Surface & Coatings Technology 325 (2017) 410–416
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Interface-induced electronic structure toughening of nitride superlattices a, b, c, a, b, c ˇ ˇ Petr Rehák , David Holec d * , Miroslav Cerný a
Central European Institute of Technology, CEITEC VUT, Brno University of Technology, Technická 3058/10, Brno CZ-616 00, Czech Republic Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 513/22, Brno CZ-616 62, Czech Republic c Institute of Engineering Physics, Faculty of Mechanical Engineering, Brno University of Technology, Technická 2896/2, Brno CZ-616 69, Czech Republic d Department of Physical Metallurgy and Materials Testing, Montanuniversität Leoben, Franz-Josef-Strasse 18, Leoben AT-8700, Austria b
A R T I C L E
I N F O
Article history: Received 12 April 2017 Received in revised form 22 June 2017 Accepted 25 June 2017 Available online 28 June 2017 Keywords: Nitride multilayers Cleavage Friedel oscillations Ab initio calculations
A B S T R A C T We report on first-principles study of uniaxial strength for brittle cleavage of AlN/VN, AlN/TiN and VN/TiN systems. In agreement with previous studies we predict that the VN/TiN exhibits interface induced toughening of VN as compared to bulk values, and a similar effect is predicted also to occur in the VN/AlN system. However, a more detailed insight reveals, that the theoretical critical stress for brittle cleavage largely oscillates (even below the critical stress for bulk) with the distance from the interface inside the VN layer, a phenomenon not present (or hugely reduced) in TiN and AlN layers. The oscillating values for critical stress well correlate with the same behavior of interplanar distances and charge density. The origin of these unexpected properties was pinpointed to structural instabilities of cubic VN. © 2017 Elsevier B.V. All rights reserved.
1. Introduction The ever more demanding application conditions call for new systems with properties over-performing current materials. In addition to alloying and self-organization, a multilayer architecture is a perspective concept to reach these goals. For example, it has been shown that TiN/VN superlattices with bi-layer period in a range of several nanometers yield hardness values higher than either of the constituent materials [1]. Similarly, AlN/CrN system has been reported to exhibit increased hardness [2] with respect to the AlN or CrN alone. Recently, Hahn et al. [3] reported on a peak in hardness and toughness as functions of the bi-layer period of TiN/CrN superlattice. No doubt, ab initio calculations are nowadays not only capable of reproducing experimental data, and hence helping with their interpretation, but also have predictive power, and can be effectively used for predicting trends of various material properties. Particularly in the field of nitride-based protective coatings, the symbiosis between experiment and quantum mechanical modeling has proven extremely successful in the past years, providing rich information about phase
* Corresponding author at: Institute of Engineering Physics, Faculty of Mechanical Engineering, Brno University of Technology, Technická 2896/2, Brno CZ-616 69, Czech Republic. ˇ E-mail address:
[email protected] (P. Rehák).
http://dx.doi.org/10.1016/j.surfcoat.2017.06.065 0257-8972/© 2017 Elsevier B.V. All rights reserved.
