Theory of the carbon vacancy in -SiC: crystal field and pseudo Jahn-Teller effects
The carbon vacancy in -SiC is a powerful minority carrier recombination center in as-grown material and a major cause of degradation of SiC-based devices. Despite the extensiveness and maturity of the literature regarding the characterization and modeling of the defect, many fundamental questions persist. Among them we have the shaky connection of the EPR data to the electrical measurements lacking sub-lattice site resolution, the physical origin of the pseudo-Jahn-Teller effect, the reasoning for the observed sub-lattice dependence of the paramagnetic states, and the severe temperature-dependence of some hyperfine signals which cannot be accounted for by a thermally-activated dynamic averaging between equivalent Jahn-Teller distorted structures. In this work we address these problems by means of semi-local and hybrid density functional calculations. We start by inventorying a total of four different vacancy structures from the analysis of relative energies. Diamagnetic states have well defined low-energy structures, whereas paramagnetic states display metastability. The reasoning for the rich structural variety is traced back to the filling of electronic states which are shaped by a crystal-field-dependent (and therefore site-dependent) pseudo-Jahn-Teller effect. From calculated minimum energy paths for defect rotation and transformation mechanisms, combined with the calculated formation energies and electrical levels, we arrived at a configuration-coordinate diagram of the defect. The diagram provides us with a detailed first-principles picture of the defect when subject to thermal excitations. The calculated acceptor and donor transitions agree well with the binding energies of electrons emitted from the Z and EH traps, respectively. From the comparison of calculated and measured -values, and correlating the site-dependent formation energies with the relative intensity of the DLTS peaks in as-grown material, we assign Z (EH) and Z (EH) signals to acceptor (donor) transitions of carbon vacancies located on the and sub-lattice sites, respectively
pacs:71.55.-i, 71.55.Cn, 71.70.Ch, 71.70.Ej
A wide and indirect band-gap, high chemical and thermal stability, as well as radiation and electrical hardness, are among the merits that make silicon carbide (SiC) an outstanding material for high-voltage and high-power electronics.Park (1998); Kimoto and Cooper (2014) Due to its superior properties, the -SiC polytype has been the material of choice of the industry. However, the presence of carbon-related point defects in SiC, particularly carbon vacancies (V), is a major cause for minority carrier recombination in n-type material and device failures like low field effect mobility.Iwamoto and Svensson (2015); Nipoti et al. (2015) These problems have been connected to a set of V-related deep traps measured by deep-level transient spectroscopy (DLTS) and labelled Z and EH.Kimoto et al. (1995); Hemmingsson et al. (1997); Hiyoshi and Kimoto (2009)
The Z has been ascribed to the superposition of Z and Z signals, each of which arising from a two-electron emission cascade at distinct sub-lattice sites of the polytype.Hemmingsson et al. (1998, 1999); Son et al. (2012) Defects behaving like that are said to possess a negative- as they show an inverted order of energy levels.Watkins (1984) This is possible thanks to a strong atomic relaxation somewhere along the emission sequence. For the case of Z this translates into the appearance of a occupancy level at about eV, implying that the formation of negatively charged vacancies () is always energetically unfavorable against the formation of any mix of neutral () and double negative () defects, no matter the position of the Fermi energy. The appearance of is most likely when the Fermi level lies at the transition energy, where its formation energy, , is lowest with respect to other charge states. Depending on the temperature and the energy difference , some states can still be populated. Alternatively, can be formed from other charge states after capture/emission of carriers upon optical excitation. The actual and levels were respectively measured at 0.67 eV and eV below , whereas and were found at about eV and eV, respectively.Hemmingsson et al. (1998, 1999); Son et al. (2012) Also noteworthy is the fact that in -SiC, a pair of electron traps located at eV and labelled E/E from DLTS measurements, were attributed to acceptor transitions from equivalent defects at different sub-lattice sites.Dmowski (1990); Aboelfotoh and Doyle (1999) More recently, high-resolution Laplace-DLTS was able to further resolve E/E into three components, and based on their similarity with Z (including their capture cross section and negative- ordering of levels), they were assigned to the carbon vacancy located on all three available sites (, and ) of the polytype.Koizumi et al. (2013)
The EH DLTS band has been a subject of discussion and surrounded by some controversy. It usually shows up with a magnitude lower than Z,Ayedh et al. (2014) and it is made of two nearly overlapping peaks, apparently with varying amplitude ratio (between 1:3 and 1:5) depending on sample conditions.Danno and Kimoto (2006); Booker et al. (2016) These facts led to suggestions that EH should not have the same origin of Z, but rather be connected to a complex involving V.Storasta et al. (2004); Reshanov et al. (2007); Zippelius et al. (2011); Alfieri and Kimoto (2013) Recently, Booker and co-workersBooker et al. (2016) analyzed the EH capacitance transients, and based on a three-charge state model they concluded that like Z, the EH band results from two correlated, two-electron emission processes from two defects. Most importantly, they found that the concentration ratio of EH:EH is 1:1 if we consider that the stronger peak actually combines EH, EH and EH transitions, while the smaller component of the band comes from EH alone. The issue of the inconsistent magnitude ratio between EH and Z was poorly addressed. For all these transitions, carrier binding energies were measured at eV, eV, eV and about eV, respectively. This suggests that EH is a negative- defect, while that cannot be said for EH due to uncertainty in the measurements.
