typo + new c ints

[lectures/latex.git] / posic / thesis / defects.tex

\chapter{Point defects in silicon}

Given the conversion mechnism of SiC in crystalline silicon introduced in \ref{section:assumed_prec} the understanding of carbon and silicon interstitial point defects in c-Si is of great interest.
Both types of defects are examined in the following both by classical potential as well as density functional theory calculations.

In case of the classical potential calculations a simulation volume of nine silicon lattice constants in each direction is used.
Calculations are performed in an isothermal-isobaric NPT ensemble.
Coupling to the heat bath is achieved by the Berendsen thermostat with a time constant of 100 fs.
The temperature is set to zero Kelvin.
Pressure is controlled by a Berendsen barostat again using a time constant of 100 fs and a bulk modulus of 100 GPa for silicon.
To exclude surface effects periodic boundary conditions are applied.

Due to the restrictions in computer time three silicon lattice constants in each direction are considered sufficiently large enough for DFT calculations.
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}
\includegraphics[width=9cm]{unit_cell_e.eps}
\end{center}
\caption[Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal  ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration.]{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal  ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration. The black dots ({\color{black}$\bullet$}) correspond to the silicon atoms and the blue lines ({\color{blue}-}) indicate the covalent bonds of the perfect c-Si structure.}
\label{fig:defects:ins_pos}
\end{figure}

The interstitial atom positions are displayed in figure \ref{fig:defects:ins_pos}.
In seperated simulation runs the silicon or carbon atom is inserted at the
\begin{itemize}
 \item tetrahedral, $\vec{p}=(0,0,0)$, ({\color{red}$\bullet$})
 \item hexagonal, $\vec{p}=(-1/8,-1/8,1/8)$, ({\color{green}$\bullet$})
 \item nearly \hkl<1 0 0> dumbbell, $\vec{p}=(-1/4,-1/4,-1/8)$, ({\color{yellow}$\bullet$})
 \item nearly \hkl<1 1 0> dumbbell, $\vec{p}=(-1/8,-1/8,-1/4)$, ({\color{magenta}$\bullet$})
 \item bond-centered, $\vec{p}=(-1/8,-1/8,-3/8)$, ({\color{cyan}$\bullet$})
\end{itemize}
interstitial position.
For the dumbbell configurations the nearest silicon atom is displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ respectively of the unit cell length to avoid too high forces.
A vacancy or a substitutional atom is realized by removing one silicon atom and switching the type of one silicon atom respectively.

From an energetic point of view the free energy of formation $E_{\text{f}}$ is suitable for the characterization of defect structures.
For defect configurations consisting of a single atom species the formation energy is defined as
\begin{equation}
E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}}
                  -E_{\text{coh}}^{\text{defect-free}}\right)N
\label{eq:defects:ef1}
\end{equation}
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-\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 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 & 4.48$^*$ & 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. 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:si_self}
\end{table}
The final configurations obtained after relaxation are presented in figure \ref{fig:defects:conf}.
\begin{figure}[h]
\begin{center}
%\hrule
%\vspace*{0.2cm}
%\begin{flushleft}
%\begin{minipage}{5cm}
%\underline{\hkl<1 1 0> dumbbell}\\
%$E_{\text{f}}=3.39\text{ eV}$\\
%\includegraphics[width=3.0cm]{si_pd_vasp/110_2333.eps}
%\end{minipage}
%\begin{minipage}{5cm}
%\underline{Hexagonal}\\
%$E_{\text{f}}=3.42\text{ eV}$\\
%\includegraphics[width=3.0cm]{si_pd_vasp/hex_2333.eps}
%\end{minipage}
%\begin{minipage}{5cm}
%\underline{Tetrahedral}\\
%$E_{\text{f}}=3.77\text{ eV}$\\
%\includegraphics[width=3.0cm]{si_pd_vasp/tet_2333.eps}
%\end{minipage}\\[0.2cm]
%\begin{minipage}{5cm}
%\underline{\hkl<1 0 0> dumbbell}\\
%$E_{\text{f}}=4.41\text{ eV}$\\
%\includegraphics[width=3.0cm]{si_pd_vasp/100_2333.eps}
%\end{minipage}
%\begin{minipage}{5cm}
%\underline{Vacancy}\\
%$E_{\text{f}}=3.63\text{ eV}$\\
%\includegraphics[width=3.0cm]{si_pd_vasp/vac_2333.eps}
%\end{minipage}
%\begin{minipage}{5cm}
%\begin{center}
%VASP\\
%calculations\\
%\end{center}
%\end{minipage}
%\end{flushleft}
%\vspace*{0.2cm}
%\hrule
\begin{flushleft}
\begin{minipage}{5cm}
\underline{Tetrahedral}\\
$E_{\text{f}}=3.40\text{ eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/tet.eps}
\end{minipage}
\begin{minipage}{10cm}
\underline{Hexagonal}\\[0.1cm]
\begin{minipage}{4cm}
$E_{\text{f}}^*=4.48\text{ eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/hex_a.eps}
\end{minipage}
\begin{minipage}{0.8cm}
\begin{center}
$\Rightarrow$
\end{center}
\end{minipage}
\begin{minipage}{4cm}
$E_{\text{f}}=3.96\text{ eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/hex.eps}
\end{minipage}
\end{minipage}\\[0.2cm]
\begin{minipage}{5cm}
\underline{\hkl<1 0 0> dumbbell}\\
$E_{\text{f}}=5.42\text{ eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/100.eps}
\end{minipage}
\begin{minipage}{5cm}
\underline{\hkl<1 1 0> dumbbell}\\
$E_{\text{f}}=4.39\text{ eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/110.eps}
\end{minipage}
\begin{minipage}{5cm}
\underline{Vacancy}\\
$E_{\text{f}}=3.13\text{ eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/vac.eps}
\end{minipage}
\end{flushleft}
%\hrule
\end{center}
\caption[Relaxed silicon self-interstitial defect configurations obtained by classical potential calculations.]{Relaxed silicon self-interstitial defect configurations obtained by classical potential calculations. The silicon atoms and the bonds (only for the interstitial atom) are illustrated by yellow spheres and blue lines.}
\label{fig:defects:conf}
\end{figure}

