Equilibrium and metastable phase transitions in silicon nitride at high pressure: A first-principles and experimental study

We have combined first-principles calculations and high-pressure experiments to study pressure-induced phase transitions in silicon nitride (Si${}_3$N${}_4$). Within the quasi-harmonic approximation, we predict that the $\alpha$ phase is always metastable relative to the $\beta$ phase over a wide pressure-temperature range. Our lattice vibration calculations indicate that there are two significant and competing phonon-softening mechanisms in the $\beta$-Si${}_3$N${}_4$, while phonon softening in the $\alpha$-Si${}_3$N${}_4$ is rather moderate. When the previously observed equilibrium high-pressure and high-temperature $\beta$ $\to$ $\gamma$ transition is bypassed at room temperature (RT) due to kinetic reasons, the $\beta$ phase is predicted to undergo a first-order structural transformation to a denser $P\overline{6}$ phase above 39 GPa. The estimated enthalpy barrier height is less than 70 meV$/$atom, which suggests that the transition is kinetically possible around RT. This predicted new high-pressure metastable phase should be classified as a “postphenacite” phase. Our high-pressure x-ray diffraction experiment confirms this predicted RT phase transition around 34 GPa. No similar RT phase transition is predicted for $\alpha$-Si${}_3$N${}_4$. Furthermore, we discuss the differences in the pressure dependencies of phonon modes among the $\alpha$, $\beta$, and $\gamma$ phases and the consequences on their thermal properties. We attribute the phonon modes with negative Grüneisen ratios in the $\alpha$ and $\beta$ phases as the cause of the predicted negative thermal expansion coefficients (TECs) at low temperatures in these two phases, and predict no negative TECs in the $\gamma$ phase.

Figure 1

Polymorphs of Si${}_3$N${}_4$ and synthesis conditions.

Figure 2

Crystal structures of (a), (b) $\alpha$-Si${}_3{\mathrm{N}}_4$; (c), (d) $\beta$-${\mathrm{Si}}_3{\mathrm{N}}_4$; and (e), (f), (g) $\gamma$-Si${}_3$N${}_4$. In the panel of $\alpha$- and $\beta$-Si${}_3$N${}_4$, the first graph illustrates the unit-cell model and the second graph is the $2\times 2\times 1$ supercell model viewed in the direction of the $c$ axis. In the panel of $\gamma$-Si${}_3$N${}_4$, the first graph shows the conventional cubic cell of the spinel structure and the following two graphs show the fourfold and sixfold coordinated Si units (SiN${}_4$ and SiN${}_6$) with tetrahedra and octahedra, respectively.

Figure 3

Energy-volume curves for $\alpha$-, $\beta$- and $\gamma$-Si${}_3$N${}_4$ in the scale of per atom. The $\beta$ phase has an equilibrium energy of 3 meV lower than that of the $\alpha$ phase. $E_0$ of the $\gamma$ phase is 93 meV higher than the $\beta$ phase.

Figure 4

Phonon dispersion curves and VDOS of (a) $\alpha$-${\mathrm{Si}}_3{\mathrm{N}}_4$; (b) $\beta$-${\mathrm{Si}}_3{\mathrm{N}}_4$; and (c) $\gamma$-Si${}_3$N${}_4$ at zero pressure.

Figure 5

Calculated dispersion curves (scattered circles) of mode Grüneisen parameter of (a) $\alpha$-Si${}_3$N${}_4$, (b) $\beta$-Si${}_3$N${}_4$, and (c) $\gamma$-Si${}_3$N${}_4$ at zero pressure. (Red) horizontal line is present to separate the positive and negative values.

Figure 6

Gibbs free energy of $\alpha$-Si${}_3$N${}_4$ relative to that of the $\beta$ phase as a function of temperature. Solid, dashed, and dotted lines represent the pressure of 0, 5, and 10 GPa, respectively.

Figure 7

$T$-$P$ phase diagram of Si${}_3$N${}_4$. Solid curve denotes the phase boundary between $\beta$- and $\gamma$-Si${}_3$N${}_4$. Dashed curve denotes the phase boundary between $\alpha$- and $\gamma$-Si${}_3$N${}_4$.

Figure 8

(a) Raman, (b) IR, and (c) silent mode frequencies as a function of pressure up to 60 GPa for $\beta$-Si${}_3$N${}_4$. Experimental pressure dependencies of Raman modes up to 30 GPa are also presented in discrete symbols as a comparison.[68] Solid squares denote measurements upon compression and open squares denote measurements upon decompression. Several low-frequency modes are found to decrease with increasing pressure. One B${}_u$ branch of the silent modes is found dropping to zero at about 60 GPa.

