Crystal Structure Prediction of the Novel Cr₂SiN₄ Compound via Global Optimization, Data Mining, and the PCAE Method

Abstract

A number of studies have indicated that the implementation of Si in CrN can significantly improve its performance as a protective coating. As has been shown, the Cr-Si-N coating is comprised of two phases, where nanocrystalline CrN is embedded in a Si₃N₄ 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 Cr₂SiN₄. 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 Cr₂SiN₄. 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 Si₃N₄ system, where some novel monoclinic phases have been predicted in the past.

1. Introduction

Tools and equipment used in metal and wood machining frequently experience severe damage during their usage, which leads to the urgent need for protective coatings to extend the lifetime of the machinery [1,2]. This particularly applies to marine equipment, where certain parts have to operate in challenging marine environments. Additionally, past nuclear accidents have led to an increase in research regarding nuclear materials and the development of coatings that can be used under such extreme conditions.

Transition metal nitrides are widely used as hard coatings, and, among them, CrN is one with many desired properties—high hardness, toughness, and corrosion resistance, good oxidation resistance, good adhesion strength, and excellent wear resistance [3,4,5,6,7,8,9,10]. However, due to its high friction coefficient, it does not perform well in some extreme conditions [2]. Delamination is considered to be critical to the wear failure of CrN coatings in seawater where microcracks play an important role in the evolution of delamination. [11].

Even though CrN is a well-established and very stable coating in wide use, the addition of alloying elements, such as Si, has been actively investigated to further improve its properties as a hard coating material [5,6,12,13,14,15,16,17,18]. Compared to CrN thin films, Cr-Si-N coatings exhibit lower friction coefficients and wear rates due to their low surface roughness. In general, their tribological and mechanical characteristics show significant improvement under different environments and conditions (dry, humid, or aqueous; dynamic or static; ambient or high temperature; etc.) compared to CrN [3,19,20,21,22,23,24,25,26,27]. Due to the finer crystal structure, the addition of Si can also lead to the elimination of large flake pits, which were previously found on the wear trace of CrN coatings [26]. This increase in silicon content also leads to a change in the crystal structure of the coating, and the particle shape changes from triangular to circular with a distribution that tends to be uniform, while the density of the coating increases [28].

Four kinds of Cr-Si-N coatings are known that are considered to be harder than the CrN coating [28]. Nanocomposite Cr-Si-N thin films (usually in the form of nanocolumns or nanoclusters of CrN embedded in an amorphous Si₃N₄ 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].

Epitaxially grown Cr-Si-N thin films that appear to constitute solid solutions of Si in CrN exhibit even higher hardness than less homogeneous nanocomposite Cr-Si-N thin films—a maximum hardness value of 51 GPa was obtained for 10 mol% Si in Cr-Si-N [34]. The increased hardness of Cr-Si-N is obtained due to the formation of a fine nanocomposite structure and the refinement of CrN crystallites [18,24] as well as to the dissolution of Si in CrN [34]. The decrease of friction is attributed to the formation of SiO₂ 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].

As its Si content increases, the structure of the Cr-Si-N coating changes from a crystalline to an amorphous state [35,36], and while the corrosion resistance constantly improves and the friction coefficient decreases, wear-resistance and hardness also improve until a certain Si content limit is reached beyond which the coatings start to deteriorate [25,34,37]. The electronic properties of Cr-Si-N thin films also change significantly depending on their chemical composition. While fcc-CrN exhibits metallic-like behavior, an increase of N or Si content leads to a more non-metallic character of the material [37].

Furthermore, investigations of the friction and wear properties of CrSi based coatings led to the conclusion that Cr-Si-N coatings showed better properties in comparison to the CrSi coatings [38]. Increasing the silicon content in these coatings led to the best friction and wear performance, which, in the case of an ATF system in a water environment, can reduce the wear volume by three orders of magnitude [38]. In summary, the Cr-Si-N coatings exhibit superior tribological performance due to their excellent toughness, high hardness, preferable adhesion, and good corrosion resistance [2], which all lead to the possibility of wide applications.

