Formation and migration energy of native defects in silicon carbide from first principles: an overview

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1 Formation and migration energy of native defects in silicon carbide from first principles: an overview Guido Roma 1,2, a, Fabien Bruneval 1, Ting Liao 1,3, Olga Natalia Bedoya Martínez 1, and Jean-Paul Crocombette 1 1 CEA, DEN, Service de Recherches de Métallurgie Physique, F Gif sur Yvette, France 2 Institut for Analytical and Inorganic Chemistry, Johannes Gutenberg Universität, Mainz, Germany 3 Centre for Computational Molecular Science,The University of Queensland, St. Lucia, Australia. a guido.roma@cea.fr Keywords: Defects, diffusion, silicon carbide, ab-initio, Density Functional Theory, kinetics, irradiation effects. Abstract. We present here an overview of native point defects calculations in silicon carbide using Density Functional Theory, focusing on defects energetics needed to understand self-diffusion. The goal is to assess the availability of data that are necessary in order to perform kinetic calculations to predict not only diffusion properties but also the evolution of defect populations under or after irradiation. We will discuss the spread of available data, comment on the main defect reactions that should be taken into account, and mention some of the most recent promising developments. Introduction Silicon carbide is used for its structural properties in a variety of fields, including nuclear applications, where high temperature and irradiation make the knowledge of defect kinetics of the utmost importance in order to understand the evolution of the properties of the material. However, most of the studies of point defects at the atomic scale were driven by its use as a functional material, namely as a wide band gap semiconductor. Density Functional Theory (DFT) has been, since the early eighties, the privileged approach at the quantum level to study the properties of solids from first principles. It is a mean field approach based on the electronic charge density. In almost all implementations of DFT the density is obtained from one-electron wave functions which are solutions of Schrödinger-like equations, the so called Kohn-Sham equations. The numerical solution of these equations is particularly demanding from the computational point of view when the number of atoms to be described is large, as the computational load scales with the third power of the number of basis functions. For this reason DFT calculations were applied to describe isolated point defects in solids only in the second half of the eighties, especially for semiconductors, driven by their importance in understanding the physics of devices for microelectronic applications. In this framework appeared the first application of DFT to intrinsic point defects in silicon carbide [1]. Then, at the end of the nineties (starting from 1997), several papers appeared dealing with first principles calculation of defect properties in SiC. This flourishing can be followed through the proceedings of International and European Conferences on Silicon Carbide and related Materials, initially published as isolated volumes and since 1997 as special issues of Materials Science Forum (or Materials Science and Engineering). Since then a few groups kept on working on point defects in SiC from first principles. Many works, especially in the last few years, have focused on spectroscopic signatures of defects, like EPR 1 spectra, local vibrational modes or positron annihilation lifetimes. Reviews discussing in more details these aspects can be found in Refs. [2,3]. Here, at variance, the focus will be on formation energies and energy barriers in various charge states. In spite of valuable works like the one by Bockstedte and co-workers [4], giving a comprehensive picture of formation and migration 1 Electron Paramagnetic Resonance

2 energies, which are of course crucial for the study of diffusion and kinetic processes, the knowledge of defect energetics still lacks completeness and/or reliability, as we will show. We will preferentially discuss results for cubic SiC, but we will cite also results for hexagonal polytypes as, in many cases, formation and migration energies are very similar, unless the defect has occupied levels close to the conduction band. DFT is a ground state technique and, as such, is perfectly suited to the study of defect energetics. However, it relies on approximations. A well known feature of the common approximations, like the LDA 2 or the GGA 3, is the fact that they are unable to describe the energy jump associated with a change in the number of electrons in the system [5]. A known consequence is the serious underestimation of the band gap in semiconductors. This point is a strong limitation when one has to cope with the formation energy of charged defects in semiconductors or insulators, because their formation energy is obtained through the energy difference with a reference charge state [6]. These energy differences, when taken at the same atomic configuration, are related to the position of oneelectron energy levels introduced by the point defect in the band gap of the material (shortly, defect levels). On the other hand, if the energy difference is taken between fully relaxed configurations for each charge, it is related to the so called "adiabatic" Charge Transition Level (CTL) of the defect. More advanced functionals, or more sophisticated many body techniques, have to be used to overcome these limitations, but at the expense of a much higher computational load. A proposal is to overcome the problem by using hybrid density-functionals, in which a variable fraction of the standard LDA or GGA exchange energy is replaced with Hartree-Fock exchange, leaving unchanged the LDA or GGA correlation part; they are known to give much better results for the band gap of several insulating/semiconducting materials as well as for excitation energies of several molecules. Applications to defects in SiC are relatively recent; as calculations with these functionals are computationally much more demanding than LDA or GGA, the corrections where calculated at the LDA (or GGA) relaxed geometry [7,8]. According to [8], which focuses on CTLs for defects in various insulating and semiconducting materials, the LDA or GGA CTLs (and so the formation energy) can be corrected, at least for defects who are not shallow donors or acceptors, simply