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Crystal Structure Prediction of the Novel Cr_{2}SiN_{4} Compound via Global Optimization, Data Mining, and the PCAE Method

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## Abstract

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_{3}N

_{4}amorphous matrix. However, these earlier experimental studies reported only Cr-Si-N in thin films. Here, we present the first investigation of possible bulk Cr-Si-N phases of composition Cr

_{2}SiN

_{4}. To identify the possible modifications, we performed global explorations of the energy landscape combined with data mining and the Primitive Cell approach for Atom Exchange (PCAE) method. After ab initio structural refinement, several promising low energy structure candidates were confirmed on both the GGA-PBE and the LDA-PZ levels of calculation. Global optimization yielded six energetically favorable structures and five modifications possible to be observed in extreme conditions. Data mining based searches produced nine candidates selected as the most relevant ones, with one of them representing the global minimum in the Cr

_{2}SiN

_{4}. Additionally, employing the Primitive Cell approach for Atom Exchange (PCAE) method, we found three more promising candidates in this system, two of which are monoclinic structures, which is in good agreement with results from the closely related Si

_{3}N

_{4}system, where some novel monoclinic phases have been predicted in the past.

## 1. Introduction

_{3}N

_{4}matrix) also have higher hardness and oxidation resistance compared with CrN thin films [5,6,15,16,17,18,29]—their hardness can be increased by up to ~9 GPa [2,3,30,31], while their oxidation resistance is significantly improved at temperatures well above the oxidation limit of ~600–700 °C of CrN [6,32,33].

_{2}or Si(OH)

_{2}layers in a humid environment, which plays a role as a self-lubricant, and by a smoother surface due to the Cr-Si-N amorphous phase [18,27]. Improved oxidation resistance can be explained by the formation of amorphous silicon oxide, which retards the diffusion of O as well as of Cr, Si, and N [29].

_{3}N

_{4}, we chose the analogous composition Cr

_{2}SiN

_{4}, where we would expect similarly high hardness as for Si

_{3}N

_{4}itself. Thus, in this study, we identify feasible structure candidates of the composition Cr

_{2}SiN

_{4}, using global optimization techniques, data-mining, and the Primitive Cell approach for Atom Exchange (PCAE) method, followed by careful re-optimizations of the candidates on the ab initio level.

## 2. Materials and Methods

_{2}SiN

_{4}system, a combination of global optimization (GO), data mining (DM), and the PCAE method has been used [39,40]. First, we performed global optimizations on the energy landscape of Cr

_{2}SiN

_{4}using simulated annealing [41], within the G42+ code [42]. A fast computable robust empirical two-body potential consisting of Lennard-Jones and exponentially damped Coulomb terms was employed to perform the GO searches with a reasonable computational effort [43]. Next, we have performed DM-based searches of the ICSD database [44,45] in order to find possible structure candidates for the unknown Cr

_{2}SiN

_{4}compound.

_{2}SiN

_{4}compound. The PCAE method is simple, fast, and computationally inexpensive compared to the supercell approach [40].

_{3}N

_{4}chemical system by first transforming the crystallographic (conventional) cell to the primitive cell, while keeping the symmetry and multiplicity of the atom positions (for more information, c.f. ref [40]). In the next stage, we replaced the number of atoms needed to obtain a certain stoichiometry (2 × Si by 2 × Cr atoms) on the symmetry-related Wyckoff positions. Subsequently, an ab initio full structure optimization without symmetry constraints was performed.

_{2}SiN

_{4}are available for comparison with the experiment [55,56,57].

## 3. Results and Discussion

#### 3.1. Global and Local Optimization Using Empirical and Ab Initio Energy Functions

_{2}SiN

_{4}phases using the GGA-PBE functional, where the α-Cr

_{2}SiN

_{4}-type is the lowest in calculated total energy, and the nf5-Cr

_{2}SiN

_{4}-type the highest one. Moreover, we grouped the resulting GO structures into energetically favorable ones, ranging from the α- to the λʹ-Cr

_{2}SiN

_{4}phase, and non-favorable Cr

_{2}SiN

_{4}modifications marked nf1 to nf5, which might be observed under extreme conditions. In addition, each favorable structure candidate has been subjected to a DFT-LDA optimization with similar results in the energetic ranking (Table A2.)

#### 3.1.1. Structural Analysis of the Most Promising Modifications Found after Global Optimization (GO)

_{2}SiN

_{4}-type and appears in the orthorhombic space group Pma2 (no. 28) with unit cell parameters of a = 5.54, b = 7.91, and c = 2.81 Å on the GGA-PBE level of calculation. The α-Cr

_{2}SiN

_{4}phase is visualized in Figure 1a, while full structural data of all favorable candidates are presented in Table 2 for calculations with the PBE functional, and in Table A1 for those computed with the LDA functional, respectively.

_{6}octahedra (with atom–atom distances of Cr(1) 2 × 1.82 Å-N, 1 × 1.89 Å-N, 1 × 1.97 Å-N, 2 × 2.13 Å-N, Cr(2) 2 × 1.91 Å-N, 2 × 1.95 Å-N, and 2 × 2.04 Å-N), while silicon is four-fold coordinated by nitrogen (with atom–atom distances of 2 × 1.77 Å-N and 2 × 1.71 Å-N) forming a tetrahedron. Moreover, the octahedra are connected by edges, while the tetrahedra are corner connected to them; additionally, the tetrahedra fall into two groups with opposite orientations.

_{2}SiN

_{4}-type. This triclinic structure appears in space group P-1 (no. 2), with cell parameters a = 7.28, b = 7.79, c = 2.74 Å, α = 93.66, β = 82.48, and γ = 120.64 calculated with GGA-PBE, and is visualized in Figure 1b. In this structure, chromium has a six-fold coordination by nitrogen forming two different CrN

_{6}octahedra similar to the α-modification, while silicon is four-fold coordinated by nitrogen with interatomic distances for chromium (Cr(1) 2 × 1.88 Å-N, 1 × 1.93 Å-N, 1 × 1.99 Å-N, 1 × 2.00 Å-N, 1 × 2.12 Å-N, Cr(2) 1 × 1.86 Å-N, 1 × 1.90 Å-N, 1 × 1.94 Å-N, 1 × 1.96 Å-N, 1 × 2.02 Å-N, and 2 × 2.05 Å-N) and silicon (2 × 1.70 Å-N, 1 × 1.78 Å-N, and 1 × 1.81 Å-N).

_{2}SiN

_{4}-type modification, SiN

_{4}tetrahedra are oriented in different directions; however, in this modification, the CrN

_{6}octahedra are face-connected and lean against each other, thus, forming a void in the center of the structure that is reminiscent of zeolite formation.

_{2}SiN

_{4}-type that crystallizes in space group P21/m (no. 11). This structure is presented in Figure 2a with unit cell parameters a = 6.21 b = 3.82 c = 5.54 Å, and β = 116.24, computed using the PBE functional (Table 2). Interestingly, in the δ-phase both, chromium and silicon are six-fold coordinated by nitrogen, with the octahedra being edge-connected, indicating that this candidate might be a high-pressure phase.

_{6}octahedra are Cr(1) 1 × 1.91 Å-N, 2 × 1.94 Å-N, 1 × 1.95 Å-N, 1 × 1.97 Å-N, 1 × 2.06 Å-N, Cr(2) 1 × 1.71 Å-N, 1 × 1.88 Å-N, 2 × 1.95 Å-N, 1 × 2.12 Å-N, and 1 × 2.27 Å-N, and the atom–atom distances in the SiN

_{6}octahedra are 1 × 1.81 Å-N, 1 × 1.93 Å-N, 2 × 1.92 Å-N, 1 × 1.85 Å-N, and 1 × 1.91 Å-N, respectively.