Gaining deeper insight into the structure–property relationship of the Cr-Si-N coating on the atomic level would greatly support the optimization of such coatings. As a first step, we focus on the possible existence of crystalline phases in the Cr/Si/N system. Considering the large hardness of Si₃N₄, we chose the analogous composition Cr₂SiN₄, where we would expect similarly high hardness as for Si₃N₄ itself. Thus, in this study, we identify feasible structure candidates of the composition Cr₂SiN₄, 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

To perform structure prediction and gain insight into the structural stability of the possible phases existing in the Cr₂SiN₄ 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 Cr₂SiN₄ 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 Cr₂SiN₄ compound.

We used the well-known KDD (knowledge discovery in databases) process, which involves selection, preprocessing, transformation, and interpretation/evaluation (or post-processing), and has been successfully used in previous studies [46,47,48]. Finally, the Primitive Cell approach for Atom Exchange (PCAE) method was employed for generating alternative structure candidates in the Cr₂SiN₄ compound. The PCAE method is simple, fast, and computationally inexpensive compared to the supercell approach [40].

In the current study, the starting modifications were taken from the related Si₃N₄ 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.

After the structure candidates have been identified using the GO, DM and PCAE methods, each structure candidate has been subjected to local optimization on the ab initio level. Density functional theory (DFT) calculations of the total energy, and local optimizations (including volume, cell parameters, and atomic positions), were performed using the CRYSTAL17 code [49,50] based on linear combinations of atomic orbitals (LCAO). For the local optimization runs, we employed analytical gradients [51,52], and equation of state (EOS) calculations using polynomial fitting [49,50].

Local optimizations were performed on the DFT level employing two different functionals: the Local Density Approximation (LDA) with the Perdew–Zunger (PZ) correlation functional [53] and the Generalized Gradient Approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional [54] for comparison. It is reasonable to choose at least two different ab initio methods, to gain better insight into the quantitative validity of the results since no compounds of the composition Cr₂SiN₄ are available for comparison with the experiment [55,56,57].

An all-electron basis set based on Gaussian-type orbitals was employed in the case of chromium (Cr_86-411d41G_catti_1995) [58,59], silicon (Si_86-311G**_pascale_2005) [60,61], and nitrogen (N_6-21G*_dovesi_1990) [62,63]. The symmetries of the analyzed structures were determined using the SFND [64] and RGS [65] algorithms, and duplicate structures were removed using the CMPZ algorithm [66]; these three algorithms were implemented in the KPLOT code [67]. The optimized structures of various modifications were visualized using the Vesta3 program [68].

3. Results and Discussion

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

Global optimizations were performed on the energy landscapes using empirical potentials for various numbers of formula units per simulation cell and for slightly varying ionic radii, yielding a total of 5000 structure candidates. Most of these global searches resulted in low-symmetry structure candidates (triclinic and monoclinic symmetry). After detailed statistical, structural, and crystallographic analysis, the most promising ones were submitted for local optimization on ab initio level. With further local optimization on the ab initio level, we have obtained the 11 most-relevant structure candidates from global optimization (GO).

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 Cr₂SiN₄ 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-Cr2SiN4-type	−5193.474	20.708
β-Cr2SiN4-type	−5193.438	43.298
δ-Cr2SiN4-type	−5193.419	55.221
ε-Cr2SiN4-type	−5193.413	58.986
λ-Cr2SiN4-type	−5193.407	62.750
λʹ-Cr2SiN4-type	−5193.404	64.634
nf1-Cr2SiN4-type	−5193.398	68.399
nf2-Cr2SiN4-type	−5193.388	74.674
nf3-Cr2SiN4-type	−5193.385	76.556
nf4-Cr2SiN4-type	−5193.364	89.734
nf5-Cr2SiN4-type	−5193.364	89.734

