1 \documentclass[prb,twocolumn,superscriptaddress,a4paper,showkeys,showpacs]{revtex4}
\r
2 \usepackage{graphicx}
\r
3 \usepackage{subfigure}
\r
5 \usepackage{booktabs}
\r
8 \usepackage{amsfonts}
\r
16 \title{Description of Defects in Carbon implanted Silicon -- a comparison of classical potentials and first principles methods}
\r
17 \author{F. Zirkelbach} \author{B. Stritzker}
\r
18 \affiliation{Experimentalphysik IV, Universit\"at Augsburg, D-86153 Augsburg, Germany}
\r
19 \author{K. Nordlund}
\r
20 \affiliation{Accelerator Laboratory, University of Helsinki, 00014 Helsinki, Finland}
\r
21 \author{J. K. N. Lindner}
\r
22 \affiliation{Experimentelle Physik, Universit\"at Paderborn, 33095 Paderborn, Germany}
\r
23 \author{W. G. Schmidt} \author{E. Rauls}
\r
24 \affiliation{Theoretische Physik, Universit\"at Paderborn, 33095 Paderborn, Germany}
\r
27 We present a comparative theoretical investigation of carbon interstitials in silicon.
\r
28 Calculations using classical potentials are put aside first principles density functional theory calculations of the geometries, formation and activation energies of the carbon dumbbell interstitial, showing the importance of a quantum mechanical description of this system.
\r
31 \keywords{point defects, migration, interstitials, first principles calculations, classical potentials }
\r
32 \pacs{ find out later... }
\r
35 % --------------------------------------------------------------------------------
\r
36 \section{Introduction}
\r
38 %Frank: Idea: description of 3C-SiC-precipitation in C-implanted silicon.\\
\r
39 % cite and describe briefly experimental work - why is this material important/better than other SiC). \\
\r
40 % Describe the precipitation process in brief.\\
\r
41 % Sum up literature where classical potentials have been used (a) successful, and (b) failed. Also add citations of Nordlunds papers. Not only on silicon or SiC!\\
\r
43 % there should be a short motivation for the material system!
\r
44 Silicon carbide (SiC) has a number of remarkable physical and chemical properties.
\r
45 The wide band gap semiconductor (2.3 eV - 3.3 eV) exhibiting a high breakdown field, saturated electron drift velocity and thermal conductivity in conjunction with its unique thermal and mechanical stability as well as radiation hardness is a suitable material for high-temperature, high-frequency and high-power devices\cite{wesch96,morkoc94}, which are moreover deployable in harsh and radiation-hard environments\cite{capano97}.
\r
46 % there are different polytpes with different properties and 3c-sic in special
\r
47 SiC, which forms fourfold coordinated covalent bonds, tends to crystallize into many different modifications, which solely differ in the one-dimensional stacking sequence of identical, close-packed SiC bilayers\cite{fischer90}.
\r
48 Different polytypes exhibit different properties, in which the cubic phase (3C-SiC) shows increased values for the thermal conductivity and breakdown field compared to other polytypes\cite{wesch96}, which is, thus, most effective for high-performance electronic devices.
\r
50 % (thin films of) 3c-sic can be produced by ibs
\r
51 Next to the fabrication of 3C-SiC layers by chemical vapor deposition (CVD) and molecular beam epitaxy (MBE) on hexagonal SiC\cite{powell90,fissel95,fissel95_apl} and Si\cite{nishino83,nishino87,kitabatake93,fissel95_apl} substrates, 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}.
\r
52 % maybe split CVD and MBE from IBS and explain remaining problems:
\r
53 % - on 6H-SiC: twin boundaries
\r
54 % - on Si: structural defects due to thermal conductivity and lattice mismatch
\r
55 Utilized and enhanced, ion beam synthesis (IBS) has become a promising method to form thin SiC layers of high quality exclusively of the 3C polytype embedded in and epitactically aligned to the Si host featuring a sharp interface\cite{lindner99,lindner01,lindner02}.
\r
56 % precipitation model
\r
57 However, only little is known on the SiC conversion in C implanted Si.
\r
58 High resolution transmission electron microscopy (HREM) studies\cite{werner96,werner97,lindner99_2} suggest the formation of C-Si dimers (dumbbells) on regular Si lattice sites, which agglomerate into large clusters indicated by dark contrasts and otherwise undisturbed Si lattice fringes in HREM.