stability, elasticity, or electronic properties (e.g., Refs. [4–7] and references therein). With the increasing available computational power, it becomes feasible to treat larger system such as superlattices also on the ab initio level [8–13]. There is, however, only a limited number of theoretical works investigating the effect of the superlattice design on their ideal strength. One such example was the pioneering study by Lazar et al. [14]. The questions which ultimately motivated the present work were: What is the weakest link in a perfect superlattice? Is it the material with smaller toughness, the interface, or is the answer yet more complicated? With this paper we aim on a detailed exploration of the interface impact on the ideal strength of nitride superlattices. We focus on a rocksalt structure (B1, NaCl prototype, space group ¯ #225, Fm3m), which is a stable configuration of TiN, a prototype hard coating material [15]. Industrially important is the metastable supersaturated solid solution Ti1−x Alx N [16], nevertheless TiN/AlN, TiN/CrN, and CrN/AlN multilayers have also attracted significant attention due to their extraordinary properties [2,3,17-20]. To avoid struggling with magnetism of CrN [13,21-25], we choose AlN as the second system for the present study. AlN crystallizes in a wurtzite B4 structure (ZnS prototype, space group #186, P63 mc), but can be stabilized in the rocksalt structure by coherency stresses present in the superlattices [2,3,17-19,26-28]. In contrast to metallic TiN, AlN (in both, wurtzite and cubic phases) is a wide band gap semiconductor with a strong covalent bonding character. Finally, a third system,
ˇ P. Rehák et al. / Surface & Coatings Technology 325 (2017) 410–416
VN, was chosen as a benchmark to compare our results with the literature [14] as well as to provide an insight into the impact of the lattice mismatch, since the lattice constant of cubic B1 VN lies between the values of TiN and AlN. Surface and formation energies of various combinations of these three nitrides were studied by Stampfl and Freeman [20] who, however, did not address their cleavage properties. 2. Methods Epitaxial superlattices consisting of various combinations of cubic AlN, TiN, and VN having a common (001) plane as the interface, were loaded along the [001] direction. Brittle cleavage along (001) planes was simulated by splitting a particular system into two rigid blocks. The cleavage energy, Ec , was obtained by fitting the thus obtained total energy of the system, E, as a function of the separation, x, with the universal binding energy relation [29] x x exp − ]. E(x) = Ec [1 − 1 + l l
(1)
The above fit also provides the critical length, l, i.e., the separation at which the stress s perpendicular to the cleavage plane (causing the separation) reaches its maximum, s c . This cohesive stress is given as sc =
dE Ec = dx x=l el
.
411
(hereafter labeled as “bi-layer”) both illustrated in Fig. 1. In the latter case, the positions of the surface atoms were fixed (in the (001) plane by the crystal symmetry) in order to simulate bulk region of a stable material in which any interface-induced displacements shall vanish. The reason for using the bi-layer configuration instead of a full superlattice one is to keep the overall simulation box dimensions constant in all calculations, and hence to minimize any numerical errors. The impact of this assumption will be discussed later. Each (001) plane contains 1 metal and 1 N atom, i.e., the supercell vectors ¯ are along the [110], [110], and [001] directions (with respect to a reference conventional cubic coordination frame). According to tests we made, twelve atomic planes, i.e. 24 atoms, (as illustrated in Fig. 1) are sufficient to build a representative model of the nitride multilayers. Unless stated otherwise, we used the same number of atoms for all the presented calculations. Nitrides for protective coating applications are typically synthesized by physical vapor deposition techniques. In order to obtain superlattices with a well defined epitaxial relationship between individual layers, the deposition process has to be adjusted with a special care. One of the important parameters is the substrate selection, acting as a seed for well orientated growth. Experimental experience shows that an MgO(001) substrate is particularly suitable to trigger the growth of (001) orientated TiN [34]. Consequently, in all our simulations we fixed the interface in-plane lattice parameter (in the (001) plane) to the experimental value of the MgO lattice parameter (4.22 Å), while the structures were relaxed in the perpendicular [001] direction.
(2) 3. Results and discussion
The total energy, E, was computed from first principles using the Density Functional Theory as implemented in the Vienna Ab initio Simulation Package (VASP) [30,31] employing a plane wave basis set. Electron-ion interaction was described with pseudopotentials supplied with VASP [32] which are compatible with the projector augmented waves method. The plane-wave cut-off energy was set to 600 eV which together with a 15 × 15 × 1 Monkhorst-Pack mesh sampling the first Brillouin zone guarantee accurate total energies and forces. The solution was considered sufficiently converged when the total energy values computed in two consecutive steps of the self-consistent cycle differed less than 10 −6 eV. Whenever we optimized the structure, forces acting on atoms were relaxed below 5 • 10 −4 eV/Å. The exchange-correlation energy was evaluated within the generalized gradient approximation (GGA) parametrized by Perdew and Wang [33]. The brittle cleavage of the cubic nitrides was modeled using a tetragonal (periodically repeated) cell consisting of either two slabs of nitrides (hereafter referred to as “superlattice”) to simulate a bulk region of a multilayer, or two slabs separated by 20 Å of vacuum
-5 -4 -3
-2 -1
0
3.1. Bulk VN, TiN, AlN phases In order to check the accuracy of our calculations, we have computed the equilibrium lattice constant, a0 , and the single crystal elastic constants, C11 , C12 and C44 , of studied cubic nitrides (see Table 1). The values are in good agreement with other theoretical [14,35,36] as well as experimental [37,38] data. Experimental values for cubic AlN are missing since its stable bulk form is the hexagonal wurtzite structure. The largest difference with respect to experimental data is 24% obtained for C12 of VN. On the other hand, the technologically more important tetragonal shear modulus, C = (C11 − C12 )/2, reaches the difference of only 11% between our and the experimental value. For predicting cleavage properties of each of the considered material systems, supercells consisting of twelve (001) layers were constructed within the periodic boundary conditions to form a bulk crystal. To stay consistent with later reported results for two-phase systems, also in this case of single-phase systems we fixed their in-plane lattice parameter to that of MgO(001), while the lattice
1
2
3
4
5
[110]
vacuum [110]
[001]
Fig. 1. Two types of cells used in our calculations. The upper one called “superlattice” consists only of stacks of (001) planes. Cell referred to as “bi-layer” contains also vacuum slab (on the right). The numbered vertical dotted lines mark considered positions of cleavage planes.