presents the energetic ranking of these new Cr₂SiN₄ phases using the GGA-PBE functional, where the α-Cr₂SiN₄-type is the lowest in calculated total energy, and the nf5-Cr₂SiN₄-type the highest one. Moreover, we grouped the resulting GO structures into energetically favorable ones, ranging from the α- to the λʹ-Cr₂SiN₄ phase, and non-favorable Cr₂SiN₄ 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. 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 Cr₂SiN₄ 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)
Al2MgO4-spinel-type_LDA	DM	−5180.729	0.0
α-Cr2SiN4-type_LDA	GO	−5180.694	21.963
Na2MnCl4-type_LDA	DM	−5180.672	35.768
γ-Cr2SiN4-type_LDA	PCAE	−5180.667	38.906
β-Cr2SiN4-type_LDA	GO	−5180.653	47.691
δ-Cr2SiN4-type_LDA	GO	−5180.653	47.691
TiMn2O4-type_LDA	DM	−5180.621	67.771
Mg2SiO4-type_LDA	DM	−5180.620	68.399
ε-Cr2SiN4-type_LDA	GO	−5180.615	71.536
λ-Cr2SiN4-type_LDA	GO	−5180.606	77.184
λ’-Cr2SiN4-type_LDA	GO	−5180.604	78.439

.)

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

The energetically most favorable modification after global optimization is denoted as α-Cr₂SiN₄-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-Cr₂SiN₄ phase is visualized in Figure 1a, while full structural data of all favorable candidates are presented in

Table 2. The modifications, space groups, unit cell parameters, and atomic positions for favorable Cr₂SiN₄ 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-Cr2SiN4-type Pma2 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-Cr2SiN4-type P-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-Cr2SiN4-type P21/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-Cr2SiN4-type P21/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-Cr2SiN4-type Pm 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’-Cr2SiN4-type Pm 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

for calculations with the PBE functional, and in

Table A1. The modifications, space groups, unit cell parameters, and atomic positions for the most favorable Cr₂SiN₄ modifications after structural optimization on the DFT-LDA level.

Modifications and Space Groups	Cell Parameters	Position of Atoms
Al2MgO4-spinel-type_LDA Fd-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-Cr2SiN4-type_LDA Pma2 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
Na2MnCl4-type_LDA Pbam 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-Cr2SiN4-type_LDA Cc 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-Cr2SiN4-type_LDA P-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-Cr2SiN4-type_LDA P21/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
TiMn2O4-type_LDA P4322 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
Mg2SiO4-type_LDA Pnma 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-Cr2SiN4-type_LDA P21/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-Cr2SiN4-type_LDA Pm 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’-Cr2SiN4-type_LDA Pm 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

for those computed with the LDA functional, respectively.

In the α-modification, chromium is six-fold coordinated by nitrogen, forming two different CrN₆ 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.

A second energetically favorable candidate obtained from GO is denoted as β-Cr₂SiN₄-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 CrN₆ 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).

As in the α-Cr₂SiN₄-type modification, SiN₄ tetrahedra are oriented in different directions; however, in this modification, the CrN₆ 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.

The next energetically favorable candidate found using global optimization is a monoclinic structure denoted as the δ-Cr₂SiN₄-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.

Additionally, the interatomic distances in the two different CrN₆ 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 SiN₆ 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.

One more energetically favorable candidate is a monoclinic structure, denoted as ε-Cr₂SiN₄-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-Cr₂SiN₄-type is composed of SiN₄ 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 CrN₄ tetrahedra and CrN₆ 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-Cr₂SiN₄-types (Table 1 and Table 2 and Figure 2).

Two other interesting favorable structure candidates are monoclinic modifications belonging to space group Pm (no. 6) and are visualized in Figure 3a,b, respectively. The first one (a) is denoted the λ-Cr₂SiN₄-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-Cr₂SiN₄-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 CrN₄ and SiN₄ tetrahedra, with CrN₆ octahedra between them.

The CrN₆ octahedra are edge-connected, with corner-connected tetrahedra. Similar to the previous structures of the α-, β-, ε-Cr₂SiN₄-type of modifications, these tetrahedra have the opposite orientations in different layers of the structure. In the λ-Cr₂SiN₄-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 SiN₄ 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).

The modification referred to as λʹ-Cr₂SiN₄-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 SiN₄ 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).

