Direct Observation of a Pressure-Induced Precursor Lattice in Silicon

Detailed knowledge of atomic-scale structural change is essential for understanding the process and mechanism of phase transitions in solids. We present the direct experimental evidence of a precursor lattice in silicon at high pressures. The precursor lattice may appear to coexist dynamically with the host lattice over a large pressure range through rapid lattice fluctuations. The first-principles calculations are used to elucidate a dynamic lattice-fluctuation mechanism that accounts for the experimental observations. This precursor lattice-fluctuation mechanism for the phase transition goes beyond previously considered reconstructive or displacive processes and provides a novel picture of the underlying dynamics.

Figure 1

Electron density distributions in silicon at high pressures. (a) and (b) the electron density maps at 1.4 GPa, where (a) is a slide image along the [110] plane as indicated in (b). The electron density data reveal a lattice in diamond structure at 1.4 GPa. (c) and (d) the electron density data at 12.4 GPa where the two active groups are clearly observed; (c) a slide image along the [110] plane as in (a). The host lattice of the $\alpha$ phase is represented by the symbols in blue. The void group is shown in red, while the vicinity group in green. The local distribution of the vicinity group has the point-group symmetry of $S_4$ as shown in (e).

Figure 2

Integrated intensities of the two active groups relative to those of the host lattice as a function of pressure (bottom). The intensities of the active groups are obtained by integrating charge densities in a volume surrounding each active group site. The local electron density maps are shown at four different pressure points as indicated in the top. The solid symbols in the intensity plot represent the data where active groups are formed and clearly visible, while open symbols represent scattered intensities from the same volumes of the vicinity or void sites where the active groups are not clearly identified.

Figure 3

(a) Atomic positions of the host diamond structure (blue) together with the two active groups (red: the void group, green: the vicinity group). A unit cell of the diamond structure is shown by connecting the blue atomic positions. (b) A new unit cell by combining the void group (red sublattice) with the vicinity group (blue sublattice which represents the averaged equivalent positions of the vicinity group). The unit cell orientation is shown in the lower-right corner of (b). The tetragonal symmetry of the $\beta$-Sn structure originates from the atomic displacements of the vicinity group which gives a stretching along the [100] and [010] directions.

Figure 4

(a) The calculated enthalpy change, $\Delta H$, relative to the value for $\alpha$-Si along the pathway for the phase transformation, which proceeds with a continual reduction of the $c/a$ ratio over the pressure range of 0 to 13 GPa (top to bottom curves at a pressure step of 1 GPa). The two structures ($\alpha$-Si and $\beta$-Si) are depicted in a common body-centered-tetragonal cell to show the change and relation of the atomic positions in the two structures. (b) The enthalpy difference between $\beta$-Si and $\alpha$-Si, $H_{\beta }-H_{\alpha }$, and the kinetic barrier as a function of pressure extracted from the first-principles results shown in (a).

Figure 5

The theoretical values compared to the experimental data for the relative intensities of (a) the void group and (b) the vicinity group as a function of reduced pressure $P/P_t$. The theoretical values in the fit are obtained using the first-principles energetic results shown in Fig. 4 (see the Supplemental Material [8] for details).