Structural optimizations, total energies, and properties are calculated within Density Functional Theory (DFT) [37] (chapter 2.4, page 10), for which the Vienna ab-initio Simulation Package (VASP) was used. It combines the total energy pseudopotential method with a plane-wave basis set [5962, 76]. The electron exchange and corre- lation energy is treated within the Local Density Approximation (LDA) [77] (chap- ter 2.4.3.1, page 13) and Generalized Gradient Approximation (GGA) [50, 78, 79] (chapter 2.4.3.3, page 15). The Projector-Augmented-Wave (PAW) method was em- ployed [64, 65] (chapter 2.6, page 20). The cut-o energy for the expansion of the wave function into the plane wave basis is 500 eV. Residual forces are converged below 5 * 10-3eV/Å. The Brillouin-Zone integration is done via the Monkhorst-Pack scheme [58].
Structure optimizations are obtained through relaxation of all structural parame- ters, atomic positions as well as cell parameters. Details on the used k-point mesh as well as the calculated cell of the relevant structures of this thesis are listed in Table B1 in Appendix B. However, it has to be noted, that a wide selection of structures with composition M3N2 and MSiN2 (M = Be, Mg, Ca, Sr, Ba) was computed and only the data for the structures ultimately discussed in this thesis are given in B1. In most cases, only transition pressures derived from GGA calculations are given. The choice of the GGA functional is based on the experience that it signicantly better describes the relative energies of structures, especially when structures with dierent environments for the constituting atoms are concerned. As the calcu-
2.8 Practical Calculations
lated transition pressure pt strongly depends on the calculated energy dierence
∆E (pt = ∂V∂E ≈ ∆∆EV), pt derived from GGA calculations give more accurate results
than those derived from LDA calculations.
Demuth et al. [80] and Zupan et al. [81] found for α-quartz and stishovite, that L(S)DA calculations seriously underestimate∆E (exp.: 0.51 - 0.54 eV (references in [80]); L(S)DA: 0.1044 eV [80]; -0.09 eV [81]; GGA: 0.6367 eV, 0.691 eV [80], 0.51 eV [81]). Zupan et al. even found in LSDA calculations an energetic preference of stishovite over α-quartz.
The transition pressure ofα-quartz to stishovite has been experimentally determined to 7.2 to 7.5 GPa (references in [80, 81]). The calculated transition pressure from LDA calculations amounts to 1.3 GPa [80], whereas it amounts to 8.0 GPa [80] and 6.2 GPa [81], respectively, for GGA calculations. Similar results were obtained for the diamond toβ-tin structure type transition in Si, which can be observed at 10.5 to 12.5 GPa (references in [81]). Zupan et al. [81] calculated pt to 6.7 GPa within
L(S)DA and to 9.2 GPa (PBE) and 10.6 GPa (PW91) within GGA.
Since target of this thesis is to study structures and structural transformations at high pressures, the GGA is the better choice in comparison to the LDA. Therefore, while all calculations were controlled within the LDA as well, all enthalpy dierences and transition pressures given in this thesis are based on GGA calculations, if not noted otherwise.
3 Silicon Nitride Si
3
N
4
Silicon nitride Si3N4, due to its extraordinary physical and chemical properties, is a very important ceramic material. Its combined hardness, exceptional chemical in- ertness, low density and high wear resistance render it a high-performance ceramic, which is used in many elds of technology. Its main applications are as a construc- tion material for engine and turbine building, as it withstands temperatures up to 1400° C. This is attributed to the fact, that Si3N4 ceramics are encoated with a thin SiO2 layer upon heating in air. Processing occurs mainly by injection or extrusion molding, slip casting or pressing combined with sintering (hot-press sintering, gas pressure sintering or pressure-less sintering) [82].
Three modications of Si3N4 are known, two at ambient pressure, α-Si3N4 and β-Si3N4 [2, 3], and a high-pressure modication,γ-Si3N4 [4, 5]. α-Si3N4 andβ-Si3N4 are three-dimensional networks of corner-sharing SiN4 tetrahedra, whereas γ-Si3N4 exhibits the cubic spinel structure with tetrahedrally as well as octahedrally coor- dinated Si in a molar ratio 1 to 2. δ-Si3N4, appearing at 160 GPa and possessing a CaTi2O4-type structure, has been theoretically predicted [10].
To synthesize pure Si3N4 three processes are of major importance: (1) direct nitri- dation of silicon powder, carbothermic reduction of SiO2 under N2 or NH3 atmo- sphere and (3) ammonolysis of SiCl4 or SiH4. Ammonolysis of reactive silicon com- pounds does not directly result in Si3N4 but intermediate silicon diimide "Si(NH)2" is formed, which gives amorphous Si3N4 upon heating [82].
