Electronic structures and phase transition characters of β-, P61-, P62- and δ-Si3N4 under extreme conditions: a density functional theory study

    Corresponding author: Dong Chen, chchendong2010@163.com
  • College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang 464000, China

Key words: phase transitionbond lengthselastic constantsdensity functional theory

Abstract: This paper describes the results of structural, electronic and elastic properties of silicon nitride (in its high-pressure P61 and P62 phases) through the first-principles calculation combined with an ultra-soft pseudo-potential. The computed equilibrium lattice constants agree well with the experimental data and the theoretical results. The strongest chemical bond (N--Si bond) shows a covalent nature with a little weaker ionic character. P61-Si3N4 is more stable than P62-Si3N4 due mainly to the fact that the shorter N--Si bond in the P61 phase allows stronger electron hybridizations. We have also predicted the phase stability of Si3N4 using the quasi-harmonic approximation, in which the lattice vibration and phonon effect are both considered. The results show that the β → P61 phase transition is very likely to occur at 42.9 GPa and 300 K. The reason why the β → P61 → δ phase transitions had never been observed is also discussed.


1.   Introduction
  • Due to its high hardness, excellent wear resistance, good ability to withstand high temperature, wide band gap and low density, the nitride-based ceramic Si$_{3}$N$_{4}$ plays a key role in the development of advanced materials[1]. Silicon nitride (Si$_{3}$N$_{4})$ belongs to the group-IV nitrides exhibiting superior physical and mechanical properties. Its tunable electrical conductivity and high dielectric constant lead to a variety of diversified applications, especially in a corrosive environment[2]. Si$_{3}$N$_{4}$ has also been applied to cutting tools, photo-electronics, solar cells, and metal-nitride-oxide-semiconductor (MNOS) memory devices[3]. Besides, its high strength makes Si$_{3}$N$_{4}$ a desirable material for hostile environments[4]. The stacking sequences along the $c$-axis of hexagonal $\alpha $- and $\beta$-Si$_{3}$N$_{4}$ are different[5]. In 1999, a new phase $\gamma$-Si$_{3}$N$_{4}$ with a cubic spinel structure has been synthesized by the diamond cell technique[6]. Upon further compression, $\gamma$-Si$_{3}$N$_{4}$ will transform into the so-called "post-spinel" phase[7]. One decade ago, the willemite-II phase (wII-Si$_{3}$N$_{4})$ was analyzed by Kroll[8]. It was found that the $\beta$-Si$_{3}$N$_{4}$ transforms into an unknown phase ($\delta$-Si$_{3}$N$_{4})$ at high pressures[9]. It is worth noting that the $\alpha$ $\to$ $\beta$[10, 11], $\beta$ $\to$ wII[8], $\beta$ $\to$ $\gamma$[12, 13, 14, 15, 16], $\alpha$ $\to$ $\gamma$[14], $\beta$ $\to$ $\delta$[14], and $\gamma$ $\to$ post-spinel[7, 17] phase transitions have been carefully investigated by researchers.

    Very recently, Xu et al.[14] investigated the new hexagonal polymorphs of silicon nitride, namely the P61 and P62 phases. Different atoms of P61-Si$_{3}$N$_{4}$ occupy the N-1c (0.3333, 0.6667, 0), N-1f (0.6667, 0.3333, 0.5), N-3j (0.269, 0.013, 0), N-3k (0.63, 0.994, 0.5), Si-3j (0.148, 0.741, 0) and Si-3k (0.853, 0.257, 0.5) Wyckoff positions. In the P62 phase, the N-1c (0.3333, 0.6667, 0), N-1e (0.6667, 0.3333, 0), N-3j (0.261, 0.005, 0), N-3k (0.623, 0.021, 0.5), Si-3j (0.136, 0.734, 0) and Si-3k (0.86, 0.293, 0.5) sites are occupied[14]. The $\beta$ $\to$ P62 $\to$ $\delta$ transitions can be observed only when the $\beta$ $\to$ $\delta$ transition is suppressed due to kinetic reasons[14]. Curiously, the $\beta$ $\to$ P61 $\to$ $\delta$ transitions had never been observed by experimental or theoretical investigations. Consequently, the physical properties of P61- and P62-Si$_{3}$N$_{4}$ are still unclear. The aim of this paper is twofold: (1) to obtain the fundamental physical properties of the high pressure P61 and P62 phases; and (2) to investigate the rules of the $\beta$ $\to$ P61 $\to$ $\delta$ phase transitions.