_{2}SiN

_{4}-type, which appears in space group P21/m (no. 11). It is visualized in Figure 2b with the unit cell parameters a = 5.09, b = 2.89, c = 8.90 Å, and β = 90.20. However, the ε-Cr

_{2}SiN

_{4}-type is composed of SiN

_{4}tetrahedra (with interatomic distances 1 × 1.77 Å-N, 2 × 1.72 Å-N, and 1 × 1.73 Å-N), while chromium is four-fold and six-fold coordinated by nitrogen, thus, forming CrN

_{4}tetrahedra and CrN

_{6}octahedra (interatomic distances are Cr(1) 1 × 1.90 Å-N, 4 × 2.00 Å-N, 1 × 2.07 Å-N, Cr(2) 1 × 1.75 Å-N, 2 × 1.78 Å-N, and 1 × 1.82 Å-N). Apart from being the same space group as the δ-phase, the monoclinic ε-modification resembles more the α- and β-Cr

_{2}SiN

_{4}-types (Table 1 and Table 2 and Figure 2).

_{2}SiN

_{4}-type with unit cell parameters a = 5.07, b = 2.88, c = 9.27 Å, and β = 99.77, while the other one (b) is called the λʹ-Cr

_{2}SiN

_{4}-type and has the unit cell parameters a = 5.06, b = 2.87, c = 9.18 Å, and β = 90.97. These two structures are structurally and energetically very similar, and both of them are composed of CrN

_{4}and SiN

_{4}tetrahedra, with CrN

_{6}octahedra between them.

_{6}octahedra are edge-connected, with corner-connected tetrahedra. Similar to the previous structures of the α-, β-, ε-Cr

_{2}SiN

_{4}-type of modifications, these tetrahedra have the opposite orientations in different layers of the structure. In the λ-Cr

_{2}SiN

_{4}-type of structure, chromium is connected to nitrogen with atom–atom distances Cr(1) 1 × 1.70 Å-N, 2 × 1.79 Å-N, 1 × 1.89 Å-N, Cr(2) 1 × 1.91 Å-N, 2 × 1.93 Å-N, 2 × 2.02 Å-N, 1 × 2.19 Å-N, Cr(3) 1 × 1.91 Å-N, 2 × 1.93 Å-N, 3 × 2.09 Å-N, Cr(4) 2 × 1.75 Å-N, 1 × 1.79 Å-N, 1 × 1.81 Å-N, and 1 × 2.64 Å-N, while there are also two different types of SiN

_{4}tetrahedra (with interatomic distances Si(1) 1 × 1.72 Å-N, 1 × 1.76 Å-N, 2 × 1.73 Å-N, Si(2) 3 × 1.74 Å-N, and 1 × 1.75 Å-N).

_{2}SiN

_{4}-type has a similar structure, with chromium being tetrahedrally and octahedrally coordinated by nitrogen in four different ways (with atom–atom distances Cr(1) 2 × 1.75 Å-N, 1 × 1.80 Å-N, 1 × 1.81 Å-N, Cr(2) 1 × 1.92 Å-N, 2 × 1.94 Å-N, 2 × 2.01 Å-N, 1 × 2.23 Å-N, Cr(3) 1 × 1.92 Å-N, 2 × 1.97 Å-N, 2 × 2.01 Å-N, 1 × 2.08 Å-N, Cr(4) 1 × 1.71 Å-N, 2 × 1.78 Å-N, and 1 × 1.88 Å-N) and two types of SiN

_{4}tetrahedra (interatomic distances Si(1) 1 × 1.73 Å-N, 2 × 1.74 Å-N, 1 × 1.75 Å-N, Si(2) 2 × 1.74 Å-N, and 2 × 1.73 Å-N).

#### 3.1.2. Structural Details of Non-Favorable Structures Found after a Global Search

_{2}SiN

_{4}-type and appears in space group P21/m (no. 11). It is visualized in Figure 4a with the unit cell parameters a = 5.03, b = 2.89, c = 9.25 Å, and β = 100.34.

_{6}octahedra positioned between two layers of nitrogen tetrahedra coordinating silicon and chromium. Hence, chromium in this structure has a four-fold as well as a six-fold coordination (with atom–atom distances Cr(1) 1 × 1.94 Å-N, 2 × 1.98 Å-N, 2 × 2.00 Å-N, 1 × 2.08 Å-N, Cr(2) 3 × 1.77 Å-N, 1 × 1.80 Å-N, and 1 × 2.72 Å-N) while silicon still remains in four-fold coordination (with the interatomic distances 1 × 1.76 Å-N and 3 × 1.74 Å-N). Both CrN

_{4}and SiN

_{4}tetrahedra in the upper and lower part of the structure are oriented in the opposite directions.

_{2}SiN

_{4}-type and crystallizes in space group Cc (no. 9). It is visualized in Figure 4b with the unit cell parameters a = 5.06, b = 14.14, c = 4.77 Å, and β = 121.05. Within this structure, chromium is five-fold coordinated by nitrogen and forms two types of polyhedra (with atom–atom distances Cr(1) 1 × 1.84 Å-N, 1 × 1.87 Å-N, 1 × 1.88 Å-N, 1 × 1.91 Å-N, 1 × 1.93 Å-N, Cr(2) 1 × 1.78 Å-N, 1 × 1.79 Å-N, 1 × 1.84 Å-N, 1 × 2.00 Å-N, and 1 × 2.11 Å-N) while silicon is four-fold coordinated by nitrogen (interatomic distances are 1 × 1.71 Å-N, 1 × 1.77 Å-N, 1 × 1.75 Å-N, and 1 × 1.76 Å-N).

_{2}SiN

_{4}-type, nf4-Cr

_{2}SiN

_{4}-type, and nf5-Cr

_{2}SiN

_{4}-type with the first two structures appearing in the space group Pm (no. 6) and the last one showing space group P-1 (no. 2), respectively. The total energies of these five modifications on ab initio level (GGA-PBE functional) are listed in Table 1. Furthermore, we note that most of the energetically favorable and non-favorable structures found after global optimization exhibit low symmetry, mostly orthorhombic and monoclinic symmetry (Table 2 and Table 3). This is in agreement with previous theoretical reports in the closely related Si

_{3}N

_{4}chemical system, where orthorhombic and monoclinic structures have been proposed [69].

#### 3.2. Data Mining (DM) Based Searches Using the ICSD Database

_{2}SiN

_{4}system using the GGA-PBE functional.

_{2}SiN

_{4}modifications found from the DM-based searches are presented in Table 5, while their corresponding total energies are listed in Table 4. The data-mining-based searches of the ICSD database resulted in many possible modifications, among which the four structures presented here are distinguished as being the energetically most favorable ones, while the others corresponded to non-favorable DM structure candidates.

#### 3.2.1. Structural Analysis of Low-Energy Candidates from the DM-Based Searches

_{2}MgO

_{4}-spinel-type modification [70,71], generated from the DM-based searches and visualized in Figure 5a, is the lowest one in the calculated total energy at both the GGA-PBE and the LDA-PZ level, for the whole energy landscape including the structures obtained from the GO and the PCAE method calculations. It exhibits space group Fd-3m (no. 227) with unit cell parameters a = 7.88 Å at the GGA-PBE level of calculation with all structural data presented in Table 5.

_{2}MgO

_{4}-type modification, chromium is six-fold coordinated forming a CrN

_{6}octahedron with the interatomic distance Cr 6 × 1.95 Å-N, while silicon is four-fold coordinated forming a SiN

_{4}tetrahedron with the atom–atom distance 4 × 1.74 Å-N. In this cubic modification, the CrN

_{6}octahedra are connected by edges while the SiN

_{4}tetrahedra are corner-connected. When performing structure optimizations of the candidates on the DFT-LDA level, the Al

_{2}MgO

_{4}-spinel-type modification remains the global minimum (Table A1 and Table A2). We note that the Al

_{2}MgO

_{4}-spinel-type of the structure appears in more than 4000 compounds (4250) with the chemical formula A1B2C4 indicating the importance of this structure-type on the energy landscape of ternary systems [44,45].

_{2}MnCl

_{4}-type [72], which is an orthorhombic structure that appears in space group Pbam (no. 55) with unit cell parameters a = 4.73, b = 8.70, and c = 2.73Å and is visualized in Figure 5b. In this modification, both chromium and silicon are six-fold coordinated by nitrogen, thus, forming distorted octahedra that are quite different from each other.

_{6}octahedra are quite similar to those in the WC structure-type, while the SiN

_{6}octahedra are “NaCl-type” octahedra. Interatomic distances in the CrN

_{6}octahedra are longer (1 × 1.92 Å-N, 2 × 1.97 Å-N, 2 × 1.98 Å-N, and 1 × 2.01 Å-N) than in the SiN

_{6}octahedra (4 × 1.85 Å-N, 2 × 1.88 Å-N). The SiN

_{6}octahedra are positioned in the center and at the edges of the cell, with the CrN

_{6}octahedra connecting them. Both edge and corner connections are observed.

