author hackbard Sun, 22 May 2011 21:34:04 +0000 (23:34 +0200) committer hackbard Sun, 22 May 2011 21:34:04 +0000 (23:34 +0200)
 posic/thesis/basics.tex patch | blob | history posic/thesis/sic.tex patch | blob | history posic/thesis/simulation.tex patch | blob | history posic/thesis/thesis.tex patch | blob | history

index bc16410..f6a8eda 100644 (file)
@@ -421,7 +421,7 @@ for the exchange-correlation energy, where $\epsilon_{\text{xc}}(\vec{r};[n(\til Expressing$n(\tilde{\vec{r}})$in a Taylor series,$\epsilon_{\text{xc}}$can be thought of as a function of coefficients, which correspond to the respective terms of the expansion. Neglecting all terms of order$\mathcal{O}(\nabla n(\vec{r}))$results in the functional equal to LDA, which requires the function of variable$n$. Including the next element of the Taylor series introduces the gradient correction to the functional, which requires the function of variables$n$and$|\nabla n|$. -This is called the generalized gradient approximation (GGA), which expresses the exchange-correlation energy density as a function of the local density and the local gradient of the density +This is called the generalized-gradient approximation (GGA), which expresses the exchange-correlation energy density as a function of the local density and the local gradient of the density E^{\text{GGA}}_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}),|\nabla n(\vec{r})|)n(\vec{r}) d\vec{r} \text{ .} @@ -532,6 +532,7 @@ Using PPs the rapid oscillations of the wave functions near the core of the atom More importantly, less accuracy is required compared to all-electron calculations to determine energy differences among ionic configurations, which almost totally appear in the energy of the valence electrons that are typically a factor$10^3$smaller than the energy of the core electrons. \subsection{Brillouin zone sampling} +\label{subsection:basics:bzs} Following Bloch's theorem only a finite number of electronic wave functions need to be calculated for a periodic system. However, to calculate quantities like the total energy or charge density, these have to be evaluated in a sum over an infinite number of$\vec{k}$points. index 59d82c2..5d1f802 100644 (file) @@ -72,7 +72,7 @@ As such, SiC will continue to play a major role in the production of future supe Especially substrates of the 3C polytype promise good quality, single crystalline GaN films~\cite{takeuchi91,yamamoto04,ito04}. The focus of SiC based applications, however, is in the area of solid state electronics experiencing revolutionary performance improvements enabled by its capabilities. -These devices include ultraviolet (UV) detectors, high power radio frequency (RF) amplifiers, rectifiers and switching transistors as well as \ac{MEMS} applications. +These devices include ultraviolet (UV) detectors, high power radio frequency (RF) amplifiers, rectifiers and switching transistors as well as microelectromechanical system (MEMS) applications. For UV dtectors the wide band gap is useful for realizing low photodiode dark currents as well as sensors that are blind to undesired near-infrared wavelenghts produced by heat and solar radiation. These photodiodes serve as excellent sensors applicable in the monitoring and control of turbine engine combustion. The low dark currents enable the use in X-ray, heavy ion and neutron detection in nuclear reactor monitoring and enhanced scientific studies of high-energy particle collisions as well as cosmic radiation. @@ -83,7 +83,7 @@ For instance, SiC based solid state transmitters hold great promise for High Def The high breakdown field of SiC compared to Si allows the blocking voltage region of a device to be designed roughly 10 times thinner and 10 times heavier doped, resulting in a decrease of the blocking region resistance by a factor of 100 and a much faster switching behavior. Thus, rectifier diodes and switching transistors with higher switching frequencies and much greater efficiencies can be realized and exploited in highly efficient power converters. Therefor, SiC constitutes a promising candidate to become the key technology towards an extensive development and use of regenerative energies and elctromobility. -Beside the mentioned electrical capabilities the mechanical stability, which is almost as hard as diamond, and chemical inertness almost suggest SiC to be used in \ac{MEMS} designs. +Beside the mentioned electrical capabilities the mechanical stability, which is almost as hard as diamond, and chemical inertness almost suggest SiC to be used in (MEMS) designs. Among the different polytypes of SiC, the cubic phase shows a high electron mobility and the highest break down field as well as saturation drift velocity. In contrast to its hexagonal counterparts 3C-SiC exhibits isotropic mechanical and electronic properties. @@ -115,7 +115,7 @@ In the following, the principal difficulties involved in the formation of crysta Though possible, melt growth processes \cite{nelson69} are complicated due to the small C solubility in Si at temperatures below \unit[2000]{$^{\circ}$C} and its small change with temperature \cite{scace59}. High process temperatures are necessary and the evaporation of Si must be suppressed by a high-pressure inert atmosphere. Crystals grown by this method are not adequate for practical applications with respect to their size as well as quality and purity. -The presented methods, thus, focus on vapor transport growth processes such as \ac{CVD} or \ac{MBE} and the sublimation technique. +The presented methods, thus, focus on vapor transport growth processes such as chemical vapor deposition (CVD) or molecular beam epitaxy (MBE) and the sublimation technique. Excellent reviews of the different SiC growth methods have been published by Wesch \cite{wesch96} and Davis~et~al. \cite{davis91}. \subsection{SiC bulk crystal growth} @@ -154,16 +154,16 @@ Further efforts have to be expended to find relations between the growth paramet \subsection{SiC epitaxial thin film growth} Crystalline SiC layers have been grown by a large number of techniques on the surfaces of different substrates. -Most of the crystal growth processes are based on \ac{CVD}, solid-source \ac{MBE} (SSMBE) and gas-source \ac{MBE} (GSMBE) on Si as well as SiC substrates. -In \ac{CVD} as well as GSMBE, C and Si atoms are supplied by C containing gases like CH$_4$, C$_3$H$_8$, C$_2$H$_2$or C$_2$H$_4$and Si containing gases like SiH$_4$, Si$_2$H$_6$, SiH$_2$Cl$_2$, SiHCl$_3$or SiCl$_4$respectively. +Most of the crystal growth processes are based on (CVD), solid-source (MBE) (SSMBE) and gas-source (MBE) (GSMBE) on Si as well as SiC substrates. +In (CVD) as well as GSMBE, C and Si atoms are supplied by C containing gases like CH$_4$, C$_3$H$_8$, C$_2$H$_2$or C$_2$H$_4$and Si containing gases like SiH$_4$, Si$_2$H$_6$, SiH$_2$Cl$_2$, SiHCl$_3$or SiCl$_4$respectively. In the case of SSMBE atoms are provided by electron beam evaporation of graphite and solid Si or thermal evaporation of fullerenes. -The following review will exclusively focus on \ac{CVD} and \ac{MBE} techniques. +The following review will exclusively focus on (CVD) and (MBE) techniques. The availability and reproducibility of Si substrates of controlled purity made it the first choice for SiC epitaxy. The heteroepitaxial growth of SiC on Si substrates has been stimulated for a long