Maximum Achievable N Content in Atom-by-Atom Growth of Amorphous Si-B-C-N Materials

Amorphous Si-B-C-N alloys can combine exceptional oxidation resistance up to 1500 °C with high-temperature stability of superior functional properties. Because some of these characteristics require as high N content as possible, the maximum achievable N content in amorphous Si-B-C-N is examined by combining extensive ab initio molecular dynamics simulations with experimental data. The N content is limited by the formation of unbonded N2 molecules, which depends on the composition (most intensive in C rich materials, medium in B rich materials, least intensive in Si-rich materials) and on the density (increasing N2 formation with decreasing packing factor when the latter is below 0.28, at a higher slope of this increase at lower B content). The maximum content of N bonded in amorphous Si-B-C-N networks of lowest-energy densities is in the range from 34% to 57% (materials which can be grown without unbonded N2) or at most from 42% to 57% (at a cost of affecting materials characteristics by unbonded N2). The results are important for understanding the experimentally reported nitrogen contents, design of stable amorphous nitrides with optimized properties and pathways for their preparation, and identification of what is or is not possible to achieve in this field.

The status of nitrogen is special: while [Si], [B] and [C] can be varied from 0 to 100%, maximum achievable values of [N] total are much lower and depend on the preparation technique (e.g., N 2 pressure during sputtering [21] or combination of sputtering with N (2) + ion beam [6]). See the literature overview [30] on CN x for an extreme case: maximum [N] total obtained by various techniques of 23-45% (atom-by-atom growth) or 23-57% (including growth from N-rich precursors or assisted by N 2 + bombardment). Furthermore, the composition is sometimes measured by techniques (e.g., X-ray fluorescence or elastic recoil detection) which cannot distinguish the content of N bonded in the amorphous networks ([N] network ) and the total N content ([N] total which may include also unbonded N 2 or other N-rich molecules, reported e.g., for TiN [31] or oxidized TiN [32]). Thus, owing to the aforementioned importance of as high [N] network as possible for some of the functional properties, it is desirable to quantify the highest achievable [N] network at a given The relationships between composition, density, structure and energy of (pure or hydrogenated) amorphous (Si)-(B)-(C)-(N) materials can be and are often studied by ab initio molecular dynamics (MD) simulations, see the examples for C [33,34], CH x [35], CN x [36,37], Si-C-H [38], Si-C-N [39,40] or Si-B-C-N [21,28]. The reliable predictions of local energy minima followed by their characterization in terms of bonding environment of individual atoms or localization of individual electronic states on these atoms allows one to obtain a lot of information not available experimentally. Recently, enhanced computing capacities have allowed the shift from modeling of few selected compositions to detailed sampling over a wide range of compositions and densities.
The main aim of this work is to predict maximum achievable [N] network in amorphous Si-B-C-N materials in a wide range of [Si]/[B]/[C] ratios. The background hypothesis, supported by the author's previous works [30,41] on simpler nitrides and the discussion therein, is that maximum [N] network in those materials which are not affected by the composition and structure of their precursors is largely given by the formation of N 2 molecules during the atom-by-atom growth (not to be confused with polymerization of N-rich precursors such as melamine or dicyandiamide [42], which is beyond the present scope). Maximum [N]  values obtained in our laboratory by reactive magnetron sputtering using equally numerous sputter target compositions. The effect of packing factor (not only atomic density) on the N 2 formation is investigated as well.

Simulation Technique
All simulations utilized density functional theory (DFT) and Car-Parrinello MD implemented in the CPMD code [43]. The amorphous structures were predicted by the liquid-quench (LQ) algorithm consisting of (1) mixing of a melt (6000 K) of a given composition and density, (2) exponential cooling to a representative deposition temperature (450 K), (3) equilibration at this temperature and (4) collecting the results. The length of each step was 0.5 ps, leading to a total length of all steps of 2 ps. This frequently used [21,28,30,[33][34][35][36]38,39,41] algorithm captures material formation conditions arising from melting and subsequent rapid quenching of a small volume of a material after an energetic particle impact.
The presented energy differences (E-E min where E min corresponds to the lowest-energy density of a given composition) were obtained using time-averaged Kohn-Sham energies collected at the aforementioned temperature of 450 K. For the purpose of a low-temperature comparison of qualitatively similar structures of materials with a band gap, the relevance of time-averaged energy differences is considered to be comparable with that of free energy differences. The bonding statistics, including the number of N 2 molecules ([N 2 ]), were calculated using a representation of pairs of valence electrons by centers of maximally localized Wannier functions (WFCs) [38,44].
Further technical details of the simulations and detailed justification of the algorithm including its time scale, ranging from analytical calculation to the comparison of calculated bonding statistics with infrared spectroscopy, are included in the previous works [30,41].