Before continuing, let us introduce some notation with the help of Figure 1(a). Here we depict the atomic structure of perfect V defects at - and -sites (with and labels referring to quasi-cubic and quasi-hexagonal sub-lattice sites of the -SiC crystal). For the sake of convenience, the atom numbering scheme was chosen in line with previous works in the literature.Bockstedte et al. (2003); Umeda et al. (2004a) Hence, for a trigonal structure we have Si (axial) and Si (basal) shells of Si atoms. For monoclinic structures we assume Si and Si to lie on the mirror plane and the Si pair to be mirror-symmetric. Hereafter, V refers to the carbon vacancy at the sub-lattice site and charge state (from double minus to double plus). Occasionally we may also distinguish a vacancy with a specific atomic geometry as V. We also introduce at this point a way to represent the vacancy electronic states using simple linear combination of atomic orbitals (LCAO). Accordingly, a state stands for , where is a normalization constant, are hybridization coefficients, and the summation runs over all four Si radical states (with ).
Many details about the electronic and atomistic structure of V in -SiC, particularly in their paramagnetic and states, could be unraveled by electron paramagnetic resonance (EPR) measurements.Son et al. (2001); Zvanut and Konovalov (2002); Konovalov et al. (2003); Umeda et al. (2004b); Bratus’ et al. (2005); Umeda et al. (2005); Son et al. (2012); Trinh et al. (2013) Among these reports, those combining experiments with first-principles calculationsUmeda et al. (2004a); Bratus’ et al. (2005); Umeda et al. (2005); Trinh et al. (2013) turn out to be particularly elucidating. Below we provide a brief summary of those results, with a special focus on the relevant issues for the purpose of this work.
In p-type material irradiated with MeV electrons at high temperatures (850C), the EPR spectrum revealed two signals, labelled as EI5 (also referred to as Ky1/Ky2/ID1) and EI6 (also Ky3/ID2), which were assigned to and , respectively.Zvanut and Konovalov (2002); Konovalov et al. (2003); Umeda et al. (2004b, a); Bratus’ et al. (2005) Below K the main line of exhibited symmetry, and was accompanied by three distinct hyper-fine (HF) signals due to interactions between the electron spin and Si nuclei in shells with 1, 1 and 2 atoms. Above 50 K the spectrum was converted to a trigonal () pattern with two HFs representative of 1 axial Si atom and 3 equivalent Si atoms on the basal plane. From the temperature dependence of the HF life-times, the conversion from monoclinic to trigonal symmetry was estimated to be limited by a barrier as low as 0.014 eV.Umeda et al. (2004a) on the other hand, always showed trigonal symmetry irrespectively of the temperature of the measurement (down to K). The HF structure consisted of two line pairs with about 1:3 intensity ratio when the magnetic field was aligned along . However, unlike for V at the cubic site, the HF principal direction of the basal radicals of V strongly deviated from the perfect tetrahedral angle, and shifted from 103 down to 98° as the temperature was lowered from 150 K to 10 K. This behavior was interpreted as an increase of the anti-bonding character between the axial and basal radicals when the temperature was lowered.Umeda et al. (2004b)
Most observations described above were accounted for by density-functional calculations. They arrived at ground state structures and HF tensors compatible with the low-temperature EPR data.Bockstedte et al. (2003); Umeda et al. (2004a); Bratus’ et al. (2005) According to the calculations, V and V defects adopt and geometries in the ground state, with their highest (semi-)occupied Kohn-Sham states (HOKS) possessing and symmetry, respectively. Within the above LCAO picture they can be approximately described as and , respectively, explaining the HF structure observed for V and V at low-temperatures ( K). They are also consistent with the measurements of V at K if we assume that above this temperature the defect assumes a motional-averaged trigonal state due to fast hopping between all three , and degenerate structures neighboring the undistorted () configuration. Note that in line with the observations, all Si radicals contribute to in V (under static and dynamic conditions), and the amplitude of the axial radical in the state accounts for about 50% of the total LCAO localization in V. The paramagnetic state of V is also consistent with the observed anti-bonding character between Si and the basal Si radicals. However, the model is still unable to account for the magnitude of the high-temperature ( K) HF signals. Another puzzle, which was noted by Bockstedte and co-workers,Bockstedte et al. (2003) is that despite being a singlet state, the trigonal V configuration is unstable against monoclinic distortion, implying the influence of a pseudo-Jahn-Teller (pJT) effect. However, neither was a justification provided for its manifestation, nor was it found why a similar effect is apparently missing in V.
