From: hackbard <hackbard@sage.physik.uni-augsburg.de>
Date: Mon, 15 Mar 2010 16:21:32 +0000 (+0100)
Subject: 110 db interaction finished
X-Git-Url: https://hackdaworld.org/gitweb/?a=commitdiff_plain;h=f3e46ec7d389cf55550c9af20acbc81285f14f94;p=lectures%2Flatex.git

110 db interaction finished
---

diff --git a/posic/thesis/defects.tex b/posic/thesis/defects.tex
index a639609..d5afa44 100644
--- a/posic/thesis/defects.tex
+++ b/posic/thesis/defects.tex
@@ -16,7 +16,7 @@ The cell volume and shape is allowed to change using the pressure control algori
 Periodic boundary conditions in each direction are applied.
 All point defects are calculated for the neutral charge state.
 
-\begin{figure}[h]
+\begin{figure}[th]
 \begin{center}
 \includegraphics[width=9cm]{unit_cell_e.eps}
 \end{center}
@@ -60,7 +60,7 @@ Point defects in silicon have been extensively studied, both experimentally and
 Quantum-mechanical total-energy calculations are an invalueable tool to investigate the energetic and structural properties of point defects since they are experimentally difficult to assess.
 
 The formation energies of some of the silicon self-interstitial configurations are listed in table \ref{tab:defects:si_self} for both methods used in this work as well as results obtained by former studies \cite{leung99}.
-\begin{table}[h]
+\begin{table}[th]
 \begin{center}
 \begin{tabular}{l c c c c c}
 \hline
@@ -79,7 +79,7 @@ The formation energies of some of the silicon self-interstitial configurations a
 \label{tab:defects:si_self}
 \end{table}
 The final configurations obtained after relaxation are presented in figure \ref{fig:defects:conf}.
-\begin{figure}[h]
+\begin{figure}[t!h!]
 \begin{center}
 %\hrule
 %\vspace*{0.2cm}
@@ -176,7 +176,7 @@ The Si interstitial atom then begins to slowly move towards an energetically mor
 The formation energy of 3.96 eV for this type of interstitial is equal to the result for the hexagonal one in the original work \cite{albe_sic_pot}.
 Obviously the authors did not carefully check the relaxed results assuming a hexagonal configuration.
 In figure \ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
-\begin{figure}[h]
+\begin{figure}[th]
 \begin{center}
 \includegraphics[width=10cm]{e_kin_si_hex.ps}
 \end{center}
@@ -188,7 +188,7 @@ The same type of interstitial arises using random insertions.
 In addition, variations exist in which the displacement is only along two \hkl<1 0 0> axes ($E_\text{f}=3.8\text{ eV}$) or along a single \hkl<1 0 0> axes ($E_\text{f}=3.6\text{ eV}$) successively approximating the tetdrahedral configuration and formation energy.
 The existence of these local minima located near the tetrahedral configuration seems to be an artifact of the analytical potential without physical authenticity revealing basic problems of analytical potential models for describing defect structures.
 However, the energy barrier is small.
-Todo: Check!
+{\color{red}Todo: Check!}
 Hence these artifacts should have a negligent influence in finite temperature simulations.
 
 The bond-centered configuration is unstable and the \hkl<1 0 0> dumbbell interstitial is the most unfavorable configuration for both, the Erhard/Albe and VASP calculations.
@@ -199,7 +199,7 @@ The length of these bonds are, however, close to the cutoff range and thus are w
 The same applies to the bonds between the interstitial and the upper two atoms in the \hkl<1 1 0> dumbbell configuration.
 
 A more detailed description of the chemical bonding is achieved by quantum-mechanical calculations by investigating the accumulation of negative charge between the nuclei.
-Todo: Plot the electron density for these types of defect to derive conclusions of existing bonds?
+{\color{red}Todo: Plot the electron density for these types of defect to derive conclusions of existing bonds?}
 
 \section{Carbon related point defects}
 
@@ -212,7 +212,7 @@ Formation energies of the most common carbon point defects in crystalline silico
 The type of reservoir of the carbon impurity to determine the formation energy of the defect was chosen to be SiC.
 This is consistent with the methods used in the articles \cite{tersoff90,dal_pino93}, which the results are compared to in the following.
 Hence, the chemical potential of silicon and carbon is determined by the cohesive energy of silicon and silicon carbide.
-\begin{table}[h]
+\begin{table}[th]
 \begin{center}
 \begin{tabular}{l c c c c c c}
 \hline
@@ -231,7 +231,7 @@ Hence, the chemical potential of silicon and carbon is determined by the cohesiv
 \caption[Formation energies of carbon point defects in crystalline silicon determined by classical potential molecular dynamics and density functional calculations.]{Formation energies of carbon point defects in crystalline silicon determined by classical potential molecular dynamics and density functional calculations. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal, B the bond-centered and S the substitutional interstitial configuration. The dumbbell configurations are abbreviated by DB.  Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
 \label{tab:defects:c_ints}
 \end{table}
-\begin{figure}[h]
+\begin{figure}[th]
 \begin{center}
 \begin{flushleft}
 \begin{minipage}{4cm}
@@ -331,7 +331,7 @@ It was first identified by infra-red (IR) spectroscopy \cite{bean70} and later o
 