\r
59 Once a critical radius of 2 nm to 4 nm is reached a topotactic transformation into a 3C-SiC precipitate occurs.
\r
60 The transformation is manifested by the disappearance of dark contrasts in favor of Moir\'e patterns due to the lattice mismatch of \unit[20]{\%} of the 3C-SiC precipitate and c-Si.
\r
61 The insignificantly lower Si density of SiC ($\approx \unit[4]{\%}$) compared to c-Si results in the emission of only a few excess Si atoms.
\r
62 % motivation to understand the precipitation and link to atomistic simulations
\r
63 A detailed understanding of the underlying processes will enable significant technological progress in 3C-SiC thin film formation and likewise offer perspectives for processes which rely upon prevention of precipitation events, e.g. the fabrication of strained pseudomorphic Si$_{1-y}$C$_y$ heterostructures\cite{strane96,laveant2002}.
\r
65 Atomistic simulations offer a powerful tool to study materials on a microscopic level providing detailed insight not accessible by experiment.
\r
66 Relevant structures consisting of $\approx 10^4$ atoms for the nanocrystal and even more atoms for a reasonably sized Si host matrix are too large to be completely described by high accuracy quantum mechanical methods.
\r
67 Directly modelling the dynamics of the processes mentioned above almost inevitably requires the atomic interaction to be described by less accurate though computationally more efficient classical potentials.
\r
68 The most common empirical potentials for covalent systems are the Stillinger-Weber\cite{stillinger85} (SW), Brenner\cite{brenner90}, Tersoff\cite{tersoff_si3} and environment-dependent interatomic (EDIP)\cite{bazant96,bazant97,justo98} potential.
\r
69 Until recently\cite{lucas10}, a parametrization to describe the C-Si multicomponent system within the mentioned interaction models did only exist for the Tersoff\cite{tersoff_m} and related potentials.
\r
70 Whether such potentials are appropriate for the description of the physical problem has, however, to be verified first by applying classical and quantum-mechanical methods to relevant processes that can be treated by both methods.
\r
71 For instance, by combination of empirical potential molecular dynamics (MD) and density functional theory (DFT) calculations, SW turned out to be best suited for simulations of dislocation nucleation processes\cite{godet03} and threshold displacement energy calculations\cite{holmstroem08} important in ion implantation, while the Tersoff potential yielded a qualitative agreement for the interaction of Si self-interstitials with substitutional C\cite{mattoni02}.
\r
72 An extensive comparison\cite{balamane92} concludes that each potential has its strengths and limitations and none of them is clearly superior to others.
\r
73 Despite their shortcomings these potentials are assumed to be reliable for large-scale simulations\cite{balamane92,huang95,godet03} on specific problems under investigation providing insight into phenomena that are otherwise not accessible by experimental or first principles methods.
\r
74 Remaining shortcomings have frequently been resolved by modifying the interaction\cite{tang95,mattoni2007} or extending it\cite{devanathan98_2} with data gained from ab initio calculations\cite{nordlund97}.
\r
76 In this work, the applicability of a Tersoff-like bond order potential\cite{albe_sic_pot} to some basic processes involved in the initially mentioned SiC precipitation mechanism is investigated by comparing results gained by classical and ab inito calculations.
\r
77 In the following a comparative investigation of density functional theory studies and classical potential calculations of the structure, energetics and mobility of carbon defects in silicon is presented.
\r
79 % --------------------------------------------------------------------------------
\r
80 \section{Methodology}
\r
82 The first-principles DFT calculations are performed with the plane-wave based Vienna Ab-initio Simulation Package (VASP)\cite{kresse96}.
\r
83 The Kohn-Sham equations are solved using the generalized-gradient XC-functional approximation proposed by Perdew and Wang (GGA-PW91)\cite{perdew86,perdew92}.
\r
84 The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials\cite{hamann79} as implemented in VASP\cite{vanderbilt90}.
\r
85 Throughout this work an energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis.
\r
86 Sampling of the Brillouin zone is restricted to the $\Gamma$-point.
\r
87 The defect structures and the migration paths are modelled in cubic supercells containing 216 Si atoms.
\r
88 The ions and cell shape are allowed to change in order to realize a constant pressure simulation.
\r
89 Spin polarization is fully accounted for.
\r
90 Only neutral defects are considered.
\r
92 % ------ Albe potential ---------
\r
93 %% Frank: Setup/short description of the potential ?