Negatively charged carbon vacancies were observed by EPR in n-type -SiC irradiated either with MeV electrons at 850C or with 250 keV electrons at room temperature.Umeda et al. (2005); Son et al. (2012) Although some traces of V and V signals could be detected above K in darkness (in heavily doped material),Son et al. (2012) most experiments were performed on illuminated samples, which gave rise to much stronger signals.Umeda et al. (2005); Son et al. (2012); Trinh et al. (2013) At the cubic site and below K, the V main line showed a monoclinic pattern and a single HF pair related to two symmetry-equivalent Si nuclei (Si).Trinh et al. (2013) Additional and weaker HF signals were related to more distant shells. Above K the Si HF signal disappeared from the spectrum and the main line acquired a trigonal pattern, accompanied by the appearance of a new axial HF pair (due to interaction between a magnetic Si nucleus and the electron spin). As the temperature further increased, the magnitude of the Si HF splitting increased and at 80-90 K a weak and broad HF pair accounting for three equivalent (Si) nuclei appeared in the spectrum as well.Trinh et al. (2013) Regarding V, the main signal is monoclinic at K and below. At these temperatures two HF signal pairs related to two inequivalent Si nuclei (Si and Si) with site symmetry were detected. Raising the temperature above K led to the broadening and disappearance of the Si HF, while the main-line and Si HF components merged into single trigonal peaks. The activation barrier for the monoclinic-trigonal conversion was estimated as 0.02 eV.Umeda et al. (2005) Within the temperature range of 70-120 K only the Si HF was detected, but when K a trigonal HF signal representative of three equivalent Si nuclei (Si) was also observed.
Again, first-principles modeling played a key role in grasping several of the above features.Zywietz et al. (1999); Umeda et al. (2005); Trinh et al. (2013) Recent density functional calculations indicated that V has a paramagnetic ground state, whereas the symmetric state was metastable by only 0.03 eV.Trinh et al. (2013) The calculated HF tensors for Si radicals accounted very well for the low-temperature ( K) experimental data. The quenching of the Si HF signal above 40 K and the observation of trigonal hyperfine structures at higher temperatures was suggested to result from the partial population of both and states. Accordingly, under these conditions they would quickly hop between three equivalent Jahn-Teller (JT) distorted alignments. While this picture aims at accounting for the observed non-zero amplitude of the wave-function on all four radicals above 90 K, it cannot be correct. Any sequential transformation between and states involves an intermittent quenching of the spin-density on the basal nuclei. Further, the model could not explain why there is a K gap between the quenching of the Si HF signal (at 40 K) and the appearance of the Si HF signal (at 80 K). Also puzzling and unexplored was the fact that the V ground state was found to be nodal (), which in principle has higher kinetic energy than the metastable state (). Finally, the symmetry lowering of V cannot simply be explained by the JT effect. In the perfect vacancy ( symmetry), the four Si radicals hybridize into a fully occupied valence state , and three gap states , and to be populated with three electrons. We calculated these states for an undistorted (trigonal) vacancy at the -site using the same method of Ref. Trinh et al., 2013 and they are depicted in Figure 1(b). The latter two are higher in energy and represent orthogonal components of a doublet which is split from due to the internal crystal field. For the case of V the doublet becomes partially populated (with a single electron) and the JT effect is expected to split (within ) into (within ), where the upward arrow stands for the paramagnetic electron. Now, while the first-principles results from Ref. Trinh et al., 2013 indicate that the metastable state has amplitude on Si, it is clear from Figure 1(b) that a JT-split component cannot account for this feature.
Turning now to V, the calculations arrived at a ground state rather different than that found for the cubic site, namely the unpaired electron was localized on the Si-Si pair as .Zywietz et al. (1999); Umeda et al. (2005) The calculated HF tensor elements for both (inequivalent) Si and Si radicals agreed very well with the measurements below K (both in magnitude and principal directions), providing compelling evidence for the correctness of the model. The disappearance of the Si HF signal together with the conversion of the -symmetric Si HF into a trigonal signal at K was justified based on a thermal activated hopping between , and equivalent states, which preserves a steady wave function amplitude only on Si.Umeda et al. (2005) Again, the reasoning for a 70-120 K temperature window where only Si HF was observed and above which another trigonal Si HF was observed, was left unaddressed. Analogously to the metastable structure in the cubic site, the electronic structure of V in the ground state involves a non-vanishing spin-density on Si. Hence, unlike suggested in Ref. Umeda et al., 2005, the model cannot be explained by the JT effect, simply because none of the -components in Fig. 1(b) shows non-zero amplitude on Si. Finally, V and V show monoclinic ground-states with opposite symmetry with respect to the mirror plane. Although the calculations were successful in accounting for the observed site-dependent ordering of electronic states,Zywietz et al. (1999); Umeda et al. (2005) again the reasonings behind this effect were left unaddressed.
