X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=posic%2Fpublications%2Femrs2008_full.tex;h=8748709ec0332b1b1b010ed1451aaf6d02a0a12a;hb=f88d7ca5b5d9e560cbd9e016600d4a2700ed945d;hp=5ee36aa2cd1698e5b4647b36755d60ef7b903607;hpb=a26586e1fd32745c018e4111b98b34e628aaf245;p=lectures%2Flatex.git diff --git a/posic/publications/emrs2008_full.tex b/posic/publications/emrs2008_full.tex index 5ee36aa..8748709 100644 --- a/posic/publications/emrs2008_full.tex +++ b/posic/publications/emrs2008_full.tex @@ -4,25 +4,32 @@ \usepackage{verbatim} \usepackage{a4} \usepackage{a4wide} -\usepackage[german]{babel} +\usepackage[german,english]{babel} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} \usepackage{amsmath} +\usepackage{latexsym} \usepackage{ae} \usepackage{aecompl} \usepackage[dvips]{graphicx} -\graphicspath{{./img/}} +\graphicspath{{../img/}} \usepackage{color} \usepackage{pstricks} \usepackage{pst-node} \usepackage{rotating} +\bibliographystyle{h-physrev3} + \setlength{\headheight}{0mm} \setlength{\headsep}{0mm} \setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm} \setlength{\oddsidemargin}{-10mm} \setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm} \setlength{\textheight}{26cm} \setlength{\headsep}{0cm} +%\linespread{2.0} + +\selectlanguage{english} + \begin{document} % header @@ -35,8 +42,8 @@ \textsc{\Large F. Zirkelbach$^1$, J. K. N. Lindner$^1$, K. Nordlund$^2$, B. Stritzker$^1$}\\ \vspace{16pt} - $^1$ Experimentalphysik IV, Institut f"ur Physik, Universit"at Augsburg,\\ - Universit"atsstr. 1, D-86135 Augsburg, Germany\\ + $^1$ Experimentalphysik IV, Institut f\"ur Physik, Universit\"at Augsburg,\\ + Universit\"atsstr. 1, D-86135 Augsburg, Germany\\ \vspace{16pt} $^2$ Accelerator Laboratory, Department of Physical Sciences, University of Helsinki,\\ @@ -46,79 +53,184 @@ } \end{center} -\vspace{24pt} +%\vspace{24pt} \section*{Abstract} -The precipitation process of silicon carbide in heavily carbon doped silicon is not yet understood for the most part. -High resolution transmission electron microscopy indicates that in a first step carbon atoms form $C-Si$ dumbbells on regular $Si$ lattice sites which agglomerate into large clusters. -In a second step, when the cluster size reaches a radius of a few $nm$, the high interfacial energy due to the $SiC$/$Si$ lattice misfit of almost $20 \, \%$ is overcome and the precipitation occurs. -By simulation details of the precipitation process can be obtained on the atomic level. -A newly parametrized Tersoff like bond-order potential is used to model the system appropriately. +The precipitation process of silicon carbide in heavily carbon doped silicon is not yet fully understood. +High resolution transmission electron microscopy observations suggest that in a first step carbon atoms form C-Si dumbbells on regular Si lattice sites which agglomerate into large clusters. +In a second step, when the cluster size reaches a radius of a few $nm$, the high interfacial energy due to the SiC/Si lattice misfit of almost 20\% is overcome and the precipitation occurs. +By simulation, details of the precipitation process can be obtained on the atomic level. +A newly parametrized Tersoff-like bond order potential is used to model the system appropriately. First results gained by molecular dynamics simulations using this potential are presented. -The influence of the amount and placement of inserted carbon atoms on the defect formation and structural changes is discussed. -Furthermore a minimal carbon concentration necessary for precipitation is examined by simulation. \\\\ -{\bf Keywords:} Silicon carbide, Nucleation, Molecular dynamics simulation. +{\bf Keywords:} Silicon, carbon, silicon carbide, nucleation, defect formation, + molecular dynamics simulations \section*{Introduction} -Understanding the precipitation process of cubic silicon carbide (3C-SiC) in heavily carbon doped silicon (Si) will enable significant technological progress in thin film formation of an important wide band gap semiconductor material. -On the other hand it will likewise offer perspectives for processes which rely upon prevention of precipitation processes, e.g. for the fabrication of strained silicon. +Understanding the precipitation process of cubic silicon carbide (3C-SiC) in heavily carbon doped silicon will enable significant technological progress in thin film formation of the important