From 547f55bc88bc3a3db92c800cf4eff1211819708c Mon Sep 17 00:00:00 2001 From: hackbard Date: Mon, 11 Jan 2010 18:22:16 +0100 Subject: [PATCH] si self ints ... more to come --- posic/thesis/defects.tex | 42 ++++++++++++++++++++++++++++++++++++++-- 1 file changed, 40 insertions(+), 2 deletions(-) diff --git a/posic/thesis/defects.tex b/posic/thesis/defects.tex index 89bd382..3e3d8d2 100644 --- 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} -- 2.20.1