The connection of (via EPR) with the Z and EH traps (via DLTS) was suggested based on the correlation between the position of the DLTS levels and the photo-EPR excitation thresholds for and , respectively (where represent a free electron at the conduction band bottom).Son et al. (2012) More recently, Kawahara et al.Kawahara et al. (2013, 2014) investigated samples irradiated by low-energy (250 keV) electrons, which could displace C atoms only. In those works they reported a good correlation between the area density of EPR active and the fraction of carriers trapped by the dominant Z on samples irradiated with different electron fluences.
It seems clear that Z is a negative- center. This is consistent with the need of optical excitation in order to observe negatively charged vacancies by EPR. However, that is not the case for the defect responsible for EH. In recent state-of-the-art electrical level calculations using many-body perturbationBockstedte et al. (2010) and hybrid density functionalHornos et al. (2011) methods, the donor levels were predicted to be separated by a small positive or essentially zero -value ( eV). While this agrees with the low-temperature EPR measurements in darkness, it is also in apparent conflict with the negative- ordering reported for EH and tentatively proposed for EH from DLTS.Booker et al. (2016) As a word of caution, we note that when periodic charge corrections were neglected, the calculations clearly indicated for both acceptors and donors.Zywietz et al. (1999); Bockstedte et al. (2010); Trinh et al. (2013)
It is clear that despite many advances, there are several fundamental puzzles to be solved. This paper aims at addressing those issues, as well as others that will become evident further ahead. In this section we wanted to introduce the reader to the main properties of the carbon vacancy in -SiC, how the EPR data has been related to the prominent Z and EH electron traps, and the importance of theory/computational modeling in providing models and checking their quality. We will now proceed with a description of the theoretical methods followed by the main results. These include the reproduction of structures and electronic levels previously reported, as well as new results like a physical description ofthe observed pseudo-Jahn-Teller distortions, the crystal-field impact on the distinct electronic structure of cubic and hexagonal vacancies, and the atomistic mechanisms behind the -dependent dynamic effects observed by EPR. Before the conclusions, we also include a section where we discuss the above issues.
The calculations were carried out using the VASP package,Kresse and Hafner (1993, 1994); Kresse and Furthmüller (1996a, b) employing the projector-augmented wave (PAW) method to avoid explicit treatment of core electrons.Blöchl (1994) A plane wave basis set with kinetic energy up to 400 eV was used to describe the electronic Kohn-Sham states. The many-body electronic potential was evaluated using the hybrid density functional of Heyd-Scuseria-Ernzerhof (HSE06),Heyd et al. (2003); Krukau et al. (2006) which mixes semi-local and exact exchange interactions at short ranges, treating the long-range interactions within the simpler generalized gradient approximation as proposed by Perdew, Burke and Ernzerhof (PBE).Perdew et al. (1996) When compared to plain PBE calculations, HSE06 has the main advantage of predicting a Kohn-Sham (indirect) band gap 3.17 eV wide for -SiC, which should be compared to the experimental value of 3.27 eV.Grivickas et al. (2007) To a large extent, this approach mitigates the well known underestimated gap syndrome affecting PBE-level calculations, which show a 2.19 eV band gap width. Although most results reported below were obtained using the HSE06 method, PBE-level results are also included and in that case they are explicitly identified.
We used 576-atom hexagonal supercells, obtained by replication of unit cells, from where a carbon atom was removed to produce a V defect. The equilibrium (calculated) lattice parameters of -SiC were Å and 2 Å. These are close to the experimental values of Å and Å extrapolated to K.Li and Bradt (1986) All defect structures were optimized within PBE-level, using a conjugate-gradient method until the forces acting on the atoms were lower than 10 meV/Å. After this step, we took the relaxed structure, and self-consistent energies, electron and spin densities were finally obtained within HSE06. Electronic relaxations were computed with a numerical accuracy of 1 , and the band structures were solved at in reciprocal lattice units. This is conventionally referred to as the -point in the hexagonal Brillouin zone (BZ). We found this particular -point to provide the best compromise between sampling accuracy and computational performance. It is representative of the -point set in non-relativistic calculations, it led to energy differences with an error bar of about 5 meV (when compared to results obtained using sampled BZ), and most importantly, it did not cause so strong hybridization between defect levels lying high in the gap and the conduction band states as in the -sampled PBE calculations of Ref. Trinh et al., 2013.