\r
94 For the classical potential calculations a supercell of 9 Si lattice constants in each direction consisting of 5832 Si atoms is used.
\r
95 A Tersoff-like bond order potential by Erhart and Albe\cite{albe_sic_pot} is utilized, which accounts for nearest neighbour interactions only realized by a cut-off function dropping the interaction to zero in between the first and second next neighbour distance.
\r
96 Constant pressure simulations are realized by the Berendsen barostat\cite{brendsen84}.
\r
97 Structural relaxation in the MD run is achieved by the verlocity verlet algorithm\cite{verlet67} and the Berendsen thermostat\cite{berendsen84} with a time constant of \unit[1]{fs} resulting in direct velocity scaling and the temperature set to zero Kelvin.
\r
101 % ... wer sagt das ...
\r
102 %After ion implantation, carbon interstitials are the most common defects in the silicon sample.
\r
103 According to the assumed SiC precipitation model described in the introductary part, carbon interstitial defects form and agglomerate into large clusters.
\r
104 Thus, it is of crucial importance to investigate the various possible structures of carbon defects and the mobility of the lowest energy, hence most probable, defect configuration in crystalline silicon.
\r
105 %Their mobility is the crucial quantity to be investigated.
\r
106 %We thus started our comparative investigations by comparing the stability and the mobility of an isolated carbon interstitial in silicon bulk in the various possible geometries it can take.
\r
108 \subsection{Carbon interstitials in various geometries}
\r
110 Table~\ref{tab:defects} summarizes the formation energies of defect structures for the Erhart/Albe and VASP calculations performed in this work as well as further results from literature.
\r
111 The formation energies are defined in the same way as in the articles used for comparison\cite{tersoff90,dal_pino93} chosing SiC as a reservoir for the carbon impurity.
\r
112 Relaxed geometries are displayed in Fig.~\ref{fig:defects}.
\r
113 Astonishingly there is only little literature present to compare with.
\r
115 \begin{ruledtabular}
\r
116 \begin{tabular}{l c c c c c c}
\r
119 & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
\r
121 Erhart/Albe & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
\r
122 VASP & unstable & unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
\r
123 Tersoff\cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
\r
124 ab initio\cite{dal_pino93,capaz94} & - & - & x & - & 1.89 \cite{dal_pino93} & x+2.1 \cite{capaz94} \\
\r
125 % there is no more ab initio data!
\r
130 \caption{Formation energies of carbon point defects in crystalline silicon determined by classical potential and ab initio methods. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal, B the bond-centered and S the substitutional interstitial configuration. The dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations obtained by classical potential MD are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\r
131 \label{tab:defects}
\r
134 \begin{minipage}[t]{0.32\columnwidth}
\r
135 \underline{Tetrahedral}\\
\r
136 \includegraphics[width=\columnwidth]{01.eps}
\r
138 \begin{minipage}[t]{0.32\columnwidth}
\r
139 \underline{Hexagonal}\\
\r
140 \includegraphics[width=\columnwidth]{02.eps}
\r
142 \begin{minipage}[t]{0.32\columnwidth}
\r
143 \underline{\hkl<1 0 0> dumbbell}\\
\r
144 \includegraphics[width=\columnwidth]{03.eps}
\r
146 \begin{minipage}[t]{0.32\columnwidth}
\r
147 \underline{\hkl<1 1 0> dumbbell}\\
\r
148 \includegraphics[width=\columnwidth]{04.eps}
\r
150 \begin{minipage}[t]{0.32\columnwidth}
\r
151 \underline{Substitutional}\\[0.05cm]
\r
152 \includegraphics[width=\columnwidth]{05.eps}
\r
154 \begin{minipage}[t]{0.32\columnwidth}
\r
155 \underline{Bond-centered}\\
\r
156 \includegraphics[width=\columnwidth]{06.eps}
\r
158 \caption{Configurations of carbon point defects in silicon. The silicon/carbon atoms and the bonds (only for the interstitial atom) are illustrated by yellow/grey spheres and blue lines. Bonds are drawn for atoms located within a certain distance and do not necessarily correspond to chemical bonds.}
\r
159 \label{fig:defects}
\r
162 Substitutional carbon (C$_{\text{sub}}$) in silicon, which is in fact not an interstitial defect, is found to be the lowest configuration with regard to energy for all potential models.