_{2}O

_{4}-type [73]. It is a tetragonal structure in space group P4322 (no.95) with unit cell parameters a = 5.64 and c = 7.74 Å and is visualized in Figure 6a. In the TiMn

_{2}O

_{4}phase, chromium is both four-fold and six-fold coordinated by nitrogen, thus, forming CrN

_{4}tetrahedra (with atom–atom distances 2 × 1.78 Å-N and 2 × 1.81 Å-N) and CrN

_{6}octahedra (with atom–atom distances 2 × 1.89 Å-N, 2 × 1.97 Å-N, and 2 × 2.05 Å-N). Silicon is six-fold coordinated by nitrogen with interatomic distances (2 × 1.86 Å-N and 4 × 1.91 Å-N). The octahedra are edge-connected, while the CrN

_{4}tetrahedra are corner-connected.

_{2}SiO

_{4}- (Forsterite) type of structure [74], and it crystallizes in space group Pnma (no. 62) with unit cell parameters a = 9.42, b = 5.45, and c = 4.82 Å. Within this modification, two types of CrN

_{6}octahedra are connected by edges and oriented in the structure in two directions with interatomic distances (Cr(1) 2 × 1.92 Å-N, 2 × 1.98 Å-N, 2 × 2.00 Å-N, Cr(2) 1 × 1.88 Å-N, 2 × 1.95 Å-N, 2 × 2.00 Å-N, and 1 × 2.02 Å-N). Silicon is four-fold coordinated by nitrogen (with atom–atom distances 1 × 1.70 Å-N, 2 × 1.76 Å-N, and 1 × 1.77 Å-N) connected by the edges; the structure is visualized in Figure 6b.

#### 3.2.2. Structural Analysis of Non-Favorable Candidates Found after Data Mining

_{2}RuO

_{4}-type [75] and is visualized in Figure 7a. This is an orthorhombic structure that appears in space group Pbca (no. 61) with the unit cell parameters a = 4.55, b = 4.88, and c = 10.31 Å. Chromium and silicon are both six-fold coordinated by nitrogen but form different octahedra. Similar to the energetically favorable Na

_{2}MnCl

_{4}-type modification, within this modification, the CrN

_{6}octahedra resemble those in the WC-type of structure with the interatomic distances 1 × 1.82 Å-N, 1 × 1.89 Å-N, 1 × 1.90 Å-N, 1 × 1.94 Å-N, 1 × 1.97 Å-N, and 1 × 2.49 Å-N.

_{6}octahedra is via edges and corners as well. The whole structure consists of layers of different octahedra, where the SiN

_{6}ones are located on the faces and in the center of the cell with the CrN

_{6}octahedra situated in-between.

_{2}O

_{4}-like type of structure and crystallizes in space group P21 (no. 4). We note that the starting HgC

_{2}O

_{4}structure [76] after DFT optimization has been structurally modified, however, within the same space group (no.4), thus, resulting in a HgC

_{2}O

_{4}-like structure. This is a monoclinic structure with the unit cell parameters a = 5.34, b = 5.09, c = 5.36, and β = 115.62, with the structural parameters with corresponding energies given in Table 4 and Table 5.

_{6}octahedra (atom–atom distance Cr(1) 1 × 1.83 Å-N, 1 × 1.88 Å-N, 1 × 1.94 Å-N, 1 × 1.98 Å-N, 1 × 2.03 Å-N, 1 × 2.19 Å-N, Cr(2) 1 × 1.91 Å-N, 1 × 1.94 Å-N, 1 × 1.95 Å-N, 1 × 1.99 Å-N, 1 × 2.00 Å-N, and 1 × 2.09 Å-N), connected by faces and corners among each other. Silicon is four-fold coordinated forming edge- and corner-connected SiN

_{4}tetrahedra with the interatomic distances 1 × 1.72 Å-N, 1 × 1.73 Å-N, 1 × 1.75 Å-N, and 1 × 1.77 Å-N. The structure is visualized in Figure 7b.

_{2}IrO

_{4}-type [77] structure; it exhibits the space group P-62m (no. 189) with the unit cell parameters a = 8.33 and c = 2.70 and is visualized in Figure 8a. Within the structure, chromium is six-fold and seven-fold coordinated by nitrogen with the interatomic distances Cr(1) 6 × 1.98 Å-N, Cr(2) 6 × 2.08 Å-N, 3 × 2.42 Å-N, Cr(3) 1 × 1.88 Å-N, 4 × 2.02 Å-N, and 2 × 2.18 Å-N. The CrN

_{6}octahedra are face-connected resembling the ones in the WC-type of a structure, while the CrN7 polyhedra are edge- and corner-connected. Similarly, the SiN

_{6}octahedra are edge- and corner-connected, with the atom–atom distances 2 × 1.76 Å-N, 2 × 1.79 Å-N, and 2 × 1.92 Å-N.

_{2}O

_{4}-like structure, visualized in Figure 8b. This modification crystallizes in space group Pccn (no. 56) with the unit cell parameters a = 7.98, b = 14.42, and c = 4.85. Similarly, as with the HgC

_{2}O

_{4}-like modification, the CaB

_{2}O

_{4}-like structure is modified during the local optimization from the original prototypic CaB

_{2}O

_{4}structure [78] but still within the same space group (no. 56).

_{5}polyhedron with the interaction distances 2 × 1.81 Å-N, 1 × 1.88 Å-N, and 2 × 1.90 Å-N.

_{2}SnS

_{4}-type [79] and is visualized in Figure 9. This orthorhombic structure appears in the space group Cmmm (no. 65) with the unit cell parameters a = 5.58, b = 7.82, and c = 2.76. In this structure, both chromium and silicon, are six-fold coordinated by nitrogen with edge-connected octahedra and the interatomic distances Cr 6 × 1.96 Å-N, Si 4 × 1.85 Å-N, and 2 × 1.93 Å-N.

_{3}N

_{4}chemical system where orthorhombic structures have also been found [69]. In this context, we would like to remark on the Cu

_{2}HgI

_{4}type of structure [80]. This structure type has been recently predicted to exist as a modification in a study of novel hard phases of Si

_{3}N

_{4}[69].

_{2}HgI

_{4}type has also been found in our DM-based searches; however, it is energetically much worse than most of the other DM or GO/PCAE-based structure candidates (E

_{tot}= −5193.312 Eh calculated using the GGA-PBE functional). Full structural optimization resulted in the original prototypic structure in tetragonal space group I-42m (no. 121) with unit cell parameters a = 4.34 and c = 8.09 Å, at the GGA-PBE level of computation (both chromium and silicon are fourfold coordinated by nitrogen forming tetrahedra with interatomic distances of Cr 4 × 1.80 Å-N and Si 4 × 1.75 Å-N).

#### 3.3. Structural Searches Using the PCAE Method

_{2}SiN

_{4}compound, starting from typical structures in the related Si

_{3}N

_{4}system, the γ-, β-, and α-phase of Si

_{3}N

_{4}. Ranking the ab initio minimized structures according to the calculated total energy using the GGA-PBE functional, the most promising candidates generated using the PCAE method are presented in Table 6.

#### Structural Details of Candidates Found Using the PCAE Method

_{2}SiN

_{4}-type modification. Figure 10a shows a prototypic γ-phase in the Si

_{3}N

_{4}system [71], which was used as starting structure for generating the γ-Cr

_{2}SiN

_{4}-type. The Si

_{3}N

_{4}γ-phase crystallizes in the cubic space group I-43d (no. 220), [71] forming corner connected tetrahedra of silicon atoms (Figure 10a). However, after local optimization in the Cr

_{2}SiN

_{4}system using both GGA-PBE and LDA-PZ functionals, it converts to the γ-Cr

_{2}SiN

_{4}-type modification. It is a low-energy candidate in the Cr

_{2}SiN

_{4}system; however, it is structurally completely different from the starting γ-phase in the Si

_{3}N

_{4}system (compare Figure 10a,b).

