Phonon anharmonicity in silicon from 100 to 1500 K

Inelastic neutron scattering was performed on silicon powder to measure the phonon density of states (DOS) from 100 to 1500 K. The mean fractional energy shifts with temperature of the modes were $\langle \Delta {\varepsilon }_i/{\varepsilon }_i\Delta T\rangle =-0.07$, giving a mean isobaric Grüneisen parameter of $+6.95\pm 0.67$, which is significantly different from the isothermal parameter of +0.98. These large effects are beyond the predictions from quasiharmonic models using density functional theory or experimental data, demonstrating large effects from phonon anharmonicity. At 1500 K the anharmonicity contributes $0.15k_{\mathrm{B}}$/atom to the vibrational entropy, compared to $0.03k_{\mathrm{B}}$/atom from quasiharmonicity. Excellent agreement was found between the entropy from phonon DOS measurements and the reference NIST-JANAF thermodynamic entropy from calorimetric measurements.

Figure 1

Phonon DOS curves of silicon $g_T\left(\varepsilon \right)$ normalized to unity. Curves are offset for clarity.

Figure 2

The negative of the fractional thermal shifts of the five features of the DOS: transverse acoustic (TA; red squares and purple circles), longitudinal acoustic (LA; blue pentagons), longitudinal acoustic and optical (LA/LO; green hexagons), and transverse and longitudinal optical modes (TO/LO; black triangles). The average between the shifts of the five features is shown by a dotted orange line.

Figure 3

Phonon dispersion and DOS of silicon at the quasiharmonic 100 K equilibrium volume (black solid line) scaled in energy to fit the experimental 100 K DOS (orange dotted line) from Fig. 1. The distinct features are labeled with the corresponding markers used in Fig. 2.

Figure 4

Silicon phonon DOS of experimental measurements at 100 K (blue solid line) and 1500  K (red solid line). A DOS at 1500 K (red dotted line) is obtained from shifting and renormalizing the measured DOS at 100 K [Eq. (2)].

Figure 5

Vibrational entropy of silicon from experimental phonon DOS curves (orange circles). Entropy from calculated DOS curves are also shown: harmonic (solid black line) and quasiharmonic (green triangles). The interpolated anharmonic approximation is shown (orange dotted line), as is the NIST-JANAF entropy data (blue diamonds) [46]. The inset enlarges the high-temperature region.