time due to the lack of suitable large substrates that could be adopted for homoepitaxial growth. Furthermore, heteroepitaxy on Si substrates enables the fabrication of the advantageous 3C polytype, which constitutes a metastable phase and, thus, can be grown as a bulk crystal only with small sizes of a few mm. The main difficulties in SiC heteroepitaxy on Si is due to the lattice mismatch of Si and SiC and the difference in the thermal expansion coefficient of \unit[8]{\%}. -Thus, in most of the applied \ac{CVD} and \ac{MBE} processes, the SiC layer formation process is split into two steps, the surface carbonization and the growth step, as proposed by Nishino~et~al. \cite{nishino83}. +Thus, in most of the applied (CVD) and (MBE) processes, the SiC layer formation process is split into two steps, the surface carbonization and the growth step, as proposed by Nishino~et~al. \cite{nishino83}. Cleaning of the substrate surface with HCl is required prior to carbonization. During carbonization the Si surface is chemically converted into a SiC film with a thickness of a few nm by exposing it to a flux of C atoms and concurrent heating up to temperatures about \unit[1400]{$^{\circ}$C}. In a next step, the epitaxial deposition of SiC is realized by an additional supply of Si atoms at similar temperatures. @@ -172,12 +172,12 @@ Next to surface morphology defects such as pits and islands, the main defects in APB defects, which constitute the primary residual defects in thick layers, are formed near surface terraces that differ in a single-atom-height step resulting in domains of SiC separated by a boundary, which consists of either Si-Si or C-C bonds due to missing or disturbed sublattice information \cite{desjardins96,kitabatake97}. However, the number of such defects can be reduced by off-axis growth on a Si \hkl(0 0 1) substrate miscut towards \hkl[1 1 0] by \unit[2]{$^{\circ}$}-\unit[4]{$^{\circ}$} \cite{shibahara86,powell87_2}. This results in the thermodynamically favored growth of a single phase due to the uni-directional contraction of Si-C-Si bond chains perpendicular to the terrace steps edges during carbonization and the fast growth parallel to the terrace edges during growth under Si rich conditions \cite{kitabatake97}. -By \ac{MBE}, lower process temperatures than these typically employed in \ac{CVD} have been realized \cite{hatayama95,henke95,fuyuki97,takaoka98}, which is essential for limiting thermal stresses and to avoid resulting substrate bending, a key issue in obtaining large area 3C-SiC surfaces. +By (MBE), lower process temperatures than these typically employed in (CVD) have been realized \cite{hatayama95,henke95,fuyuki97,takaoka98}, which is essential for limiting thermal stresses and to avoid resulting substrate bending, a key issue in obtaining large area 3C-SiC surfaces. In summary, the almost universal use of Si has allowed significant progress in the understanding of heteroepitaxial growth of SiC on Si. However, mismatches in the thermal expansion coefficient and the lattice parameter cause a considerably high concentration of various defects, which is responsible for structural and electrical qualities that are not yet statisfactory. The alternative attempt to grow SiC on SiC substrates has shown to drastically reduce the concentration of defects in deposited layers. -By \ac{CVD}, both, the 3C \cite{kong88,powell90} as well as the 6H \cite{kong88_2,powell90_2} polytype could be successfully grown. +By (CVD), both, the 3C \cite{kong88,powell90} as well as the 6H \cite{kong88_2,powell90_2} polytype could be successfully grown. In order to obtain the homoepitaxially grown 6H polytype, off-axis 6H-SiC wafers are required as a substrate \cite{kimoto93}. %In the so called step-controlled epitaxy, lateral growth proceeds from atomic steps without the necessity of preceding nucleation events. Investigations indicate that in the so-called step-controlled epitaxy, crystal growth proceeds through the adsorbtion of Si species at atomic steps and their carbonization by hydrocarbon molecules. @@ -190,7 +190,7 @@ Additionally, 6H-SiC was observed on clean substrates even for a tilt angle as l Thus, 3C nucleation is assumed as a result of migrating Si and C cointaining molecules interacting with surface disturbances by a yet unknown mechanism, in contrast to a model \cite{ueda90}, in which the competing 6H versus 3C growth depends on the density of surface steps. Combining the fact of a well defined 3C lateral growth direction, i.e. the tilt direction, and an intentionally induced dislocation enables the controlled growth of a 3C-SiC film mostly free of DPBs \cite{powell91}. -Lower growth temperatures, a clean growth ambient, in situ control of the growth process, layer-by-layer deposition and the possibility to achieve dopant profiles within atomic dimensions due to the reduced diffusion at low growth temperatures reveal \ac{MBE} as a promising technique to produce SiC epitaxial layers. +Lower growth temperatures, a clean growth ambient, in situ control of the growth process, layer-by-layer deposition and the possibility to achieve dopant profiles within atomic dimensions due to the reduced diffusion at low growth temperatures reveal (MBE) as a promising technique to produce SiC epitaxial layers. Using alternating supply of the gas beams Si$_2$H$_6$and C$_2$H$_2$in GSMBE, 3C-SiC epilayers were obtained on 6H-SiC substrates at temperatures between \unit[850]{$^{\circ}$C} and \unit[1000]{$^{\circ}$C} \cite{yoshinobu92}. On \hkl(000-1) substrates twinned \hkl(-1-1-1) oriented 3C-SiC domains are observed, which suggest a nucleation driven rather than step-flow growth mechanism. On \hkl(0-11-4) substrates, however, single crystalline \hkl(001) oriented 3C-SiC grows with the c axes of substrate and film being equal. @@ -204,7 +204,7 @@ To realize single monolayer growth precise control of the gas supply to form the However, accurate layer-by-layer growth is achieved under certain conditions, which facilitate the spontaneous desorption of an additional layer of one atom species by supply of the other species \cite{hara93}. Homoepitaxial growth of the 6H polytype has been realized on off-oriented substrates utilizing simultaneous supply of the source gases \cite{tanaka94}. Depending on the gas flow ratio either 3C island formation or step flow growth of the 6H polytype occurs, which is explained by a model including aspects of enhanced surface mobilities of adatoms on a$(3\times 3)$reconstructed surface. -Due to the strong adsorption of atomic hydrogen \cite{allendorf91} decomposited of the gas phase reactants at low temperatures, however, there seems to be no benefit of GSMBE compared to \ac{CVD}. +Due to the strong adsorption of atomic hydrogen \cite{allendorf91} decomposited of the gas phase reactants at low temperatures, however, there seems to be no benefit of GSMBE compared to (CVD). Next to lattice imperfections, incorporated hydrogen effects the surface mobility of the adsorbed species \cite{eaglesham93} setting a minimum limit for the growth temperature, which would preferably be further decreased in order to obtain sharp doping profiles. Thus, growth rates must be adjusted to be lower than the desorption rate of hydrogen, which leads to very low deposition rates at low temperatures. SSMBE, by supplying the atomic species to be