_{2}SiN

_{4}-type modification crystallizes in the monoclinic space group Cc (no. 9) with unit cell parameters a = 5.62, b = 8.96, c = 5.36 Å, and β = 117.93, with both cations—chromium and silicon—being six-fold coordinated by nitrogen where these octahedra are edge- and face-connected (Figure 10b). We deal with two different types of CrN

_{6}octahedra with the interatomic distances Cr(1) 1 × 1.84 Å-N, 1 × 1.85 Å-N, 1 × 1.92 Å-N, 1 × 1.96 Å-N, 1 × 2.01 Å-N, 1 × 2.13 Å-N, Cr(2) 1 × 1.83 Å-N, 1 × 1.90 Å-N, 1 × 1.92 Å-N, 1 × 1.96 Å-N, 1 × 2.01 Å-N, and 1 × 2.03 Å-N; the atom–atom distances in the SiN

_{6}octahedra are 1 × 1.74 Å-N, 1 × 1.75 Å-N, 1 × 1.81 Å-N, 1 × 2.08 Å-N, 1 × 2.20 Å-N, and 1 × 2.34 Å-N). In the closely related Si

_{3}N

_{4}compound, there has been a prediction of novel monoclinic phases from first-principles calculations [69].

_{2}SiN

_{4}-PCAE-1 phase, which is energetically less favorable than the γ-phase (Table 6). The Cr

_{2}SiN

_{4}-PCAE-1 type has been generated similarly to the previous one, starting from the β-phase of Si

_{3}N

_{4}in the hexagonal P63/m (no. 176) space group [81,82]. After ab initio structural optimization in the Cr

_{2}SiN

_{4}system, the structure converted to a monoclinic modification denoted Cr

_{2}SiN

_{4}-PCAE-1 that crystallizes in space group Pm (no. 6) with unit cell parameters a = 7.04465, b = 3.03610, c = 7.03270, β = 110.7203 (Table 7).

_{4}tetrahedra and two different CrN

_{5}polyhedra, with the atom–atom distances Cr(1) 2 × 1.86 Å-N, 1 × 1.89 Å-N, 1 × 1.95 Å-N, 1 × 2.01 Å-N, Cr(2) 2 × 1.81 Å-N, 1 × 1.88 Å-N, 1 × 1.92 Å-N, 1 × 1.93 Å-N, Cr(3) 2 × 1.74 Å-N, 1 × 1.75 Å-N, 1 × 1.99 Å-N, 1 × 2.18 Å-N, Cr(4) 2 × 1.77 Å-N, and 2 × 1.78 Å-N.

_{4}tetrahedra with interatomic distances of 1 × 1.74 Å-N, 2 × 1.76 Å-N, and 1 × 1.83 Å-N. Additionally, there is one SiN

_{5}polyhedron corner connected to one of the tetrahedra (atom–atom distances of 1 × 1.72 Å-N, 1 × 1.77 Å-N, 2 × 1.78 Å-N, and 1 × 2.49 Å-N), where chromium is completely located in the inner part of the unit cell, and the connection between polyhedra is formed via edges and corners (Figure 11a).

_{2}SiN

_{4}-PCAE-2-type of structure presented as a final non-favorable structure candidate generated using the PCAE method. In this case, the α-type structure of Si

_{3}N

_{4}with trigonal P31c (no. 159) space group [82,83] was used as starting point. However, after full structural optimization on the GGA-PBE level, the symmetry of the Cr

_{2}SiN

_{4}-PCAE-2-type is completely reduced to space group P1 (no. 1) with unit cell parameters of a = 7.87856, b = 7.96102, c = 5.78635, α = 89.9706, β = 89.8616, and γ = 120.2646 (Table 7).

_{4}tetrahedra, with the atom–atom distances Cr(1) 1 × 1.77 Å-N, 2 × 1.80 Å-N, 1 × 1.86 Å-N, Cr(2) 2 × 1.78 Å-N, 1 × 1.79 Å-N, 1 × 1.81 Å-N, Cr(3) 2 × 1.79 Å-N, 2 × 1.80 Å-N, Cr(4) 1 × 1.77 Å-N, 1 × 1.78 Å-N, 1 × 1.80 Å-N, 1 × 1.81 Å-N, Cr(5) 1 × 1.77 Å-N, 1 × 1.79 Å-N, 1 × 1.81 Å-N, 1 × 1.87 Å-N, Cr(6) 1 × 1.78 Å-N, 1 × 1.80 Å-N, 2 × 1.82 Å-N, Cr(7) 1 × 1.78 Å-N, 2 × 1.79 Å-N, 1 × 1.81 Å-N, Cr(8) 1 × 1.78 Å-N, 2 × 1.79 Å-N, and 1 × 1.80 Å-N).

_{4}tetrahedra with the interatomic distances Si1 1 × 1.73 Å-N, 1 × 1.74 Å-N, 1 × 1.75 Å-N, 1 × 1.76 Å-N, Si2 2 × 1.74 Å-N, and 2 × 1.75 Å-N. The SiN

_{4}tetrahedra are mostly positioned at the corners of the cell with three tetrahedra located inside along with CrN

_{4}tetrahedra located entirely inside the cell. Nevertheless, all tetrahedra within this phase are corner-connected (Figure 11b). A summary of the structural data of the presented PCAE structures is shown in Table 7; other structure candidates generated using the PCAE method were energetically much less favorable and, thus, have not been included.

#### 3.4. Energy Landscape of Cr_{2}SiN_{4} on the Ab Initio Level

_{2}SiN

_{4}modifications. Table 8 presents the energetic ranking of these modifications, where the Al2MgO4-spinel-type appears lowest in calculated total energy with the value of −5193.507 E

_{h}, thus, representing the global minimum among the candidates obtained in the various searches. If the calculations are performed using DFT-LDA (Table A2), the spinel structure remains the global minimum, and the energetic ranking of the other modifications is very similar, with few exceptions.

_{2}SiN

_{4}modifications. We note that the global minimum in the Cr

_{2}SiN

_{4}system is the Al

_{2}MgO

_{4}-spinel phase. This prominence of the Al

_{2}MgO

_{4}-type candidate on the energy landscape of Cr

_{2}SiN

_{4}is not unreasonable, since several earlier calculations in the related (binary) Si

_{3}N

_{4}system have also found a Al

_{2}MgO

_{4}-spinel-like phase [71,84,85].

_{2}SiN

_{4}-type to possibly become competitive, as well as the TiMn

_{2}O

_{4}-type and Mg

_{2}SiO

_{4}-type modifications from the DM-based searches. Similarly, the most relevant modifications that might appear in the high-pressure region are the Na

_{2}MnCl

_{4}-type, and the α-, γ-, and δ-Cr

_{2}SiN

_{4}-types. Therefore, enthalpy vs. pressure, H(p), curves were computed for these five modifications (Figure 13).

_{2}MnCl

_{4}-type at a pressure of ~33 GPa (Figure 13). In addition, there was a phase transition between the Na

_{2}MnCl

_{4}-type and the metastable α-Cr

_{2}SiN

_{4}-type modifications at ~15 GPa (Figure 13), i.e., the α-phase was more stable than the Na

_{2}MnCl

_{4}-type modification below 15 GPa in the Cr

_{2}SiN

_{4}system.

_{2}SiN

_{4}is highly complex with a wide range of structurally different modifications possible. On the ab initio level, the global minimum corresponds to the AlMg

_{2}O

_{4}-type of structure. This modification is also known as a spinel structure, formulated as AB

_{2}X

_{4}, with a fcc close-packed array of anions X, and A and B cations occupying some or all of the octahedral and tetrahedral sites in the lattice, respectively. The structural features found in this structure type are the most dominant ones in the low-energy region of the landscape of Cr

_{2}SiN

_{4}at standard pressure, where most of the structures are found to exhibit an octahedral coordination of chromium by nitrogen and a tetrahedral one for silicon, respectively.

_{2}SiN

_{4}-type, the Na

_{2}MnCl

_{4}-type, and the Mn

_{2}SnS

_{4}-type), while there is only one stable modification (Cr

_{2}SiN

_{4}-PCAE-2-type) exhibiting tetrahedral coordination by nitrogen for both cations. In addition, a few structures show unusual five-fold and seven-fold coordination, but these are energetically non-favorable.

