is created and the silicon dumbbell atom recaptures the dumbbell lattice site.
+With a binding energy of -5.39 eV, this is the energetically most favorable configuration observed.
+A great amount of strain energy is reduced by removing the silicon atom at position 3, which is illustrated in figure \ref{fig:defects:comb_db_06} b).
+The dumbbell structure shifts towards the position of the vacancy which replaces the silicon atom usually bound to and at the same time strained by the silicon dumbbell atom.
+Due to the displacement into the \hkl<1 -1 0> direction the bond of the dumbbell silicon atom to the silicon atom on the top left breaks and instead forms a bond to the silicon atom located in \hkl<1 -1 1> direction which is not shown in the figure.
+A binding energy of -3.14 eV is obtained for this structure composing another energetically favorable configuration.
+A vacancy ctreated at position 2 enables a relaxation of the silicon atom number 1 mainly in \hkl<0 0 -1> direction.
+The bond to silicon atom number 5 breaks.
+Hence, the silicon dumbbell atom is not only displaced along \hkl<0 0 -1> but also and to a greater extent in \hkl<1 1 0> direction.
+The carbon atom is slightly displaced in \hkl<0 1 -1> direction.
+A binding energy of -0.59 eV indicates the occurrence of much less strain reduction compared to that in the latter configuration.
+Evidently this is due to a smaller displacement of silicon atom number 1, which would be directly bound to the replaced silicon atom at position 2.
+In the case of a vacancy created at position 4, even a slightly higher binding energy of -0.54 eV is observed, while the silicon atom at the bottom left, which is bound to the carbon dumbbell atom, is vastly displaced along \hkl<1 0 -1>.
+However the displacement of the carbon atom along \hkl<0 0 -1> is less than it is in the preceding configuration.
+Although expected due to the symmetric initial configuration silicon atom number 1 is not displaced correspondingly and also the silicon dumbbell atom is displaced to a greater extent in \hkl<-1 0 0> than in \hkl<0 -1 0> direction.
+The symmetric configuration is, thus, assumed to constitute a local maximum, which is driven into the present state by the conjugate gradient method used for relaxation.
+Figure \ref{fig:defects:comb_db_06} d) shows the relaxed structure of a vacancy created at position 5.
+The silicon dumbbell atom is largely displaced along \hkl<1 1 0> and somewaht less along \hkl<0 0 -1>, which corresponds to the direction towards the vacancy.
+The silicon dumbbell atom approaches silicon number 1.
+Indeed a non-zero charge density is observed inbetween these two atoms exhibiting a cylinder-like shape superposed with the charge density known from the dumbbell itself.
+Strain reduced by this huge displacement is partially absorbed by tensile strain on silicon atom number 1 originating from attractive forces of the carbon atom and the vacancy.
+A binding energy of -0.50 eV is observed.
+{\color{red}Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities. Due to the initial defect, symmetries are broken. The system should have relaxed into the minumum energy configuration!?}
+
+{\color{blue}Todo: Si int + vac and C sub ...?
+Investigation of vacancy, Si and C interstitital.
+As for the ground state of the single Si self-int, a 110 is also assumed as the lowest possibility in combination with other defects (which is a cruel assumption)!
+}
+
+\section{Migration in systems of combined defects}
+
+During carbon implantation into crystalline silicon the energetic carbon atoms may kick out silicon atoms from their lattice sites.
+A vacancy accompanied by a silicon self-interstitial is generated.
+The silicon self-interstitial may migrate to the surface or recombine with other vacancies.
+Once a vacancy and a carbon interstitial defect exist the energetically most favorable configuration is the configuration of a substitutional carbon atom, that is the carbon atom occupying the vacant site.
+In addition, it is a conceivable configuration the system might experience during the silicon carbide precipitation process.