 Figure \ref{fig:defects:100db_cmp} schematically shows the \hkl<1 0 0> dumbbell structure and table \ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by analytical potential and quantum-mechanical calculations.
 For comparison, the obtained structures for both methods visualized out of the atomic position data are presented in figure \ref{fig:defects:100db_vis_cmp}.
-\begin{figure}[h]
+\begin{figure}[th]
 \begin{center}
 \includegraphics[width=12cm]{100-c-si-db_cmp.eps}
 \end{center}
@@ -339,7 +339,7 @@ For comparison, the obtained structures for both methods visualized out of the a
 \label{fig:defects:100db_cmp}
 \end{figure}
 %
-\begin{table}[h]
+\begin{table}[th]
 \begin{center}
 Displacements\\
 \begin{tabular}{l c c c c c c c c c}
@@ -383,7 +383,7 @@ VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
 \caption[Atomic displacements, distances and bond angles of the \hkl<1 0 0> dumbbell structure obtained by the Erhard/Albe potential and VASP calculations.]{Atomic displacements, distances and bond angles of the \hkl<1 0 0> dumbbell structure obtained by the Erhard/Albe potential and VASP calculations. The displacements and distances are given in nm and the angles are given in degrees. Displacements, distances and angles are schematically displayed in figure \ref{fig:defects:100db_cmp}. In addition, the equilibrium lattice constant for crystalline silicon is listed.}
 \label{tab:defects:100db_cmp}
 \end{table}
-\begin{figure}[h]
+\begin{figure}[t!h!]
 \begin{center}
 \begin{minipage}{6cm}
 \begin{center}
@@ -401,7 +401,7 @@ VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
 \caption{Comparison of the visualized \hkl<1 0 0> dumbbel structures obtained by Erhard/Albe potential and VASP calculations.}
 \label{fig:defects:100db_vis_cmp}
 \end{figure}
-\begin{figure}[h]
+\begin{figure}[th]
 \begin{center}
 \includegraphics[height=10cm]{c_pd_vasp/eden.eps}
 \includegraphics[height=12cm]{c_pd_vasp/100_2333_ksl.ps}
@@ -432,7 +432,7 @@ An acceptor level arises at approximately $E_v+0.35\text{ eV}$ while a band gap
 \subsection{Bond-centered interstitial configuration}
 \label{subsection:bc}
 
-\begin{figure}[h]
+\begin{figure}[t!h!]
 \begin{center}
 \begin{minipage}{8cm}
 \includegraphics[width=8cm]{c_pd_vasp/bc_2333.eps}\\
@@ -470,7 +470,7 @@ In addition, the energy level diagram shows a net amount of two spin up electron
 In the following the problem of interstitial carbon migration in silicon is considered.
 Since the carbon \hkl<1 0 0> dumbbell interstitial is the most probable hence most important configuration the migration simulations focus on this defect.
 
-\begin{figure}[h]
+\begin{figure}[t!h!]
 \begin{center}
 \begin{minipage}{15cm}
 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
@@ -548,9 +548,7 @@ As a last migration path, the defect is only changing its orientation.
 Thus, it is not responsible for long-range migration.
 The silicon dumbbell partner remains the same.
 The bond to the face-centered silicon atom at the bottom of the unit cell breaks and a new one is formed to the face-centered atom at the forefront of the unit cell.
-Todo: \hkl<1 1 0> to \hkl<1 0 0> and bond-centerd configuration (in progress).
-Todo: \hkl<1 1 0> to \hkl<0 -1 0> (rotation of the DB, in progress).
-Todo: Comparison with classical potential simulations or explanation to only focus on ab initio calculations.
+{\color{red}Todo: Comparison with classical potential simulations or explanation to only focus on ab initio calculations.}
 