by aligning the bulk valence band top (VBT) position of the LDA (GGA) calculation with that obtained by a hybrid calculation for the perfect bulk material. A different approach consists in correcting the LDA or GGA defect levels by comparison with quasi-particle energies based on many-body perturbation theory, namely using the so-called GW approximation; applications of this emerging technique to point defects can be found in refs. [9-11]. Apart from the reliability of the approximations to calculate total energies, it should be recalled that most of the available works neglect any entropic contributions to the free energy of defects. We recently discussed in detail this point by calculating with DFT the vibrational entropies of formation and migration of self-interstitials [12] in cubic SiC; the results suggest that the empirical formula known as the Meyer-Neldel rule can provide reasonable estimations. The paper is organized as follows: the first section will give an overview of defect structures and formation energy calculations. This section is further subdivided in four subsections devoted to vacancies, interstitials, antisites, and defect complexes, respectively. Section two will describe migration energies and other relevant energy barriers. A short summary will conclude. Basic native defects: vacancies, interstitials, antisites The first DFT calculations of point defects in silicon carbide [1], dating back to 1988, were performed with relatively small supercells (16 and 32 atoms), limited basis sets (plane waves with energy up to 28 ry) and further approximations, namely for the relaxation of atomic positions. Moreover, they were limited to high symmetry configurations. The results were inaccurate, however 2 Local Density Approximation 3 Generalized Gradient Approximation

3 it was already clear that vacancies and antisites could be relatively abundant with respect to selfinterstitials. The authors dared to approach some defect complexes and could predict that antisite pairs and divacancies were bound. The predicted CTLs for the carbon vacancy are also not so different from more recent results. The usual expression of the formation energy of a charged defect [6] depends on the electronic chemical potential; for simplicity, as different authors use different conventions for presenting their results, we choose to present the formation energy of neutral defects and the CTLs, that allow one to recover the formation energies of charged species if one recalls that the formation energy for a defect i of charge Q can be expressed as E f (Q)=E f (0)+Q[µ e -ε(0/q)], (1) where µ e and ε(0/q) are respectively the electron chemical potential (Fermi energy) and the CTL between charge 0 and charge Q. As for the atomic chemical potential, which affects the formation energy of all non-stoichiometric defects, we will always show results in silicon-rich conditions. Vacancies. Silicon and carbon vacancies were studied in detail at the end of the nineties in a few studies which also underlined the role of spin in determining the ground state configuration [13-15]. Carbon vacancies are expected to be much more abundant than silicon vacancies (this remains true in carbon-rich conditions). Table 1 summarises some results in the literature on the formation energy of vacancies. The spread in formation energies, especially for charged vacancies, can reach 10%. The silicon vacancy is known to be metastable and to turn into the carbon vacancy-carbon antisite pair (V C -C Si ). In a recent work [11] we showed, using a more advanced approach, that LDA or GGA are even qualitatively wrong on the relative stability of charge states. For example, the charge state of the silicon vacancy passes from 2+ to 1- at a charge transition level of 0.84 ev above VBT, while all other works cited in Table 1 predict a certain range of stability for the neutral defect. Interstitials. If we exclude the pioneering work by Wang et al. [1], works on interstitial defects started to appear later than those on vacancies [16,17]. In 2003 was published the comprehensive work by Bockstedte et al. [4] covering formation and migration energetics of basic intrinsic defects in silicon carbide (vacancies, interstitials, antisites), shortly followed, in 2004, by a paper from the Helsinki group [18] devoted to carbon and silicon interstitials in silicon carbide. Both carbon and silicon interstitials have high formation energies. The most stable configuration for the carbon interstitial was found to be the dumbbell along the <100> direction on a carbon site (I C sp<100>), followed by the I C spsi<110> (split interstitial on a silicon site), with formation energy a few tenth of an ev higher. The I C sp<100> configuration is almost ideal in the +2 state (expected in p-type SiC) but is quite distorted in the neutral and negative state [4]. Comparing the results (see Table 2) it is clear that the charge transition levels are quite sensitive to image charge interaction corrections (or lack of them) and on the details of the calculations for charge states (supercell size, k-point sampling). However, it seems that carbon interstitials can be positively charged (1+ or 2+) for Fermi energies corresponding to relatively low p-type doping conditions. The conclusions of [4] and [18] are in disagreement concerning the stability of configurations in the neutral state for the silicon interstitial: the first finds the silicon split interstitials in the <110> direction (I Si sp<110>) as the most stable one, with a formation energy of 8.6 ev, while the second finds the tetrahedral carbon coordinated silicon interstitial (I Si TC) as the most stable one, with a much lower formation energy (less than 6 ev). This major discrepancy has to be ascribed to the insufficient k-point sampling of the second work, leading to a gross error in formation energies. There are disagreements between the two papers also concerning the stability of positive charge states for p-type SiC, where the first predicts the dominant species to be the I Si TC in charge state 4+, while Lento and co-workers don t go beyond the charge state 2+. Apart from the fact that corrections for image interactions for charge states as high as 4+ are large (and questionable), we would like to stress a point that both papers almost overlook: I Si TC in the neutral state shows up as metallic in LDA or GGA (there are two doubly occupied electronic levels in the conduction band).