2.   Calculation models and methods
  • All calculations in this work were performed by employing the plane-wave pseudo-potential (PW-PP) method[18] within the frame of density functional theory (DFT)[19]. The exchange-correlation functional was calculated by using the generalized gradient approximation in the form of Perdew, Burke and Ernzerhof (PBE)[20]. The N-2s$^{2}$2p$^{3}$ and Si-3s$^{2}$3p$^{2}$ states were treated as valence electrons. Plane-wave cutoff energies of 500 eV ($\beta$-Si$_{3}$N$_{4})$, 450 eV (P61-Si$_{3}$N$_{4})$, 450 eV (P62-Si$_{3}$N$_{4})$ and 450 eV ($\delta$-Si$_{3}$N$_{4})$ were employed according to our convergence tests. The Monkhorst-Pack meshes of 4 $\times$ 4 $\times$ 12, 5 $\times$ 5 $\times$ 12, 5 $\times$ 5 $\times$ 12 and 6 $\times$ 6 $\times$ 15 $k$-points were used for the Brillouin zones of $\beta$-, P61-, P62- and $\delta$-Si$_{3}$N$_{4}$, respectively[21]. The internal coordinates were relaxed using the Broyden-Fletcher-Goldfard-Shanno (BFGS) minimization technique[22]. The self-consistent computations were considered to be converged with a criterion of 10$^{-6}$ eV/atom. The phase transition characters were determined through the quasi-harmonic Debye approximation (QHD)[23, 24]. In this work, the calculation parameters of QHD are the same as the parameters given in our previous paper[25].

3.   Results and discussion

    3.1.   Crystal structures

  • The currently known polymorphs of Si$_{3}$N$_{4}$ are plotted in Figure 1. Each path shown in Figure 1 indicates a reversible or non-reversible phase transition, which has been verified by experiments or theoretical investigations. $\beta$-Si$_{3}$N$_{4}$ is set as a benchmark system in order to testify the validity of our calculation since the reliable investigations for the P61, P62 and $\delta$ phases are not available.

    As shown in Table 1, the calculated lattice constants of $\beta$-Si$_{3}$N$_{4}$ agree well with the experimental data[27, 28] and the theoretical results[25, 26, 29, 30]. The hexagonal lattice satisfies Born's criteria $C_{12} >0$, $C_{33} >0$, $C_{11} -\left| {C_{12} } \right|>0$, $C_{44} >0$, $(C_{11} +C_{12} )C_{33} -2C_{13}^2 >0$[31]. The results indicate that the P61-, P62- and $\delta$-Si$_{3}$N$_{4}$ can retain their stability at 45, 45 and 60 GPa, respectively. The bulk modulus of P61-Si$_{3}$N$_{4}$ is larger than that of P62-Si$_{3}$N$_{4}$, which means that the P62 phase is more compressible than the P61 phase. Although the atomic coordinations are different, P61- and P62-Si$_{3}$N$_{4}$ have similar lattice parameters and elastic constants (except for $C_{44})$. It is well known that $C_{44}$ represents the transverse sound velocity along the [001] plane of a hexagonal lattice. The value of $C_{44}$ for P61-Si$_{3}$N$_{4}$ is only one half of the P62 phase, which yields a smaller shear modulus. For example, the $C_{44}$ values are 88.3 and 169.2 GPa for P61- and P62-Si$_{3}$N$_{4}$, respectively. The results show that in the P62 phase, the transverse acoustic velocity along the [001] plane is much faster than the velocity in the P61 phase. The crystal structures of P61 and P62-Si$_{3}$N$_{4}$ are shown in our previous paper [25]. In Reference [25], it is clearly seen that the atom distributions of the two phases are quite different on the [001] plane. The reason for the discrepancy of $C_{44}$ may be: (1) the different atom distributions on the [001] plane; or (2) the larger cell volume of the P61 phase. However, these results are all predictions and need to be identified by future experiments.

  • 3.2.   Electronic structures

  • We now turn our attention to the density of states (DOS) and the Mulliken populations[32, 33]. At 45 GPa, the calculated total DOS (TDOS) and partial DOS (PDOS) of Si$_{3}$N$_{4}$ are drawn in Figure 2.