wide band gap semiconductor material SiC \cite{edgar92}. +On the other hand it will likewise offer perspectives for processes which rely upon prevention of precipitation events, e.g. the fabrication of strained, pseudomorphic $\text{Si}_{1-y}\text{C}_y$ heterostructures \cite{}. -Epitaxial growth of 3C-SiC films is achieved either by ion implantation or chemical vapour deposition techniques. -Surface effects dominate the CVD process while for the implantation process carbon is introduced into bulk silicon. -This work tries to realize conditions which hold for the ion implantation process. +Epitaxial growth of 3C-SiC films is achieved either by ion beam synthesis (IBS) \cite{lindner02} and chemical vapour deposition (CVD) or molecular beam epitaxy (MBE) techniques. +While in CVD and MBE surface effects need to be taken into account, SiC formation during IBS takes place in the bulk of the Si crystal. +In the present work the simulation tries to realize conditions which hold for the ion implantation process. -First of all a suitable model is considered. -After that the realization by simulation is discussed. -First results gained by simulation are presented in a next step. -Finally these results are outlined and conclusions are infered. +First of all a picture of the supposed precipitation event is presented. +Afterwards the applied simulation sequences are discussed. +Finally first results gained by simulation are presented. \section*{Supposed conversion mechanism} -Silicon (Si) nucleates in diamond structure. -Contains of two fcc lattices, on displaced one quarter of volume diagonal compared to the first. -3C-SiC nucleates in zincblende structure where the shifted fcc lattice sites are composed of carbon atoms. -The length of four lattice constants of Si is approximately equal to the length of five 3C-SiC lattice constants ($4a_{Si}\approx 5a_{3C-SiC}$), which means that there is a lattice misfit of almost 20\%. -Due to this the silicon density of 3C-SiC is slightly lower than the one of silicon. - -There is a supposed conversion mechanism of heavily carbon doped Si into SiC. -Fig. 1 schematically displays the mechanism. -As indicated by high resolution transmission microscopy \ref{hrem_ind} introduced carbon atoms (red dots) form C-Si dumbbells on regular Si (black dots) lattice sites. -The dumbbells agglomerate int large clusters, so called embryos. -Finally, when the cluster size reaches a critical radius of 2 to 4 nm, the high interfacial energy due to the lattice misfit is overcome and the precipitation occurs. -Due to the slightly lower silicon density of 3C-SiC residual silicon atoms exist. -The residual atoms will most probably end up as self interstitials in the silicon matrix since there is more space than in 3C-SiC. - -Taking this into account, it is important to understand both, the configuration and dynamics of carbon interstitials in silicon and silicon self interstitials. -Additionaly the influence of interstitials on atomic diffusion is investigated. - -\section*{Simulation} +Silicon has diamond structure and thus is composed of two fcc lattices which are displaced by one quarter of the volume diagonal. +3C-SiC grows in zincblende structure, i.e. is also composed of two fcc lattices out of which one is occupied by Si the other by C atoms. +The length of four lattice constants of Si is approximately equal to the length of five 3C-SiC lattice constants ($4a_{\text{Si}}\approx 5a_{\text{3C-SiC}}$) resulting in a lattice misfit of almost 20\%. +Due to this the silicon atomic density of 3C-SiC is slightly lower than the one of pure Si. + +%\begin{figure}[!h] +% \begin{center} +% \begin{minipage}{5.5cm} +% \includegraphics[width=5cm]{sic_prec_seq_01_s.eps} +% \end{minipage} +% \begin{minipage}{5.5cm} +% \includegraphics[width=5cm]{sic_prec_seq_02_s.eps} +% \end{minipage} +% \begin{minipage}{5.5cm} +% \includegraphics[width=5cm]{sic_prec_seq_03_s.eps} +% \end{minipage} +% \caption{Schematic of the supposed conversion mechanism of highly C (${\color{red}\Box}$) doped Si (${\color{black}\bullet}$) into SiC ($_{\color{black}\bullet}^{{\color{red}\Box}}$) and residual Si atoms ($\circ$). The figure shows the dumbbell formation (left), the agglomeration into clusters (middle) and the situation after precipitation (right).