_{2}SiN

_{4}and the observation of analogous coordination environments of Cr and Si in most of the structures found as low-energy minima on the landscape are strong indications that this compound should be synthetically accessible. Furthermore, a spinel-type modification of Cr

_{2}SiN

_{4}could be of great importance, since ferrite spinels and related structures are of technological interest due to their magnetic ordering, which can be ferrimagnetic or antiferromagnetic depending on the structure and the nature of the metal ions.

_{2}SiN

_{4}structure candidates [86,87,88]. Another interesting point is that the global minimum AlMg

_{2}O

_{4}-spinel-type of the structure shows the highest cubic Fd-3m symmetry, while most of the predicted structure candidates show much lower symmetry.

_{2}SiN

_{4}-type structure was obtained instead. Quite generally, data-mining-based searches produced high-symmetry candidates (except for the HgC

_{2}O

_{4}-type, which is monoclinic), while global optimization and the PCAE method mostly produced structures with lower symmetry, with the orthorhombic space group Pma2 (no. 28) of the α-Cr

_{2}SiN

_{4}-type as the highest symmetry space group.

_{2}SiN

_{4}system, the global optimization resulted in a large number of new unknown but kinetically stable low-energy structures and provided a general overview over the broad structural variety present in the system at low energies, while the PCAE method generated the most diverse modifications, ranging from structures consisting of networks with only tetrahedral coordination polyhedra of the cations by nitrogen to modifications with only octahedral coordination environments.

## 4. Conclusions

_{2}SiN

_{4}and to perform structure prediction in the Cr

_{2}SiN

_{4}system. Global optimization was performed using simulated annealing and a fast computable robust empirical two-body potential. Data-mining-based searches reduced a large number of crystal structures from the ICSD database to four energetically favorable structures and five structure candidates that might be feasible modifications at extreme conditions for the not-yet-synthesized Cr

_{2}SiN

_{4}compound.

_{2}SiN

_{4}system with starting modifications taken from the related Si

_{3}N

_{4}chemical system. Every structure candidate found in these searches was subjected to crystallographic analysis and all the promising ones to a local optimization on the ab initio level.

_{2}SiN

_{4}system.

_{2}SiN

_{4}system resulted in numerous predicted new structures/modifications not yet observed in the experiment. Within each method used in this study, we grouped our results into energetically favorable and non-favorable Cr

_{2}SiN

_{4}modifications, with the latter ones perhaps accessible for extreme thermodynamic conditions. Among the eleven energetically most favorable candidates, the global minimum was the AlMg

_{2}O

_{4}-spinel-type exhibiting the cubic Fd-3m symmetry.

_{2}MnCl

_{4}-type modifications should take place. Thus, we present a large number of feasible structures for modifications in the synthetically not-yet-explored Cr

_{2}SiN

_{4}system, which might be suitable for many technological applications, such as high-speed cutting, wood machining applications, hydraulic piston pumps, valves, gears, shafts, and propellers, as corrosion-resistant coatings and low-cost water-based lubricating systems.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**The modifications, space groups, unit cell parameters, and atomic positions for the most favorable Cr

_{2}SiN

_{4}modifications after structural optimization on the DFT-LDA level.

Modifications and Space Groups | Cell Parameters | Position of Atoms |
---|---|---|

Al_{2}MgO_{4}-spinel-type_LDAFd-3m No 227 | a = 7.76 | Cr 0.000000 0.000000 0.000000 Si 0.625000 0.625000 0.625000 N 0.752556 0.752556 0.752556 |

α-Cr_{2}SiN_{4}-type_LDAPma2 No 28 | a = 5.45 b = 7.80 c = 2.76 | Cr 0.750000 0.241721 0.552936 Cr 0.500000 0.000000 0.898141 Si 0.750000 0.614777 0.000000 N 0.750000 0.006165 0.421951 N 0.750000 0.489793 0.499478 N 0.501351 0.244258 0.983987 |

Na_{2}MnCl_{4}-type_LDAPbam No 55 | a = 4.67 b = 8.58 c = 2.69 | Cr 0.432611 0.175929 0.500000 Si 0.000000 0.000000 0.000000 N 0.133271 0.203741 0.000000 N 0.257704 0.966687 0.500000 |

γ-Cr_{2}SiN_{4}-type_LDACc No 9 | a = 5.56 b = 8.82 c = 5.25 β = 117.96 | Cr 0.499339 0.356416 0.521849 Cr 0.490863 0.639616 0.502192 Si 0.000000 0.578114 0.000000 N 0.817340 0.495466 0.665199 N 0.336682 0.499101 0.681972 N 0.677149 0.750383 0.851076 N 0.670732 0.238989 0.855745 |

β-Cr_{2}SiN_{4}-type_LDAP-1 No 2 | a = 7.18 b = 7.69 c = 2.70 α = 94.01 β = 82.69 γ = 121.02 | Cr 0.350127 0.864148 0.734578 Cr 0.863665 0.518329 0.302902 Si 0.625158 0.737213 0.910010 N 0.050410 0.680530 0.762885 N 0.352905 0.668370 0.167669 N 0.662487 0.971411 0.739846 N 0.713401 0.667644 0.365727 |

δ-Cr_{2}SiN_{4}-type_LDAP21/m No 11 | a = 6.14 b = 3.76 c = 5.41 β = 115.88 | Cr 0.091491 0.750000 0.659687 Cr 0.148173 0.750000 0.246670 Si 0.610570 0.750000 0.913336 N 0.638880 0.250000 0.419381 N 0.142755 0.250000 0.721438 N 0.367862 0.750000 0.044580 N 0.131910 0.250000 0.172992 |

TiMn_{2}O_{4}-type_LDAP4322 No 95 | a = 5.56 c = 7.61 | Cr 0.500000 0.290089 0.000000 Cr 0.233800 0.233800 0.625000 Si 0.000000 0.260505 0.000000 N 0.238872 0.500821 0.998335 N 0.247363 0.028667 0.009638 |

Mg_{2}SiO_{4}-type_LDAPnma No 62 | a = 9.23 b = 5.35 c = 4.77 | Cr 0.500000 0.500000 0.500000 Cr 0.752388 0.750000 0.005094 Si 0.911386 0.750000 0.581141 N 0.915976 0.750000 0.228570 N 0.580994 0.750000 0.754642 N 0.831042 0.496221 0.748887 |

ε-Cr_{2}SiN_{4}-type_LDAP21/m No 11 | a = 5.07 b = 2.86 c = 8.45 β = 90.40 | Cr 0.780604 0.250000 0.510721 Cr 0.923710 0.750000 0.845133 Si 0.427115 0.250000 0.804711 N 0.570806 0.750000 0.868258 N 0.885468 0.750000 0.109395 N 0.589494 0.750000 0.396942 N 0.052827 0.250000 0.362060 |

λ-Cr_{2}SiN_{4}-type_LDAPm No 6 | a = 4.96 b = 2.84 c = 8.88 β = 98.40 | Cr 0.914225 0.500000 0.418382 Cr 0.707639 0.500000 0.700020 Cr 0.303954 0.000000 0.721261 Cr 0.508118 0.500000 0.052804 Si 0.000000 0.000000 0.000000 Si 0.419874 0.000000 0.378878 N 0.545716 0.000000 0.572397 N 0.460020 0.500000 0.846268 N 0.704067 0.000000 0.081337 N 0.551407 0.500000 0.308625 N 0.057263 0.500000 0.603299 N 0.068995 0.000000 0.346274 N 0.943534 0.000000 0.807418 N 0.155450 0.500000 0.072091 |

λ’-Cr_{2}SiN_{4}-type_LDAPm No 6 | a = 4.99 b = 2.82 c = 8.84 β = 90.95 | Cr 0.175453 0.500000 0.329781 Cr 0.898546 0.500000 0.677506 Cr 0.311343 0.000000 0.668615 Cr 0.483688 0.500000 0.973214 Si 0.000000 0.000000 0.000000 Si 0.681904 0.000000 0.380834 N 0.683473 0.000000 0.573821 N 0.506564 0.500000 0.783113 N 0.663254 0.000000 0.037342 N 0.818343 0.500000 0.305474 N 0.181392 0.500000 0.535694 N 0.363881 0.000000 0.300294 N 0.023309 0.000000 0.806317 N 0.153474 0.500000 0.070996 |

**Table A2.**The total energy and relative energy values compared to the global minimum (spinel structure taken as the zero of energy) of the energetically most favorable Cr

_{2}SiN

_{4}modifications found using various search methods and later locally optimized on the ab initio level using the LDA-PZ functional. DM stands for data mining, GO stands for global optimization, and PCAE stands for primitive cell approach for atom exchange methods.

