Lattice Instability during Solid-Solid Structural Transformations under a General Applied Stress Tensor: Example of $\mathrm{Si}\mathrm{I}\to \mathrm{Si}\mathrm{II}$ with Metallization

The density functional theory was employed to study the stress-strain behavior and elastic instabilities during the solid-solid phase transformation (PT) when subjected to a general stress tensor, as exemplified for semiconducting Si I and metallic Si II, where metallization precedes the PT, so stressed Si I can be a metal. The hydrostatic PT occurs at 76 GPa, while under uniaxial loading it is 11 GPa (3.7 GPa mean pressure), 21 times lower. The $\mathrm{Si}\mathrm{I}\to \mathrm{Si}\mathrm{II}$ PT is described by a critical value of the phase-field’s modified transformation work, and the PT criterion has only two parameters given six independent stress elements. Our findings reveal novel, more practical synthesis routes for new or known high-pressure phases under predictable nonhydrostatic loading, where competition of instabilities can serve for phase selection rather than free energy minima used for equilibrium processing.

Figure 1

DFT energy of Si versus lattice parameters $c$ and $a$. Insets: Tetragonal ($a=b$, $c$) unit cells for stress-free Si II (left), unstable SP (middle), and Si I (right).

Figure 2

(a) True (Cauchy) stress ${\sigma }_3$ versus Lagrangian strain $E_3$ for compression or tension along $c$ for lateral stresses ${\sigma }_1={\sigma }_2$ for $\mathrm{Si}\mathrm{I}\leftrightarrow \mathrm{Si}\mathrm{II}$ PTs; (b) corresponding transformation paths in the ($F_1=F_2$, $F_3$) plane. Hollow (solid) symbols mark instability points for forward (reverse) PT. The dashed line shows hydrostatic loading.

Figure 3

Elastic instability versus ${\sigma }_3$ and ${\sigma }_1={\sigma }_2$ for direct ($D$) $\mathrm{Si}\mathrm{I}\to \mathrm{Si}\mathrm{II}$ and reverse ($R$) $\mathrm{Si}\mathrm{II}\to \mathrm{Si}\mathrm{I}$ PTs from DFT and TP-based results [11, 12] and the metallization curve from DFT. Hydrostatic condition (${\sigma }_1={\sigma }_2={\sigma }_3$) is a diagonal (dashed black) line.

Figure 4

Criterion for $\mathrm{Si}\mathrm{I}\to \mathrm{Si}\mathrm{II}$ PT for triaxial stresses. (a) DFT results (points) lie very close to a plane described by a constant value of modified transformation work in Eq. (3). (b) The result in (a) rotated to visualize an approximate plane.