Regarding stability, below 1500° C theα-phase is favored, while above theβ-phase is formed. α-Si3N4 transforms into β-Si3N4 at temperatures exceeding 1650° C [2, 83]. A retransformation of β-Si3N4 into α-Si3N4 could not be observed, which is at- tributed to kinetic inhibition [82]. Due to the unobserved retransformation and frequent oxygen-impurity in α-Si3N4, it has been discussed, that α-Si3N4 does not constitute a true polymorph of Si3N4. However, todayα-Si3N4 is generally accepted
3 Silicon Nitride Si3N4
as an independent Si3N4 modication [82]. γ-Si3N4 is synthesized from α-Si3N4, β-Si3N4 or amorphous Si3N4 at pressures between 10 to 13 GPa and at tempera- tures of 1600 to 1800° C [4, 5].
For the purpose of high-pressure studies on nitridosilicates the two phases of Si3N4 to be considered areβ-Si3N4 andγ-Si3N4, since α-Si3N4 diers only within the error of the method used in this thesis in energy fromβ-Si3N4 [6] and solely eitherβ-Si3N4 or γ-Si3N4 are obtained from high-pressure experiments.
3.1 Structure Optimization of
β-Si
3N
4and
γ-Si
3N
4The structures of β-Si3N4 (Figure 3.1) and γ-Si3N4 (Figure 3.2) were optimized within LDA and GGA.
β-Si3N4 crystallizes in the hexagonal space group P63/m (no. 176) and is isotypic to phenakite Be2SiO4 [3]. Along [001] sechser and vierer ring channels penetrate the three-dimensional network of corner-sharing SiN4 tetrahedra. Every N atom is connecting three Si atoms. Therefore, every SiN4 tetrahedra is connected to eight other SiN4 tetrahedra.
γ-Si3N4 crystallizes in Fd 3m (no. 227) in the spinel structure [4, 5]. Si occupies half of the octahedral voids and 1/
8th of the tetrahedral voids in the cubic close packing of N atoms, forming a dense network of SiN4 tetrahedra and SiN6 octahedra. Every SiN4 tetrahedron is connected to 12 SiN6 octahedra via common corners and every SiN6 octahedron shares a common corner with six SiN4 tetrahedra and a common edge with six SiN6 octahedra.
Data on calculated bond lengths compared to the experimental data are given in Table 3.2. The calculated crystallographic data compared to experimental results are given in Appendix A. The results of the zero-pressure optimization of the crystal structures for both β- and γ-Si3N4 are in agreement with experimental values and previous calculations [4, 6]. For β-Si3N4 the calculated volume per formula unit Si3N4 diers by 1.5 % (LDA) and 1.7 % (GGA) from the experimentally found vol- ume and are in good accordance with previously calculated data (Table 3.1).
3.1 Structure Optimization of β-Si3N4 and γ-Si3N4
Figure 3.1: Crystal Structure ofβ-Si3N4, view along [001] (Si atoms are depicted dark gray, N
atoms are depicted black, SiN4 tetrahedra are drawn).
Figure 3.2: Crystal Structure ofγ-Si3N4 (Si atoms are depicted gray, N atoms black, SiN4tetra-
3 Silicon Nitride Si3N4
Regardingγ-Si3N4, the calculated volumes are 1.7 % smaller (LDA) and 1.9 % larger (GGA) than the experimantal volume. An overview of calculated lattice parameters in comparison to the experimetal data as well as previously calculated data is given in Table 3.1.
Table 3.1: Comparrison of calculated and experimental lattice parameter ofβ-Si3N4 andγSi3N4. β-Si3N4, P63/m, Z = 2 V / 106 pm3 c / pm a / pm Ref. 145.90 291.07 760.80 exp. [3] 146.42 291.02 762.20 [9] 146 291 761 [84] 142.70 288.52 755.75 [85] 142.68 / / [6] 147.76 / / [6]
143.74 289.14 757.65 this thesis (LDA) 148.43 292.38 765.63 this thesis (GGA)
γ-Si3N4, Fd 3m, Z = 8 V / 106 pm3 a / pm Ref. 463.81 773.81 exp. [5] 481.34 783.7 [9] 467 776 [4] 452.64 767.8 [6] 469.52 777.2 [6] 468.8 776.8 [10]
455.64 769.50 this thesis (LDA) 472.02 778.62 this thesis (GGA)
The calculated Si-N bonds in β-Si3N4 are almost identical, as can be expected for solely Si[4]-N[3] bonds. They are in good agreement with the experimental values (see Table 3.2). For γ-Si3N4, the Si[6]-N bonds are longer than the Si[4]-N bonds, as is to be expected for an increased coordination number of Si. This corresponds well to the experimental ndings (see Table 3.2).
Comparing the calculated densities of the two considered Si3N4 modications to the experimental values, reveals that LDA overestimates the experimental density and GGA underestimates it, as can be expected from the corresponding volume under- and overestimation. Nevertheless, the density trend of γ-Si3N4 being denser than β-Si3N4 is correctly reproduced (β-Si3N4: 3.24 (LDA), 3.14 (GGA); γ-Si3N4: 4.09