4 This fact, on one side, calls for a better description of the exchange correlation potential for these configurations; on the other it makes the convergence with k-points and cell size very slow, as we have recently been pointed out [21]. The formation energies found for the silicon interstitials by the two mentioned groups are compared in Table 2. For comparison: our calculations gave 8.3 ev for I Si sp<110> and 8.1 for I Si TC using a large supercell [21]. The 0/4+ CTL is at 1.4 ev above the VBT, i.e. close to the DFT band gap, as expected from the filling of the conduction band in the neutral state. Table 1: Formation energies for neutral vacancies and their charge transition levels according to various authors. The values are for the 3C-polytype in silicon-rich conditions. Units are ev. V C 0 0/+ 0/++ +/++ 0 0/++ 0/+ 0/- -/-- Wang88 [1] Zywietz99 [13] Wimbauer97 [14] Bockstedte03 [4] Bernardini04 [19] Torpo01 [20] Torpo99 [15] Bruneval11 [11] This drawback of standard DFT-LDA/GGA supercell calculations is common to other defects in SiC. The results for the carbon interstitial are, conversely, in relatively good agreement between the two papers, giving the split configuration in a strongly distorted <100> direction as the most stable one. Both papers apply the widely used monopole correction for charged defects, whose reliability, however, is still a matter of debate [22]. We ought to mention a further paper, which dealt with all simple native defects [19] and disagrees with the two previous ones concerning interstitials, while for vacancies and antisites the results are closer to those obtained by others, as can be seen from Tables 1-4. Bernardini and co-workers report a transition from the I C sp<100> configuration in charge 2+ to the I C spsi<110> configuration at charge 1+ then to neutral, then to -1 at, respectively, 0.81,1.81 and 2.16 ev above the VBT, for carbon interstitials. For silicon interstitials, if we understand correctly their somewhat oscillating notation, they find, as in [4] and [18], the I Si TC 4+ as most stable configuration for p-type and compensated SiC; for n-type SiC they find, in sequence, +3,+2 and +1 charged states of the I Si spc<110> configurations, up to the neutral one, whose formation energy is Table 2: Formation energies for neutral carbon interstitials and CTLs according to various authors. The values are for the 3C-SiC in Si-rich conditions. Units are ev. V Si I C sp<100> I C spsi<110> 0 0/++ 0/+ +/++ 0/- 0 0/++ 0/+ +/++ 0/- -/-- Lento04 [18] Lento04 [18] with Madelung correction Bockstedte03 [4] Bernardini04 [19] Gali03 [23] reported to be as high as ev. These findings are probably linked to the peculiar treatment of defects having electronic levels close to (or hybridised with) the conduction band. Indeed Bernardini and co-workers recognised, on one side, the need for a sufficient sampling of the BZ and the problem of defect level dispersion even in a relatively large supercell, but, on the other, they 4 This figure refers to a transition from the I C++ sp<100> to the I C0 spsi<110> configuration.