deposited by evaporation of a solid, presumably constitutes the preffered method in order to avoid the problems mentioned above. @@ -220,11 +220,11 @@ Solving this issue remains a challenging problem necessary to drive SiC for pote Although tremendous progress has been achieved in the above-mentioned growth methods during the last decades, available wafer dimensions and crystal qualities are not yet statisfactory. Thus, alternative approaches to fabricate SiC have been explored. -The \ac{IBS} technique, i.e. high-dose ion implantation followed by a high-temperature annealing step, turned out to constitute a promising method to directly form compound layers of high purity and accurately controllable depth and stoichiometry. -A short chronological summary of the \ac{IBS} of SiC and its origins is presented in the following. +The ion beam synthesis (IBS) technique, i.e. high-dose ion implantation followed by a high-temperature annealing step, turned out to constitute a promising method to directly form compound layers of high purity and accurately controllable depth and stoichiometry. +A short chronological summary of the (IBS) of SiC and its origins is presented in the following. -High-dose carbon implantation into \ac{c-Si} with subsequent or in situ annealing was found to result in SiC microcrystallites in Si \cite{borders71}. -\ac{RBS} and \ac{IR} spectroscopy investigations indicate a \unit[10]{at.\%} C concentration peak and the occurence of disordered C-Si bonds after implantation at \ac{RT} followed by crystallization into SiC precipitates upon annealing demonstrated by a shift in the \ac{IR} absorption band and the disappearance of the C profile peak in \ac{RBS}. +High-dose carbon implantation into crystalline silicon (c-Si) with subsequent or in situ annealing was found to result in SiC microcrystallites in Si \cite{borders71}. +Rutherford backscattering spectrometry (RBS) and infrared (IR) spectroscopy investigations indicate a \unit[10]{at.\%} C concentration peak and the occurence of disordered C-Si bonds after implantation at room temperature (RT) followed by crystallization into SiC precipitates upon annealing demonstrated by a shift in the (IR) absorption band and the disappearance of the C profile peak in (RBS). Implantations at different temperatures revealed a strong influence of the implantation temperature on the compound structure \cite{edelman76}. Temperatures below \unit[500]{$^{\circ}$C} result in amorphous layers, which is transformed into polycrystalline 3C-SiC after \unit[850]{$^{\circ}$C} annealing. Otherwise single crystalline 3C-SiC is observed for temperatures above \unit[600]{$^{\circ}$C}. @@ -241,7 +241,7 @@ In order to avoid extreme annealing temperatures close to the melting temperatur It was shown that a thick buried layer of SiC is directly formed during implantation, which consists of small, only slightly misorientated but severely twinned 3C-SiC crystallites. The authors assumed that due to the auxiliary heating rather than ion beam heating as employed in all the preceding studies, the complexity of the remaining defects in the synthesized structure is fairly reduced. Even better qualities by direct synthesis were obtained for implantations at \unit[950]{$^{\circ}$C} \cite{nejim95}. -Since no amorphous or polycrystalline regions have been identified, twinning is considered to constitute the main limiting factor in the \ac{IBS} of SiC. +Since no amorphous or polycrystalline regions have been identified, twinning is considered to constitute the main limiting factor in the (IBS) of SiC. Further studies revealed the possibility to form buried layers of SiC by IBS at moderate substrate and anneal temperatures \cite{lindner95,lindner96}. Different doses of C ions with an energy of \unit[180]{keV} were implanted at \unit[330-440]{$^{\circ}$C} and annealed at \unit[1200]{$^{\circ}$C} or \unit[1250]{$^{\circ}$C} for \unit[5-10]{h}. @@ -273,7 +273,7 @@ To further improve the interface quality and crystallinity a two-temperature imp To form a narrow, box-like density profile of oriented SiC nanocrystals \unit[93]{\%} of the total dose of \unit[$8.5\cdot 10^{17}$]{cm$^{-2}$} is implanted at \unit[500]{$^{\circ}$C}. The remaining dose is implanted at \unit[250]{$^{\circ}$C}, which leads to the formation of amorphous zones above and below the SiC precipitate layer and the desctruction of SiC nanocrystals within these zones. After annealing for \unit[10]{h} at \unit[1250]{$^{\circ}C} a homogeneous, stoichiometric SiC layer with sharp interfaces is formed. -Fig. \ref{fig:sic:hrem_sharp} shows the respective \ac{HREM} micrographs. +Fig. \ref{fig:sic:hrem_sharp} shows the respective high resolution transmission electron microscopy (HREM) micrographs. \begin{figure}[t] \begin{center} \includegraphics[width=0.6\columnwidth]{ibs_3c-sic.eps} @@ -282,8 +282,8 @@ Fig. \ref{fig:sic:hrem_sharp} shows the respective \ac{HREM} micrographs. \label{fig:sic:hrem_sharp} \end{figure} -To summarize, by understanding some basic processes, \ac{IBS} nowadays has become a promising method to form thin SiC layers of high quality exclusively of the 3C polytype embedded in and epitaxially aligned to the Si host featuring a sharp interface. -Due to the high areal homogeneity achieved in \ac{IBS}, the size of the layers is only limited by the width of the beam-scanning equipment used in the implantation system as opposed to deposition techniques, which have to deal with severe wafer bending. +To summarize, by understanding some basic processes, (IBS) nowadays has become a promising method to form thin SiC layers of high quality exclusively of the 3C polytype embedded in and epitaxially aligned to the Si host featuring a sharp interface. +Due to the high areal homogeneity achieved in (IBS), the size of the layers is only limited by the width of the beam-scanning equipment used in the implantation system as opposed to deposition techniques, which have to deal with severe wafer bending. This enables the synthesis of large area SiC films. \section{Substoichiometric concentrations of carbon in crystalline silicon} @@ -345,20 +345,20 @@ The tensile strain induced by the C atoms is found to compensates the compressiv Studies on the thermal stability of Si_{1-y}$C$_y$/Si heterostructures formed in the same way and equal C concentrations showed a loss of substitutional C accompanied by strain relaxation for temperatures ranging from \unit[810-925]{$^{\circ}C} and the formation of spherical 3C-SiC precipitates with diameters of \unit[2-4]{nm}, which are incoherent but aligned to the Si host \cite{strane94}. During the initial stages of precipitation C-rich clusters are assumed, which maintain coherency with the Si matrix and the associated biaxial strain. Using this technique a metastable solubility limit was achieved, which corresponds to a C concentration exceeding the solid solubility limit at the Si melting point by nearly three orders of magnitude and, furthermore, a reduction of the defect denisty near the metastable solubility limit is assumed if the regrowth temperature is increased by rapid thermal annealing \cite{strane96}. -Since high temperatures used in the solid-phase epitaxial regrowth method promotes SiC precipitation, other groups realized substitutional C incorporation for strained Si_{1-y}$C$_y$/Si heterostructures \cite{iyer92,fischer95,powell93,osten96,osten99,laveant2002} or partially to fully strain-compensated (even inversely distorted \cite{osten94_2}) Si$_{1-x-y}$Ge$_x$C${_y}$layers on Si \cite{eberl92,powell93_2,osten94,dietrich94} by \ac{MBE}. +Since high temperatures used in the solid-phase epitaxial regrowth method promotes SiC precipitation, other groups realized substitutional C incorporation for strained Si$_{1-y}$C$_y$/Si heterostructures \cite{iyer92,fischer95,powell93,osten96,osten99,laveant2002} or partially to fully strain-compensated (even inversely distorted \cite{osten94_2}) Si$_{1-x-y}$Ge$_x$C${_y}$layers on Si \cite{eberl92,powell93_2,osten94,dietrich94} by (MBE). Investigations reveal a strong dependence of the growth temperature on the amount of substitutionally incorporated C, which is increased for decreasing temperature accompanied by deterioration of the crystal quality \cite{osten96,osten99}. While not being compatible to very-large-scale integration technology, C concentrations of \unit[2]{\%} and more have been realized \cite{laveant2002}. \section{Assumed silicon carbide conversion mechanisms} \label{section:assumed_prec} -Although high-quality films of single-crystalline 3C-SiC can be produced by means of \ac{IBS} the precipitation mechanism in bulk Si is not yet fully understood. +Although high-quality films of single-crystalline 3C-SiC can be produced by means of (IBS) the precipitation mechanism in bulk Si is not yet fully understood. Indeed, closely investigating the large amount of literature pulled up in the last two sections and a cautios combination of some of the findings reveals controversial ideas of SiC formation, which are reviewed in more detail in the following. -\ac{HREM} investigations of C-implanted Si at room temperature followed by \ac{RTA} show the formation of C-Si dumbbell agglomerates, which are stable up to annealing temperatures of about \unit[700-800]{$^{\circ}$C}, and a transformation into 3C-SiC precipitates at higher temperatures \cite{werner96,werner97}. +High resolution transmission electron microscopy (HREM) investigations of C-implanted Si at room temperature followed by rapid thermal annealing (RTA) show the formation of C-Si dumbbell agglomerates, which are stable up to annealing temperatures of about \unit[700-800]{$^{\circ}$C}, and a transformation into 3C-SiC precipitates at higher temperatures \cite{werner96,werner97}. The precipitates with diamateres between \unit[2]{nm} and \unit[5]{nm} are incorporated in the Si matrix without any remarkable strain fields, which is explained by the nearly equal atomic density of C-Si agglomerates and the SiC unit cell. Implantations at \unit[500]{$^{\circ}$C} likewise suggest an initial formation of C-Si dumbbells on regular Si lattice sites, which agglomerate into large clusters \cite{lindner99_2}. -The agglomerates of such dimers, which do not generate lattice strain but lead to a local increase of the lattice potential \cite{werner96}, are indicated by dark contrasts and otherwise undisturbed Si lattice fringes in \ac{HREM}, as can be seen in Fig.~\ref{fig:sic:hrem:c-si}. +The agglomerates of such dimers, which do not generate lattice strain but lead to a local increase of the lattice potential \cite{werner96}, are indicated by dark contrasts and otherwise undisturbed Si lattice fringes in (HREM), as can be seen in Fig.~\ref{fig:sic:hrem:c-si}. \begin{figure}[t] \begin{center} \subfigure[]{\label{fig:sic:hrem:c-si}\includegraphics[width=0.25\columnwidth]{tem_c-si-db.eps}} @@ -392,19 +392,19 @@ With increasing dose and proceeding time the highly mobile dumbbells agglomerate Finally, when the cluster size reaches a critical radius, the high interfacial energy due to the 3C-SiC/c-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 c-Si matrix since there is more space than in 3C-SiC. -In contrast, \ac{IR} spectroscopy and \ac{HREM} investigations on the thermal stability of strained Si$_{1-y}$C$_y$/Si heterostructures formed by \ac{SPE} \cite{strane94} and \ac{MBE} \cite{guedj98}, which finally involve the incidental formation of SiC nanocrystallites, suggest a coherent initiation of precipitation by agglomeration of substitutional instead of interstitial C. +In contrast, (IR) spectroscopy and (HREM) investigations on the thermal stability of strained Si$_{1-y}$C$_y$/Si heterostructures formed by solid-phase epitaxy (SPE) \cite{strane94} and (MBE) \cite{guedj98}, which finally involve the incidental formation of SiC nanocrystallites, suggest a coherent initiation of precipitation by agglomeration of substitutional instead of interstitial C. These experiments show that the C atoms, which are initially incorporated substitutionally at regular lattice sites, form C-rich clusters maintaining coherency with the Si lattice during annealing above a critical temperature prior to the transition into incoherent 3C-SiC precipitates. Increased temperatures in the annealing process enable the diffusion and agglomeration of C atoms. Coherency is lost once the increasing strain energy of the stretched SiC structure surpasses the interfacial energy of the incoherent 3C-SiC precipitate and the Si substrate. Estimates of the SiC/Si interfacial energy \cite{taylor93} and the consequent critical size correspond well with the experimentally observed precipitate radii within these studies. This different mechanism of precipitation might be attributed to the respective method of fabrication. -While in \ac{CVD} and \ac{MBE} surface effects need to be taken into account, SiC formation during IBS takes place in the bulk of the Si crystal. -However, in another \ac{IBS} study Nejim et~al. \cite{nejim95} propose a topotactic transformation that is likewise based on substitutional C, which replaces four of the eight Si atoms in the Si unit cell accompanied by the generation of four Si interstitials. +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. +However, in another (IBS) study Nejim et~al. \cite{nejim95} propose a topotactic transformation that is likewise based on substitutional C, which replaces four of the eight Si atoms in the Si unit cell accompanied by the generation of four Si interstitials. Since the emerging strain due to the expected volume reduction of \unit[48]{\%} would result in the formation of dislocations, which, however, are not observed, the interstitial Si is assumed to react with further implanted C atoms in the released volume. The resulting strain due to the slightly lower Si density of SiC compared to Si of about \unit[3]{\%} is sufficiently small to legitimate the absence of dislocations. Furthermore, IBS studies of Reeson~et~al. \cite{reeson87}, in which implantation temperatures of \unit[500]{$^{\circ}$C} were employed, revealed the necessity of extreme annealing temperatures for C redistribution, which is assumed to result from the stability of substitutional C and consequently high activation energies required for precipitate dissolution. -The results support a mechanism of an initial coherent precipitation based on substitutional C that is likewise valid for the \ac{IBS} of 3C-SiC by C implantation into Si at elevated temperatures. +The results support a mechanism of an initial coherent precipitation based on substitutional C that is likewise valid for the (IBS) of 3C-SiC by C implantation into Si at elevated temperatures. The fact that the metastable