The above two-step recipe to obtain hybrid density-functional energies using structures that were previously relaxed within PBE (hereafter referred to as pseudo-relaxed energies), casts doubts regarding its accuracy when compared to fully-relaxed HSE06-energies obtained by minimizing HSE06-forces. To clarify this issue, we compared energies and forces of pseudo- and fully-relaxed V and V states. These two charge states have rather different structures (to be discussed below), and while V does not have electrons occupying gap levels, V has two fully occupied gap states, one of them being close to the conduction band edge. These tests were carried out using 256-atom supercells ( unit cells) with a -centered grid of -points for BZ sampling. Accordingly, we obtained fully-relaxed HSE06-energies 13 meV and 11 meV below the energy of pseudo-relaxed V and V calculations, respectively. Despite these small relaxation energies, the average HSE06-force acting on Si-atoms on PBE-relaxed structures were 0.28 eV/Å and 0.10 eV/Å for for V and V, respectively, and therefore cannot be neglected. However, the energy difference was 39.384 eV and 39.386 eV for pseudo-relaxed and fully-relaxed calculations, respectively, suggesting that the error of pseudo-relaxed energy differences is of the order of a few meV.
The energy of a charged defect, when calculated using periodic boundary conditions, is actually the energy of a supercell contaminated by artificial Coulomb interactions across an array of charged defects embedded on a compensating background charge.Makov and Payne (1995) These interactions are long-ranged and difficult to remove. Several post-processing recipes have been proposed to mitigate this problem (see for example Ref. Komsa et al., 2012 and references therein). Here we use the method by Freysoldt, Neugebauer and Van de Walle,Freysoldt et al. (2009) recently generalized for anisotropic materials.Kumagai and Oba (2014) Accordingly, the total energy of a defect in an infinite crystal is approximately , where is a localized net charge on the defect, is the total energy of the periodic problem and the charge correction,
where is a point charge correction, which for isotropic materials reduces to the Madelung energy and depends on the ratio between the Madelung constant and a characteristic length (usually a lattice constant), the net charge and the dielectric constant . Further details about the explicit calculation of for anisotropic materials (like -SiC) can be found in Ref. Kumagai and Oba, 2014.
is the offset between the defect induced average potential and that produced by a point-charge, .Kumagai and Oba (2014) The space-averaged potentials and are obtained from first-principles from the Hartree (electrostatic) potential considering defective and pristine (bulk) supercells. The averaging is done at remote locations from the defect, more precisely at all atomic sites outside the largest sphere inscribed by the Wigner-Seitz supercell (see Figure 2(a) of Ref. Kumagai and Oba, 2014). For the 576-atom supercell employed in this work, that meant a 15.1 Å radius sphere leaving a total of 382 outer atomic sites to be sampled.
Since defects distort and polarize the surrounding lattice, besides the electronic (ion-clamped) component, , the dielectric tensor employed in the calculation of has to account for the ionic contribution as well, .Komsa et al. (2012) We calculated using density-functional perturbation theory with local field effects within PBE.Baroni and Resta (1986) The ionic part was evaluated from the Born effective charges and eigen-frequencies of the Hessian matrix.Cockayne and Burton (2000) Accordingly, we obtained and for the dielectric constant parallel and perpendicular to the crystallographic -axis. These figures account well for the values and obtained from refractive index measurements and Raman scattering data.Patrick and Choyke (1970)
Figure 2 depicts the values of obtained for a defect in -SiC (with symmetry) and -SiC (with symmetry) as a function of the number of atoms in the supercell (). The calculations shown in this particular figure were carried out within PBE-level. The results were essentially the same when using the HSE06 functional. Integer triplets in the figure are scaling factors applied to the lattice vectors of the conventional cell (8 atoms in both polytypes) to obtain the respective supercell. For example, the largest hexagonal cell (-SiC) consisted of atoms, whereas the smallest cubic cell (-SiC) had atoms. can be expanded in a power series of , with the first term and the second term scaling as .Makov and Payne (1995); Castleton et al. (2006) The data were therefore used to fit a function of the form , which is shown by the dashed line. The solid straight lines represent the leading term, , as a function of . It is clear that the simple Madelung (point-like) correction overestimates the spurious Coulomb energy. Also as expected, asymptotically converges to for supercells of infinite size. The error of the Madelung correction is always above the statistical error of obtained from averaging and shown as error bars. In the case of the -SiC supercells (used in this work to study the carbon vacancy), our best estimate for the correction of is eV, whereas for a singly charged vacancy (not shown in the graph) we obtained eV.
For the calculation of formation energies we follow the usual procedure, introduced by Qian, Martin and Chadi.Qian et al. (1988) Here the formation energy of a carbon vacancy is
where is the charge-corrected total energy of the defective supercell as defined above. Besides depending on the charge state , may refer to more than one atomic structure . is the energy of a perfect supercell, is the carbon chemical potential (see below), is the valence band edge and the Fermi energy which may vary within . The upper limit, eV, is the calculated forbidden gap width, here obtained within the delta self-consistent (SCF) method,Hedin and Lundqvist (1970) where eV and eV are ionization potentials of neutral and negatively charged supercells. They are negative as their reference (zero-energy) is ill-defined for a periodic calculation. According to this method , allowing to consistently express the calculated transition levels with respect to both and without having to rely on the experimental band gap.