List of Simulations
All simulations were performed with 70 atoms of random initial coordinates in a cubic periodical cell. The presented data span 16      Si-B-C-N materials with different energies and N 2 contents were obtained using 13-19 inverse atomic densities (volumes per atom, V) of 6, 6.5, 7, 7.5, 8, and then a step of 1 up to 16, 18, 20 and 22 Å 3 /at. for [Si]/[Si+B+C] = 0-22%, 33-66%, 100% except pure Si and pure Si, respectively. These V ranges capture all lowest-energy densities calculated in this paper as well as all densities of crystalline or amorphous materials from the Si-B-C-N system found in the literature. Furthermore, the statistical noise has been reduced by performing each simulation 5× with different random initial coordinates and averaging all quantities (consequently, [N 2 ] is not necessarily integer). Thus, the calculated data are based on 16 × 13 × 13 − 19 × 5 ≈ 15,000 liquid-quench simulations.

Reliability of the Simulation Protocol
While the LQ algorithm in itself is well established, it is worth discussing the decision to allow the N atoms to form unbonded N 2 molecules which are not instantly lost to the atmosphere. Figure 1d,e show lowest-energy V of CN x and SiN x (binary systems chosen because of the availability of corresponding experimental data) predicted using the described simulation protocol in a wide [N] total range. The calculated V is compared with measured V (converted from mass density, ρ, using the Avogadro number, N A , and the molar masses, M Si,C,N ): and ρ SiNx (g/cm 3 ) = 2.3 + 0.01575 × [N] total (own fit on experimental data reported in [45]) It can be seen that the agreement shown in Figure 1d,e is excellent, supporting the correctness of the simulation protocol in general and the decision concerning allowed presence of unbonded N 2 in particular. Indeed, V of CN x increases with [N] total because of more/larger voids occupied by unbonded N 2 (not e.g., because of covalent radii of both elements: that of N is actually slightly lower). It is crucial that the agreement was achieved without any fitting parameters, only by reproducing the material formation process on the experimentally relevant time scale.
The total pressure was 0.500 Pa and the composition of the discharge gas mixture (except the series which examines its effect) was 50% (0.250 Pa) N 2 + 50% (0.250 Pa) Ar. Varying the substrate surface temperature (up to 700 • C) and substrate bias voltage (down to −500 V) did not visibly affect the measured [N] total values (the effect of both these parameters on other film characteristics such as content of implanted argon, stress or hardness is examined elsewhere [17,19]).
The film composition was measured by Rutherford back-scattering spectroscopy (evaluated by the code GISA [47]; all elements except H) and elastic recoil detection (evaluated by the code SIMNRA [48]; H) using a Van

Maximum Achievable N Content
The N 2 formation in itself and its dependence on V is illustrated in Figure 2a , including ≈34% for CN x , just above 50% for BN x (allowing BN) and 57% for SiN x (allowing Si 3 N 4 ). The first of these numbers corresponds to C 3 N 1.5 , far from C 3 N 4 which was predicted to be superhard in its β phase [50] but which seems to be impossible to prepare at least by the low-pressure atom-by-atom growth. The N-rich compositions allowed by the figure include SiBCN 3 in the middle of the triangle, one of the early examples [2] of extremal thermal stability and oxidation resistance of Si-B-C-N ceramics. Figure 2c shows the aforementioned saturation values of [N] network which are achievable when the network formation is accompanied by N 2 formation. These values are of course higher than those in Figure 2b, e.g., ≈42% (C 3 N 2.2 ) instead of ≈34% (C 3 N 1.5 ) in the case of CN x , but arguably at a cost of properties affected by the lower densification resulting from the presence of voids occupied (either permanently, or temporarily until a loss to the atmosphere) by unbonded N 2 . for networks with N 2 -containing voids (Figure 2c). This constitutes the key result of the present paper, which can be tested experimentally (below). the triangle, one of the early examples [2] of extremal thermal stability and oxidation resistance of Si-B-C-N ceramics. Figure 2c shows the aforementioned saturation values of [N]network which are achievable when the network formation is accompanied by N2 formation. These values are of course higher than those in Figure 2b, e.g., ≈42% (C3N2.2) instead of ≈34% (C3N1.5) in the case of CNx, but arguably at a cost of properties affected by the lower densification resulting from the presence of voids occupied (either permanently, or temporarily until a loss to the atmosphere) by unbonded N2.