    As one of the most important findings, Ren and Ching[34] found that the empty Si-3d orbital has almost no effect on the DOS of Si$_{3}$N$_{4}$. Hence the Si-3d state is not included in this work. As shown in Figure 2(a), the valence band (VB) DOS can be divided into two parts, namely the lower region ($-21$ to -14 eV) and the upper region (-11 to 0 eV). Compared with the DOS of P62-Si$_{3}$N$_{4}$, the DOS of P61-Si$_{3}$N$_{4}$ has a larger band gap and a steeper conduction band (CB) edge. The DOSs of P62- and P61-Si$_{3}$N$_{4}$ are remarkably similar with only slight differences in very subtle structures in both the CB and VB regions. The proximity of electronic structures is due to the similarity of local atomic arrangements in the P62- and P61-Si$_{3}$N$_{4}$ crystals. Besides, the DOSs of P62-Si$_{3}$N$_{4}$ are more localized than those of P61-Si$_{3}$N$_{4}$.

    It is clearly seen in Figure 2 that the lower VB band is made up by the N-2s state hybrid with the Si-3s, 3p states. The upper VB band is predominantly composed of bonding N-2p and Si-3p states; the contributions of Si-3s and N-2s bands are small but non-negligible. We conclude that the bonding interaction between Si and N plays a key role in the stability of P62- and P61-Si$_{3}$N$_{4}$. The band gaps between CB and VB are 3.03 and 1.50 eV for P61- and P62-Si$_{3}$N$_{4}$, respectively. This indicates that these two phases are semiconductors. To the authors' knowledge, there are no experimental data or theoretical results are available for us to compare with. It is well known that the GGA or local density approximation (LDA) usually underestimate the band gaps of semiconductors, which is due mainly to the fact that the excited states are not considered in the DFT calculations. More accurate estimation of band gaps requires a quasi-particle approach. Besides, the electronic specific heat coefficient $\gamma$ can be calculated by $\gamma =\frac{\pi ^2}{3}k_{\rm B}^2 n(E_{\rm F})$[35], where $k_{\rm B}$ is Boltzmann's constant and $n(E_{\rm F})$ represents the total DOS at Fermi energy level $E_{\rm F}$. As a consequence, the total DOSs at $E_{\rm F}$($\gamma$) are 0.58 (1.37 mJ/mol cell-K$^{2})$ and 0.72 (1.69 mJ/mol cell-K$^{2})$ for P61- and P62-Si$_{3}$N$_{4}$, respectively.

    Bond population is an important parameter which reflects the nature of chemical bonding. The results are listed in Table 2, where BL is the average bond length, BOP is the bond overlap population, and BOP$^{\rm s}$ ($=$ BOP/BL) is the scaled bond overlap population. Besides, the population ionicity $P_{\rm i}$ can be defined as[36]

    As shown in Table 2, the BOP of the N-Si bonds are positive, which indicates that there are strong chemical bonding interactions between the N and Si atoms. The negative BOP$_{\rm N-N}$ and BOP$_{\rm Si-Si}$ values reflect the anti-bonding states among different Si/N atoms. The N-Si bond of P61-Si$_{3}$N$_{4}$ is stronger than that of P62-Si$_{3}$N$_{4}$. This is due mainly to the shorter N-Si bond of P61-Si$_{3}$N$_{4}$, which allows stronger electron hybridizations. Since $P_{\rm i}$ $=$ 0 (or 1) reflects a pure covalent (or ionic) bond, the N-Si bond shows a covalent nature with a little weaker ionic character. Our results show that the bonding characters of P61- and P62-Si$_{3}$N$_{4}$ are different from that of $\gamma$-Si$_{3}$N$_{4}$ (covalent bonds)[37]. It is worth noting that the populations of the two phases are very similar, except the Si-Si bond.

  • 3.3.   Phase transitions

  • A simple way to determine the transition pressure from one phase to another is from a common tangent of the Gibbs free energy-volume curves. Another way to obtain the critical pressure of a phase transition is from the usual condition of equal Gibbs free energies at a given temperature. Obviously, the second way is more convenience. In order to compare the $\beta$ $\to$ P61 $\to$ $\delta$ and $\beta$ $\to$ P62 $\to$ $\delta$ phase transformations, our calculation parameters (energy cut-off, $k$-points, etc.) are exactly the same as the parameters given in our previous paper[25].