We note that, after optimization on the ab-initio level with the LDA-PZ functional and with the GGA-PBE functional, both structures remain distinct but stay in the same space group.

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

At extreme conditions of temperature and/or pressure, the global optimization (GO) yielded several additional candidate structures, and

Table 3. The modifications, space groups, unit cell parameters, and atomic positions for non-favorable Cr₂SiN₄ 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-Cr2SiN4-type P21/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-Cr2SiN4-type Cc 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-Cr2SiN4-type Pm 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-Cr2SiN4-type Pm 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-Cr2SiN4-type P-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

presents the structural data for the five energetically non-favorable yet structurally promising GO modifications. The first modification, lowest in the energy compared to the other structures within this group is denoted as nf1-Cr₂SiN₄-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.

It is composed of CrN₆ 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 CrN₄ and SiN₄ tetrahedra in the upper and lower part of the structure are oriented in the opposite directions.

The next modification according to the total energy ranking is referred to as nf2-Cr₂SiN₄-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).

Other relevant modifications are denoted as nf3-Cr₂SiN₄-type, nf4-Cr₂SiN₄-type, and nf5-Cr₂SiN₄-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 Si₃N₄ chemical system, where orthorhombic and monoclinic structures have been proposed [69].

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

The DM-based searches were performed within the ICSD database, which, in the latest release contains 242,828 inorganic structures, out of which more than 80% have already been assigned to distinct structure types (up to now, 9724 structure types are listed in the ICSD) [44,45]. Using the final prototype criterion as part of the KDD approach [46,47,48] to eliminate quasi-duplicate structures, the number of structures was reduced to 66 unique structure candidates in the A1B2C4 chemical system. After performing full structural optimization on the ab initio level, the number of structure candidates was further reduced.

Table 4. The total energy and relative energy values compared to the global minimum (spinel structure taken as the zero of energy) of Cr₂SiN₄ modifications obtained from DM-based searches and calculated using GGA-PBE.

Modification	Total Energy (Eh)	Relative Energy (kcal/mol)
Al2MgO4-spinel-type	−5193.507	0.0
Na2MnCl4-type	−5193.436	44.553
TiMn2O4-type	−5193.414	58.358
Mg2SiO4-type	−5193.403	65.261
Ca2RuO4-type	−5193.402	65.889
HgC2O4-like	−5193.400	67.144
Ca2IrO4-type	−5193.349	99.147
CaB2O4-like	−5193.347	100.402
Mn2SnS4-type	−5193.342	103.539

shows the total energy ranking of the DM-based structure candidates in the Cr₂SiN₄ system using the GGA-PBE functional.

Full structural data for all Cr₂SiN₄ modifications found from the DM-based searches are presented in

Table 5. The modifications, space groups, unit cell parameters, and atomic positions for Cr₂SiN₄ modifications obtained from data-mining-based searches and local optimization at the GGA-PBE level.

Modifications and Space Group	Cell Parameters	Position of Atoms
Al2MgO4-spinel-type Fd-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
Na2MnCl4-type Pbam 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
TiMn2O4-type P4322 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
Mg2SiO4-type Pnma 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
Ca2RuO4-type Pbca 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
HgC2O4-like P21 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
Ca2IrO4-type P-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
CaB2O4-like Pccn 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
Mn2SnS4-type Cmmm 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

, 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

The Al₂MgO₄-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.

In the Al₂MgO₄-type modification, chromium is six-fold coordinated forming a CrN₆ octahedron with the interatomic distance Cr 6 × 1.95 Å-N, while silicon is four-fold coordinated forming a SiN₄ tetrahedron with the atom–atom distance 4 × 1.74 Å-N. In this cubic modification, the CrN₆ octahedra are connected by edges while the SiN₄ tetrahedra are corner-connected. When performing structure optimizations of the candidates on the DFT-LDA level, the Al₂MgO₄-spinel-type modification remains the global minimum (Table A1 and Table A2). We note that the Al₂MgO₄-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].

The next favorable modification found after DM is in the Na₂MnCl₄-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.