} +% \end{center} +%\end{figure} +There is a supposed conversion mechanism of heavily carbon doped Si into SiC \cite{werner97}. +As concluded by high resolution transmission electron microscopy introduced carbon atoms form C-Si dumbbells on regular Si lattice sites. +The dumbbells agglomerate into large clusters, called embryos. +Finally, when the cluster size reaches a critical radius of 2 to 4 nm, the high interfacial energy due to the 3C-SiC/Si lattice misfit is overcome and precipitation occurs. +Due to the slightly lower silicon density of 3C-SiC excessive silicon atoms exist which will most probably end up as self-interstitials in the silicon matrix since there is more space than in 3C-SiC. + +Thus, in addition to the precipitation event itself, knowledge of C and Si interstitials in Si are of great interest in order to investigate the precipitation of heavily C doped Si into SiC. +%Additionaly the influence of interstitials on atomic diffusion is investigated. + +\section*{Simulation sequences} A molecular dynamics simulation approach is used to examine the steps involved in the precipitation process. -For integrating the equations of motion the velocity verlet algorithm \ref{} with a timestep of 1 fs is deployed. -The interaction of the silicon and carbon atoms is realized by a newly parametrized Tersoff like bond order potential \ref{}. +For integrating the equations of motion the velocity verlet algorithm \cite{verlet67} with a timestep of $1\, fs$ is adopted. +The interaction of the silicon and carbon atoms is realized by a newly parametrized Tersoff-like bond order potential \cite{albe_sic_pot}. Since temperature and pressure of the system is kept constant in experiment the isothermal-isobaric NPT ensemble is chosen for the simulation. -Coupling to the heat bath is achieved by the Berendsen thermostat \ref{} with a time constant $\tau_T=100\, fs$. -The pressure is scaled by the Berendsen barostat \ref{} again using a timeconstant of $\tau_P=100\, fs$ and a bulk modulus of $100\, GPa$ for silicon. +Coupling to the heat bath is achieved by the Berendsen thermostat \cite{berendsen84} with a time constant $\tau_T=100\, fs$. +The pressure is scaled by the Berendsen barostat \cite{berendsen84} again using a timeconstant of $\tau_P=100\, fs$ and a bulk modulus of $100\, GPa$ for silicon. To exclude surface effects periodic boundary conditions are applied. -\\\\ -To investigate the intesrtitial configurations of C and Si in Si, a simulation volume of 9 silicon unit cells is each direction used. -The temperature is set to $T=0\, K$. -The insertion positions are illustrated in Fig 2. -In separated simulation runs a carbon and a silicon atom respectively is inserted at the tetrahedral $(0,0,0)$ (red), hexagonal $(-1/8,-1/8,1/8)$ (green), supposed dumbbell $(-1/8,-1/8)$ (purple) and at random positions (in units of the silicon lattice constant) where the origin is located in the middle of the unit cell. -In order to avoid too high kinetic energies in the case of the dumbbell configuration the nearest silicon neighbour atom is shifted to $(-1/4,-1/4,-1/4)$ (dashed border). -The introduced kinetic energy is scaled out by a relaxation time of $2\, ps$. -\\\\ -The same volume is used to investigate diffusion. -A certain amount of silicon atoms are inserted at random positions in a centered region of $11 \, \AA$ in each direction. -The insertion is taking place step by step in order to maintain a constant system temeprature. -Finally a carbon atom is inserted at a random position in the unit cell located in the middle of the simulation volume. -The simulation continues for another $30\, ps$. -\\\\ -Simulation runs /unt - +\begin{figure}[!h] + \begin{center} + \includegraphics[width=8cm]{unit_cell_s.eps} + \caption{Insertion positions for the tetrahedral (${\color{red}\triangleleft}$), hexagonal (${\color{green}\triangleright}$) and <110> dumbbell (${\color{magenta}\Box}$) interstitial configuration.