We note that using the -point for BZ sampling, the eV obtained by the SCF method is 0.25 eV wider than the indirect gap from the Kohn-Sham energies of the highest-occupied and lowest-unoccupied states at and , respectively. This compares with eV and eV from analogous SCF calculations using -centered (simple -point) and -point sampling grids. The -sampled energy coincides with the Kohn-Sham gap simply because of band-folding, which for a -supercell brings the -point into the origin of the BZ. These results indicate that -sampled energy differences (like - and -point calculations) may suffer from insufficient sampling density. This effect is expected to be more severe for energy differences involving the occupation (or emptying) of highly dispersive states. The calculation of is perhaps an extreme case. It involves emptying the top-most valence band and filling the bottom-most conduction band, both showing considerable dispersion amplitudes. On the other hand, for sufficiently large supercells, localized defect states show little dispersion and sampling errors tend to cancel when considering energy differences. This is confirmed by -point, -point and -grid calculations of for the vacancy at the -site, which gives an average value and maximum deviation of eV.
In Eq. 3, represents the energy per carbon atom in the SiC crystal, which is subject to
where the upper and lower bounds represent C-rich and C-poor SiC crystals, which are in equilibrium with standard carbon and silicon phases, respectively. For crystals grown under stoichiometric conditions we have . Here is the heat of formation of SiC estimated as eV, with being the energy per SiC formula unit in a perfect crystal, while and are chemical potentials (energy per atom) of C and Si in diamond and silicon crystals, respectively. The value calculated for is close to eV as obtained from calorimetry measurements.Greenberg et al. (1970)
An important use of Eq. 3 is in locating the value of for which two different charge states, say and , have the same energy, and therefore the same probability to occur. The transition level with respect to the valence band top is found at such that ,
where we distinguish eventual different structures and for charge states and , respectively. It is also useful to calculate transition levels with respect to the conduction band minimum. For that we have,
We also investigated the transformation of V defects between different structures and also between different symmetry-equivalent alignments. We assume the adiabatic approximation, and the potential energy surface governing the atomic motion was calculated using the climbing-image nudged elastic band (NEB) method.Henkelman et al. (2000) The NEB algorithm allows to find saddle points and minimum energy paths separating known initial and final structures. The method optimizes a number of intermediate structures along the reaction path while maintaining equal spacing between them. This is possible thanks to the introduction of spring forces connecting neighboring structures (the elastic band) and projecting out the component of the force due to the potential perpendicular to the band. The NEB relaxations were carried out within the PBE-level, used 7 intermediate structures, and the forces acting on the atoms were also converged within eV/Å. The initial, final and saddle-point structures (, and , respectively) were used to obtain their respective total energies (, and ) using the HSE06 functional.
iii.1 Ground-state results for the carbon vacancy
We start by reporting on the structural properties of the defect on different charge states. The V defect was always found to have the lowest energy in low-spin states. We identified four different atomistic configurations, which we label with the letters A (with symmetry), and B, C and D (with symmetry). They are distinguished by the shape of the tetrahedron with volume and with edge lengths connecting Si-Si nuclei. Some edges are shorter/longer than others and they are schematically represented by solid/dashed edges, respectively, in the middle of Figure 3. By defining an effective length as the geometric average length of the edges , A-D structures may be defined by simple distortion coordinates with magnitudes,
where elongations have pre-factors that depend on the number of symmetry-equivalent edges. Hence, structure A forms a triangular pyramid with a Si apex and a contracted Si base, whereas structures B, C and D form monoclinic tetrahedrons with a mirror plane and mirror-symmetric Si and Si. On these three structures, we found 2, 1 and 3 contracted edges (4, 5 and 3 elongated ones), respectively. Below, we show how these shapes are intimately related to the occupation of the one-electron orbitals.
In line with previous works,Zywietz et al. (1999); Bockstedte et al. (2003); Umeda et al. (2004a); Bratus’ et al. (2005); Trinh et al. (2013) we found that V and V defects are trigonal (structure A). Both introduce three empty states deep in the gap, namely a singlet level below a doubly degenerate level (). Their separation, , results from the local crystal field. Figure 3 represents the (hybrid) Kohn-Sham energies of ground-state carbon vacancies as a function of the level occupancy and sub-lattice site. The level energies are reported with respect to the eigenvalue of bulk. To make the interpretation easier, we present a spin-averaged picture, although the filling of levels is represented with upward/downward arrows.
From the eigenvalues we obtain crystal-field energies eV and eV for V and V, respectively. We will show that this site-dependence of the crystal-field confers rather distinct electronic structures on V and V. Table 1 reports the geometrical details regarding the evolution of ground-state structures as we fill in the manifold with electrons. Also included are the results for metastable structures (starred structures). These were only found for charge states and will be discussed in Section III.2. Besides edge elongations , distortion magnitudes and the volume expansion of the vacancy tetrahedron (with respect to the analogous quantity in bulk) are also shown. It is clear that positive charge states are compressive (), while negatively charged ones are tensile (). This is a consequence of the breaking/formation of reconstructed bonds between the four Si radicals edging the vacancy as we respectively remove/add electrons from/to defect states in the gap. For the same reason, distortions () tend to increase in magnitude as we go from V to V. We also note that structures A and B were concurrently found for the positive charge state, whereas structures C and D were found for the negative charge state. This is emphasized in Figure 3 by two shaded regions.
