X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=blobdiff_plain;f=posic%2Fpublications%2Femrs2008_full.tex;h=a2dc8f48f7ae71898213f8bcca54bff05cc2ac23;hb=cf759aa2ad773ce165e27084ba66bc66c54aa445;hp=7f7aa62f7a37030bdd8b794ed18902201412dfdb;hpb=ada42e841e4f5e3d38779e7c0583e8e1fd3f00d6;p=lectures%2Flatex.git diff --git a/posic/publications/emrs2008_full.tex b/posic/publications/emrs2008_full.tex index 7f7aa62..a2dc8f4 100644 --- a/posic/publications/emrs2008_full.tex +++ b/posic/publications/emrs2008_full.tex @@ -8,6 +8,7 @@ \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} \usepackage{amsmath} +\usepackage{latexsym} \usepackage{ae} \usepackage{aecompl} \usepackage[dvips]{graphicx} @@ -25,6 +26,10 @@ \setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm} \setlength{\textheight}{26cm} \setlength{\headsep}{0cm} +%\linespread{2.0} + +\selectlanguage{english} + \begin{document} % header @@ -48,70 +53,63 @@ } \end{center} -\selectlanguage{english} - -\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. +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 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 takein 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. - -\begin{figure}[!h] - \begin{center} - \begin{minipage}{5.5cm} - \includegraphics[width=5cm]{sic_prec_seq_01.eps} - \end{minipage} - \begin{minipage}{5.5cm} - \includegraphics[width=5cm]{sic_prec_seq_02.eps} - \end{minipage} - \begin{minipage}{5.5cm} - \includegraphics[width=5cm]{sic_prec_seq_03.eps} - \end{minipage} - \caption{Schematic of the supposed conversion mechanism of highly C doped Si into SiC. C is represented by red dots, Si by black dots and residual Si atoms by white dots with black border.} - \end{center} -\end{figure} -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 \cite{} 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 \cite{verlet67} with a timestep of 1 fs is deployed. +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 \cite{berendsen84} with a time constant $\tau_T=100\, fs$. @@ -120,72 +118,79 @@ To exclude surface effects periodic boundary conditions are applied. \begin{figure}[!h] \begin{center} - \includegraphics[width=8cm]{unit_cell.eps} - \caption{Distinguished interstitial configurations.} + \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} - \label{} \end{figure} -To investigate the intesrtitial configurations of C and Si in Si, a simulation volume of 9 silicon unit cells in each direction is used. +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 $(0,0,0)$ (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 $11 \,\textrm{\AA}$ 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 $30\, ps$. - -For the simulations aiming to reproduce a precipitation process the simulation is 31 silicon lattice constants in each direction. +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 $(-1/8,-1/8,-1/4)$ (${\color{magenta}\bullet}$) 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 minimal 3C-SiC precipitation) are consecutively inserted in a way to keep constant the system temperature. +$6000$ 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, the volume corresponding to a minimal SiC precipitation volume and the volume containing the necessary amount of silicon to form such a precipitation. -After the insertion procedure the system is cooled down to $20\, ^{\circ} \textrm{C}$. +The whole simulation volume, the volume corresponding to the size of a minimal SiC precipitate and the volume containing the amount of silicon necessary for the formation of such a minimal precipitate. +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} -The tetrahedral and the <110> dumbbell self interstitial configurations can be reproduced as observed in \cite{albe_sic_pot}. +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 a energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes. +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 is observed within the random insertion runs. -Variations exist where the displacement is along two axes ($E_f=3.8\, eV$) or along one axis ($E_f=3.6\, eV$) succesively approximating the tetrahedral configuration and formation energy. +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 not stable. -The interstitial atom moves to the more favorable <100> dumbbell position, which has a formation energy of $0.5\, eV$. +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. -This type of configuration is frequently observed for the random insertion runs. +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]{../plot/foo150.ps} - \caption{Diffusion constants} + \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} - \label{} \end{figure} -The influence of interstitials on the diffusion of a single carbon atom is displayed in Fig. 4. -\ldots +%\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} - \begin{minipage}{8.25cm} - \includegraphics[width=8cm]{../plot/foo150.ps} - \end{minipage} - \begin{minipage}{8.25cm} - \includegraphics[width=8cm]{../plot/foo_end.ps} - \end{minipage} - \caption{Pair correlation functions for C-C and Si-C bonds. - Carbon atoms are introduced into the whole simulation volume (red), the region which corresponds to the size of a minimal SiC precipitation (green) and the volume which contains the necessary amount of silicon for a minimal precipitation (blue).} + \includegraphics[width=12cm]{../plot/foo_end.ps} + \includegraphics[width=12cm]{../plot/foo150.ps} + \caption{Pair correlation functions for Si-C and C-C bonds. + Carbon atoms are introduced into the whole simulation volume ({\color{red}-}), the region which corresponds to the size of a minimal SiC precipitate ({\color{green}-}) and the volume which contains the necessary amount of silicon for such a minimal precipitate ({\color{blue}-}).} \end{center} \end{figure} -Fig. 5 shows results of the simulation runs targeting the observation of a precipitation event. +Fig. 4 shows results of the simulation runs targeting the observation of precipitation events. The C-C pair correlation function suggests carbon nucleation for the simulation runs where carbon was inserted into the two smaller regions. The peak at $1.5\, \textrm{\AA}$ fits quite well the next neighbour distance of diamond. 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. @@ -194,9 +199,11 @@ This gives suspect to the competition of Si-C and C-C bond formation in which th \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 proposed. -The <100> C-Si dumbbel is reproducable by simulation and is the energetically most favorable configuration. -The influence of silicon self interstitials on the diffusion of a single carbon atom is demonstrated. +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}