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{albe_sic_pot}.
In the first two pico seconds while kinetic energy is decoupled from the system the Si interstitial seems to condense at the hexagonal site.
The formation energy of 4.48 eV is determined by this low kinetic energy configuration shortly before the relaxation process starts.
The Si interstitial atom then begins to slowly move 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{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{center}
\includegraphics[width=12cm]{e_kin_si_hex.ps}
\end{center}
\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the Erhard/Albe classical potential.}
\label{fig:defects:kin_si_hex}
\end{figure}
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 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 artefact of the analytical potential without physical authenticity revealing basic problems of analytical potential models for describing defect structures.
However, the energy barrier is small (DAS MAL DURCHRECHNEN).
Hence these artefacts 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.

In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, that is if the distance of two atoms is within the cutoff region $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
For the tetrahedral and the slightly displaced configurations four bonds to the atoms located in the center of the planes of the unit cell exist in addition to the four tetrahedral bonds.
The length of these bonds are, however, close to the cutoff range and thus are weak interactions not constituting actual chemical bonds.
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 ...

\section{Carbon related point defects}

Carbon is a common and technologically important impurity in silicon.
Concentrations as high as $10^{18}\text{ cm}^{-3}$ occur in Czochralski-grown silicon samples.
It is well established that carbon and other isovalent impurities prefer to dissolve substitutionally in silicon.
However, radiation damage can generate carbon interstitials \cite{watkins76} which have enough mobility at room temeprature to migrate and form defect complexes.

Formation energies of the most common carbon point defects in crystalline silicon are summarized in table \ref{tab:defects:c_ints}.
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{center}
\begin{tabular}{l c c c c c c}
\hline
\hline
 & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
\hline
 Erhard/Albe MD & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
 %VASP & unstable & unstable & 3.15 & 3.60 & 1.39 & 4.10 \\
 VASP & unstable & unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
 Tersoff \cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
 ab initio & - & - & x & - & 1.89 \cite{dal_pino93} & x+2.1 \cite{capaz94} \\
\hline
\hline
\end{tabular}
\end{center}
\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}

Substitutional carbon in silicon is found to be the lowest configuration in energy for all potential models.
An experiemntal value of the formation energy of substitutional carbon was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\text{ eV}/k_{\text{B}}T)} \text{ cm}^{-3}$ \cite{bean71}.
However, there is no particular reason for treating the prefactor as a free parameter in the fit to the experimental data.
It is simply given by the atomic density of pure silicon, which is $5\times 10^{22}\text{ cm}^{-3}$.
Tersoff \cite{tersoff90} and Dal Pino et. al. \cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from 1.6 to 1.89 eV an excellent agreement with the experimental solubility data within the entire temeprature range of the experiment is obtained.
This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal Pino et. al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.

Except for Tersoff's tedrahedral configuration results the \hkl<1 0 0> dumbbell is the energetically most favorable interstital configuration.
The low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cutoff set to 2.5 \AA (see ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between 3 and 10 eV.
Keeping these considerations in mind, the \hkl<1 0 0> dumbbell is the most favorable interstitial configuration for all interaction models.
In addition to the theoretical results compared to in table \ref{tab:defects:c_ints} there is experimental evidence of the existence of this configuration \cite{watkins76}.
It is frequently generated in the classical potential simulation runs in which carbon is inserted at random positions in the c-Si matrix.
In quantum-mechanical simulations the unstable tetrahedral and hexagonal configurations undergo a relaxation into the \hkl<1 0 0> dumbbell configuration.
Thus, this configuration is of great importance and discussed in more detail in section \ref{subsection:100db}.

The highest energy is observed for the hexagonal interstitial configuration using classical potentials.
Quantum-mechanical calculations reveal this configuration to be unstable, which is also reproduced by the Erhard/Albe potential.
In both cases a relaxation towards the \hkl<1 0 0> dumbbell configuration is observed.
The tetrahedral is the second most unfavorable interstitial configuration using classical potentials and keeping in mind the abrupt cutoff effect in the case of the Tersoff potential as discussed earlier.
Again, quantum-mechanical results reveal this configuration unstable.
The fact that the tetrahedral and hexagonal configurations are the two most unstable configurations in classical potential calculations and, thus, are less likely to arise in MD simulations acts in concert with the fact that these configurations are found to be unstable in the more accurate quantum-mechanical calculations.

Bond-centered ...

\subsection[\hkl<1 0 0> dumbbell interstitial configuration]{\boldmath\hkl<1 0 0> dumbbell interstitial configuration}
\label{subsection:100db}

\subsection{Bond-centered interstitial configuration}

\section[Migration of the carbon \hkl<1 0 0> interstitial]{\boldmath Migration of the carbon \hkl<1 0 0> interstitial}

\section{Combination of point defects}