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Table 1 Calculated and experimental values of the equilibrium lattice constant (a0 ) and single crystal elastic constants (C11 , C12 , C44 ) of bulk crystals of VN, TiN, AlN in the cubic B1 structure. Crystal
VN
TiN
AlN
Property
Present Calc.
Other works Exp.
Calc.
a0 (Å) C11 (GPa) C12 (GPa) C44 (GPa) a0 (Å) C11 (GPa) C12 (GPa) C44 (GPa) a0 (Å) C11 (GPa) C12 (GPa) C44 (GPa)
4.121 615 168 123 4.256 633 139 160 4.069 424 167 309
4.139 [37] 533 [38] 135 [38] 133 [38] 4.241 [37] 625 [38] 165 [38] 163 [38] 4.045 [37] – – –
4.128 [14] , 4.127 [36] 636 [14] , 660 [36] 162 [14] , 144 [36] 126 [14] , 120 [36] 4.270 [14] , 4.253 [36] 604 [14] , 575 [36] 136 [14] , 130 [36] 162 [14] , 163 [36] 4.069 [35] , 4.069 [36] 423 [35] , 418 [36] 167 [35] , 169 [36] 306 [35] , 308 [36]
Refs. [14,36] : VASP, GGA, Perdew and Wang [33] . Ref. [35] : VASP, GGA, Perdew, Burke and Ernzerhof [39] .
parameter in the [001] direction perpendicular to the cleavage plane, was optimized. The total energy of the system was calculated as a function of the separation of two equally large rigid blocks and fitted using the universal binding energy relation Eq. (1), hence yielding the cleavage energy Ec , critical length l and the critical stress s c (Table 2). On the one hand, the two-parameter approximation (Eq. (1)) has been tested and found reliable for description of brittle cleavage in many cases (various cleavage planes in bulk crystals, at grain boundaries or other interfaces). On the other hand, such a simple expression cannot fit the E(x) dependence perfectly for all systems with different bonding characteristics. In such cases, we primarily assured that Ec well reproduces the cleavage energy (for large separations 5 Å energy of the studied systems changes by no more than 0.02 J/m2 which is negligible) and then we found the l parameter yielding the best coefficient of determination. This way we got reliable Ec (insensitive to the number and distribution of fitted data points) whereas the l values might change slightly for different set of computed data points. The values of s c thus also could be affected by the choice of the fitting data set. For this reason we also computed the cleavage stress using finite differences of the computed energy (for sufficiently small x increment) and compared the thus obtained s c values with results of Eq. (2). Both results differed less than is the uncertainty expressed by rounding the numbers in Table 2. Nevertheless, we decided to use the s c values supplied by the finite difference method in all further diagrams. Our predictions of cleavage characteristics in Table 2 for VN and TiN compare favorably with the results of isolated bulk phases studied by Lazar et al. [14]. The higher values of strength and cleavage energy of VN predicted here is caused by the biaxial stresses due to
Table 2 Cleavage energy (Ec ), critical stress (s c ), and critical length (l) of the bulk nitrides loaded along the [001] direction with the transverse lattice parameters equal to those of the MgO(001) substrate. Values reported by Lazar et al. [14] are given for comparison. Structure
Property
Present
Ref. [14]
VN
Ec (J/m2 ) s c (GPa) l (Å) Ec (J/m2 ) s c (GPa) l (Å) Ec (J/m2 ) s c (GPa) l (Å)
3.2 32 0.37 3.0 28 0.39 3.4 33 0.38
2.6 25 0.37 3.0 28 0.39 – – –
TiN
AlN