    In Figure 3(a), we illustrate the calculated Gibbs free energies for P61- and $\beta$-Si$_{3}$N$_{4}$. The curves show that $\beta$-Si$_{3}$N$_{4}$ will undergo a structural phase transition to the P61 phase. The critical pressure is 42.9 GPa (at 300 K), which is a little higher than that of the $\beta$ $\to$ P62 transition (40.0 GPa at 300 K)[25]. The reason why Xu et al.[14] did not observe the $\beta$ $\to$ P61 transformation may be that $\beta$-Si$_{3}$N$_{4}$ would transform into P62-Si$_{3}$N$_{4}$ before it could transform into P61-Si$_{3}$N$_{4}$. If, the $\beta$ $\to$ P62 transition was suppressed by some kinetic reasons, the $\beta$ $\to$ P61 transition would be observed. The Gibbs free energies obtained, $G$, as functions of pressure $P$, are plotted in Figure 3(b). It can be seen from Figure 3(b) that $G_{\rm P61}$ remains minimum up to 47.7 GPa, which indicates that the P61 phase is stable below 47.7 GPa. At 47.7 GPa and 300 K, P61- and $\delta$-Si$_{3}$N$_{4}$ have equal free energies, showing that a phase transition occurs at this point. Upon further compression, the $\delta $ phase becomes stable.

    Figure 3(c) shows the phase diagram of Si$_{3}$N$_{4}$ including the two phases P61 and $\delta$. A positive Clapeyron slope of the phase boundary is found, which suggests that the $\delta$ phase has lower entropy compared with the P61 phase. Higher temperatures will require higher pressures for inducing the P61 $\to$ $\delta$ phase transition. According to the Clausius-Clapeyron relation (d$T$/d$P$ $=$ $\Delta S/\Delta V$, where $\Delta S$ and $\Delta V$ are entropy and volume variations, respectively), the P61 $\to$ $\delta$ transition is accompanied by the shrinkage of volume. This transition (at 47.7~GPa and 300 K) occurs earlier than the P62 $\to$ $\delta$ transition (at 53.1~GPa and 300 K)[25]. The reason why the $\beta$ $\to$ P61 phase transition was not observed may be: (1) $\beta$-Si$_{3}$N$_{4}$ will transform into the P62 phase (40.0 GPa and 300 K)[25] before it transforms into the P61 phase; or (2) the $\beta$ $\to$ P62 phase transition can be interpreted as a low-barrier transition[14] compared with the $\beta$ $\to$ P61 transition. Generally speaking, the exact reason explaining why the $\beta$ $\to$ P61 phase transition had never been observed is still unclear. More experiments may be required to evaluate the high temperature $\beta$ $\to$ P61 $\to$ $\delta$ phase transitions of Si$_{3}$N$_{4}$ with higher reliability. This paper introduces a first-principles method plus a QHD model to study the phase transition characters of semi-conducting nitrides like Si$_{3}$N$_{4}$, which can avoid the huge cost in experiments when generating high temperature and high pressure.

4.   Conclusions
  • We have investigated the lattice parameters, elastic moduli and phase stabilities of P61-, P62- and $\delta$-Si$_{3}$N$_{4}$ through \textit{ab initio} calculations. The density of states and Mulliken populations are also analyzed in order to give a deep insight into the electronic structures of P61-and P62-Si$_{3}$N$_{4}$. The main results are as follows.

    (1) The lattice constants of $\beta$-Si$_{3}$N$_{4}$ are in accordance with the experimental and theoretical results. Since P61- and P62-Si$_{3}$N$_{4}$ have similar structures, elastic constants and electronic structures, the $\beta$ $\to$ P61 $\to$ $\delta$ phase transitions are likely to occur at high pressures since the $\beta$ $\to$ P62 $\to$ $\delta$ transitions have already been observed.

    (2) The band gaps (electronic specific heat) for P61 and P62 phases are 3.03 (1.37) and 1.50 eV (1.69 mJ/mol cell-K$^{2})$, respectively. The N-Si bond shows a covalent nature with a little weaker ionic character, while the Si-Si and N-N bonds indicate the anti-bonding states. The shorter N-Si bond in P61-Si$_{3}$N$_{4}$ allows stronger electron hybridizations, which reflects that P61-Si$_{3}$N$_{4}$ is more stable than P62-Si$_{3}$N$_{4}$.

    (3) The reason why the $\beta$ $\to$ P61 transition had never been observed may be that $\beta$-Si$_{3}$N$_{4}$ would transform into P62-Si$_{3}$N$_{4}$ (at 40.0 GPa and 300 K) before it could transform into P61-Si$_{3}$N$_{4}$ (at 42.9 GPa and 300 K). Only when the $\beta$ $\to$ P62 transition was suppressed by kinetic causes, would we observe the $\beta$ $\to$ P61 phase transition. The P61 $\to$ $\delta$ transition occurs at 47.7 GPa and 300 K, which is accompanied by the cell volume shrinkage.

Figure (3)  Table (2) Reference (37) Relative (20)

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