All paramagnetic ground-states differ in their atomic geometries, namely V, V, V and V, and show , , and symmetry, respectively. Inspection of the paramagnetic (highest occupied) one-electron wave-functions allowed us to identify their symmetry and LCAO representations. For the monoclinic structures we have two mirror-symmetric states (see Figure 3), and they are distinguished by and subscripts, standing for low- and high-energy symmetric states. Hence, we found that and for V and V, respectively, while for negative charge states we found and for V and V, respectively. Besides being compatible with the low-temperature EPR and HF data, the above results reproduce earlier density-functional findings.Bockstedte et al. (2003); Umeda et al. (2004a); Bratus’ et al. (2005); Trinh et al. (2013)
We went on and explored the symmetry and wave-function character of non-paramagnetic states. The results are shown in Figure 3. Here we can see that the evolution of the one-electron states, as they become occupied, exhibits a rich picture, which includes crossing and mixing (anti-crossing) features. These effects are responsible for the structural variety that is observed, and to understand them we have to invoke the JT and pJT effects.
iii.2 The pseudo-Jahn-Teller effect on the carbon vacancy in -SiC
While the JT theorem asserts the existence of spontaneous symmetry breaking of degenerate electronic states, “the pJT effect is the only source of instability and distortions of high-symmetry configurations of polyatomic systems in non-degenerate states, and it contributes significantly to the instability of degenerate states”.Bersuker (2006)
The pJT effect results in the softening of the adiabatic potential energy surface (APES) around a reference configuration with non-degenerate ground state , and it is due to overlap with excited states via electron-phonon coupling. Should this softening be severe enough to make unstable against atomic distortion towards structure , the curvature of the APES along , which transforms as some irreducible representation , must be negative, . Here is the total energy and the Hamiltonian. It may be shownBersuker (2006, 2013) that the APES softening comes from a negative vibronic contribution to the total curvature , where
is the harmonic curvature and from second-order perturbation theory,
where are off-diagonal vibronic coupling constants between the reference state and excited states with energies and , respectively. represents the force constant resisting the motion of atoms along , whereas is always negative and represents the change in that force constant that results from adapting the electron distribution to one more suited to the new nuclear coordinates,
corresponding to a lower energy . We note that unlike the Jahn-Teller effect, the pJT effect mixes the ground state with excited states to create new bonds and distort the structure. We may actually state that the driving force of the pJT effect is the increase of covalent bonding.Bersuker (2013)
Given that the product , which is the linear term in the expansion of in powers of , is fully symmetric, must also have the same symmetry as . This implies that only excited states which transform as the same irreducible representation of , such that , will lead to non-vanishing coupling constants and contribute to the softening of the APES.Bader and Bandrauk (1968); Pearson (1969, 1986)
Besides the symmetry restrictions imposed to the integral, it is often assumed that only a few low-energy states contribute to due to the increasing energy in denominator of Eq. 12.Bersuker et al. (2002) This premise has justified the replacement of the infinite sums in Eqs. 12 and 13 by a finite set of interacting states, or indeed by a two-level paradigm where a single excited state couples to the ground state via an effective vibronic coupling constant , where is the effective energy separation between the mixing states.Bersuker et al. (2002)
For an accurate description of the pJT effect one would have to solve the many-body Hamiltonian by accounting for dynamic correlation effects (e.g. by means of configuration interaction methods), the electron-phonon coupling would have to be included as well, considering all phonons obeying the above selection rule. Although this has been realized for small molecules using sophisticated quantum chemistry methods,Bersuker (2013) severe approximations have to be made in order to study defects in solids. By using a single-determinant density functional approach we may still arrive at a sufficiently detailed picture of the problem. For instance, García-Fernández and co-workersGarcía-Fernández et al. (2006) were able to explain the off-center displacement of the Fe interstitial ion in SrCl using local density functional theory. On the contrary, the wave-function-based complete active space second-order perturbation method was applied to the same problem and was unable to reproduce the observations. This failure was attributed to the insufficient number of states included in the active space.García-Fernández et al. (2006) The case of a vacancy in SiC would be much more demanding since the active space spans many ligands to the vacancy site.
We investigated the pJT effect on the V defect in -SiC, restricting our approach to a single-electron picture. Although we do not have access to important parameters such as accurate many-body gap energies and electron-phonon coupling strengths, we will arrive at an instructive and reasonable picture for the observed distortions. To that end we monitored the change of the one-electron wave-functions and respective energies, while the atomic structure was progressively changed from the high-symmetry V(A) configuration towards lower-symmetry structures B, C and D with symmetry (). Therefore, according to the selection rules, only fully symmetric states () have to be considered as the source of a pJT effect in V. The calculations reported within this Subsection were done using the spin-averaged density-functional method within the PBE level. Some tests using a spin-polarized HSE06 functional were also carried out, and apart from the expected differences regarding the energy separation of levels, the conclusions drawn below apply equally.