5 claimed that the correct filling of the electronic levels was determined according to their ordering at the Γ point, due to the correct degeneracy of defect levels. They adopted then a somewhat cumbersome, though interesting, procedure whose influence on the final results has probably not been sufficiently analysed. The concern is that the final formation energy is actually not obtained from the DFT ground state. Table 3: Formation energies for neutral silicon interstitials and their charge transition levels according to various authors. The values are for 3C-SiC in Si-rich conditions. Units are ev. I Si <sp110> 0 ++/+ +/0 0 0/++ 0/+ 0/- 0/-- 0/++++ Lento04 [18] Lento04 [18] with Madelung correction Bockstedte03 [4] I Si TC Antisites. The relatively low formation energy for antisites, after having qualitatively been predicted by Wang et al. [1] was confirmed with more accurate calculations by Torpo and coworkers [24], in spite of the still relatively small supercells (32 atoms) and plane waves cutoff (20 Ry). The formation energy for both neutral antisites was predicted to vary around 4 ev according to Si-rich/C-rich conditions. In the latter paper, where no charge interaction correction was applied, several charge states (from +4 to neutral) were reported as possible, with two close pairs of transition levels around 0.2 and 0.4 ev. For the carbon antisite no defect levels in the band gap were found. The transition levels for the 2H polytype were found to be higher in energy. The results for Si C in the 3C polytype were considered as strong evidence for the identification of this defect as the H centre, according to the previously measured DLTS 6 spectra. Later, the same authors [20] applied Madelung corrections to defect energies and found as only stable levels the 2+ and neutral ones, with a transition level around 0.2 ev above the VBT. Apart from these discrepancies, concerning relatively strong p-type conditions, various authors are generally in agreement on the fact that antisites are neutral on wide doping ranges. Table 4: Formation energies and charge transition levels for antisites and for the carbon vacancy-carbon antisite complex (V C -C Si ) according to various authors. The values are for the 3C-polytype in silicon-rich conditions. Units are ev. C Si Si C V C -C Si /++ ++/0 0 ++/+ +/0 0/- Mattausch05 [25] Bockstedte03 [4] Bernardini04 [19] Torpo98 [24] Bruneval11 [11] Defect complexes. Several defect complexes in silicon carbide have high binding energy. The divacancy has been shown to be strongly bound [26,27] so that the formation energy of a divacancy is close to that of a single silicon vacancy. The antisite pair is also strongly bound [28-31] (E b >2 ev). Another complex featuring antisites is the V C -C Si complex, which we mentioned in the subsection about vacancies, because it is the stable form of a silicon vacancy. Several authors dealt with this complex using standard DFT LDA or GGA approach; however, at least for certain charge states, the formation energies are strongly affected by the wrong position of defect levels in the band 5 Transition from I Si TC 4+ and I Si sp<110> in the neutral state 6 Deep Level Transient Spectroscopy

6 gap, as we have recently shown using the GW approach [11]. These recent results are compared with LDA or GGA calculations in Table 4. Various kinds of carbon clusters were studied theoretically in detail in 3C- and 4H-SiC [23, 32-34]. It was shown that the aggregation of carbon interstitials with carbon antisites can lead to various bound configurations. In particular two, three or even four carbon atoms can substitute one silicon atom forming very stable (C n ) Si (n=2,3,4) structures, called di- (or tri- or tetra-) carbon antisites. The binding energy of these structures are high: from 3.9 to 5 ev, according to the charge state, for the (C 2 ) Si, and further energy is gained when adding further carbon atoms. Other carbon clusters on carbon sites were also found to be more or less strongly bound; the most stable configuration for a di-carbon interstitial is reported to be higher than 5 ev (ladder structures), while metastable (pentagon and ring) structures have binding energies around 3 ev [34]. Similar interstitial clusters were studied by Mattausch and co-workers [32], as well as pairs or four-membered rings of dicarbon antisites, [(C 2 ) Si ] 2 and [(C 2 ) Si ] 4. Silicon clusters did not raise as much interest as carbon ones; however, a recent work [35] dealt with their stability and that of mixed carbon-silicon clusters in 4H-SiC; actually this paper is limited to di-interstitials. Silicon di-interstitials have been found to have maximum binding energy slightly higher than 3 ev, while I C -I Si complexes can be bound by more than 4 ev. We have obtained similar results that are briefly discussed in an internal report [36]. We have also found a bound configuration for a neutral tri-interstitial whose binding energy is 1 ev per interstitial. Defect kinetics: migration and recombination of point defects The knowledge of energy barriers for migration or for defect reactions is of course crucial to understand kinetic properties. Before the cited work by Bockstedte and co-workers [4] almost no work was devoted to migration properties of point defects in SiC. We should, however, cite previous preliminary works by the same group [37,38], a work on the mechanisms of formation of antisite pairs [39], as well as a work on vacancy migration published in 2003 [39]. The comprehensive study of migration barriers in [4] showed, first