The CrN₆ octahedra are quite similar to those in the WC structure-type, while the SiN₆ octahedra are “NaCl-type” octahedra. Interatomic distances in the CrN₆ 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 SiN₆ octahedra (4 × 1.85 Å-N, 2 × 1.88 Å-N). The SiN₆ octahedra are positioned in the center and at the edges of the cell, with the CrN₆ octahedra connecting them. Both edge and corner connections are observed.

The next favorable modification obtained via DM is denoted as TiMn₂O₄-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 TiMn₂O₄ phase, chromium is both four-fold and six-fold coordinated by nitrogen, thus, forming CrN₄ tetrahedra (with atom–atom distances 2 × 1.78 Å-N and 2 × 1.81 Å-N) and CrN₆ 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 CrN₄ tetrahedra are corner-connected.

The next modification obtained from the data mining search is a structure denoted as Mg₂SiO₄- (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 CrN₆ 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

The first non-favorable energy minimum found after data mining appears in the Ca₂RuO₄-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 Na₂MnCl₄-type modification, within this modification, the CrN₆ 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.

These octahedra are connected by edges to each other; however, the connection to the SiN₆ octahedra is via edges and corners as well. The whole structure consists of layers of different octahedra, where the SiN₆ ones are located on the faces and in the center of the cell with the CrN₆ octahedra situated in-between.

The next non-favorable modification found after data mining is denoted as a HgC₂O₄-like type of structure and crystallizes in space group P21 (no. 4). We note that the starting HgC₂O₄ structure [76] after DFT optimization has been structurally modified, however, within the same space group (no.4), thus, resulting in a HgC₂O₄-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.

Within this structure, chromium is six-fold coordinated by nitrogen forming CrN₆ 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 SiN₄ 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.

The next modification by energy shows a Ca₂IrO₄-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 CrN₆ 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 SiN₆ 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.

Another interesting modification obtained from the data mining search is the orthorhombic structure denoted as a CaB₂O₄-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 HgC₂O₄-like modification, the CaB₂O₄-like structure is modified during the local optimization from the original prototypic CaB₂O₄ structure [78] but still within the same space group (no. 56).

In this structure, chromium is five-fold and six-fold coordinated by nitrogen forming two different polyhedra with the atom–atom distances Cr(1) 1 × 1.83 Å-N, 1 × 1.87 Å-N, 1 × 1.98 Å-N, 1 × 1.99 Å-N, 1 × 2.01 Å-N, 1 × 2.05 Å-N, Cr(2) 1 × 1.81 Å-N, 1 × 1.82 Å-N, 1 × 1.88 Å-N, 1 x 1.92 Å-N, and 1 x 2.07 Å-N, and one has a SiN₅ polyhedron with the interaction distances 2 × 1.81 Å-N, 1 × 1.88 Å-N, and 2 × 1.90 Å-N.

A final interesting modification from the data mining search is denoted as Mn₂SnS₄-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.

We note that most of the structure candidates found using DM-based searches show orthorhombic symmetry (Table 5), in some cases reminiscent of the structurally related Si₃N₄ chemical system where orthorhombic structures have also been found [69]. In this context, we would like to remark on the Cu₂HgI₄ type of structure [80]. This structure type has been recently predicted to exist as a modification in a study of novel hard phases of Si₃N₄ [69].

The same Cu₂HgI₄ 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

Finally, the Primitive Cell approach for Atom Exchange (PCAE) method was employed for generating alternative structure candidates in the Cr₂SiN₄ compound, starting from typical structures in the related Si₃N₄ system, the γ-, β-, and α-phase of Si₃N₄. 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. The total energy and relative energy values compared to the global minimum (spinel structure taken as the zero of the energy) of Cr₂SiN₄ modifications found using the PCAE method and locally optimized using the GGA-PBE functional.

Modification	Total Energy (Eh)	Relative Energy (kcal/mol)
γ-Cr2SiN4-type	−5193.435	45.181
Cr2SiN4-PCAE-1-type	−5193.385	76.556
Cr2SiN4-PCAE-2-type	−5193.374	83.459

.