Modification and Space Group | Search Method | Total Energy (Eh) | Relative Energy (kcal/mol) |
---|---|---|---|

Al_{2}MgO_{4}-spinel-type_LDA | DM | −5180.729 | 0.0 |

α-Cr_{2}SiN_{4}-type_LDA | GO | −5180.694 | 21.963 |

Na_{2}MnCl_{4}-type_LDA | DM | −5180.672 | 35.768 |

γ-Cr_{2}SiN_{4}-type_LDA | PCAE | −5180.667 | 38.906 |

β-Cr_{2}SiN_{4}-type_LDA | GO | −5180.653 | 47.691 |

δ-Cr_{2}SiN_{4}-type_LDA | GO | −5180.653 | 47.691 |

TiMn_{2}O_{4}-type_LDA | DM | −5180.621 | 67.771 |

Mg_{2}SiO_{4}-type_LDA | DM | −5180.620 | 68.399 |

ε-Cr_{2}SiN_{4}-type_LDA | GO | −5180.615 | 71.536 |

λ-Cr_{2}SiN_{4}-type_LDA | GO | −5180.606 | 77.184 |

λ’-Cr_{2}SiN_{4}-type_LDA | GO | −5180.604 | 78.439 |

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**Figure 1.**Visualization of favorable Cr

_{2}SiN

_{4}modifications: (

**a**) α-Cr

_{2}SiN

_{4}-type in space group Pma2 (no. 28); (

**b**) β-Cr

_{2}SiN

_{4}-type in space group P-1 (no. 2). Green, red, and yellow spheres denote Cr, Si, and N atoms, respectively.

**Figure 2.**Visualization of favorable Cr

_{2}SiN

_{4}modifications: (

**a**) δ-Cr

_{2}SiN

_{4}-type in space group P21/m (no. 11). (

**b**) ε-Cr2SiN4-type in space group P21/m (no. 11). Green, red, and yellow spheres denote Cr, Si, and N atoms, respectively.

**Figure 3.**Visualization of favorable Cr2SiN4 modifications found in space group Pm (no. 6): (

**a**) λ- Cr2SiN4-type; (

**b**) λʹ- Cr2SiN4-type. Green, red, and yellow spheres denote Cr, Si, and N atoms, respectively.

**Figure 4.**Visualization of non-favorable Cr

_{2}SiN

_{4}modifications: (

**a**) nf1-Cr

_{2}SiN

_{4}-type in space group P21/m (no. 11); (

**b**) nf2-Cr

_{2}SiN

_{4}-type in space group Cc (no. 9). Green, red, and yellow spheres denote Cr, Si, and N atoms, respectively.

**Figure 5.**Visualization of favorable Cr2SiN4 modifications: (

**a**) Al2MgO4-spinel-type in space group Fd-3m (no. 227); (

**b**) Na

_{2}MnCl

_{4}-type that appears in space group Pbam (no. 55). Green, red, and yellow spheres denote Cr, Si, and N atoms, respectively.

**Figure 6.**Visualization of favorable Cr2SiN4 modifications: (

**a**) TiMn2O4-type in space group P4322 (no. 95); (

**b**) Mg2SiO4-type in space group Pnma (no.62). Green, red, and yellow spheres denote Cr, Si, and N atoms, respectively.

**Figure 7.**Visualization of non-favorable Cr

_{2}SiN

_{4}modification from data-mining: (

**a**) Ca

_{2}RuO

_{4}-type in space group Pbca (no. 61), (

**b**) HgC

_{2}O

_{4}-type in space group P21 (no. 4).

**Figure 8.**Visualization of non-favorable Cr

_{2}SiN

_{4}modifications obtained from data-mining: (

**a**) Ca

_{2}IrO

_{4}-type in space group P-62m (no. 189); (

**b**) CaB

_{2}O4-type in space group Pccn (no. 56). Green, red, and yellow spheres denote Cr, Si, and N atoms, respectively.

**Figure 9.**Visualization of non-favorable Cr

_{2}SiN

_{4}modification obtained from data-mining referred to as Mn

_{2}SnS

_{4}-type in space group Cmmm (no 65).

**Figure 10.**Visualization of: (

**a**) the Si

_{3}N

_{4}γ-phase in space group I-43d (no. 220); (

**b**) γ- Cr

_{2}SiN

_{4}-type in space group Cc (no. 9). Green, red, and yellow spheres denote Cr, Si, and N atoms, respectively.

**Figure 11.**Visualization of the non-favorable PCAE structures: (

**a**) Cr

_{2}SiN

_{4}-PCAE-1-type in space group Pm (no. 6); (

**b**) Cr

_{2}SiN

_{4}-PCAE-2-type in space group P1 (no. 1). Green, red, and yellow spheres denote Cr, Si, and N atoms, respectively.

**Figure 12.**Energy vs. volume, E(V), curves for the most favorable Cr

_{2}SiN

_{4}modifications calculated using the GGA-PBE functional. Energies per formula unit are given in Hartree (E

_{h}).

**Figure 13.**(

**a**) Enthalpy vs. pressure, H(p), curves computed for the five most relevant structures, including the high-pressure region. (

**b**) Plots on the right side show H(p) curves focused on two transitions. Top: H(p) curves for the spinel- and the Na

_{2}MnCl

_{4}-type (transition pressure was found to be ~33 GPa); Bottom: H(p) curves between the Na

_{2}MnCl

_{4}-type and the α-Cr

_{2}SiN

_{4}-type modifications (transition pressure would be ~15 GPa). The calculations were performed using the GGA-PBE functional and the energies per formula unit are given in Hartree (E

_{h}).

**Table 1.**The total energy (in Eh) and relative energy values (in kcal/mol) compared to the global minimum (spinel structure taken as the zero of energy) of Cr

_{2}SiN

_{4}modifications found after global optimization on an empirical potential level and locally optimized using the GGA-PBE functional.

Modification | Total Energy (Eh) | Relative Energy (kcal/mol) |
---|---|---|

α-Cr_{2}SiN_{4}-type | −5193.474 | 20.708 |

β-Cr_{2}SiN_{4}-type | −5193.438 | 43.298 |

δ-Cr_{2}SiN_{4}-type | −5193.419 | 55.221 |

ε-Cr_{2}SiN_{4}-type | −5193.413 | 58.986 |

λ-Cr_{2}SiN_{4}-type | −5193.407 | 62.750 |

λʹ-Cr_{2}SiN_{4}-type | −5193.404 | 64.634 |

nf1-Cr_{2}SiN_{4}-type | −5193.398 | 68.399 |

nf2-Cr_{2}SiN_{4}-type | −5193.388 | 74.674 |

nf3-Cr_{2}SiN_{4}-type | −5193.385 | 76.556 |

nf4-Cr_{2}SiN_{4}-type | −5193.364 | 89.734 |

nf5-Cr_{2}SiN_{4}-type | −5193.364 | 89.734 |

**Table 2.**The modifications, space groups, unit cell parameters, and atomic positions for favorable Cr

_{2}SiN

_{4}modifications found after global optimization and later locally optimized on the ab initio level using the GGA-PBE functional.

