First-principles predictions of temperature-dependent infrared dielectric function of polar materials by including four-phonon scattering and phonon frequency shift

Recently, first-principles calculations based on density functional theory have been widely used to predict the temperature-dependent infrared spectrum of polar materials, but the calculations are usually limited to the harmonic frequency (0 K) and three-phonon scattering damping for the zone-center infrared-active optical phonon modes, and fail to predict the high-temperature infrared optical properties of materials such as sapphire ($\alpha -{\text{Al}}_2{\text{O}}_3$), GaAs, $\text{Ti}{\text{O}}_2$, etc., due to the neglect of high-order phonon scattering damping and phonon frequency shift. In this work, we implemented first-principles calculations to predict the temperature-dependent infrared dielectric function of polar materials by including four-phonon scattering and phonon frequency shift. The temperature-dependent phonon damping by including three- and four-phonon scattering as well as the phonon frequency shift by including cubic and quartic anharmonicity and the thermal expansion effect are calculated based on anharmonic lattice dynamics method. The infrared dielectric function of $\alpha -{\text{Al}}_2{\text{O}}_3$ is parameterized, and then the temperature-dependent infrared optical reflectance is determined. We find that our predictions agree better with the experimental data than the previous density functional theory-based methods. This work will help to effectively predict the thermal radiative properties of polar materials at elevated temperature, which is generally difficult to measure, and will enable predictive design of new materials for radiative applications.

Figure 1

(a) Crystal structure of the primitive unit cell of $\alpha -{\text{Al}}_2{\text{O}}_3$, which contains four aluminum atoms (gray) and six oxygen atoms (orange). The ${\mathit{a}}_{\text{1}}, {\mathit{a}}_{\text{2}}$, and ${\mathit{a}}_{\text{3}}$ are the primitive lattice vectors. (b) First Brillouin zone of $\alpha -{\text{Al}}_2{\text{O}}_3$. The high-symmetry notations are referred to Ref. [41]. (c) Temperature-dependent phonon dispersion curve of $\alpha -{\text{Al}}_2{\text{O}}_3$ calculated from first principles. The harmonic frequency (dashed pink line, at 0 K) corresponding to ${\omega }_{\lambda }^0$ and the anharmonic frequency (solid blue line, at 1800 K) corresponding to ${\omega }_{\lambda }^0+\mathrm{\Delta }{\omega }_{\lambda }^{3\text{ph}}+\mathrm{\Delta }{\omega }_{\lambda }^{4\text{ph}}+\mathrm{\Delta }{\omega }_{\lambda }^{\text{quasi}}$ are shown to compare the temperature effect.

Figure 2

The temperature-dependent phonon frequency calculated from first principles as compared with experimental data. The IR-active optical mode with a symmetry of $E_u$, which includes four species in $\alpha -{\text{Al}}_2{\text{O}}_3$ at the $\mathrm{\Gamma }$ point, is calculated in this work. The frequency shift effect on harmonic frequency (solid blue line labeled ${\omega }_{\lambda }^0$) due to three-phonon scattering (dot-dashed brown line labeled ${\omega }_{\lambda }^0+\mathrm{\Delta }{\omega }_{\lambda }^{3\text{ph}}$), four-phonon scattering (dotted yellow line labeled ${\omega }_{\lambda }^0+\mathrm{\Delta }{\omega }_{\lambda }^{4\text{ph}}$), thermal expansion (dashed purple line labeled ${\omega }_{\lambda }^0+\mathrm{\Delta }{\omega }_{\lambda }^{\text{quasi}}$), and total effects (solid green line labeled ${\omega }_{\lambda }^0+\mathrm{\Delta }{\omega }_{\lambda }^{3\text{ph}}+\mathrm{\Delta }{\omega }_{\lambda }^{4\text{ph}}+\mathrm{\Delta }{\omega }_{\lambda }^{\text{quasi}}$) is calculated for (a)–(d) TO and (e)–(h) LO modes. The experimental data [28] (triangles) fitted from the experimental measurements of reflectance of $\alpha -{\text{Al}}_2{\text{O}}_3$ are shown for comparison.

Figure 3

The temperature-dependent phonon damping $\gamma$ calculated from first-principles as compared with experimental data. The $\gamma$ of the IR-active optical mode with a symmetry of $E_u$, which includes four species in $\alpha -{\text{Al}}_2{\text{O}}_3$ at the $\mathrm{\Gamma }$ point, is calculated in this work. The ${\gamma }_{\lambda }^{3\text{ph}}$ (dotted red line) and ${\gamma }_{\lambda }^{4\text{ph}}$ (dashed yellow line) due to three- and four-phonon scattering are shown as a function of temperature for (a)–(d) TO and (e)–(h) LO modes. The experimental data [28] (triangles) fitted from the experimental measurements of reflectance of $\alpha -{\text{Al}}_2{\text{O}}_3$ are shown for comparison with the calculated total phonon damping ${\gamma }_{\lambda }^{3+4\text{ph}}$ (solid purple line).

Figure 4

The semi-infinite normal reflectance calculated from first principles as compared with experimental data. The temperature-dependent reflectance spectrum (ordinary ray) of $\alpha -{\text{Al}}_2{\text{O}}_3$ is shown (a) at 77 K and (b) at 1775 K. The reflectances calculated using ${\gamma }_{\lambda }^{3\text{ph}}$ with ${\omega }_{\lambda }^0$ (dotted red line), ${\gamma }_{\lambda }^{3+4\text{ph}}$ with ${\omega }_{\lambda }^0$ (dashed yellow line), and ${\gamma }_{\lambda }^{3+4\text{ph}}$ with ${\omega }_{\lambda }^0+\mathrm{\Delta }{\omega }_{\lambda }^{3\text{ph}}+\mathrm{\Delta }{\omega }_{\lambda }^{4\text{ph}}+\mathrm{\Delta }{\omega }_{\lambda }^{\text{quasi}}$ (solid purple line) are plotted for comparison with experimental data [28] (blue circles).

Figure 5

The semi-infinite normal reflectance calculated from first principles as compared with experimental data. The temperature-dependent reflectance spectrum (extraordinary ray) of $\alpha -{\text{Al}}_2{\text{O}}_3$ is shown at 300 K. The reflectances calculated using ${\gamma }_{\lambda }^{3\text{ph}}$ with ${\omega }_{\lambda }^0$ (dotted red line), ${\gamma }_{\lambda }^{3+4\text{ph}}$ with ${\omega }_{\lambda }^0$ (dashed yellow line), and ${\gamma }_{\lambda }^{3+4\text{ph}}$ with ${\omega }_{\lambda }^0+\mathrm{\Delta }{\omega }_{\lambda }^{3\text{ph}}+\mathrm{\Delta }{\omega }_{\lambda }^{4\text{ph}}+\mathrm{\Delta }{\omega }_{\lambda }^{\text{quasi}}$ (solid purple line) are plotted for comparison with experimental data [2] (blue circles).