Densification and Packing Factor
The formation of N 2 -containing voids affects the lowest-energy atomic density and packing factor (p; calculated using the covalent radii r Si = 1.11 Å, r B = 0.82 Å, r C = 0.77 Å, r N = 0.75 Å) of the materials. An example of the dependence of these two quantities on  Figure 3a,b, respectively. Indeed, Figure 3a shows that V of 10.5-11.0 Å 3 /at. can be found in the two bottom corners, i.e., for CN x and SiN x . However, as confirmed in Figure 3b, the reasons behind these two maxima are fundamentally different.
In the case of CN x , the enhanced V (despite the low radii of C and N) is due to the large volume occupied by N 2 -containing voids and leading to a low packing factor down to ≈16%. In the case of SiN x , there are no unbonded N 2 molecules which allows more than 2× higher packing factor of ≈34%, and the enhanced V is due to the high radius of Si atoms. Contrary to CN x and SiN x which represent local maxima of V, BN x represents the minimum V of ≈7.5 Å 3 /at. resulting from a combination of medium packing factor (nonzero but low number of N 2 molecules) of ≈25% with the low radius of B.

Experimental Verification
The experimentally measured [N]total obtained by sputtering 15 different targets at fixed process parameters such as total N2+Ar pressure, N2 partial pressure or sputtering current, are shown in Figure 4a  While the minimum V leading to N 2 formation is different for different compositions, the maximum p leading to N 2 formation is almost independent of the composition. In particular, Figure 3c shows that p ≥ 0.28 leads to almost zero [N 2 ], while p < 0.28 leads to increasing [N 2 ] with decreasing p. Note that while this finding is shown graphically only for the highest [N] total investigated, it is independent of [N] total . The limit p ≥ 0.28 is consistent with the densities of covalent nitrides such as β-Si 3 N 4 (p = 0.33), β-C 3 N 4 (predicted p = 0.30) or c-BN (p = 0.34; the bonds between hexagonal planes of h-BN with p = 0.21 are not covalent). The results obtained at [B] = 0 almost exactly reproduce the linear dependence which connects [N 2 ] = 0 at p = 0.28 with the other obvious limit [N] network = 0 (nitrogen only in the form of N 2 molecules) at p = 0. The results obtained at [B] > 0 exhibit the same critical p = 0.28, but lead to slower increase of [N 2 ] with p decreasing below 28%, by a factor which is actually very close to (1 − 0.01 × [B]). This indicates, beyond the present scope, a qualitative difference between compositionally stable low-density B doped by N ([B] close to 100%, i.e., 1 − 0.01 × [B] close to zero) and unstable low-density Si or C doped by N.

Experimental Verification
The experimentally measured [N] total obtained by sputtering 15 different targets at fixed process parameters such as total N 2 +Ar pressure, N 2 partial pressure or sputtering current, are shown in Figure 4a Figure 4b from the diagonal line is less than 1% of N. Second, the figure confirms that the measured [N] total almost monotonically increases with the predicted maximum [N] network . Third, quantitatively, the measured [N] total values are well below the predicted maxima when the prediction is low, but converge to the predicted maxima when the prediction is high. In other words, the figure shows that less intensive N 2 formation and loss does not only mean that the maximum N content is higher, it also means that this higher N content can be achieved in a wider range of experimental conditions. The single [N] total value which exceeds the predicted maximum [N] network may not only be due to the presence of unbonded molecules (unlikely in this particular case, taking the thermal stability [25,28] into account) but also due to a higher than estimated measurement error (the four rightmost datapoints include two almost exactly reproducing the predicted maximum, one above it and one below it) and/or the role of impurities (e.g., compressive stress and changes in network topology resulting from implantation of Ar + ions [49]: even when [N]/([Si+B+C+N] = 61% is above the prediction, [N]/[Si+B+C+N+Ar+O+H] = 54% is below it). The role of impurities is important in various field of materials science (e.g., [51,52]) but beyond the present scope. Fourth, the empty balls in Figure 4b confirm that increasing the N 2 partial pressure from 0.125 Pa to 0.500 Pa can move the measured [N] total from a value well below the prediction almost all the way toward the prediction.  [4,19,46]) or in 0.125-0.500 Pa of N2 at a fixed total pressure of 0.500 Pa using the Si40C60 target (empty balls [53]; this dependence is not vertical because of different ratios of sputtering yields of individual elements by N2 + and by Ar + ).