} + \end{center} +\end{figure} +To investigate the interstitial configurations of C and Si in Si, a simulation volume of 9 silicon unit cells in each direction is used. +The temperature is set to $T=0\, K$. +The insertion positions are illustrated in Fig. 2. +In separated simulation runs a carbon and a silicon atom respectively is inserted at the tetrahedral $(0,0,0)$ (${\color{red}\triangleleft}$), hexagonal $(-1/8,-1/8,1/8)$ (${\color{green}\triangleright}$), nearby dumbbell $(-1/8,-1/8,-1/4)$ (${\color{magenta}\Box}$) and at random positions (in units of the silicon lattice constant) where the origin is located in the centre of the unit cell. +In order to avoid too high potential energies in the case of the dumbbell configuration the nearest silicon neighbour atom is shifted to $(-3/8,-3/8,-1/4)$ ($\circ$). +The energy introduced into the system is scaled out within a relaxation phase of $2\, ps$. + +%The same volume is used to investigate diffusion. +%Different amounts of silicon atoms are inserted at random positions within a centered region of $11 \,\textrm{\AA}$ in each direction. +%Insertion events are carried out step by step maintaining a constant system temperature of $450\, ^{\circ} \textrm{C}$. +%Finally a single carbon atom is inserted at a random position within the unit cell located in the middle of the simulation volume. +%The simulation is proceeded for another $30\, ps$. + +For the simulations aiming to reproduce a precipitation process the volume is 31 silicon lattice constants in each direction. +The system temperature is set to $450\, ^{\circ} \textrm{C}$. +$6000$ carbon atoms (the amount necessary to form a 3C-SiC precipitate with a radius of 3 nm) are consecutively inserted in a way to keep constant the system temperature. +Precipitation is examined for three insertion volumes which differ in size. +The whole simulation volume $V_1$, the volume corresponding to the size of a minimal SiC precipitate $V_2$ and the volume containing the amount of silicon necessary for the formation of such a minimal precipitate $V_3$ are examined. +The two latter ones are accomplished since no long range diffusion of the carbon atoms is expected at this temperature. +Following the insertion procedure the system is cooled down to $20\, ^{\circ} \textrm{C}$. \section*{Results} -\section*{Conclusion} +The tetrahedral and the <110> dumbbell self-interstitial configurations can be reproduced as observed in \cite{albe_sic_pot}. +The formation energies are $3.4\, eV$ and $4.4\, eV$ respectively. +However the hexagonal one is not stable opposed to what is presented in \cite{albe_sic_pot}. +The 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 $4.0\, eV$ of this type of interstitial equals the result obtained in the reference for the hexagonal one. +The same type of interstitial may arise using random insertions. +In addition variations exist in which the displacement is only along two axes ($E_f=3.8\, eV$) or along a single axis ($E_f=3.6\, eV$) succesively approximating the tetrahedral configuration and formation energy. + +The tetrahedral and <110> dumbbel carbon interstitial configurations are stable. +The formation energies are $2.7\, eV$ and $1.8\, eV$ respectively. +Again the hexagonal one is found to be unstable. +The interstitial atom moves to the more favorable <100> dumbbell position which has a formation energy of $0.5\, eV$. +The interstitial configuration is shown in Fig. 2. +There is experimental evidence \cite{watkins76} of the existence of this configuration. +It is frequently generated and has the lowest formation energy of all the defects observed in all the simulation runs in which carbon is inserted at random positions. + +\begin{figure}[!h] + \begin{center} + \includegraphics[width=8cm]{c_in_si_int_001db_0.eps} + \caption{Position of a <100> dumbbell carbon interstitial in silicon. + Only bonds of the carbon interstitial atom are shown.} + \end{center} +\end{figure} + +%\begin{figure}[!h] +% \begin{center} +% \includegraphics[width=12cm]{../plot/diff_dep.ps} +% \caption{Diffusion coefficients of a single carbon atom for different amount of Si selft interstitials} +% \end{center} +%\end{figure} +%The influence of Si self-interstitials on the diffusion of a single carbon atom is displayed in Fig. 3. +%Diffusion coefficients for different amount of Si self-interstitials are shown. +%A slight increase is first observed in the case of 30 interstitial atoms. +%Further increasing the amount of interstitials leads to a tremendous decay of the diffusion coeeficient. +%Generally there is no long range diffusion of the carbon atom for a temperature of $450\, ^{\circ} \textrm{C}$. +%The maximal displacement of the carbon atom relativ to its insertion position is between 0.5 and 0.7 \AA. + +\begin{figure}[!h] + \begin{center} + \includegraphics[width=12cm]{pc_si-c_c-c.ps} + \caption{Pair correlation functions for Si-C and C-C bonds. + Carbon atoms are introduced into the whole simulation volume $V_1$, the region which corresponds to the size of a minimal SiC precipitate $V_2$ and the volume which contains the necessary amount of silicon for such a minimal precipitate $V_2$ respectively.