cubic phase instead of the thermodynamically more favorable hexagonal$\alpha-SiC structure is formed and the alignment of these cubic precipitates within the Si matrix, which can be explained by considering a topotactic transformation by C atoms occupying substitutionally Si lattice sites of one of the two fcc lattices that make up the Si crystal, reinforce the proposed mechanism. To conclude, a controversy with respect to the precipitation of SiC in Si exists in literature. index 2a9e057..2f098d5 100644 (file) @@ -1,6 +1,172 @@ -\chapter{Simulation parameters and test calculations} +\chapter{Details of simulation parameters and test calculations} \label{chapter:simulation} +All calculations are carried out utilizing the supercell approach, which means that the simulation cell contains a multiple of unti cells and periodic boundary conditions are imposed on the boundaries of that simulation cell. +Strictly, these supercells become the unit cells, which, by a periodic sequence, compose the bulk material that is actually investigated by this approach. +Thus, importance need to be attached to the construction of the supercell. +Three basic types of supercells to compose the initial Si bulk lattice, which can be scaled by integers in the different directions, are considered. +The basis vectors of the supercells are shown in Fig. \ref{fig:simulation:sc}. +\begin{figure}[t] +\begin{center} +\subfigure[]{\label{fig:simulation:sc1}\includegraphics[width=0.3\textwidth]{sc_type0.eps}} +\subfigure[]{\label{fig:simulation:sc2}\includegraphics[width=0.3\textwidth]{sc_type1.eps}} +\subfigure[]{\label{fig:simulation:sc3}\includegraphics[width=0.3\textwidth]{sc_type2.eps}} +\end{center} +\caption{Basis vectors of three basic types of supercells used to create the initial Si bulk lattice.} +\label{fig:simulation:sc} +\end{figure} +Type 1 (Fig. \ref{fig:simulation:sc1}) constitutes the primitive cell. +The basis is face-centered cubic (fcc) and is given byx_1=(0.5,0.5,0)$,$x_2=(0,0.5,0.5)$and$x_3=(0.5,0,0.5)$. +Two atoms, one at$(0,0,0)$and the other at$(0.25,0.25,0.25)$with respect to the basis, generate the Si diamond primitive cell. +Type 2 (Fig. \ref{fig:simulation:sc2}) covers two primitive cells with 4 atoms. +The basis is given by$x_1=(0.5,-0.5,0)$,$x_2=(0.5,0.5,0)$and$x_3=(0,0,1)$. +Type 3 (Fig. \ref{fig:simulation:sc3}) contains 4 primitive cells with 8 atoms and corresponds to the unit cell shown in Fig. \ref{fig:sic:unit_cell}. +The basis is simple cubic. + +In the following an overview of the different simulation procedures and respective parameters is presented. +These procedures and parameters differ depending on whether classical potentials or {\em ab initio} methods are used and on what is going to be investigated. + +\section{DFT calculations} +\label{section:simulation:dft_calc} + +The first-principles DFT calculations are performed with the plane-wave-based Vienna {\em ab initio} simulation package ({\textsc vasp}) \cite{kresse96}. +The Kohn-Sham equations are solved using the GGA utilizing the exchange-correlation functional proposed by Perdew and Wang (GGA-PW91) \cite{perdew86,perdew92}. +The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials as implemented in {\textsc vasp} \cite{vanderbilt90}. +An energy cut-off of \unit[300]{eV} is used to expand the wave functions into the plane-wave basis. +Sampling of the Brillouin zone is restricted to the$\Gamma$point. +Spin polarization has been fully accounted for. +The electronic ground state is calculated by an interative Davidson scheme \cite{davidson75} until the difference in total energy of two subsequent iterations is below \unit[$10^{-4}$]{eV}. + +Defect structures and the migration paths have been modeled in cubic supercells of type 3 containing 216 Si atoms. +The conjugate gradiant algorithm is used for ionic relaxation. +The cell volume and shape is allowed to change using the pressure control algorithm of Parinello and Rahman \cite{parrinello81} in order to realize constant pressure simulations. +Due to restrictions by the {\textsc vasp} code, {\em ab initio} MD could only be performed at constant volume. +In MD simulations the equations of motion are integrated by a fourth order predictor corrector algorithm for a timestep of \unit[1]{fs}. + +% todo +% All point defects are calculated for the neutral charge state. + +Most of the parameter settings, as determined above, constitute a tradeoff regarding the tasks that need to be addressed. +These parameters include the size of the supercell, cut-off energy and$k$point mesh. +The choice of these parameters is considered to reflect a reasonable treatment with respect to both, computational efficiency and accuracy, as will be shown in the next sections. +Furthermore, criteria concerning the choice of the potential and the exchange-correlation (XC) functional are being outlined. +Finally, the utilized parameter set is tested by comparing the calculated values of the cohesive energy and the lattice constant to experimental data. + +\subsection{Supercell} + +Describing defects within the supercell approach runs the risk of a possible interaction of the defect with its periodic, artificial images. +Obviously, the interaction reduces with increasing system size and will be negligible from a certain size on. +\begin{figure}[t] +\begin{center} +\includegraphics[width=0.7\textwidth]{si_self_int_thesis.ps} +\end{center} +\caption{Defect formation energies of several defects in c-Si with respect to the size of the supercell.} +\label{fig:simulation:ef_ss} +\end{figure} +To estimate a critical size the formation energies of several intrinsic defects in Si with respect to the system size are calculated. +An energy cut-off of \unit[250]{eV} and a$4\times4\times4$Monkhorst-Pack$k$-point mesh \cite{monkhorst76} is used. +The results are displayed in Fig. \ref{fig:simulation:ef_ss}. +The formation energies converge fast with respect to the system size. +Thus, investigating supercells containing more than 56 primitive cells or$112\pm1$atoms should be reasonably accurate. + +\subsection{Brillouin zone sampling} + +Throughout this work sampling of the BZ is restricted to the$\Gamma$point. +The calculation is usually two times faster and half of the storage needed for the wave functions can be saved since$c_{i,q}=c_{i,-q}^*$, where the$c_{i,q}$are the Fourier coefficients of the wave function. +As discussed in section \ref{subsection:basics:bzs} this does not pose a severe limitation if the supercell is large enough. +Indeed, it was shown \cite{dal_pino93} that already for calculations involving only 32 atoms energy values obtained by sampling the$\Gamma$point differ by less than \unit[0.02]{eV} from calculations using the Baldereschi point \cite{baldereschi73}, which constitutes a mean-value point in the BZ. +Thus, the calculations of the present study on supercells containing$108$primitive cells can be considered sufficiently converged with respect to the$k$-point mesh. + +\subsection{Energy cut-off} + +To determine an appropriate cut-off energy of the plane-wave basis set a$2\times2\times2$supercell of type 3 containing$32$Si and$32$C atoms in the 3C-SiC structure is equilibrated for different cut-off energies in the LDA. +% todo +% mention that results are within lda +\begin{figure}[t] +\begin{center} +\includegraphics[width=0.7\textwidth]{sic_32pc_gamma_cutoff_lc.ps} +\end{center} +\caption{Lattice constants of 3C-SiC with respect to the cut-off energy used for the plane-wave basis set.