si self ints ... more to come

diff --git a/posic/thesis/defects.tex b/posic/thesis/defects.tex
index 89bd382..3e3d8d2 100644 (file)
--- a/posic/thesis/defects.tex
+++ b/posic/thesis/defects.tex
@@ -14,6 +14,7 @@ Due to the restrictions in computer time three silicon lattice constants in each
 The ions are relaxed by a conjugate gradient method.
 The cell volume and shape is allowed to change using the pressure control algorithm of Parinello and Rahman \cite{}.
 Periodic boundary conditions in each direction are applied.
+All point defects are calculated for the neutral charge state.
 
 \begin{figure}[h]
 \begin{center}
@@ -46,15 +47,52 @@ E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}}
 where $N$ and $E_{\text{coh}}^{\text{defect}}$ are the number of atoms and the cohesive energy per atom in the defect configuration and $E_{\text{coh}}^{\text{defect-free}}$ is the cohesive energy per atom of the defect-free structure.
 The formation energy of defects consisting of two or more atom species is defined as
 \begin{equation}
-E_{\text{f}}=E-N_1\mu_1-N_2\mu_2 - \ldots
+E_{\text{f}}=E-\sum_i N_i\mu_i
 \label{eq:defects:ef2}
 \end{equation}
 where $E$ is the free energy of the interstitial system and $N_i$ and $\mu_i$ are the amount of atoms and the chemical potential of species $i$.
 The chemical potential is determined by the cohesive energy of the structure of the specific type in equilibrium at zero Kelvin.
-For a defect configuration of a single species equation \ref{eq:defects:ef2} is equivalent to equation \ref{eq:defects:ef1}.
+For a defect configuration of a single atom species equation \ref{eq:defects:ef2} is equivalent to equation \ref{eq:defects:ef1}.
 
 \section{Silicon self-interstitials}
 
+Point defects in silicon have been extensively studied, both experimentally and theoretically \cite{fahey89,leung99}.
+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{center}
+\begin{tabular}{l c c c c c}
+\hline
+\hline
+ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & V \\
+\hline
+ Erhard/Albe MD & 3.40 & unstable & 5.42 & 4.39 & 3.13 \\
+ VASP & 3.77 & 3.42 & 4.41 & 3.39 & 3.63 \\
+ LDA \cite{leung99} & 3.43 & 3.31 & - & 3.31 & - \\
+ GGA \cite{leung99} & 4.07 & 3.80 & - & 3.84 & - \\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption[Formation energies of silicon self-interstitials in crystalline silicon determined by classical potential molecular dynamics and density functional calculations.]{Formation energies of silicon self-interstitials 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 V the vacancy interstitial configuration. The dumbbell configurations are abbreviated by DB.}
+\label{tab:defects:si_self}
+\end{table}
+
+There are differences between the various results of the quantum-mechanical calculations but the consesus view is that the \hkl<1 1 0> dumbbell followed by the hexagonal and tetrahedral defect is the lowest in energy.
+This is nicely reproduced by the DFT calculations performed in this work.
+
+It has turned out to be very difficult to capture the results of quantum-mechanical calculations in analytical potential models.
+Among the established analytical potentials only the EDIP \cite{} and Stillinger-Weber \cite{} potential reproduce the correct order in energy of the defects.
+However, these potenitals show shortcomings concerning the description of other physical properties and are unable to describe the C-C and C-Si interaction.
+In fact the Erhard/Albe potential calculations favor the tetrahedral defect configuration.
+The hexagonal configuration is not stable opposed to results of the authors of the potential \cite{}.
+The Si interstitial atom moves towards an energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes.
+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{}.
+Obviously the authors did not carefully check the relaxed results assuming a hexagonal configuration.
+To exclude failures in the implementation of the potential or the MD code itself the hexagonal defect structure was double-checked with the PARCAS MD code \cite{}.
+
+The bond-centered configuration is unstable for both, the Erhard/Albe and VASP calculations.
 
 
 \section{Carbon related point defects}