Modifications and Space Group | Cell Parameters | Position of Atoms |
---|---|---|

α-Cr_{2}SiN_{4}-typePma2 No 28 | a = 5.54 b = 7.91 c = 2.81 | Cr 0.750000 0.245688 0.560883 Cr 0.500000 0.000000 0.891696 Si 0.750000 0.616760 0.000000 N 0.750000 0.011076 0.423540 N 0.750000 0.493745 0.499960 N 0.501383 0.244129 0.983362 |

β-Cr_{2}SiN_{4}-typeP-1 No 2 | a = 7.28 b = 7.79 c = 2.74 α = 93.66 β = 82.48 γ = 120.64 | Cr 0.347419 0.861114 0.730714 Cr 0.862711 0.520348 0.305878 Si 0.627078 0.740684 0.913120 N 0.050229 0.677562 0.759435 N 0.353428 0.666353 0.160724 N 0.660109 0.973354 0.749444 N 0.717565 0.676085 0.369516 |

δ-Cr_{2}SiN_{4}-typeP21/m No 11 | a = 6.21 b = 3.82 c = 5.54 β = 116.24 | Cr 0.088537 0.750000 0.660564 Cr 0.155145 0.750000 0.253988 Si 0.613292 0.750000 0.917590 N 0.639615 0.250000 0.413635 N 0.137168 0.250000 0.721296 N 0.368220 0.750000 0.042876 N 0.136099 0.250000 0.176958 |

ε-Cr_{2}SiN_{4}-typeP21/m No 11 | a = 5.09 b = 2.89 c = 8.90 β = 90.20 | Cr 0.778189 0.250000 0.506779 Cr 0.918790 0.750000 0.827219 Si 0.419972 0.250000 0.798720 N 0.567333 0.750000 0.861544 N 0.897872 0.750000 0.121156 N 0.589340 0.750000 0.395794 N 0.060536 0.250000 0.368680 |

λ-Cr_{2}SiN_{4}-typePm No 6 | a = 5.07 b = 2.88 c = 9.27 β = 99.77 | Cr 0.951900 0.500000 0.415607 Cr 0.698619 0.500000 0.708230 Cr 0.291904 0.000000 0.722310 Cr 0.507805 0.500000 0.040389 Si 0.000000 0.000000 0.000000 Si 0.455229 0.000000 0.394275 N 0.535616 0.000000 0.585846 N 0.455096 0.500000 0.844293 N 0.704956 0.000000 0.076886 N 0.586474 0.500000 0.329108 N 0.050793 0.500000 0.599924 N 0.108035 0.000000 0.351718 N 0.939400 0.000000 0.811337 N 0.163111 0.500000 0.068461 |

λ’-Cr_{2}SiN_{4}-typePm No 6 | a = 5.06 b = 2.87 c = 9.18 β = 90.97 | Cr 0.166652 0.500000 0.355602 Cr 0.889145 0.500000 0.685518 Cr 0.307908 0.000000 0.681019 Cr 0.485319 0.500000 0.980324 Si 0.000000 0.000000 0.000000 Si 0.670339 0.000000 0.395017 N 0.676765 0.000000 0.583978 N 0.498298 0.500000 0.794695 N 0.665262 0.000000 0.039248 N 0.812605 0.500000 0.324037 N 0.176521 0.500000 0.552065 N 0.351051 0.000000 0.317489 N 0.015000 0.000000 0.810727 N 0.151227 0.500000 0.066773 |

**Table 3.**The modifications, space groups, unit cell parameters, and atomic positions for non-favorable Cr

_{2}SiN

_{4}modifications found using global optimization and later optimized at the GGA-PBE level of calculation.

Modifications and Space Group | Cell Parameters | Position of Atoms |
---|---|---|

nf1-Cr_{2}SiN_{4}-typeP21/m No 11 | a = 5.03 b = 2.89 c = 9.25 β = 100.34 | Cr 0.780740 0.750000 0.501443 Cr 0.025858 0.250000 0.824580 Si 0.517842 0.750000 0.797882 N 0.959527 0.250000 0.630184 N 0.552291 0.250000 0.393501 N 0.609701 0.750000 0.135709 N 0.136923 0.250000 0.122932 |

nf2-Cr_{2}SiN_{4}-typeCc No 9 | a = 5.06 b = 14.14 c = 4.77 β = 121.05 | Cr 0.622331 0.095134 0.360180 Cr 0.380879 0.098904 0.725602 Si 0.000000 0.191719 0.000000 N 0.487148 0.004091 0.555289 N 0.511786 0.374204 0.196387 N 0.720274 0.147401 0.064041 N 0.358846 0.192090 0.368624 |

nf3-Cr_{2}SiN_{4}-typePm No 6 | a = 6.79 b = 3.09 c = 6.88 β = 109.29 | Cr 0.319643 0.500000 0.397294 Cr 0.308694 0.500000 0.793094 Cr 0.034585 0.000000 0.406174 Cr 0.733922 0.500000 0.593240 Si 0.000000 0.000000 0.000000 Si 0.588865 0.000000 0.169164 N 0.515186 0.500000 0.666342 N 0.819399 0.000000 0.531146 N 0.551036 0.500000 0.283640 N 0.215615 0.000000 0.242977 N 0.384133 0.000000 0.929487 N 0.000403 0.500000 0.870324 N 0.122011 0.500000 0.540245 N 0.829057 0.000000 0.140470 |

nf4-Cr_{2}SiN_{4}-typePm No 6 | a = 7.37 b = 3.05 c = 7.56 β = 115.96 | Cr 0.779088 0.000000 0.218259 Cr 0.459516 0.500000 0.327876 Cr 0.253478 0.000000 0.544973 Cr 0.396471 0.500000 0.935322 Si 0.000000 0.500000 0.000000 Si 0.852505 0.000000 0.603621 N 0.330961 0.500000 0.474928 N 0.889506 0.500000 0.738026 N 0.613308 0.000000 0.401256 N 0.355218 0.000000 0.802744 N 0.932400 0.000000 0.070113 N 0.262965 0.500000 0.082041 N 0.643042 0.500000 0.129399 N 0.988794 0.000000 0.464379 |

nf5-Cr_{2}SiN_{4}-typeP-1 No 2 | a = 7.17 b = 3.06 c = 7.41 α = 89.69 β = 66.68 γ = 88.06 | Cr 0.654478 0.292004 0.523373 Cr 0.288987 0.742844 0.838266 Si 0.865354 0.748679 0.767528 N 0.255163 0.244316 0.972502 N 0.804235 0.255276 0.687399 N 0.477326 0.210250 0.368818 N 0.127792 0.732554 0.709190 |

**Table 4.**The total energy and relative energy values compared to the global minimum (spinel structure taken as the zero of energy) of Cr

_{2}SiN

_{4}modifications obtained from DM-based searches and calculated using GGA-PBE.

Modification | Total Energy (Eh) | Relative Energy (kcal/mol) |
---|---|---|

Al_{2}MgO_{4}-spinel-type | −5193.507 | 0.0 |

Na_{2}MnCl_{4}-type | −5193.436 | 44.553 |

TiMn_{2}O_{4}-type | −5193.414 | 58.358 |

Mg_{2}SiO_{4}-type | −5193.403 | 65.261 |

Ca_{2}RuO_{4}-type | −5193.402 | 65.889 |

HgC_{2}O_{4}-like | −5193.400 | 67.144 |

Ca_{2}IrO_{4}-type | −5193.349 | 99.147 |

CaB_{2}O_{4}-like | −5193.347 | 100.402 |

Mn_{2}SnS_{4}-type | −5193.342 | 103.539 |

**Table 5.**The modifications, space groups, unit cell parameters, and atomic positions for Cr

_{2}SiN

_{4}modifications obtained from data-mining-based searches and local optimization at the GGA-PBE level.