} +\label{fig:simulation:lc_ce} +\end{figure} +Fig. \ref{fig:simulation:lc_ce} shows the respective lattice constants of the relaxed 3C-SiC structure with respect to the cut-off energy. +As can be seen, convergence is reached already for low energies. +Obviously, an energy cut-off of \unit[300]{eV}, although the minimum acceptable, is sufficient for the plane-wave expansion. + +\subsection{Potential and exchange-correlation functional} + +To find the most suitable combination of potential and XC functional for the C/Si system a$2\times2\times2$supercell of type 3 of Si and C, both in the diamond structure, as well as 3C-SiC is equilibrated for different combinations of the available potentials and XC functionals. +To exclude a possibly corrupting influence of the other parameters highly accurate calculations are performed, i.e. an energy cut-off of \unit[650]{eV} and a$6\times6\times6$Monkhorst-Pack$k$-point mesh is used. +Next to the ultra-soft pseudopotentials \cite{vanderbilt90} {\textsc vasp} offers the projector augmented-wave method (PAW) \cite{bloechl94} to describe the ion-electron interaction. +The two XC functionals included in the test are of the LDA \cite{ceperley80,perdew81} and GGA \cite{perdew86,perdew92} type as implemented in {\textsc vasp}. + +\begin{table}[t] +\begin{center} +\begin{tabular}{l r c c c c c} +\hline +\hline + & & USPP, LDA & USPP, GGA & PAW, LDA & PAW, GGA & Exp. \\ +\hline +Si (dia) &$a$[\AA] & 5.389 & 5.455 & - & - & 5.429 \\ + &$\Delta_a$[\%] & \unit[{\color{green}0.7}]{\%} & \unit[{\color{green}0.5}]{\%} & - & - & - \\ + &$E_{\text{coh}}$[eV] & -5.277 & -4.591 & - & - & -4.63 \\ + &$\Delta_E$[\%] & \unit[{\color{red}14.0}]{\%} & \unit[{\color{green}0.8}]{\%} & - & - & - \\ +\hline +C (dia) &$a$[\AA] & 3.527 & 3.567 & - & - & 3.567 \\ + &$\Delta_a$[\%] & \unit[{\color{green}1.1}]{\%} & \unit[{\color{green}0.01}]{\%} & - & - & - \\ + &$E_{\text{coh}}$[eV] & -8.812 & -7.703 & - & - & -7.374 \\ + &$\Delta_E$[\%] & \unit[{\color{red}19.5}]{\%} & \unit[{\color{orange}4.5}]{\%} & - & - & - \\ +\hline +3C-SiC &$a$[\AA] & 4.319 & 4.370 & 4.330 & 4.379 & 4.359 \\ + &$\Delta_a$[\%] & \unit[{\color{green}0.9}]{\%} & \unit[{\color{green}0.3}]{\%} & \unit[{\color{green}0.7}]{\%} & \unit[{\color{green}0.5}]{\%} & - \\ + &$E_{\text{coh}}$[eV] & -7.318 & -6.426 & -7.371 & -6.491 & -6.340 \\ + &$\Delta_E$[\%] & \unit[{\color{red}15.4}]{\%} & \unit[{\color{green}1.4}]{\%} & \unit[{\color{red}16.3}]{\%} & \unit[{\color{orange}2.4}]{\%} & - \\ +\hline +\hline +\end{tabular} +\end{center} +\caption[Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials and XC functionals.]{Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials (ultr-soft PP and PAW) and XC functionals (LDA and GGA). Deviations of the respective values from experimental values are given. Values are in good (green), fair (orange) and poor (red) agreement.} +\label{table:simulation:potxc} +\end{table} +Table \ref{table:simulation:potxc} shows the lattice constants and cohesive energies obtained for the fully relaxed structures with respect to the utilized potential and XC functional. +As expected, cohesive energies are poorly reproduced by the LDA whereas the equilibrium lattice constants are in good agreement. +Using GGA together with the ultra-soft pseudopotential yields improved lattice constants and, more importantly, a very nice agreement of the cohesive energies to the experimental data. +The 3C-SiC calculations employing the PAW method in conjunction with the LDA suffers from the general problem inherent to LDA, i.e. overestimated binding energies. +Thus, the PAW \& LDA combination is not pursued. +Since the lattice constant and cohesive energy of 3C-SiC calculated by the PAW method using the GGA are not improved compared to the ultra-soft pseudopotential calculations using the same XC functional, this concept is likewise stopped. +To conclude, the combination of ultr-soft pseudopotentials and the GGA XC functional are considered the optimal choice for the present study. + +\subsection{Lattice constants and cohesive energies} + +As a last test, the equilibrium lattice parameters and cohesive energies of Si, C (diamond) and 3C-SiC are again compared to experimental data. +However, in the current calculations, the entire parameter set as determined in the beginning of this section is applied. +\begin{table}[t] +\begin{center} +\begin{tabular}{l r c c c c c} +\hline +\hline + & Si (dia) & C (dia) & 3C-SiC \\ +$a$[\AA] & 5.458 & 3.562 & 4.365 \\ +$\Delta_a$[\%] & 0.5 & 0.1 & 0.1 \\ +\hline +$E_{\text{coh}}$[eV] & -4.577 & -7.695 & -6.419 \\ +$\Delta_E[\%] & 1.1 & 4.4 & 1.2 \\ +\hline +\hline +\end{tabular} +\end{center} +\caption{Equilibrium lattice constants and cohesive energies of Si, C (diamond) and 3C-SiC using the entire parameter set as determined in the beginning of this section.} +\label{table:simulation:paramf} +\end{table} +Table \ref{table:simulation:paramf} shows the respective results and deviations from experiment. +A nice agreement with experimental results is achieved. +Clearly, a competent parameter set is found, which is capabale of describing the C/Si system by {\em ab initio} calculations. + + +% todo +% rewrite dft chapter +% ref for experimental values! + \section{Classical potential MD} fast method, amoun tof atoms ... @@ -9,24 +175,191 @@ fast method, amoun tof atoms ... \subsection{Temperature and volume control} + \subsection{Test calculations} Give cohesive energies of Si, C (Dia) and (3C-)SiC and the respective lattice parameters ... -\section{DFT calculations / MD} +\subsection{3C-SiC precipitate in crystalline silicon} +\label{section:simulation:prec} -\subsection{Supercell} +A spherical 3C-SiC precipitate enclosed in a c-Si surrounding is constructed. + + as it is expected from IBS experiments and from simulations that finally succeed in simulating the precipitation event. +On the one hand this sheds light on characteristic values like the radial distribution function or the total amount of free energy for such a configuration that is aimed to be reproduced by simulation. +On the other hand, assuming a correct alignment of the precipitate with the c-Si matrix, properties of such precipitates and the surrounding as well as the interface can be investiagted. +Furthermore these investigations might establish the prediction of conditions necessary for the simulation of the precipitation process. + +To construct a spherical and topotactically aligned 3C-SiC precipitate in c-Si, the approach illustrated in the following is applied. +A total simulation volumeV$consisting of 21 unit cells of c-Si in each direction is created. +To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary. +This corresponds to a spherical 3C-SiC precipitate with a radius of approximately 3 nm. +The initial precipitate configuration is constructed in two steps. +In the first step the surrounding silicon matrix is created. +This is realized by just skipping the generation of silicon atoms inside a sphere of radius$x$, which is the first unknown variable. +The silicon lattice constant$a_{\text{Si}}$of the surrounding c-Si matrix is assumed to not alter dramatically and, thus, is used for the initial lattice creation. +In a second step 3C-SiC is created inside the empty sphere of radius$x$. +The lattice constant$y$, the second unknown variable, is chosen in such a way, that the necessary amount of carbon is generated and that the total amount of silicon atoms corresponds to the usual amount contained in the simulation volume. +This is entirely described by the system of equations \eqref{eq:md:constr_sic_01} + +\frac{8}{a_{\text{Si}}^3}( +\underbrace{21^3 a_{\text{Si}}^3}_{=V} +-\frac{4}{3}\pi x^3)+ +\underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500} +=21^3\cdot 8 +\label{eq:md:constr_sic_01} +\text{ ,} + +which can be simplified to read + +\frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500 +\Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}} +\label{eq:md:constr_sic_02} + +and + +%x^3=\frac{16\pi}{5500 \cdot 3}y^3= +%\frac{16\pi}{5500 \cdot 3}\frac{5500 \cdot 3}{32 \pi}a_{\text{Si}}^3 +%\Rightarrow +y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}} +\text{ .