the simulated coherency to the (001) MgO substrate, which act in directions perpendicular to the tensile axis. As was already pointed out by Lazar et al. [14], such biaxial deformation reduces the interatomic spacing in the [001] direction, and hence increases the value of cleavage energy Ec as well as the critical stress s c . Our calculations for the equilibrium volume gave the values Ec = 2.7 J/m2 and s c = 26 GPa that are in remarkably better agreement with the results by Lazar et al. [14]. TiN exhibits an opposite effect which is, however, not so pronounced as the lattice mismatch to MgO (less than 1%) is smaller than in the case of VN (above 2%). Lattice parameter of AlN is even lower than that of VN (yielding the mismatch of 4%) which lets us expect more remarkable overestimating of Ec with respect to optimized bulk than in the case of VN. 3.2. Brittle cleavage of VN/TiN, TiN/AlN, AlN/VN systems The ultimate goal of our study is to quantify the strength of interfaces and its impact on the strength of neighboring phases. The results are summarized in Fig. 2 in terms of critical stress s c and cleavage energy Ec for all three combinations of the considered systems. The position of interface is indicated by thick dashed vertical line. For grasping faster the important features, we also show in each region the corresponding bulk value of s c (cf. Table 2). The cleavage energy for the VN(001)TiN(001) interface was determined to be 2.88 J/m2 . This value as well as the associated critical stress s c = 28.6 GPa agree well with the values Ec = 2.91 J/m2 and s c = 27.1 GPa calculated by Lazar et al. [14]. Nonetheless, a discussion of some general trends is complicated as the overall behavior turns out to be strongly system- as well as location-dependent. VN exhibits strong oscillations of s c ranging from ≈25 GPa to ≈45 GPa, which, moreover, do not seem to vanish with the increasing distance from the interface. Although the upper bound is higher than the corresponding bulk value s bulk = 32 GPa, there are locac tions with lower critical stress, and hence the interface to either TiN (Fig. 2a) or AlN (Fig. 2c) makes VN weaker with respect to its single-phase form at cleavage planes that have not been considered in former studies. Similarly, also TiN exhibits oscillatory behavior (Fig. 2a and b). The calculated minimum values are in both cases slightly lower TiN/VN than the bulk value (s bulk = 28.3 GPa vs. sc,min = 26.3 GPa and c TiN/AlN
= 28.0 GPa), hence again leading to weakening the matesc,min rial with respect to the single-phase TiN case. It is worth noting, however, that the oscillations in TiN tend to vanish away from the interface (or free surface), unlike in the case of VN as we verified for a larger computational cell. This suggests that the origin of these oscillations is different in VN on the one hand and in TiN on the other hand, as will be discussed in the next section. Contrary to TiN and VN, no oscillations are predicted in AlN. Moreover, the s c values are almost identical to the single-phase bulk AlN value. Regarding the interface, its strength is reduced in the TiN/AlN and VN/AlN cases compared with the single-phase values of the respective materials, while it is almost the same as TiN in the TiN/VN system, i.e., the weaker component. Consequently, the interface is predicted to be the weakest link in the TiN/AlN superlattice, while in the other cases, i.e., TiN/VN and VN/AlN, the weakest component is predicted to be VN. However, this weakening of VN is induced by the presence of the interface. Therefore, a brittle fracture for loading perpendicular to the superlattice interfaces, is predicted to take place at the interface for TiN/AlN system, and inside of the VN layers in TiN/VN and VN/AlN. We note that analogous behavior as for the critical stress, s c , is also obtained for the cleavage energy, Ec . 3.3. Origin of the oscillations in the VN layer A closer analysis of the relaxed structures reveals that also distances between atoms are, in some cases, periodically changing,
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413
(a)
(b)
(c) Fig. 2. Critical stress s c and cleavage energy Ec computed for individual cleavage planes in (a) TiN/VN, (b) TiN/AlN, and (c) VN/AlN systems. Numbers on the horizontal axis describe positions of the cleavage planes (defined in Fig. 1).