Modifications and Space Group | Cell Parameters | Position of Atoms |
---|---|---|

Al_{2}MgO_{4}-spinel-typeFd-3m No 227 | a = 7.88 | Cr 0.000000 0.000000 0.000000 Si 0.625000 0.625000 0.625000 N 0.752483 0.752483 0.752483 |

Na_{2}MnCl_{4}-typePbam No 55 | a = 4.74 b = 8.70 c = 2.73 | Cr 0.433387 0.175989 0.500000 Si 0.000000 0.000000 0.000000 N 0.133667 0.203146 0.000000 N 0.256351 0.966597 0.500000 |

TiMn_{2}O_{4}-typeP4322 No 95 | a = 5.64 c = 7.74 | Cr 0.500000 0.288490 0.000000 Cr 0.234522 0.234522 0.625000 Si 0.000000 0.259306 0.000000 N 0.239859 0.499775 0.998379 N 0.246452 0.027106 0.010564 |

Mg_{2}SiO_{4}-typePnma No 62 | a = 9.42 b = 5.45 c = 4.82 | Cr 0.500000 0.500000 0.500000 Cr 0.751225 0.750000 0.005741 Si 0.911733 0.750000 0.580414 N 0.915086 0.750000 0.227101 N 0.579248 0.750000 0.754925 N 0.831733 0.498905 0.747339 |

Ca_{2}RuO_{4}-typePbca No 61 | a = 4.55 b = 4.88 c = 10.32 | Cr 0.951271 0.088328 0.314395 Si 0.000000 0.000000 0.000000 N 0.189814 0.299961 0.071415 N 0.821277 0.910789 0.170556 |

HgC_{2}O_{4}-likeP21 No 4 | a = 5.34 b = 5.09 c = 5.36 β = 115.62 | Cr 0.009785 0.463302 0.266681 Cr 0.242643 0.853311 0.472408 Si 0.651871 0.000000 0.959796 N 0.021251 0.353294 0.916553 N 0.626651 0.296871 0.103741 N 0.597719 0.030584 0.610720 N 0.079595 0.179181 0.498154 |

Ca_{2}IrO_{4}-typeP-62m No 189 | a = 8.33 c = 2.70 | Cr 0.000000 0.000000 0.000000 Cr 0.333333 0.666667 0.500000 Cr 0.699230 0.000000 0.500000 Si 0.337748 0.000000 0.000000 N 0.173727 0.000000 0.500000 N 0.473996 0.000000 0.500000 N 0.448912 0.247641 0.000000 |

CaB_{2}O_{4}-likePccn No 56 | a = 7.98 b = 14.42 c = 4.85 | Cr 0.931376 0.565538 0.393557 Cr 0.170647 0.535703 0.989804 Si 0.857355 0.679567 0.867484 N 0.864237 0.698217 0.499220 N 0.303715 0.441717 0.754883 N 0.009972 0.618966 0.076401 N 0.377123 0.568169 0.113939 |

Mn_{2}SnS_{4}-typeCmmm No 65 | a = 5.58 b = 7.82 c = 2.76 | Cr 0.750000 0.750000 0.500000 Si 0.000000 0.000000 0.000000 N 0.000000 0.247324 0.000000 N 0.220861 0.000000 0.500000 |

**Table 6.**The total energy and relative energy values compared to the global minimum (spinel structure taken as the zero of the energy) of Cr

_{2}SiN

_{4}modifications found using the PCAE method and locally optimized using the GGA-PBE functional.

Modification | Total Energy (Eh) | Relative Energy (kcal/mol) |
---|---|---|

γ-Cr_{2}SiN_{4}-type | −5193.435 | 45.181 |

Cr_{2}SiN_{4}-PCAE-1-type | −5193.385 | 76.556 |

Cr_{2}SiN_{4}-PCAE-2-type | −5193.374 | 83.459 |

**Table 7.**The modifications, space groups, unit cell parameters, and atomic positions for favorable Cr

_{2}SiN

_{4}modifications found using the PCAE method and the GGA-PBE functional.

Modification and Space Group | Cell Parameters | Position of Atoms |
---|---|---|

γ-Cr_{2}SiN_{4}-type Cc (no. 9) | a = 5.62 b = 8.96 c = 5.36 Å, β = 117.93 | Cr 0.505134 0.354319 0.539894 Cr 0.490988 0.640587 0.506788 Si 0.000000 0.574728 0.000000 N 0.820626 0.494637 0.668512 N 0.339534 0.498028 0.687817 N 0.683793 0.745757 0.858424 N 0.673275 0.237591 0.860703 |

Cr_{2}SiN_{4}-PCAE-1-type Pm(no. 6) | a = 7.04 b = 3.04 c = 7.03 β = 110.72 | Cr 0.338406 0.500000 0.381358 Cr 0.068281 0.000000 0.404484 Cr 0.778353 0.500000 0.589017 Cr 0.381419 0.500000 0.771064 Si 0.000000 0.000000 0.000000 Si 0.613136 0.000000 0.165992 N 0.569205 0.500000 0.274738 N 0.859037 0.000000 0.522005 N 0.985060 0.500000 0.869495 N 0.164949 0.500000 0.544984 N 0.227376 0.000000 0.231306 N 0.437810 0.000000 0.912940 N 0.551129 0.500000 0.636169 N 0.853139 0.000000 0.153320 |

Cr_{2}SiN_{4}-PCAE-2-type P1(no. 1) | a = 7.88 b = 7.96 c = 5.79 α = 89.97 β = 89.86 γ = 120.26 | Cr 0.179061 0.774468 0.214657 Cr 0.576899 0.834251 0.216822 Cr 0.514893 0.179271 0.216823 Cr 0.596707 0.344636 0.708045 Cr 0.010451 0.684706 0.713360 Cr 0.678545 0.756495 0.700070 Cr 0.343219 0.429431 0.991834 Cr 0.923504 0.343769 0.996260 Si 0.000000 0.000000 0.000000 Si 0.263900 0.513623 0.496139 Si 0.839296 0.185951 0.496185 Si 0.163373 0.091041 0.497914 N 0.743253 0.861393 0.985305 N 0.495442 0.322138 0.989374 N 0.052018 0.604045 0.985425 N 0.711195 0.932465 0.477493 N 0.419187 0.211059 0.484630 N 0.145337 0.648381 0.483835 N 0.406296 0.578413 0.243870 N 0.773280 0.267856 0.250745 N 0.088745 0.948405 0.247062 N 0.418188 0.578819 0.734852 N 0.770928 0.262416 0.742269 N 0.078587 0.945416 0.743265 N 0.427572 0.932217 0.132893 N 0.761872 0.590844 0.634583 N 0.087863 0.251396 0.996965 N 0.090039 0.265519 0.498530 |

**Table 8.**The total energies and relative energy values compared to the global minimum (spinel structure taken as the zero of energy) of the energetically most favorable Cr

_{2}SiN

_{4}modifications found using various search methods and later locally optimized on the ab initio level using the GGA-PBE functional. DM stands for data mining, GO stands for global optimization, and PCAE stands for Primitive Cell approach for Atom Exchange method.

Modification | Search Method | Total Energy (E_{h}) | Relative Energy (kcal/mol) |
---|---|---|---|

Al_{2}MgO_{4}-spinel-type | DM | −5193.507 | 0.0 |

α-Cr_{2}SiN_{4}-type | GO | −5193.474 | 20.708 |

β-Cr_{2}SiN_{4}-type | GO | −5193.438 | 43.298 |

Na_{2}MnCl_{4}-type | DM | −5193.436 | 44.553 |

γ-Cr_{2}SiN_{4}-type | PCAE | −5193.435 | 45.181 |

δ-Cr_{2}SiN_{4}-type | GO | −5193.419 | 55.221 |

TiMn_{2}O_{4}-type | DM | −5193.414 | 58.358 |

ε-Cr_{2}SiN_{4}-type | GO | −5193.413 | 58.986 |

λ-Cr_{2}SiN_{4}-type | GO | −5193.407 | 62.750 |

λʹ-Cr_{2}SiN_{4}-type | GO | −5193.404 | 64.634 |

Mg_{2}SiO_{4}-type | DM | −5193.403 | 65.261 |

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**MDPI and ACS Style**

Škundrić, T.; Zagorac, D.; Schön, J.C.; Pejić, M.; Matović, B.
Crystal Structure Prediction of the Novel Cr_{2}SiN_{4} Compound via Global Optimization, Data Mining, and the PCAE Method. *Crystals* **2021**, *11*, 891.
https://doi.org/10.3390/cryst11080891

**AMA Style**

Škundrić T, Zagorac D, Schön JC, Pejić M, Matović B.
Crystal Structure Prediction of the Novel Cr_{2}SiN_{4} Compound via Global Optimization, Data Mining, and the PCAE Method. *Crystals*. 2021; 11(8):891.
https://doi.org/10.3390/cryst11080891

**Chicago/Turabian Style**

Škundrić, Tamara, Dejan Zagorac, Johann Christian Schön, Milan Pejić, and Branko Matović.
2021. "Crystal Structure Prediction of the Novel Cr_{2}SiN_{4} Compound via Global Optimization, Data Mining, and the PCAE Method" *Crystals* 11, no. 8: 891.
https://doi.org/10.3390/cryst11080891