} +\label{eq:md:constr_sic_03} + +By this means values of 2.973 nm and 4.309 \AA{} are obtained for the initial precipitate radius and lattice constant of 3C-SiC. +Since the generation of atoms is a discrete process with regard to the size of the volume the expected amounts of atoms are not obtained. +However, by applying these values the final configuration varies only slightly from the expected one by five carbon and eleven silicon atoms, as can be seen in table \ref{table:md:sic_prec}. +\begin{table}[!ht] +\begin{center} +\begin{tabular}{l c c c c} +\hline +\hline + & C in 3C-SiC & Si in 3C-SiC & Si in c-Si surrounding & total amount of Si\\ +\hline +Obtained & 5495 & 5486 & 68591 & 74077\\ +Expected & 5500 & 5500 & 68588 & 74088\\ +Difference & -5 & -14 & 3 & -11\\ +Notation &$N^{\text{3C-SiC}}_{\text{C}}$&$N^{\text{3C-SiC}}_{\text{Si}}$+ &$N^{\text{c-Si}}_{\text{Si}}$&$N^{\text{total}}_{\text{Si}}$\\ +\hline +\hline +\end{tabular} +\caption{Comparison of the expected and obtained amounts of Si and C atoms by applying the values from equations \eqref{eq:md:constr_sic_02} and \eqref{eq:md:constr_sic_03} in the 3C-SiC precipitate construction approach.} +\label{table:md:sic_prec} +\end{center} +\end{table} -\subsection[$k$-point sampling]{\boldmath$k$-point sampling} +After the initial configuration is constructed some of the atoms located at the 3C-SiC/c-Si interface show small distances, which results in high repulsive forces acting on the atoms. +Thus, the system is equilibrated using strong coupling to the heat bath, which is set to be$20\,^{\circ}\mathrm{C}$. +Once the main part of the excess energy is carried out previous settings for the Berendsen thermostat are restored and the system is relaxed for another 10 ps. -\subsection{Energy cutoff} +\begin{figure}[!ht] +\begin{center} +\includegraphics[width=12cm]{pc_0.ps} +\end{center} +\caption[Radial distribution of a 3C-SiC precipitate embeeded in c-Si at$20\,^{\circ}\mathrm{C}$.]{Radial distribution of a 3C-SiC precipitate embeeded in c-Si at$20\,^{\circ}\mathrm{C}$. The Si-Si radial distribution of plain c-Si is plotted for comparison. Green arrows mark bumps in the Si-Si distribution of the precipitate configuration, which do not exist in plain c-Si.} +\label{fig:md:pc_sic-prec} +\end{figure} +Figure \ref{fig:md:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration. +The Si-Si radial distribution for both, plain c-Si and the precipitate configuration show a maximum at a distance of 0.235 nm, which is the distance of next neighboured Si atoms in c-Si. +Although no significant change of the lattice constant of the surrounding c-Si matrix was assumed, surprisingly there is no change at all within observational accuracy. +Looking closer at higher order Si-Si peaks might even allow the guess of a slight increase of the lattice constant compared to the plain c-Si structure. +A new Si-Si peak arises at 0.307 nm, which is identical to the peak of the C-C distribution around that value. +It corresponds to second next neighbours in 3C-SiC, which applies for Si as well as C pairs. +The bumps of the Si-Si distribution at higher distances marked by the green arrows can be explained in the same manner. +They correspond to the fourth and sixth next neighbour distance in 3C-SiC. +It is easily identifiable how these C-C peaks, which imply Si pairs at same distances inside the precipitate, contribute to the bumps observed in the Si-Si distribution. +The Si-Si and C-C peak at 0.307 nm enables the determination of the lattic constant of the embedded 3C-SiC precipitate. +A lattice constant of 4.34 \AA{} compared to 4.36 \AA{} for bulk 3C-SiC is obtained. +This is in accordance with the peak of Si-C pairs at a distance of 0.188 nm. +Thus, the precipitate structure is slightly compressed compared to the bulk phase. +This is a quite surprising result since due to the finite size of the c-Si surrounding a non-negligible impact of the precipitate on the materializing c-Si lattice constant especially near the precipitate could be assumed. +However, it seems that the size of the c-Si host matrix is chosen large enough to even find the precipitate in a compressed state. -\subsection{Other parameters} +The absence of a compression of the c-Si surrounding is due to the possibility of the system to change its volume. +Otherwise the increase of the lattice constant of the precipitate of roughly 4.31 \AA{} in the beginning up to 4.34 \AA{} in the relaxed precipitate configuration could not take place without an accompanying reduction of the lattice constant of the c-Si surrounding. +If the total volume is assumed to be the sum of the volumes that are composed of Si atoms forming the c-Si surrounding and Si atoms involved forming the precipitate the expected increase can be calculated by + + \frac{V}{V_0}= + \frac{\frac{N^{\text{c-Si}}_{\text{Si}}}{8/a_{\text{c-Si of precipitate configuration}}}+ + \frac{N^{\text{3C-SiC}}_{\text{Si}}}{4/a_{\text{3C-SiC of precipitate configuration}}}} + {\frac{N^{\text{total}}_{\text{Si}}}{8/a_{\text{plain c-Si}}}} + +with the notation used in table \ref{table:md:sic_prec}. +The lattice constant of plain c-Si at$20\,^{\circ}\mathrm{C}$can be determined more accurately by the side lengthes of the simulation box of an equlibrated structure instead of using the radial distribution data. +By this a value of$a_{\text{plain c-Si}}=5.439\text{ \AA}$is obtained. +The same lattice constant is assumed for the c-Si surrounding in the precipitate configuration$a_{\text{c-Si of precipitate configuration}}$since peaks in the radial distribution match the ones of plain c-Si. +Using$a_{\text{3C-SiC of precipitate configuration}}=4.34\text{ \AA}$as observed from the radial distribution finally results in an increase of the initial volume by 0.12 \%. +However, each side length and the total volume of the simulation box is increased by 0.20 \% and 0.61 \% respectively compared to plain c-Si at$20\,^{\circ}\mathrm{C}$. +Since the c-Si surrounding resides in an uncompressed state the excess increase must be attributed to relaxation of strain with the strain resulting from either the compressed precipitate or the 3C-SiC/c-Si interface region. +This also explains the possibly identified slight increase of the c-Si lattice constant in the surrounding as mentioned earlier. +As the pressure is set to zero the