Phonons in ${\mathbf{Si}}_{\mathbf{24}}$ at simultaneously elevated temperature and pressure

Raman spectroscopy was used to measure the frequencies of phonons in ${\mathrm{Si}}_{24}$ with an open clathrate structure at temperatures from 80 to 400 K with simultaneous pressures of 0 to 8 GPa. The frequency shifts of the different phonons were substantially different under either temperature or pressure. The quasiharmonic behavior was isolated by varying pressure at low temperatures, and the anharmonic behavior was isolated by varying temperature at low pressures. Phonon modes dominated by bond bending were anomalous, showing stiffening with temperature and softening with pressure. Both the quasiharmonic behavior and the anharmonic behavior changed markedly with simultaneous changes in temperature $\mathrm{\Delta }T$ and pressure $\mathrm{\Delta }P$. With $\mathrm{\Delta }T=320\mathrm{K}$ and $\mathrm{\Delta }P=8\mathrm{GPa}$, some frequency shifts that scaled with the product $\mathrm{\Delta }T\mathrm{\Delta }P$ were as large as the shifts from $\mathrm{\Delta }T$ and $\mathrm{\Delta }P$ alone. The thermodynamic entropy of this material likely has a dependence on $\mathrm{\Delta }T$ and $\mathrm{\Delta }P$ that cannot be obtained by adding effects from quasiharmonicity and phonon-phonon anharmonicity.

Figure 1

Unit cell of ${\mathrm{Si}}_{24}$ with colors indicating different crystallographic sites.

Figure 2

Raman spectra of ${\mathrm{Si}}_{24}$ at selected temperatures and pressures.

Figure 3

(a) Fractional frequency shifts of Raman modes with pressure. The $A_g$ modes stiffen under higher pressures as generally expected, but the $B_g$ modes soften. The arrows at the ends of the curves show the change in $\omega \left(P\right)/\omega \left(0\mathrm{GPa}\right)$ between 80 and 400 K. (b) Fractional frequency shifts of Raman modes with temperature. The red curve is common for all four modes at a pressure of 0 GPa. The black curves are for 8 GPa. Note the anomalous thermal stiffening of the $B_{3g}^{\left(1\right)}$ and $B_{1g}^{\left(1\right)}$ modes at 8 GPa.

Figure 4

Fitted mode Grüneisen parameters ${\gamma }_i$ for the selected seven modes as a function of pressure at 0 K, where the $B_g$ and $A_g$ modes are denoted with the solid markers and open markers, respectively.

Figure 5

Fitted anharmonicity parameters $A_i$ for the selected seven modes as a function of pressure at 0 K, where the $B_g$ and $A_g$ modes are denoted with the solid markers and open markers, respectively.

Figure 6

Contributions to frequency shifts from five terms in Eq. (9) with $\mathrm{\Delta }P=8\mathrm{GPa},\mathrm{\Delta }T=320\mathrm{K}$.

Figure 7

Vibrational motions of (a) $B_{3g}^{\left(1\right)}$ and (b) $A_g^{\left(6\right)}$.