while in other cases remain almost constant. This can be seen in Fig. 3 displaying optimized TiN/VN interface (in bilayer configuration). By symmetry, there are two different planes of atoms and the structural optimization can yield different values for d1 and d2 as defined in the figure. These interatomic distances are plotted in Fig. 4 together with the cleavage energy Ec . Values of d1 and d2 are mostly very alike and therefore may represent also interplanar distances. It turns out that cleavage energy and interatomic distances are anti-correlated: bonds become stronger and the cleavage energy increases with shrinking distance. This is reflected also by changes in valence electron density which is plotted in Fig. 3 in (110) planes perpendicular to the figure as indicated. It clearly follows that V–N pairs of atoms with increased charge density in between, i.e., with stronger bonds, exhibit smaller distance (cf. Figs. 4 and 3). Similar oscillations in charge density, and related shrinking of interatomic distances accompanied with strengthening the bonds, was formerly predicted, e.g., in the TiN/SiN sandwich structure by Zhang et al. [9]. These authors argued that presence of a surface or any interface acts as a source of Friedel oscillations [40]. This is a plausible explanation also for the present case where we provide evidence of interplay between charge density fluctuations, interatomic distances and cleavage properties. However, the Friedel oscillations decrease with increasing distance from the interface and vanish in the bulk region of stable materials as also shown by Zhang et al. [9] for the TiN/SiN system. Indeed, also in the present case such material response is predicted for TiN, and the effect of diminishing the oscillatory behavior becomes even more pronounced for systems with thicker layers. Friedel oscillations are related to screening of a charge perturbation caused by a defect, e.g., the free surface or interface, and hence are related to
the availability of mobile electrons taking part in the screening. In general, screening is much more effective in metals than in semiconductors [41,42] which can explain why the effect is almost negligible in AlN, as this is a wide band-gap semiconductor [43]. Nevertheless, the situation remains puzzling for VN. Lazar et al. [14] reported on similar oscillations for interplanar distance (they evaluated averaged interplanar distances) and concluded that stoichiometric VN was metastable against a tetragonal distortion. According to the study of dynamic and structural stability performed by Mei et al. [44], vanadium nitride is stable in B1 structure at room temperature but the structure becomes unstable for temperatures below 250 K, i.e., also at 0 K as is the case of standard DFT calculations. Their results agree with our own calculations of phonon dispersions for cubic VN at 0 K (Fig. 5) that were obtained by so-called direct approach employing the codes VASP [30,31] for calculation of forces and PHON [45] for generating the phonon spectra. In these calculations we employed a 4 × 4 × 4 fcc supercell and the Brillouin zone was sampled using 5 × 5 × 5 Monkhorst-Pack grid. The cut-off energy was set to 600 eV also in these calculations. The most profound instability is that indicated by the longitudinal soft mode at the X point. The related displacements of atoms in the lattice corresponding to the three possible phonon modes are indicated by the arrows in the lower part of Fig. 5. The two transverse modes (in the middle and on the right) are degenerated. Importantly, the longitudinal mode (on the left) corresponds with the interplanar oscillations observed in the VN/TiN and AlN/VN systems (Fig. 4). On the one hand, the same behavior is predicted also for a single-phase VN slab (several VN planes separated by vacuum region) we made (not shown here). On the other hand, calculations of the cleavage properties in a single-phase bulk VN (no vacuum, i.e., “superlattice”-like configuration) did not reveal
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Fig. 3. Optimized TiN/VN interface and valence electrons density distribution in two parallel (110) planes (perpendicular to the figure).