Structural Details of Candidates Found Using the PCAE Method

The lowest energy minimum found using the PCAE method was denoted the γ-Cr₂SiN₄-type modification. Figure 10a shows a prototypic γ-phase in the Si₃N₄ system [71], which was used as starting structure for generating the γ-Cr₂SiN₄-type. The Si₃N₄ 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 Cr₂SiN₄ system using both GGA-PBE and LDA-PZ functionals, it converts to the γ-Cr₂SiN₄-type modification. It is a low-energy candidate in the Cr₂SiN₄ system; however, it is structurally completely different from the starting γ-phase in the Si₃N₄ system (compare Figure 10a,b).

The γ-Cr₂SiN₄-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 CrN₆ 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 SiN₆ 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 Si₃N₄ compound, there has been a prediction of novel monoclinic phases from first-principles calculations [69].

The next minimum found using the PCAE method is marked as Cr₂SiN₄-PCAE-1 phase, which is energetically less favorable than the γ-phase (Table 6). The Cr₂SiN₄-PCAE-1 type has been generated similarly to the previous one, starting from the β-phase of Si₃N₄ in the hexagonal P63/m (no. 176) space group [81,82]. After ab initio structural optimization in the Cr₂SiN₄ system, the structure converted to a monoclinic modification denoted Cr₂SiN₄-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. The modifications, space groups, unit cell parameters, and atomic positions for favorable Cr₂SiN₄ modifications found using the PCAE method and the GGA-PBE functional.

Modification and Space Group	Cell Parameters	Position of Atoms
γ-Cr2SiN4-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
Cr2SiN4-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
Cr2SiN4-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

).

In this structure type, both chromium and silicon are four-fold and five-fold coordinated by nitrogen forming different types of polyhedra (Figure 11a). There are two different CrN₄ tetrahedra and two different CrN₅ 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.

In the corners of the cell, there are four SiN₄ 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 SiN₅ 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).

Structurally and energetically related is the Cr₂SiN₄-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 Si₃N₄ 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 Cr₂SiN₄-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).

Within this triclinic modification, both chromium and silicon are four-fold coordinated by nitrogen thus forming tetrahedra (Figure 11b). In this modification there are eight different CrN₄ 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).

Silicon is also four-fold coordinated by nitrogen resulting in two different SiN₄ 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 SiN₄ tetrahedra are mostly positioned at the corners of the cell with three tetrahedra located inside along with CrN₄ 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 Cr₂SiN₄ on the Ab Initio Level

The global optimization, data mining, and PCAE based searches resulted in structure candidates that were, after detailed structural and crystallographic analysis, reduced to the eleven energetically most favorable Cr₂SiN₄ modifications.

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 Cr₂SiN₄ 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 (Eh)	Relative Energy (kcal/mol)
Al2MgO4-spinel-type	DM	−5193.507	0.0
α-Cr2SiN4-type	GO	−5193.474	20.708
β-Cr2SiN4-type	GO	−5193.438	43.298
Na2MnCl4-type	DM	−5193.436	44.553
γ-Cr2SiN4-type	PCAE	−5193.435	45.181
δ-Cr2SiN4-type	GO	−5193.419	55.221
TiMn2O4-type	DM	−5193.414	58.358
ε-Cr2SiN4-type	GO	−5193.413	58.986
λ-Cr2SiN4-type	GO	−5193.407	62.750
λʹ-Cr2SiN4-type	GO	−5193.404	64.634
Mg2SiO4-type	DM	−5193.403	65.261

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.

Figure 12 presents the energy versus volume (E(V)) curves on the ab initio level using the GGA-PBE functional for the energetically most favorable Cr₂SiN₄ modifications. We note that the global minimum in the Cr₂SiN₄ system is the Al₂MgO₄-spinel phase. This prominence of the Al₂MgO₄-type candidate on the energy landscape of Cr₂SiN₄ is not unreasonable, since several earlier calculations in the related (binary) Si₃N₄ system have also found a Al₂MgO₄-spinel-like phase [71,84,85].

