From 1567fb559f941ea5576455f20f151acfbfbaa288 Mon Sep 17 00:00:00 2001 From: hackbard Date: Fri, 9 Jul 2010 19:08:49 +0200 Subject: [PATCH] init for publication in prb --- posic/publications/c_defects_in_si.tex | 188 +++++++++++++++++++++++++ 1 file changed, 188 insertions(+) create mode 100644 posic/publications/c_defects_in_si.tex diff --git a/posic/publications/c_defects_in_si.tex b/posic/publications/c_defects_in_si.tex new file mode 100644 index 0000000..6386eea --- /dev/null +++ b/posic/publications/c_defects_in_si.tex @@ -0,0 +1,188 @@ +\documentclass[prb,twocolumn,superscriptaddress,a4paper,showkeys,showpacs]{revtex4} +\usepackage{graphicx} +\usepackage{subfigure} +\usepackage{dcolumn} +\usepackage{booktabs} +\usepackage{units} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} + + +\begin{document} + +\title{Description of Defects in Carbon implanted Silicon -- a comparison of classical potentials and first principles methods} +\author{F. Zirkelbach} \author{B. Stritzker} +\affiliation{Experimentalphysik IV, Universit\"at Augsburg, D-86153 Augsburg, Germany} +\author{K. Nordlund} +\affiliation{Accelerator Laboratory, University of Helsinki, 00014 Helsinki, Finland} +\author{J. K. N. Lindner} +\affiliation{Experimentelle Physik, Universit\"at Paderborn, 33095 Paderborn, Germany} +\author{W. G. Schmidt} \author{E. Rauls} +\affiliation{Theoretische Physik, Universit\"at Paderborn, 33095 Paderborn, Germany} + +\begin{abstract} +We present a comparative theoretical investigation of carbon interstitials in silicon. +Calculations using classical potentials are put aside first principles density functional theory calculations of the geometries, formation and activation energies of the carbon dumbbell interstitial, showing the importance of a quantum mechanical description of this system. +\end{abstract} + +\keywords{point defects, migration, interstitials, first principles calculations, classical potentials } +\pacs{ find out later... } +\maketitle + +% -------------------------------------------------------------------------------- +\section{Introduction} + +%Frank: Idea: description of 3C-SiC-precipitation in C-implanted silicon.\\ +% cite and describe briefly experimental work - why is this material important/better than other SiC). \\ +% Describe the precipitation process in brief.\\ +% Sum up literature where classical potentials have been used (a) successful, and (b) failed. Also add citations of Nordlunds papers. Not only on silicon or SiC!\\ + +% there should be a short motivation for the material system! +Silicon carbide (SiC) has a number of remarkable physical and chemical properties. +The wide band gap semiconductor (2.3 eV - 3.3 eV) exhibiting a high breakdown field, saturated electron drift velocity and thermal conductivity in conjunction with its unique thermal and mechanical stability as well as radiation hardness is a suitable material for high-temperature, high-frequency and high-power devices\cite{wesch96}, which are moreover deployable in harsh environments\cite{capano97}. +% there are different polytpes with different properties and 3c-sic in special +SiC, which forms fourfold coordinated covalent bonds, tends to crystallize into many different modifications, which solely differ in the one-dimensional stacking sequence of identical, close-packed SiC bilayers\cite{fischer90}. +Different polytypes exhibit different properties, where the only cubic phase (3C-SiC) shows increased values for the thermal conductivity and breakdown field compared to other polytypes\cite{wesch96}, which is of special interest for highly efficient and high-power electronic device applications. + +% (thin films of) 3c-sic can be produced by ibs +Next to epitaxial layer growth by chemical vapor deposition\cite{powell90} and molecular beam epitaxy\cite{mbe}, ion beam synthesis (IBS) constitutes a promising method to produce 3C-SiC epitaxial layers of high quality in silicon\cite{ibs}. + + + + +The relevant structures are with $\approx$ 20000 atoms/nanocrystal way too large to be completely be described with high accuracy +quantum mechanical methods. Modelling the processes described above require the use of less accurate methods, like e.g. classical +potentials (Erhard/Albe\cite{albe},Stillinger-Weber\cite{stillinger},...). Whether such potentials are appropriate for the description of the +physical problem has, however, to be verified first by applying both methods to relevant processes that can be treated by both methods. +In this work, we have implemented and compared the applicability of several (?) classical potentials to ab initio results for one +of the most important processes of our original question. + +In the following we will present a comparative investigation of density functional theory (DFT) studies and +classical potential calculations of the structure, energetics and mobility of carbon defects in silicon. + +% -------------------------------------------------------------------------------- +\section{Methodology} +% ----- DFT ------ +The first-principles DFT calculations were performed with the plane-wave based +Vienna Ab-initio Simulation Package (VASP)\cite{kresse96}. The Kohn-Sham equations were solved +using the generalized-gradient XC-functional approximation proposed by Perdew and +Wang (GGA-PW91)\cite{perdew92}. +The electron-ion interaction was described by the projector-augmented wave (PAW) method\cite{bloechel94,kresse99}. +In the PAW data scalar relativistic corrections are contained. Throughout this work an +energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis. +For the sampling of the Brillouin zone, only the $\Gamma$-point was used. +The defect structures and the migration paths were modelled in cubic supercells containing 216 Si-atoms. Spin polarization has been +fully accounted for. + +% ------ Albe potential --------- +%% Frank: Setup/short description of the potential ? +For the calculations with the classical potentials... + + +\section{Results} + +After ion implantation, carbon interstitials are the most common defects in the silicon sample. Their mobility is the +crucial quantity to be investigated. We thus started our comparative investigations by comparing the stability and the +mobility of an isolated carbon interstitial in silicon bulk in the various possible geometries it can take. + +\subsection{Carbon interstitials in various geometries} + +Several geometries have been calculated to be stable for the carbon interstitial. Fig.\ref{fig:interstitials} shows all these +structures. However, there are some discrepancies between the results from classical potential calculations and those obtained +from first principles. +Table \ref{table:formation} summarizes the formation energies of the interstitial geometries for both methods used in this work +and compares the results to literature values. (...check references for more data, ..) + +% Tables: like in the talk, but add further literature data and give the references/citations (also to bibliography +% at the end!) +%\begin{figure} +%\includegraphics[width=1.0\columnwidth]{models.eps} +%\caption{\label{fig:interstitials} Molecular model of the possible carbon interstitials. } +%\end{figure} + +While the Albe potential predicts ... as stable, DFT does not. ...(further comparisons, trend "too high/low" E-formation,...)... + Nevertheless, both methods predict the (110) dumb bell configuration to be the most stable... (?) + + +\subsection{Mobility} +A measure for the mobility of the interstitial carbon is the activation energy for the migration path from one stable +position to another. The stable defect geometries have been discussed in the previous subsection. We now investigate +the migration from the most stable structure (...should be named somehow...) on one site of the silicon host lattice to +a neighbored site. +On the lowest energy path (first principles), the carbon atom starts to move along (110)..(check that!)... The center of the line connecting +initial and final structure has been found to be a local minimum and not a saddle point as could be expected. The two +saddle points shortly before and behind this local minimum are slightly displaced out of the (110) plane by ... {\AA}. ..