Figure 9

Phonon dispersion of $\beta$-Si${}_3$N${}_4$ at a pressure of 48 GPa. Two competing soft-phonon modes are found: one TA branch at the M point and one optic branch at the $\Gamma$ point. No LO-TO splitting correction is added for the interests of low-frequency modes only.

Figure 10

The square of vibrational frequency (${\omega }^2$) as a function of pressure for two competing soft-phonon branches: one TA branch at the M point and one B${}_u$ branch at the $\Gamma$ point. Solid squares and circles represent data from calculation. Solid and dashed lines are from a linear fitting.

Figure 11

The total energy of $P{6}_3/m$ ($\beta$), $P\overline{6}$, $P{2}_1/m$, $P3$, and $P{\overline{6}}^{\prime }$ structures as a function of volume.

Figure 12

Ball-stick models of (a) $P{6}_3/m$, (b) $P\overline{6}$, (c) $P{2}_1/m$, and (d) $P3$ structures viewed along the $z$ axis. Balls in a dark color (blue) represent N atoms, and Si atoms are in a light color (yellow).

Figure 13

Enthalpy landscape and its contour plot as a function of $f_{x-y}$ and $f_z$ at the transition pressure of 38.5 GPa.

Figure 14

Enthalpy barrier (relative to the $\beta$ phase) as a function of linearly interpreted transition parameter at 30 GPa (red, gray) and the transition pressure of 38.5 GPa (black). The solid curves denote the $\beta$ $\to$ $P{3}^{\prime }$ $\to$ $P{\overline{6}}^{\prime }$ path, and the dashed curves denote direct $\beta$ $\to$ $P{\overline{6}}^{\prime }$ path. The horizontal axis is defined as qualitative structural similarity. The left end represents the $\beta$ structure, and the right end represents the $P{\overline{6}}^{\prime }$ structure.

Figure 15

x-ray diffraction patterns of the compression of the $\beta$-Si${}_3$N${}_4$ phase from 26 GPa and the formation of the $\delta$-Si${}_3$N${}_4$ phase emerging at 37 GPa and continuing to 41 GPa. The arrows highlight the positions of the peaks associated with the formation of the $\delta$ phase. The wavelength used in the monochromatic synchrotron x-ray diffraction was $\lambda =0.444$ Å 

Figure 16

Rietveld refinements of diffraction data obtained of both the $\delta$ and $\beta$ phases of Si${}_3$N${}_4$ in a lithium fluoride pressure-transmitting medium at 41 GPa. Data points and Rietveld fit are overlaid in black and red (gray), respectively, the difference plot is shown. The red (gray) tick marks indicate peaks for $\delta$-Si${}_3$N${}_4$, the top black tick marks indicate peaks for $\beta$-Si${}_3$N${}_4$, and the middle black tick marks are of the Rhenium gasket, and the bottom black ones are of the LiF pressure medium. (a) Data obtained at 41 GPa of both the $\delta$ and $\beta$ phases of Si${}_3$N${}_4$. The internal atomic coordinates used for the refinement are fixed according to the values generated from the DFT calculations. (b) The same data set as (a) at 41 GPa but with the atomic coordinates refined. The wavelength used in the monochromatic synchrotron x-ray diffraction was $\lambda =0.444$ Å 

Figure 17

Raman spectra of (a) the compression and decompression of the $\beta$-Si${}_3$N${}_4$ phase up to 43 GPa, using 4:1 methanol-ethanol as the PTM. (b) The compression of the $\beta$-Si${}_3$N${}_4$ phase up to 52 GPa, using N${}_2$ as the PTM.

Figure 18

Pressure dependencies of the observed Raman peaks for compression of the $\beta$-Si${}_3$N${}_4$ up to 43 GPa. Additional Raman peaks above $\sim$34 GPa are associated with the formation of the $\delta$ phase.

Figure 19

Temperature dependence of volume TEC of bulk $\beta$-Si${}_3$N${}_4$ at zero pressure. Solid line denotes present work, dashed line (red) denotes the first-principles calculation from Kuwabara et al.,[22] and discrete symbols denote experimental data.[34, 35, 36, 37]

Figure 20

Temperature dependence of bulk Grüneisen parameter of $\beta$-Si${}_3$N${}_4$ at zero pressure. Solid line denotes present work, and discrete symbols denote experimental data.[36]

Figure 21

Temperature dependence of volume TEC of bulk $\beta$-Si${}_3$N${}_4$ at pressures of 0, 10, 20, and 30 GPa.

Figure 22

Temperature dependencies of volume TECs of $\alpha$-, $\beta$-, and $\gamma$-Si${}_3$N${}_4$ at zero pressure. The inset shows the low-temperature TEC of the $\gamma$ phase from 0 to 300 K.