of all, that vacancies have much higher migration energies than interstitials: higher than 3 ev for the formers in the neutral state, around 1 ev for the latter (0.5 for I C, 1.4 for I Si ). Another remarkable finding is the strong variation of the migration energy with the charge state; indeed the migration energy for the carbon vacancy is raised by almost 2 ev going from the neutral to the 2+ charged state, while the silicon vacancy finds its migration barrier reduced by 1 ev when its charge goes from neutral to 2-. Neutral interstitials are reported to have their lowest migration barriers in the neutral state, except for the I Si TC configuration, which is expected to have hardly any energy barrier to migration in the 2+ and 3+ charge states. The behaviour looks similar to the so-called Bourgoin mechanism for accelerated diffusion, where the saddle point for a charge state corresponds to the stable one for another. Here, in reality, the I Si TC configuration is an intermediate position for the migration of the I Si sp<110> one, but not actually the saddle point. Coming back to the neutral state, more recently we showed that the migration barrier of I Si <sp110> can be as low as 0.8 ev using a slightly different path [21]; we should remind that all mentioned results, at least for silicon interstitials in the neutral state, are certainly seriously affected from the LDA/GGA band gap error. Almost at the same time as the work by Bockstedte and co-workers two works were published [39,40] about the migration of vacancies. The migration energies and paths were calculated with SCC-DFTB 7 and constrained relaxations; the vacancy assisted antisite migration is discussed and entropy effects on diffusion are evaluated through the calculation of vibrational entropies. The barriers for vacancy migration were found to be larger than those found by Bockstedte (especially for the carbon vacancy) due, probably, to the larger value of the band gap for this model. The authors claim that this approach is similar to applying a scissor correction to LDA results. 7 Self Consistent Charge DFT based Tight Binding approach

7 Diffusion properties of defects are crucial in understanding the evolution of defect populations under irradiation and to which extent defects created by irradiation can be healed by recombinations or other defect reactions. In a recent paper [41] we discussed previous works dealing with the annealing of vacancy and interstitials through their mutual recombination [42-44]; we showed that disagreement can derive from the complexity of the recombination paths and also that interstitialvacancy recombinations leading to antisites (heterogeneous recombinations) should be taken into account. Especially when most of the silicon vacancies are found in their stable form, i.e., they are in fact V C -C Si complexes, the recombination with silicon interstitials is going to lead to the production of antisite pairs. The antisite pair recombination was studied by several authors [29-31]. The height of the recombination barrier is around 3 ev, but some metastable configurations occur along the path and could influence the behaviour during low temperature irradiation. The barrier is lowered to 2 ev or less in the -2 or +2 charge states; the latter figures should be further investigated with more advanced mehods in order to assess the influence of the band gap underestimation. For the V Si V C -C Si transformation, the barrier obtained using LDA or GGA functionals ( ev) [43] is indeed overestimated; when the same barrier is calculated with a more sophisticated scheme relying on a GW correction at the saddle point and at the stable configurations [11], the result (2.3 ev) is in much better agreement with the experimental annealing stage at 2.2 ev. No calculation of dissociation barriers have been done for carbon or silicon interstitials clusters, so that one can only give a lower limit equal to their binding energy. Only for the silicon di-interstitials in the neutral state we have calculated some reorientation and migration barriers that were given in an internal report [36]; the results suggest that various reorientations can take place with barriers between 0.3 and 0.8 ev, while the migration of a di-interstitial as a whole has a barrier of 1.25 ev. Summary We have summarised the main results of several papers devoted to defect energetics in SiC. Antisites and carbon vacancies are the most abundant defects at equilibrium; however, interstitials are the most mobile species; vacancies are much less mobile and other defect reactions/transformations have also much higher barriers. The dominant charge state for p or n doping are roughly understood; however, many uncertainties remain, because of the limitations of current functionals and slow convergence with supercell size, especially for charged defects. Hybrid functionals and GW approaches are promising improvements. Acknowledgements We acknowledge computing time from GENCI-CINES and GENCI-CCRT (grant 2011-gen6018). References [1] C. Wang et al.: Phys. Rev. B Vol. 38 (1988), p [2] M. Bockstedte et al.: Phys. Stat. Sol. B Vol. 245 (2008), p [3] J.-P. Crocombette and G. Roma: CEA Report DEN/DANS/DMN/SRMP/NT/ /A [4] M. Bockstedte et al.: Phys. Rev. B Vol. 68 (2003), p [5] F. Bruneval: Phys. Rev. Lett. Vol. 103 (2009), p [6] S. B. Zhang and J. E. Northrup: Phys. Rev. Lett. Vol. 67 (1991), p [7] P. Déak et al.: Mater. Sci. Eng. B Vol (2008), p [8] A. Alkauskas et al.: Phys. Rev. Lett. Vol. 101 (2008), p

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