\r
163 VASP calculations performed in this work are in good agreement with results obtained by classical potential simulations by Tersoff\cite{tersoff90} and ab initio calculations done by Dal Pino et~al\cite{dal_pino93}.
\r
164 However, the Erhart/Albe potential dramatically underestimtes the C$_{\text{sub}}$ formation energy, which is a definite drawback of the potential.
\r
166 Except for the Tersoff potential the \hkl<1 0 0> dumbbell is the energetically most favorable interstital configuration, in which the C and Si dumbbell atoms share a Si lattice site.
\r
167 This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94} and experimental\cite{watkins76,song90} investigations.
\r
168 Tersoff as well, considers the \hkl<1 0 0> to be the ground state configuration and believes an artifact due to the abrupt C-Si cut-off used in the potential to be responsible for the small value of the tetrahedral formation energy\cite{tersoff90}.
\r
169 A qualitative difference is observed investigating the dumbbell structures.
\r
170 While the C-Si distance of the dumbbell atoms is almost equal for both methods, the vertical position of the dumbbell inside the tetrahedra spanned by the four next neighboured Si atoms differs significantly.
\r
171 The dumbbell based on the Erhart/Albe potential is almost centered around the regular Si lattice site as can be seen in Fig. \ref{fig:defects} whereas for VASP calculations it is translated upwards with the C atom forming an almost collinear bond to the two Si atoms of the top face of the tetrahedra and the bond angle of the Si dumbbell atom to the two bottom face Si atoms approaching \unit[120]{$^\circ$}.
\r
172 This indicates predominant sp and sp$^2$ hybridization for the C and Si dumbbell atom respectively.
\r
173 Obviously the classical potential is not able to reproduce the clearly quantum-mechanical character of bonding.
\r
174 % empirical potential adjusts according to minimum of angular function, no QM!
\r
177 % pick up again later, that this is why erhart/albe is more promising for the specific problem under investigation
\r
182 While the Erhart/Albe potential predicts ... as stable, DFT does not. ...(further comparisons, trend "too high/low" E-formation,...)...
\r
184 To conclude, discrepancies are observed between the results from classical potential calculations and those obtained from first principles.
\r
185 Nevertheless, both methods predict the \hkl<1 0 0> dumbbell configuration to be most stable.
\r
187 \subsection{Mobility}
\r
188 A measure for the mobility of the interstitial carbon is the activation energy for the migration path from one stable
\r
189 position to another. The stable defect geometries have been discussed in the previous subsection. We now investigate
\r
190 the migration from the most stable structure (...should be named somehow...) on one site of the silicon host lattice to
\r
191 a neighbored site.
\r
192 On the lowest energy path (first principles), the carbon atom starts to move along (110)..(check that!)... The center of the line connecting
\r
193 initial and final structure has been found to be a local minimum and not a saddle point as could be expected. The two
\r
194 saddle points shortly before and behind this local minimum are slightly displaced out of the (110) plane by ... {\AA}. ..(check that!)..
\r
195 This path is not surprising -- a similar behavior was e.g. found earlier for the carbon split interstitial \cite{rauls03a} and the phosphorus
\r
196 interstitial \cite{rauls03b,gerstmann03} in SiC. However, an interesting effect is the change of the spin state from zero at the (110) dumb bell
\r
197 configuration to one at the local minimum. By this, the energy of the local minimum is lowered by 0.3 eV (... check it!!..).
\r
199 %\includegraphics[width=1.0\columnwidth]{path-DFT.eps}
\r
200 %\caption{\label{fig:path-DFT} Energy of the carbon interstitial during migration from ... to ... calculated from first principles. The
\r
201 % activation energy of 0.9 eV (?) agrees well with experimental findings (0.7-0.9 eV?). }
\r
203 Fig.\ref{fig:path-DFT} shows the energy along this lowest energy migration path. The activation energy of 0.9 eV (?) agrees well
\r
204 with experimental findings (0.7-0.9 eV?).