(check that!).. +This path is not surprising -- a similar behavior was e.g. found earlier for the carbon split interstitial \cite{rauls03a} and the phosphorus +interstitial \cite{rauls03b,gerstmann03} in SiC. However, an interesting effect is the change of the spin state from zero at the (110) dumb bell +configuration to one at the local minimum. By this, the energy of the local minimum is lowered by 0.3 eV (... check it!!..). +%\begin{figure} +%\includegraphics[width=1.0\columnwidth]{path-DFT.eps} +%\caption{\label{fig:path-DFT} Energy of the carbon interstitial during migration from ... to ... calculated from first principles. The +% activation energy of 0.9 eV (?) agrees well with experimental findings (0.7-0.9 eV?). } +%\end{figure} +Fig.\ref{fig:path-DFT} shows the energy along this lowest energy migration path. The activation energy of 0.9 eV (?) agrees well +with experimental findings (0.7-0.9 eV?). + +Calculations with the Albe potential yield a different picture. +%\begin{figure} +%\includegraphics[width=1.0\columnwidth]{path-Albe.eps} +%\caption{\label{fig:path-Albe} Energy of the carbon interstitial during migration from ... to ... calculated using the classical potential +% method. Here, the activation energy is 2.2 eV (?). } +%\end{figure} +Fig.\ref{fig:path-Albe} shows the energy along the lowest energy migration path found by this method. The activation energy of 2.2 eV (?) +is way too high to explain the experimental findings (0.7-0.9 eV?). (...further discussion...) + +\section{Discussion} +The first principles results are in good agreement to previous work on this subject \cite{joannopoulos,xyz} (...add some references!...). With an +activation energy of 0.9 eV, the carbon interstitial can be expected to be mobile at temperatures in the range of... (?). +The description of the same processes obviously fails if we use the classical potential method. +Already the geometry of the most stable dumb bell configuration differs considerably from that of the first principles calculated +structure. (..... add description, the two main angles and bond lengths and an explanation...) +Formation energies are throughout too high... (...reason?...) + +A reason for this failure of the classical description is most likely... (cut-off, neglect of quantum mechanical effects,...) + + +\section{Summary} +In summary, we have shown that ab initio calculations are very close to the results expected from experimental data. +Furthermore, they agree well with other theoretical results. (...some results - later...) The classical potentials, however, fail to describe the +selected processes. This has been shown to have two reasons, i.e. the artificial cut-off of the next nearest neighbor +interaction on the one hand, on the other hand the quantum mechanical effects which are crucial in the problem under study. + +% ---------------------------------------------------- +\section*{Acknowledgment} +The calculations were done using grants of computer time from the +Paderborn Center for Parallel Computing (PC$^2$) and the +H\"ochstleistungs-Rechenzentrum Stuttgart. The Deutsche +Forschungsgemeinschaft is acknowledged for financial support. + +% --------------------------------- references ------------------- +\begin{thebibliography}{99} +\bibitem{kresse96} G. Kresse and J. Furthm\"uller, + Comput. Mater. Sci. {\bf 6}, 15 (1996). +\bibitem{perdew92} J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais, + Phys. Rev. B {\bf 46}, 6671 (1992). +\bibitem{ceperley80} D. M. Ceperley and B. J. Alder, + Phys. Rev. Lett. {\bf 45}, 556 (1980). +\bibitem{perdew81} J. P. Perdew and A. Zunger, + Phys. Rev. B {\bf 23}, 5048 (1981). +\bibitem{bloechel94} P. E. Bl\"ochl, + Phys. Rev. B {\bf 50}, 17953 (1994). +\bibitem{kresse99} G. Kresse and D. Joubert, + Phys. Rev. B {\bf 59}, 1758 (1999). +\bibitem{monk76} H. J. Monkhorst and J. D. Pack, + Phys. Rev. B {\bf 13}, 5188 (1976). +\bibitem{albe} Albe potential +\bibitem{stillinger} Stillinger-Weber potential +\bibitem{joannopoulos} Joannopoulos +\bibitem{xyz} who else? +\bibitem{rauls03a} E. Rauls, A. Gali, P. De´ak, and Th. Frauenheim, Phys. Rev. B, 68, 155208 (2003). +\bibitem{rauls03b} E. Rauls, U. Gerstmann, H. Overhof, and Th. Frauenheim, Physica B, Vols. 340-342, p. 184-189 (2003). +\bibitem{gerstmann03} U. Gerstmann, E. Rauls, Th. Frauenheim, and H. Overhof, Phys. Rev. B, 67, 205202, (2003). + +\end{thebibliography} + + +\end{document} + -- 2.20.1