We begin with neutral and positively charged defects. On the left hand side of Figure 4 we depict the highest occupied and lowest unoccupied Kohn-Sham states (HOKS and LUKS, respectively) for the ground state neutral vacancy at the site, V. Both HOKS and LUKS transform according to the irreducible representation of the point group, so we differentiate them by their energy order, i.e. the one with lower energy is referred to as while the higher energy state is . If we consider all three states in the gap, the electronic structure of V is , where the number of electrons on a specific orbital is superscripted. Comparing the ground state in Figure 4 with from V shown in Figure 1(b), it is evident that the lower symmetry state increases the covalent bonding between all four atoms, and that leads to shorter Si-Si and Si-Si distances in structure B. Furthermore, considering that V is a non-degenerate ground state (), we conclude that the states exhibited in Figure 4 must result from a pJT effect.
On the right hand side of Figure 4 we find an electronic structure diagram, showing how the three gap states develop between the trigonal V state with electronic configuration and the monoclinic V state with electronic configuration . Energies of filled and empty states are represented with closed and open symbols, respectively. The same graph also shows the total energy change as crosses, from which we conclude that the high-symmetry configuration A is unstable against relaxation to B. The corresponding pseudo-Jahn-Teller relaxation energy, eV, relates to the added covalence. The distortion transforms as within (couples to electronic states), and consists in the compression of Si-Si and Si-Si distances, along with the expansion of the remaining tetrahedron edges (see Table 1). Looking again at Figure 1(b), it becomes evident that when subject to a distortion, the state , which transforms as within and shows a strong anti-bonding character between Si-Si, should raise in energy, while the doublet component , also transforming as within and showing a bonding character between Si-Si, is expected to be stabilized and lower its energy. This opposite coupling leads to the typical pJT anti-crossing pattern shown in Figure 4 for and states.
We may estimate the relative contribution (mixing) from and states to the pJT distorted and states using our simple LCAO model. From inspection of Figures 1(b) and 4, and considering normalization coefficients and we arrive at,
Like the isosurfaces shown in Figure 4, the ground state in Eq. 15 has the same phase (bonding character) on Si and Si atom pairs, and that mostly comes from . Conversely, is an anti-bonding state between Si atoms with vanishing amplitude on Si, and most of its character comes from . The above discussion and conclusions can be applied to the neutral vacancy at the hexagonal site as well. However, the stronger crystal-field splitting leads to a larger energy gap , and therefore to a weaker mixing effect.
For positively charged vacancies on both - and -sites, the shape of the electronic structure diagrams (and wave-functions) were found to be close to those of Figure 4, although for V(A) and V(B) indicated that these were both minima in the APES of and sites. From Table 1 we see that the distortion magnitudes of positively charged defects are considerably smaller than in neutral defects. We may conclude that the pJT coupling is weaker for V(B), particularly in the hexagonal site where the crystal field is stronger. Here, the V state with two (weak) Si-Si bonds sharing a single electron, is essentially degenerate with the V state . Their energy difference is estimated below meV.
For the negatively charged vacancies, the picture is dramatically different. We start by analyzing the double negative charge state, where structure D was found to be the most stable for both and sites. For the trigonal structure on the -site (the structure was relaxed by symmetrizing the forces), we found that the state with spin-0 was less stable than the spin-1 configuration by 0.20 eV, but the latter was still metastable by 0.15 eV when compared to the non-degenerate ground state. On the left hand side of Figure 5 we depict the (fully occupied) levels found within the gap for V. Comparing these wave-functions with those shown in Figure 1(b) for the symmetric structure, we realize that although the HOKS state of structure D is rather similar to from the trigonal structure, the does not find a good match, although one could suggest some resemblance with . Considering that (i) V is non-degenerate, and therefore not vulnerable to a JT distortion, and (ii) that is a mixed state with a major contribution from , the wave-functions exhibited in Figure 5 must result from a pJT effect. In fact, looking at the right hand side of Figure 5, it becomes evident that has been converted into under distortion , whereas (derived from ) seems to have merged into the conduction band (uppermost state close to 0.8 eV) before the ground state was attained.
In fact, shows bonding character for Si-Si and Si-Si pairs, and can be described approximately as like in Eq. 15. It becomes now clear that structure D results from structure B (occupation of leads to the shortening of Si-Si and Si-Si distances) combined with the occupation of , which is anti-bonding on Si-Si. The result is a tetrahedral structure with short Si-Si, Si-Si and Si-Si edges. We finally note that V shows a similar behavior to V, with the metastability of the trigonal structure by 0.5 eV being worthy of mentioning.