Just as for the Si self-interstitial a carbon \hkl<1 1 0> dumbbell configuration exists.
For the Erhard/Albe potential the formation energy is situated in the same order as found by quantum-mechanical results.
-Similar structures arise in both types of simulations with the silicon and carbon atom sharing a silicon lattice site aligned along \hkl[1 1 0] where the carbon atom is localized slightly closer to the next nearest silicon atom located in the opposite direction to the site-sharing silicon atom even forming a bond to the next but one silicon atom in this direction.
+Similar structures arise in both types of simulations with the silicon and carbon atom sharing a silicon lattice site aligned along \hkl<1 1 0> where the carbon atom is localized slightly closer to the next nearest silicon atom located in the opposite direction to the site-sharing silicon atom even forming a bond to the next but one silicon atom in this direction.
The bond-centered configuration is unstable for the Erhard/Albe potential.
The system moves into the \hkl<1 1 0> interstitial configuration.
However, this fact could not be reproduced by spin polarized VASP calculations performed in this work.
Present results suggest this configuration to be a real local minimum.
In fact, an additional barrier has to be passed to reach this configuration starting from the \hkl<1 0 0> interstitital configuration, which is investigated in section \ref{subsection:100mig}.
-After slightly displacing the carbon atom along the \hkl<1 0 0> (equivalent to a displacement along \hkl<0 1 0>), \hkl<0 0 1> and \hkl<1 -1 0> direction the resulting structures relax back into the bond-centered configuration.
+After slightly displacing the carbon atom along the \hkl<1 0 0> (equivalent to a displacement along \hkl<0 1 0>), \hkl<0 0 1>, \hkl<0 0 -1> and \hkl<1 -1 0> direction the resulting structures relax back into the bond-centered configuration.
As we will see in later migration simulations the same would happen to structures where the carbon atom is displaced along the migration direction, which approximately is the \hkl<1 1 0> direction.
These relaxations indicate that the bond-cenetered configuration is a real local minimum instead of an assumed saddle point configuration.
Figure \ref{img:defects:bc_conf} shows the structure, the charge density isosurface and the Kohn-Sham levels of the bond-centered configuration.
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.
-There are different methods of computing migration paths and energies.
-Methods and shortcomings.
-
-Three different migration paths are accounted in this work.
-In the first path the carbon atom
-
-Results and comparison with diffusion experiments.
+\begin{figure}[h]
+\begin{center}
+\begin{minipage}{15cm}
+\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/bc_2333.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/100_next_2333.eps}
+\end{minipage}
+\end{minipage}\\
+\begin{minipage}{15cm}
+\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/00-1-0-10_2333.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/0-10_2333.eps}
+\end{minipage}
+\end{minipage}\\
+\begin{minipage}{15cm}
+\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/0-10_ip_2333.eps}
+\end{minipage}
+\end{minipage}
+\end{center}
+\caption{Migration pathways of the carbon \hkl<1 0 0> interstitial dumbbell in silicon.}
+\label{img:defects:c_mig_path}
+\end{figure}
+Three different migration paths are accounted in this work, which are shown in figure \ref{img:defects:c_mig_path}.
+The first migration investigated is a transition of a \hkl<0 0 -1> into a \hkl<0 0 1> dumbbell interstitial configuration.
+During this migration the carbon atom is changing its silicon dumbbell partner.
+The new partner is the one located at $\frac{a}{4}\hkl<1 1 -1>$ relative to the initial one.
+Two of the three bonds to the next neighboured silicon atoms are preserved while the breaking of the third bond and the accompanying formation of a new bond is observed.
+The carbon atom resides in the \hkl(1 1 0) plane.
+This transition involves an intermediate bond-centerd configuration.
+Due to symmetry it is enough to consider the transition from the bond-centered to the \hkl<1 0 0> configuration or vice versa.
+In the second path, the carbon atom is changing its silicon partner atom as in path one.
+However, the the trajectory of the carbon atom is no longer proceeding in the \hkl(1 1 0) plane.
+The orientation of the new dumbbell configuration is transformed from \hkl<0 0 -1> to \hkl<0 -1 0>.
+Again one bond is broken while another one is formed.
+As a last migration path, the defect is only changing its orientation.
+The silicon dumbbell partner remains.
+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.
+
+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.
+No constraints are applied to the remaining atoms in order to allow relaxation of the surrounding lattice.
+To prevent the remaining lattice to migrate according to the displacement of the defect an atom far away from the defect region is fixed in all three coordinate directions.
+However, it turned out, that this method tremendously failed applying it to the present migration pathways and structures.
+Abrupt changes in structure and free energy occured among relaxed structures of two successive displacement steps.
+For some structures even the expected final configurations were never obtained.
+Thus, the method mentioned above was adjusted adding further constraints in order to obtain smooth transitions, either in energy as well as structure is concerned.
+In this new method all atoms are stepwise displaced towards their final positions.
+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}.
+
+In figure ... results of the \hkl<0 0 -1> to \hkl<0 0 1> migration are presented.
\section{Combination of point defects}
projects / lectures / latex.git / blobdiff