 Since the starting and final structure, which are both local minima of the potential energy surface, are known, the aim is to find the minimum energy path from one local minimum to the other one.
 One method to find a minimum energy path is to move the diffusing atom stepwise from the starting to the final position and only allow relaxation in the plane perpendicular to the direction of the vector connecting its starting and final position.
@@ -564,9 +562,9 @@ In this new method all atoms are stepwise displaced towards their final position
 Relaxation of each individual atom is only allowed in the plane perpendicular to the last individual displacement vector.
 The modifications used to add this feature to the VASP code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
 Due to these constraints obtained activation energies can effectively be higher.
-Todo: To refine the migration barrier one has to find the saddle point structure and recalculate the free energy of this configuration with a reduced set of constraints.
+{\color{red}Todo: To refine the migration barrier one has to find the saddle point structure and recalculate the free energy of this configuration with a reduced set of constraints.}
 
-\begin{figure}[h]
+\begin{figure}[t!h!]
 \begin{center}
 \includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[1.5cm]
 \begin{picture}(0,0)(150,0)
@@ -593,7 +591,7 @@ To reach the bond-centered configuration, which is 0.94 eV higher in energy than
 This amount of energy is needed to break the bond of the carbon atom to the silicon atom at the bottom left.
 In a second process 0.25 eV of energy are needed for the system to revert into a \hkl<1 0 0> configuration.
 
-\begin{figure}[h]
+\begin{figure}[t!h!]
 \begin{center}
 \includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[1.6cm]
 \begin{picture}(0,0)(140,0)
@@ -618,7 +616,7 @@ In a second process 0.25 eV of energy are needed for the system to revert into a
 Figure \ref{fig:defects:00-1_0-10_mig} shows the migration barrier and structures of the \hkl<0 0 -1> to \hkl<0 -1 0> dumbbell transition.
 The resulting migration barrier of approximately 0.9 eV is very close to the experimentally obtained values of 0.73 \cite{song90} and 0.87 eV \cite{tipping87}.
 
-\begin{figure}[h]
+\begin{figure}[t!h!]
 \begin{center}
 \includegraphics[width=13cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[1.8cm]
 \begin{picture}(0,0)(140,0)
@@ -657,7 +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}[h]
+\begin{figure}[th]
 \begin{center}
 \begin{minipage}{7.5cm}
 \includegraphics[width=7cm]{comb_pos.eps}
@@ -686,7 +684,7 @@ Relative silicon neighbour positions:
 \caption[\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighboured silicon atoms used for the second defect.]{\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighboured silicon atoms used for the second defect. Two possibilities exist for red numbered atoms and four possibilities exist for blue numbered atoms.}
 \label{fig:defects:pos_of_comb}
 \end{figure}
-\begin{table}[h]
+\begin{table}[th]
 \begin{center}
 \begin{tabular}{l c c c c c c}
 \hline
@@ -734,7 +732,7 @@ Investigating the first part of table \ref{tab:defects:e_of_comb}, namely the co
 Surprisingly the most favorable configurations are the ones with the second defect created at the very next silicon neighbour and a change in orientation compared to the initial one.
 This leads to the conclusion that an agglomeration of C-Si dumbbell interstitials as proposed by the precipitation model introduced in section \ref{section:assumed_prec} is indeed an energetically favored configuration of the system.
 The reason for nearby interstitials being favored compared to isolated ones is most probably the reduction of strain energy enabled by combination in contrast to the strain energy created by two individual defects.
-\begin{figure}[h]
+\begin{figure}[t!h!]
 \begin{center}
 \begin{minipage}[t]{7cm}
 a) \underline{$E_{\text{b}}=-2.25\text{ eV}$}
@@ -759,16 +757,16 @@ After relaxation the initial configuration is still evident.
 As expected by the initialization conditions the two carbon atoms form a bond.
 This bond has a length of 1.38 \AA{} close to the nex neighbour distance in diamond or graphite, which is approximately 1.54 \AA.
 The minimum of binding energy observed for this configuration suggests prefered C clustering as a competing mechnism to the C-Si dumbbell interstitial agglomeration inevitable for the SiC precipitation.
-Todo: Activation energy to obtain a configuration of separated C atoms again or vice versa to obtain this configuration from separated C confs?
+{\color{red}Todo: Activation energy to obtain a configuration of separated C atoms again or vice versa to obtain this configuration from separated C confs?}
 However, for the second most favorable configuration, presented in figure \ref{fig:defects:comb_db_01} a), the amount of possibilities for this configuration is twice as high.
 In this configuration the initial Si (I) and C (I) dumbbell atoms are displaced along \hkl<1 0 0> and \hkl<-1 0 0> in such a way that the Si atom is forming tetrahedral bonds with two silicon and two carbon atoms.
 The carbon and silicon atom constituting the second defect are as well displaced in such a way, that the carbon atom forms tetrahedral bonds with four silicon neighbours, a configuration expected in silicon carbide.
 The two carbon atoms spaced by 2.70 \AA{} do not form a bond but anyhow reside in a shorter distance as expected in silicon carbide.
 The Si atom numbered 2 is pushed towards the carbon atom, which results in the breaking of the bond to atom 4.
 The breaking of the $\sigma$ bond is indeed confirmed by investigating the charge density isosurface of this configuration.
-Todo: Is this conf really benificial for SiC prec?
+{\color{red}Todo: Is this conf really benificial for SiC prec?}
 