free energy is minimized with respect to the volume enabled by the Berendsen barostat algorithm. +Apparently the minimized structure with respect to the volume is a configuration of a small compressively stressed precipitate and a large amount of slightly stretched c-Si in the surrounding. -Symmetry, spin, smearing method, real space projection, choice of ensemble and convergence criteria for electronic and ionic relaxation ... +In the following the 3C-SiC/c-Si interface is described in further detail. +One important size analyzing the interface is the interfacial energy. +It is determined exactly in the same way than the formation energy as described in equation \eqref{eq:defects:ef2}. +Using the notation of table \ref{table:md:sic_prec} and assuming that the system is composed out of$N^{\text{3C-SiC}}_{\text{C}}$C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by + + E_{\text{f}}=E- + N^{\text{3C-SiC}}_{\text{C}} \mu_{\text{SiC}}- + \left(N^{\text{total}}_{\text{Si}}-N^{\text{3C-SiC}}_{\text{C}}\right) + \mu_{\text{Si}} \text{ ,} +\label{eq:md:ife} + +with$E$being the free energy of the precipitate configuration at zero temperature. +An interfacial energy of 2267.28 eV is obtained. +The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of 29.93 \AA. +Thus, the interface tension, given by the energy of the interface devided by the surface area of the precipitate is$20.15\,\frac{\text{eV}}{\text{nm}^2}$or$3.23\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$. +This is located inside the eperimentally estimated range of$2-8\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$\cite{taylor93}. -\subsection{Test calculations / convergence tests} +Since the precipitate configuration is artificially constructed the resulting interface does not necessarily correspond to the energetically most favorable configuration or to the configuration that is expected for an actually grown precipitate. +Thus annealing steps are appended to the gained structure in order to allow for a rearrangement of the atoms of the interface. +The precipitate structure is rapidly heated up to$2050\,^{\circ}\mathrm{C}$with a heating rate of approximately$75\,^{\circ}\mathrm{C}/\text{ps}$. +From that point on the heating rate is reduced to$1\,^{\circ}\mathrm{C}/\text{ps}$and heating is continued to 120 \% of the Si melting temperature, that is 2940 K. +\begin{figure}[!ht] +\begin{center} +\includegraphics[width=12cm]{fe_and_t_sic.ps} +\end{center} +\caption{Free energy and temperature evolution of a constructed 3C-SiC precipitate embedded in c-Si at temperatures above the Si melting point.} +\label{fig:md:fe_and_t_sic} +\end{figure} +Figure \ref{fig:md:fe_and_t_sic} shows the free energy and temperature evolution. +The sudden increase of the free energy indicates possible melting occuring around 2840 K. +\begin{figure}[!ht] +\begin{center} +\includegraphics[width=12cm]{pc_500-fin.ps} +\end{center} +\caption{Radial distribution of the constructed 3C-SiC precipitate embedded in c-Si at temperatures below and above the Si melting transition point.} +\label{fig:md:pc_500-fin} +\end{figure} +Investigating the radial distribution function shown in figure \ref{fig:md:pc_500-fin}, which shows configurations below and above the temperature of the estimated transition, indeed supports the assumption of melting gained by the free energy plot. +However the precipitate itself is not involved, as can be seen from the Si-C and C-C distribution, which essentially stays the same for both temperatures. +Thus, it is only the c-Si surrounding undergoing a structural phase transition, which is very well reflected by the difference observed for the two Si-Si distributions. +This is surprising since the melting transition of plain c-Si is expected at temperatures around 3125 K, as discussed in section \ref{subsection:md:tval}. +Obviously the precipitate lowers the transition point of the surrounding c-Si matrix. +This is indeed verified by visualizing the atomic data. +% ./visualize -w 640 -h 480 -d saves/sic_prec_120Tm_cnt1 -nll -11.56 -0.56 -11.56 -fur 11.56 0.56 11.56 -c -0.2 -24.0 0.6 -L 0 0 0.2 -r 0.6 -B 0.1 +\begin{figure}[!ht] +\begin{center} +\begin{minipage}{7cm} +\includegraphics[width=7cm,draft=false]{sic_prec/melt_01.eps} +\end{minipage} +\begin{minipage}{7cm} +\includegraphics[width=7cm,draft=false]{sic_prec/melt_02.eps} +\end{minipage} +\begin{minipage}{7cm} +\includegraphics[width=7cm,draft=false]{sic_prec/melt_03.eps} +\end{minipage} +\end{center} +\caption{Cross section image of atomic data gained by annealing simulations of the constructed 3C-SiC precipitate in c-Si at 200 ps (top left), 520 ps (top right) and 720 ps (bottom).} +\label{fig:md:sic_melt} +\end{figure} +Figure \ref{fig:md:sic_melt} shows cross section images of the atomic structures at different times and temperatures. +As can be seen from the image at 520 ps melting of the Si surrounding in fact starts in the defective interface region of the 3C-SiC precipitate and the c-Si surrounding propagating outwards until the whole Si matrix is affected at 720 ps. +As predicted from the radial distribution data the precipitate itself remains stable. -Lattice parameter and cohesive energies as in former section! -Also, test convergence here for supercell size for some defects +For the rearrangement simulations temperatures well below the transition point should be used since it is very unlikely to recrystallize the molten Si surrounding properly when cooling down. +To play safe the precipitate configuration at 100 \% of the Si melting temperature is chosen and cooled down to$20\,^{\circ}\mathrm{C}$with a cooling rate of$1\,^{\circ}\mathrm{C}/\text{ps}\$.
+However, an energetically more favorable interface is not obtained by quenching this structure to zero Kelvin.
+Obviously the increased temperature run enables structural changes that are energetically less favorable but can not be exploited to form more favorable configurations by an apparently yet too fast cooling down process.

index 4c7c577..c6c3f2a 100644 (file)
% smaller captions ...
\usepackage[small,bf]{caption}

-% acronyms
-\usepackage{acronym}
-\acrodef{ALE}{atomic layer epitaxy}
-\acrodef{APB}{anti phase boundary}
-\acrodef{c-Si}{crystalline silicon}
-\acrodef{CVD}{chemical vapor deposition}
-\acrodef{HDTV}{high definition television}
-\acrodef{HREM}{high resolution transmission electron microscopy}
-\acrodef{IBS}{ion beam synthesis}
-\acrodef{LED}{light emitting diode}
-\acrodef{MBE}{molecular beam epitaxy}
-\acrodef{MEMS}{microelectromechanical system}
-\acrodef{PVT}{physical vapor transport}
-\acrodef{RT}{room temeprature}
-\acrodef{RTA}{rapid thermal annealing}
-\acrodef{RBS}{Rutherford backscattering spectrometry}
-\acrodef{SPE}{solid-phase epitaxy}
-\acrodef{SPEG}{solid-phase epitaxial regrowth}
-\acrodef{IR}{infrared}
-
% units
\usepackage{units}

% author & title
\author{Frank Zirkelbach}
-\title{Simulation study of the precipitation process of silicon carbide in carbon doped silicon}
+\title{Atomistic simulation study on the silicon carbide precipitation in silicon}

\begin{document}

\mainmatter{}
\include{intro}
-% reset all acronyms
-\acresetall
\include{sic}
\include{basics}
%\include{exp_findings}
\include{simulation}
\include{defects}
\include{md}
-\include{const_sic}
+%\include{const_sic}
%\include{results}
\include{summary_outlook}