} + \end{center} +\end{figure} +\begin{figure}[!h] + \begin{center} + \includegraphics[width=12cm]{pc_si-si.ps} + \caption{Si-Si pair correlation function for pure Si and Si with 3000 inserted C atoms. + The inset shows a magnified region between 0.28 and 0.36 nm.} + \end{center} +\end{figure} +Fig. 3 shows resulting pair correlation functions of the simulation runs targeting the observation of precipitation events. +The contributions of Si-C and C-C pairs are presented separately each of them displaying the pair correlation for the three different volumes $V_1$, $V_2$ and $V_3$ (as explained above) exposed to carbon insertion. +Results show no signigicant difference between $V_1$ and $V_2$. +Si-Si pairs for the case of 3000 inserted C atoms inserted into $V_2$ and a reference function for pure Si are displayed in Fig. 4. + +The amount of C-C bonds for $V_1$ are much smaller than for $V_2$ and $V_3$ since carbon atoms are spread over the total simulation volume which means that there are only 0.2 carbon atoms per silicon unit cell on average. +The first C-C peak appears at about 0.15 nm. +This is comparable to the nearest neighbour distance for graphite or diamond. +It is assumed that these carbon atoms form strong C-C bonds, which is supported by a decrease of the total energy during carbon insertion for the $V_2$ and $V_3$ in contrast to the $V_3$ simulation. + +The peak at 0.31 nm perfectly matches the distance of two carbon atoms in the SiC lattice which in SiC is also expected for the Si-Si bonds. +After insertion of carbon atoms the Si-Si pair correlation function in fact shows non-zero values in the range of the C-C peak width while the amount of Si pairs at the regular distances at 0.24 and 0.38 nm decreases. +However no clear peak is observed and random analyses of configurations in which distances around 0.3 nm appear, i.e. visualization of such atom pairs, identify <100> C-Si dumbbells to be responsible for stretching the Si-Si next neighbour distance for low concentrations of carbon, i.e. for the $V_1$ and early stages of $V_2$ and $V_3$ simulation runs. +For higher carbon concentrations the defect concentration is likewise increased and a considerable amount of damage is introduced into the inserted volume. +Damage and superposition of defects generate new displacement arrangements which become hard to categorize and trace. +The slightly higher amount and intense increase of Si-Si pairs at distances smaller 0.31 nm is probably due to the Si-Si cutoff radius of 0.296 nm. +The cutoff function causes artificial forces pushing the Si atoms out of the cutoff region. + +Wieder durch visuelle untersuchungen -> c-c 0.31 paare durch aufeinandertreffen unterschiedlich orientierter 100 dumbbells bzw mit 110. +C fuer sic schon besser arrangiert. Vorstellung, dass diese zuerst anordnen und spaeter dann evtl am Si ziehen ... +Nevertheless this might indicate that carbon arranges first, then 'pulls' the Si ... +\\\\ +On the other hand the Si-C pair correlation function indicates formation of SiC bonds with an increased crystallinity for the simulation in which carbon is inserted into the whole simulation volume. +There is more carbon forming Si-C bonds than C-C bonds. +This gives suspect to the competition of Si-C and C-C bond formation in which the predominance of either of them depends on the method handling carbon insertion. + +\section*{Summary} +The supposed conversion mechanism of heavily carbon doped silicon into silicon carbide is presented. +Molecular dynamics simulation sequences to investigate interstitial configurations +%, the influence of interstitials on the atomic diffusion +and the precipitation of SiC are explained. +The <100> C-Si dumbbel is reproduced and is the energetically most favorable configuration observed by simulation. +%The influence of silicon self-interstitials on the diffusion of a single carbon atom is demonstrated. +Two competing bond formations, either Si-C or C-C, seem to coexist, where the strength of either of them depends on the size of the region in which carbon is introduced. + +\bibliography{../../bibdb/bibdb} \end{document}