-\begin{figure}[h]
+\begin{figure}[t!h!]
 \begin{center}
 \begin{minipage}[t]{5cm}
 a) \underline{$E_{\text{b}}=-2.16\text{ eV}$}
@@ -826,7 +824,7 @@ Both configurations are unfavorable compared to far-off isolated dumbbells.
 Nonparallel orientations, that is the \hkl<0 1 0>, \hkl<0 -1 0> and its equivalents, result in binding energies of -0.12 eV and -0.27 eV, thus, constituting energetically favorable configurations.
 The reduction of strain energy is higher in the second case where the carbon atom of the second dumbbell is placed in the direction pointing away from the initial carbon atom.
 
-\begin{figure}[h]
+\begin{figure}[t!h!]
 \begin{center}
 \begin{minipage}[t]{7cm}
 a) \underline{$E_{\text{b}}=-1.53\text{ eV}$}
@@ -866,33 +864,46 @@ The symmetric configuration is energetically more favorable ($E_{\text{b}}=-1.66
 In figure \ref{fig:defects:comb_db_05} c) and d) the nonparallel orientations, namely the \hkl<0 -1 0> and \hkl<1 0 0> dumbbells are shown.
 Binding energies of -1.88 eV and -1.38 eV are obtained for the relaxed structures.
 In both cases the silicon atom of the initial interstitial is pulled towards the near by atom of the second dumbbell so that both atoms form fourfold coordinated bonds to their next neighbours.
-In case c) it is the carbon and in case d) the silicon atom of the second interstitial to form the additional bond with the silicon atom of the initial interstitial.
+In case c) it is the carbon and in case d) the silicon atom of the second interstitial which forms the additional bond with the silicon atom of the initial interstitial.
 The atom of the second dumbbell, the carbon atom of the initial dumbbell and the two interconnecting silicon atoms again reside in a plane.
 A typical C-C distance of 2.79 \AA{} is, thus, observed for case c).
 The far-off atom of the second dumbbell resides in threefold coordination.
 
-Assuming that it is possible for the system to minimize free energy by reorientation of the dumbbell in any position ... we now give the minimum energies of dumbbells alomg the \hkl<1 1 0> direction ...
-\begin{table}[h]
+Assuming that it is possible for the system to minimize free energy by an in place reorientation of the dumbbell at any position the minimum energy orientation of dumbbells along the \hkl<1 1 0> direction and the resulting C-C distance is shown in table \ref{tab:defects:comb_db110}.
+\begin{table}[t!h!]
 \begin{center}
 \begin{tabular}{l c c c c c c}
 \hline
 \hline
  & 1 & 2 & 3 & 4 & 5 & 6\\
 \hline
-$E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & - & - \\
-Type & \hkl<-1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<> & \hkl<> \\
+$E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
+C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
+Type & \hkl<-1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0>, \hkl<0 -1 0>\\
 \hline
 \hline
 \end{tabular}
 \end{center}
-\caption{Binding energy and type of the minimum energy configuration of an additional dumbbell with respect to the separation distance in bonds along the \hkl<1 1 0> direction.}
+\caption{Binding energy and type of the minimum energy configuration of an additional dumbbell with respect to the separation distance in bonds along the \hkl<1 1 0> direction and the C-C distance.}
 \label{tab:defects:comb_db110}
 \end{table}
+\begin{figure}[t!h!]
+\begin{center}
+\includegraphics[width=12.5cm]{db_along_110.ps}\\
+\includegraphics[width=12.5cm]{db_along_110_cc.ps}
+\end{center}
+\caption{Minimum binding energy of dumbbell combinations with respect to the separation distance in bonds along \hkl<1 1 0> and C-C distance.}
+\label{fig:defects:comb_db110}
+\end{figure}
+Figure \ref{fig:defects:comb_db110} shows the corresponding plot of the data including a cubic spline interplation and a suitable fitting curve.
+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 was ignored in the fitting procedure.
 
-Todo: DB mig along 110?
+{\color{red}Todo: DB mig along 110?}
 
-Todo: Si int and C sub ...
+{\color{red}Todo: Si int and C sub ...}
 
-Todo: Model of kick-out and kick-in mechnism?
+{\color{red}Todo: Model of kick-out and kick-in mechnism?}
 
-Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities! :(
+{\color{red}Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities! :(}