Phonon anharmonicity of rutile SnO${}_2$ studied by Raman spectrometry and first principles calculations of the kinematics of phonon-phonon interactions

Raman spectra of rutile tin dioxide (SnO${}_2$) were measured at temperatures from 83 to 873 K. The pure anharmonicity from phonon-phonon interactions was found to be large and comparable to the quasiharmonicity. First-principles calculations of phonon dispersions were used to assess the kinematics of three-phonon and four-phonon processes. These kinematics were used to generate Raman peak widths and shifts, which were fit to measured data to obtain the cubic and quartic components of the anharmonicity for each Raman mode. The $B_{2g}$ mode had a large quartic component, consistent with the symmetry of its atom displacements. The broadening of the $B_{2g}$ mode with temperature showed an unusual concave-downwards curvature. This curvature is caused by a change with temperature in the number of down-conversion decay channels, originating with the wide band gap in the phonon dispersions.

Figure 1

Rutile structure and oxygen atom displacements for Raman-active modes.

Figure 2

Raman spectra of powder samples of rutile SnO${}_2$ at selected temperatures.

Figure 3

Temperature dependence of (a) frequency shifts and (b) breadths as FWHMs of the Raman modes $E_g$, $A_{1g}$, and $B_{2g}$. Filled and open symbols represent experimental data from powder and single-crystal samples, respectively. Solid curves are the theoretical fittings with a full calculation of the kinematics of three- phonon and four-phonon processes. The dashed curve was calculated without considering the frequency dependence of $D_{0\downarrow }\left(\Omega \right)$, the number of decay channels, at elevated temperatures.

Figure 4

(a) Calculated phonon dispersion along high-symmetry directions of rutile SnO${}_2$. $\Gamma (0,0,0)$, $X(0.5,0,0)$, $M(0.5,0.5,0)$, $Z(0,0,0.5)$, $R(0.5,0,0.5),$ and $A(0.5,0.5,0.5)$. At the $\Gamma$ point, the frequencies from Table are presented as upward-pointing triangles (Raman) and downward-pointing triangles (infrared). At the $X$, $M$, $Z$, $R$, and $A$ points, the mode frequencies from Ref. 9 (all doubly degenerate) are presented as squares. (b) Total phonon DOS (black curve) and oxygen-projected DOS [shaded (green) area].

Figure 5

(a) Two-phonon density of states $D^{\omega }\left(\Omega \right)$ for 0 and 800 K. Arrows mark the positions of the three Raman modes, $E_g$, $A_{1g}$, and $B_{2g}$, respectively. The up-conversion and down-conversion contributions at 800 K are shown by green and black dashed curves, respectively. There is no up-conversion process at 0 K. (b) $P^{\omega }\left(\Omega \right)$ at 800 K.

Figure 6

Fittings of the temperature dependence of the frequency shift of (a) the $E_g$ mode, (b) the $A_{1g}$ mode, and (c) the $B_{2g}$ mode. Solid curves are the final fittings to ${\Delta }^Q+{\Delta }^{\left(3\right)}+{\Delta }^{\left(3^{\prime }\right)}+{\Delta }^{\left(4\right)}$. Different contributions are indicated individually.