Conclusions
Ab initio simulations of structures and energies of amorphous Si-B-C-N materials have been performed in a wide range of compositions and densities, and compared with the preparation of the same materials in the form of thin films by reactive magnetron sputtering in a wide range of sputter target compositions. The maximum content of N bonded in the Si-B-C-N networks is limited by the formation of unbonded N2 molecules, and can be well represented by a quadratic dependence (for N2-free materials; in %) 34 While various compositions correspond to various lowest-energy packing factors both below and above 0.28, the N2 formation is non-negligible when the packing factor is below 0.28 regardless of the composition. The driving force toward N2 formation at a given packing factor (lower than 0.28) decreases with increasing B content. The theoretical densities and especially the maximum N contents (predicted without any fitting parameters, only by reproducing the material formation process) are in very good agreement with the experiment. Because as high N content as possible is crucial for numerous functional properties of Si-B-C-N, the results are useful for the design of these technologically important nitrides (including knowing when the content of bonded N can be enhanced without any tradeoff/can be enhanced at a cost of unbonded N2/cannot be en-  [46], Si x B 20 C 80-x targets [4] and Si x (B 4 C) 100-x targets [19] (bottom, middle and top dashed oval, respectively). Panel (b) compares these experimental values with those given by Equation (3). The squares are calculated maxima of [N] network (Figure 2b). The diagonal line represents Equation (3) (fitted using these calculated data). The balls are the aforementioned values of [N] total in Si-B-C-N (neglecting the impurities) obtained by reactive magnetron sputtering in 0.250 Pa of N 2 + 0.250 Pa of Ar using Si x C 100-x , Si x B 20 C 80-x and Si x (B 4 C) 100-x targets (full balls [4,19,46]) or in 0.125-0.500 Pa of N 2 at a fixed total pressure of 0.500 Pa using the Si 40 C 60 target (empty balls [53]; this dependence is not vertical because of different ratios of sputtering yields of individual elements by N 2 + and by Ar + ).

Conclusions
Ab initio simulations of structures and energies of amorphous Si-B-C-N materials have been performed in a wide range of compositions and densities, and compared with the preparation of the same materials in the form of thin films by reactive magnetron sputtering in a wide range of sputter target compositions. The maximum content of N bonded in the Si-B-C-N networks is limited by the formation of unbonded N 2 molecules, and can be well represented by a quadratic dependence (for N 2 -free materials; in %) 34 . While various compositions correspond to various lowest-energy packing factors both below and above 0.28, the N 2 formation is non-negligible when the packing factor is below 0.28 regardless of the composition. The driving force toward N 2 formation at a given packing factor (lower than 0.28) decreases with increasing B content. The theoretical densities and especially the maximum N contents (predicted without any fitting parameters, only by reproducing the material formation process) are in very good agreement with the experiment. Because as high N content as possible is crucial for numerous functional properties of Si-B-C-N, the results are useful for the design of these technologically important nitrides (including knowing when the content of bonded N can be enhanced without any tradeoff/can be enhanced at a cost of unbonded N 2 /cannot be enhanced), their preparation (e.g., conditions for N 2 implantation and diffusion), knowing when to use characterization techniques (e.g., vibrational spectroscopy) which can distinguish bonded and unbonded N, and identification of what is or is not possible to achieve (including preparation of crystalline phases such as β-C 3 N 4 ) in this field.

Data Availability Statement:
The data that support the findings of this study are available from the author upon reasonable request.