\r
206 Calculations with the Albe potential yield a different picture.
\r
208 %\includegraphics[width=1.0\columnwidth]{path-Albe.eps}
\r
209 %\caption{\label{fig:path-Albe} Energy of the carbon interstitial during migration from ... to ... calculated using the classical potential
\r
210 % method. Here, the activation energy is 2.2 eV (?). }
\r
212 Fig.\ref{fig:path-Albe} shows the energy along the lowest energy migration path found by this method. The activation energy of 2.2 eV (?)
\r
213 is way too high to explain the experimental findings (0.7-0.9 eV?). (...further discussion...)
\r
215 \section{Discussion}
\r
216 The first principles results are in good agreement to previous work on this subject \cite{joannopoulos,xyz} (...add some references!...). With an
\r
217 activation energy of 0.9 eV, the carbon interstitial can be expected to be mobile at temperatures in the range of... (?).
\r
218 The description of the same processes obviously fails if we use the classical potential method.
\r
219 Already the geometry of the most stable dumb bell configuration differs considerably from that of the first principles calculated
\r
220 structure. (..... add description, the two main angles and bond lengths and an explanation...)
\r
221 Formation energies are throughout too high... (...reason?...)
\r
223 A reason for this failure of the classical description is most likely... (cut-off, neglect of quantum mechanical effects,...)
\r
227 In summary, we have shown that ab initio calculations are very close to the results expected from experimental data.
\r
228 Furthermore, they agree well with other theoretical results. (...some results - later...) The classical potentials, however, fail to describe the
\r
229 selected processes. This has been shown to have two reasons, i.e. the artificial cut-off of the next nearest neighbor
\r
230 interaction on the one hand, on the other hand the quantum mechanical effects which are crucial in the problem under study.
\r
232 % ----------------------------------------------------
\r
233 \section*{Acknowledgment}
\r
234 %The calculations were done using grants of computer time from the
\r
235 %Paderborn Center for Parallel Computing (PC$^2$) and the
\r
236 %H\"ochstleistungs-Rechenzentrum Stuttgart. The Deutsche
\r
237 %Forschungsgemeinschaft is acknowledged for financial support.
\r
238 One of us (F.Z.) wants to acknowledge financial support by the Bayerische Forschungsstiftung (DPA-61/05).
\r
240 % --------------------------------- references -------------------
\r
241 \bibliography{../../bibdb/bibdb}{}
\r
242 %\begin{thebibliography}{99}
\r
243 %\bibitem{kresse96} G. Kresse and J. Furthm\"uller,
\r
244 % Comput. Mater. Sci. {\bf 6}, 15 (1996).
\r
245 %\bibitem{perdew92} J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais,
\r
246 % Phys. Rev. B {\bf 46}, 6671 (1992).
\r
247 %\bibitem{ceperley80} D. M. Ceperley and B. J. Alder,
\r
248 % Phys. Rev. Lett. {\bf 45}, 556 (1980).
\r
249 %\bibitem{perdew81} J. P. Perdew and A. Zunger,
\r
250 % Phys. Rev. B {\bf 23}, 5048 (1981).
\r
251 %\bibitem{bloechel94} P. E. Bl\"ochl,
\r
252 % Phys. Rev. B {\bf 50}, 17953 (1994).
\r
253 %\bibitem{kresse99} G. Kresse and D. Joubert,
\r
254 % Phys. Rev. B {\bf 59}, 1758 (1999).
\r
255 %\bibitem{monk76} H. J. Monkhorst and J. D. Pack,
\r
256 % Phys. Rev. B {\bf 13}, 5188 (1976).
\r
257 %\bibitem{albe} Albe potential
\r
258 %\bibitem{stillinger} Stillinger-Weber potential
\r
259 %\bibitem{joannopoulos} Joannopoulos
\r
260 %\bibitem{xyz} who else?
\r
261 %\bibitem{rauls03a} E. Rauls, A. Gali, P. De´ak, and Th. Frauenheim, Phys. Rev. B, 68, 155208 (2003).
\r
262 %\bibitem{rauls03b} E. Rauls, U. Gerstmann, H. Overhof, and Th. Frauenheim, Physica B, Vols. 340-342, p. 184-189 (2003).
\r
263 %\bibitem{gerstmann03} U. Gerstmann, E. Rauls, Th. Frauenheim, and H. Overhof, Phys. Rev. B, 67, 205202, (2003).
\r
264 %\end{thebibliography}
\r
projects / lectures / latex.git / blob