+Energies needed to overcome the migration barrier of the transformation into this configuration enable predictions concerning the feasibility of a silicon carbide conversion mechanism derived from these microscopic processes.
+This is especially important for the case, in which the vacancy is created at position 3, as discussed in the last section and figure \ref{fig:defects:comb_db_06} b).
+Due to the low binding energy this configuration might constitute a trap, which it is hard to escape from.
+However, migration simulations show that only a low amount of energy is necessary to transform the system into the energetically most favorable configuration.
+\begin{figure}[!t!h]
+\begin{center}
+\includegraphics[width=13cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[2.0cm]
+\begin{picture}(0,0)(170,0)
+\includegraphics[width=3.5cm]{vasp_mig/comb_2-1_init.eps}
+\end{picture}
+\begin{picture}(0,0)(60,0)
+\includegraphics[width=3.5cm]{vasp_mig/comb_2-1_seq.eps}
+\end{picture}
+\begin{picture}(0,0)(-120,0)
+\includegraphics[width=3.5cm]{vasp_mig/comb_2-1_final.eps}
+\end{picture}
+\begin{picture}(0,0)(25,20)
+\includegraphics[width=2.5cm]{100_arrow.eps}
+\end{picture}
+\begin{picture}(0,0)(230,0)
+\includegraphics[height=2.2cm]{001_arrow.eps}
+\end{picture}
+\end{center}
+\caption{Transition vacancy-interstitial combinations into the configuration of substitutional carbon.}
+\label{fig:defects:comb_mig_01}
 \end{figure}
-The creation of the vacancy at position 1 ... c interstitital moves to acancy position ending up in a configuration of a substitutional carbon which explains the highbinding energy.
-At position 3 a great amount of strain energy is reduced, since the the vacancy replaces a silicon atom usually bond to and thus starined by the silicon dumbbell atom.
-db moves towards the vacancy in \hkl<1 -1 0> direction.
-Vac at position 2 and 4 have similar results.
-Less strain is reduced, since the displacement of the bottom silicon atom, whcih would be directly bond to the silicon atom replaced by the vacancy, is less.
-In the second case, there is even less strain reduction since the second next neighbour is replaced by the vacancy.
-A symmetric configuration is expected, but it is not!
-jahn-Teller distortion ... check this!
-In both cases the db is tilted in such a way, that the carbon atom moves towards the vacancy.
-At position 5 the silicon dumbbell atom moves in \hkl<1 1 0> direction, the same direction where the vacancy is located.
-Strain reducde by this is partialy absorbed by strain originating from the fact that si atom bound to and pulled by the carbon atom is also pulled by the vacancy.
+Figure \ref{fig:defects:comb_mig_01} shows the migration barriers and structures for transitions of the vacancy-interstitial configurations examined in figure \ref{fig:defects:comb_db_06} a) and b) into the configuration of substitutional carbon.
+
 
-CHECK C-C DIST AND SI-C DIST !!! of all!!!
 
-{\color{red}Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities? Due to the initial defect symmetries are broken. It should have relaxed into the minumum energy configuration!?}
-Once a vacancy exists the minimal e conf is the c sub conf and ofcourse necessary for formation of SiC.
-The question is whether the migration into this conf is possible.
-Fig shows the migration of the 2 and 3 conf into the c sub conf.
 Low migration barriers, which means that SiC will modt probably form ... and so on ...
 
-{\color{red}Todo: Si int and C sub ...}
-The existance of a vacancy is most often accompanied by an interstitial.
-The silicon interstitital might diffuse to the surface or recombine with other vacancy defects and tus is out of the interested simulation region.
-However, investigation of near by vacancy, Si and C interstititla is necessary, too.
-As for the ground state of the single Si self-int a 110 this is also assumed as the lowest possibility in combination with other defects, which is a cruel assumption!!!
+
+{\color{red}Todo: DB mig along 110 (at the starting of this section)?}
+
+{\color{red}Todo: Migration of Si int + vac and C sub ...?}
 
 {\color{red}Todo: Model of kick-out and kick-in mechnism?}
 
 
-\section{Summary}
-...
+\section{Conclusions for SiC preciptation}