At high-temperature conditions, one might expect structures, like the β-, ε-, λ-, and λʹ-Cr₂SiN₄-type to possibly become competitive, as well as the TiMn₂O₄-type and Mg₂SiO₄-type modifications from the DM-based searches. Similarly, the most relevant modifications that might appear in the high-pressure region are the Na₂MnCl₄-type, and the α-, γ-, and δ-Cr₂SiN₄-types. Therefore, enthalpy vs. pressure, H(p), curves were computed for these five modifications (Figure 13).

A high-pressure phase transition was predicted between the spinel and the Na₂MnCl₄-type at a pressure of ~33 GPa (Figure 13). In addition, there was a phase transition between the Na₂MnCl₄-type and the metastable α-Cr₂SiN₄-type modifications at ~15 GPa (Figure 13), i.e., the α-phase was more stable than the Na₂MnCl₄-type modification below 15 GPa in the Cr₂SiN₄ system.

In summary, the energy landscape of Cr₂SiN₄ is highly complex with a wide range of structurally different modifications possible. On the ab initio level, the global minimum corresponds to the AlMg₂O₄-type of structure. This modification is also known as a spinel structure, formulated as AB₂X₄, 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 Cr₂SiN₄ 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.

However, some of the structure candidates observed exhibit six-fold coordination with an octahedral environment for both Cr and Si atoms (e.g., the γ- and the δ-Cr₂SiN₄-type, the Na₂MnCl₄-type, and the Mn₂SnS₄-type), while there is only one stable modification (Cr₂SiN₄-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.

The appearance of spinel as a global minimum in Cr₂SiN₄ 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 Cr₂SiN₄ 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.

Similarly, the results of recent investigations demonstrating that spinel coatings can be used to protect metals, such as chromium, from oxidation or corrosion, serve as another indication for the possible technological and industrial usefulness of the proposed Cr₂SiN₄ structure candidates [86,87,88]. Another interesting point is that the global minimum AlMg₂O₄-spinel-type of the structure shows the highest cubic Fd-3m symmetry, while most of the predicted structure candidates show much lower symmetry.

Here, we note that the spinel structure contains so many atoms that it was not feasible to be realized in the periodic simulation cell employed for the global optimizations; as a consequence, the closely related α-Cr₂SiN₄-type structure was obtained instead. Quite generally, data-mining-based searches produced high-symmetry candidates (except for the HgC₂O₄-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-Cr₂SiN₄-type as the highest symmetry space group.

We note that each of the methods used brought us a new perspective: data mining found the global minimum and suggested known structure types that can be stable in the Cr₂SiN₄ 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

A combination of global optimization, data mining, and the PCAE method was used to explore the energy landscape of Cr₂SiN₄ and to perform structure prediction in the Cr₂SiN₄ 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 Cr₂SiN₄ compound.

Additionally, the Primitive Cell approach for Atom Exchange (PCAE) method was employed for generating alternative structure candidates in the Cr₂SiN₄ system with starting modifications taken from the related Si₃N₄ 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.

The local optimizations were performed on the DFT level employing the LDA-PZ and the GGA-PBE functional, for comparison, since there exists no experimental data that one could compare the results with. There was good agreement between the results from the two chosen functionals regarding the total energy ranking, space group symmetry, and other structural data of the various structure candidates after relaxation, which strongly supports the plausibility of the proposed candidates to correspond to actual (meta)stable modifications in the Cr₂SiN₄ system.

The explorations of the energy landscape of the Cr₂SiN₄ 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 Cr₂SiN₄ modifications, with the latter ones perhaps accessible for extreme thermodynamic conditions. Among the eleven energetically most favorable candidates, the global minimum was the AlMg₂O₄-spinel-type exhibiting the cubic Fd-3m symmetry.

At high pressures of ca. ~33 GPa, a phase transition between the spinel-type and the Na₂MnCl₄-type modifications should take place. Thus, we present a large number of feasible structures for modifications in the synthetically not-yet-explored Cr₂SiN₄ 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.