Limits of the quasiharmonic approximation in MgO: Volume dependence of optical modes investigated by infrared reflectivity and ab initio calculations

Experimental and numerical investigation of phonon optical modes of MgO as a function of temperature (from 300 to 1400 K) and pressure (from 0 to 21 GPa) are here presented. Infrared reflectivity measurements were performed to probe energies and widths of the optical phonons, as well as of the multiphonon processes affecting the spectral shape, over a variation of the unit-cell volume exceeding 20%. Calculations within quasiharmonic approximation (QHA) account well for the volume dependence of the optical phonon energies observed in high-pressure experiments, while they fail at larger volumes, corresponding to the highest investigated temperatures. Moreover, QHA calculations more closely predict energies of transverse optical (TO) modes than those of longitudinal optical (LO) ones. This can be ascribed to known limitations in the modeling of the effective charges ($Z^{*}$) and dielectric constant (${\varepsilon }_{\infty }$) that lead to an underestimation of the LO-TO splitting. Based on the comparison of our experimental and theoretical results, we propose an empirical analytical expression for ${Z^{*}}^2/{\varepsilon }_{\infty }$ as a function of the atomic cell volume. Density-functional perturbation theory including phonon-phonon scattering up to the third order of the lattice potential expansion is used to calculate phonon widths. These calculations reproduce and explain remarkably well the nontrivial volume dependence of both TO and LO phonons linewidths determined by the experiments.

Figure 1

Reflectivity spectra of MgO single crystal as a function of temperature (top) and pressure (bottom). The curves are all rigidly vertically shifted by the same amount (0.2) for better visibility. The red dashed curves are representative fit results. Red, blue, and gray symbols highlight the position of TO and LO phonons and the ESW feature, respectively. Circles and squares are for different experimental runs.

Figure 2

Experimental and computed optical function of MgO. Panels (a), (c), (e), and (g) show the experimental data analysis results: Lorentz fit (black curves) and Kramers-Kronig analysis (red curves). Panel (a) and panel (c) show the imaginary part of the dielectric function ${\varepsilon }_2$ at high temperature (a) and high pressure (c). Panel (e) and panel (g) shows the energy loss function at high temperature (e) and high pressure (g). Panels (b), (d), (f), and (h) show the calculated spectral function of TO and LO at high temperature (b), (f) and high pressure (d), (h). N.B. The ordinate axes are in logarithmic scale. Stars (*) indicate the ESW structure. The main features due to the transverse and to the longitudinal optical phonon are indicated by arrows. Temperature (top panels) increases from the bottom to the top. Pressure (bottom panels) increases from the top to the bottom. Curves are staggered for better visibility.

Figure 3

(a) Phonon energies calculated in the quasiharmonic approximation (lines) and extracted from the Lorentz fit of recorded spectra (circles and squares). (b) Phonon widths calculated including the third order (lines) and extracted from the Lorentz fit of recorded spectra (circles and squares). Literature data from Ref. [17] are reported as open diamonds. Colored markers correspond to P-T conditions at which we performed phonon DOS and final state calculations reported in Fig. 7. To assign an experimental error, the Lorentz fits were repeated after a rescaling of the measured reflectivity of ±1%. This procedure reduces any errors caused by the uncertainty on the absolute intensity of the single spectrum and minimizes the effects of misalignment intrinsic to the acquisition of the background. This approach produced the reported error bars for the phonon widths, while no significant effect was produced on the phonon energies.

Figure 4

Ratio ${Z^{*}}^2/{\varepsilon }_{\infty }$ as a function of volume. The estimation from experimental results (dots) is compared to estimations from the ab initio simulations (thin lines) performed in this study (DFPT) and available in the literature [5, 34]. Inset (a): Lorentzian fit (blue line) of the high-energy trend of the mean reflectivity curve $\langle R\rangle$ (red line). Inset (b): effective charge number $Z^{*}$.

Figure 5

Natural logarithm of the phonon energy ${\omega }_{\mathrm{LO}}$ (red) and ${\omega }_{\mathrm{TO}}$ (blue) as a function of ln($V$) for high-pressure and high-temperature measurements (dots) and QHA calculations (solid lines). Linear fits for the high-volume (HV) experimental data, there where the QHA calculations do not account for the experimental observations anymore, are displayed as dashed lines with consistent color coding.

Figure 6

Grüneisen parameters for LO (blue) and TO (red) phonons as a function of volume. Continuous lines are QHA results; dashed lines are obtained by the derivative of the fits to the experimental data at the high volumes (dashed lines in Fig. 5). The shaded region highlights the HV range over which QHA calculations progressively deviate from experimental observations. Inset: TO/LO Grüneisen parameters ratio.

Figure 7

Evolution of the phonon DOS with the thermodynamic conditions. The color coding is consistent with marks in Fig. 3. The blue- (gray)-shaded zone represents the final states of the LO (ESW) decay process.