any such oscillations despite straining the supercell in-plane to fit the lattice constant of MgO. The reason is, that these calculations do not include any mechanisms that can displace the atoms from their regular symmetry positions. Here we note, that the effect of the MgO substrate is only to impose bi-axial strain, but the actual interface to substrate is not considered. We therefore conclude that the interface and/or free surface induced Friedel oscillations in charge density trigger the periodic lattice distortions — displacements of atoms from their symmetry-dictated positions in rocksalt structure in accordance with the soft phonon mode in Fig. 5. We note that similar oscillations of interplanar spacings related to the superlattice geometry in the CrN/AlN system were reported experimentally by Zhang et al. [46]. Similar to the present study, they ascribed their origin to the interface-induced Friedel oscillations of the charge density. The oscillations were almost undetectable
in AlN (in agreement with the present predictions) and tend to slowly vanish in the CrN, similar to the TiN in our study. VN has been previously used in superlattices exhibiting excellent mechanical properties [1]. The present study, however, suggests that VN would become extremely prone to brittle cleavage in such superlattice geometry once cooled below the phase transformation temperature of ∼250 K [44]. Such dramatic deterioration of mechanical properties is not expected for bulk polycrystalline VN due to different orientations of the tetragonal phases existing below the phase transformation temperature. 3.4. Impact of the simulation cell geometry As was mentioned in Section 2, we considered several distinct structural models and performed many more test calculations than
(a)
(b)
(c) Fig. 4. Interatomic distances d1 and d2 (defined in Fig. 3) and cleavage energy Ec for (a) TiN/VN, (b) TiN/AlN, and (c) VN/AlN systems.
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[010]
[001]
[100]
[010]
[001] [100] Fig. 5. Phonon spectrum of bulk cubic VN shows some imaginary frequencies (plotted as negative values). Distortions corresponding to the unstable modes at the X-point are illustrated below; arrows indicate direction of the atomic displacements.
presented here, in order to check a possible effect of parameters (size, degrees of freedom) of the particular model on computed cleavage properties. Selected representative results for bi-layers and multilayers with several different numbers of atomic planes of TiN and VN are plotted in Fig. 6. In the case of bi-layer with 10 + 10 atomic planes we tested also the effect of fixing atoms in first two atomic planes from the surface. These results clearly demonstrate that there is no significant dependence of the Ec behavior in a VN layer as a consequence of the chosen structural model; the small quantitative differences are unimportant for the appearance of non-vanishing oscillations. Finally, it is worth mentioning that here considered bi-layer periods of these multilayers already reach the experimentally accessible geometries [2,3,17,28], hence our predictions are directly relevant for practical design of novel high-performance superlattice coatings.
4. Conclusions Density Functional Theory calculations were applied to study tensile strength of TiN/VN, TiN/AlN, and AlN/VN multilayers with the aim to determine what is the weakest link, whether the material with lower tensile strength in a bulk single-phase form, or the interface. It turned out that it is not possible to provide a universal answer. While the critical stress, s c , at the TiN/AlN and AlN/VN interfaces is lower than the corresponding bulk values, the interface is the weakest link only in the case of TiN/AlN. When VN layer is present, it exhibits strong oscillatory variations reaching values even lower than that at the interface, hence becoming the limiting factor. A thorough analysis revealed that the cleavage properties are strongly related to the interplanar distances and charge density: the higher is the accumulated charge density, the smaller is the interplanar distance and the higher is the cleavage strength. These charge redistribution, known as Friedel oscillations, were linked to the presence of defects, i.e., interfaces or surfaces, and should diminish with the increasing distance from their source. This is indeed the case of TiN and AlN (in the latter practically no oscillation are apparent due to a small capability for charge screening), but not of VN. There, the non-disappearing oscillations originate from the interface-induced relaxation of VN along a soft-phonon direction. Since the [111] direction (L point in the phonon dispersion curve) does not exhibit such instabilities, we predict that (111)-oriented multilayers with VN will not show this behavior. Acknowledgments
Fig. 6. Cleavage energy Ec of VN (in cleavage planes parallel to the VN/TiN interface) calculated by different models. The individual cases are labeled by the numbers of VN + TiN (001) planes considered in the periodically repeated cell. Positions of atoms were fixed in two (001) boundary planes (on both sides) of our bi-layer: 10 + 10.
ˇ acknowledge the financial support by the Czech Sciˇ and M.C. P.R. ence Foundation (Project No. GA 16-24711S) and by the Ministry of Education, Youths and Sports of the Czech Republic within the project CEITEC 2020 (LQ1601). The computational resources were provided by the Ministry of Education, Youths and Sports of the Czech Republic under the Project IT4Innovations National Supercomputer Center (Project No. LM2015070) within the program Projects
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of Large Research, Development and Innovations Infrastructures. The results presented have been also partially achieved using the Vienna Scientific Cluster (VSC).
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