Interpretation of monoclinic hafnia valence electron energy-loss spectra by time-dependent density functional theory

We present the valence electron energy-loss spectrum and the dielectric function of monoclinic hafnia ($\mathrm{m}-{\mathrm{HfO}}_2$) obtained from time-dependent density-functional theory (TDDFT) predictions and compared to energy-filtered spectroscopic imaging measurements in a high-resolution transmission-electron microscope. Fermi's golden rule density-functional theory (DFT) calculations can capture the qualitative features of the energy-loss spectrum, but we find that TDDFT, which accounts for local-field effects, provides nearly quantitative agreement with experiment. Using the DFT density of states and TDDFT dielectric functions, we characterize the excitations that result in the $\mathrm{m}-{\mathrm{HfO}}_2$ energy-loss spectrum. The sole plasmon occurs between 13 and 16 eV, although the peaks $\sim 28$ and above 40 eV are also due to collective excitations. We furthermore elaborate on the first-principles techniques used, their accuracy, and remaining discrepancies among spectra. More specifically, we assess the influence of Hf semicore electrons ($5p$ and $4f$) on the energy-loss spectrum, and find that the inclusion of transitions from the $4f$ band damps the energy-loss intensity in the region above 13 eV. We study the impact of many-body effects in a DFT framework using the adiabatic local-density approximation (ALDA) exchange-correlation kernel, as well as from a many-body perspective using “scissors operators” matched to an ab initio $GW$ calculation to account for self-energy corrections. These results demonstrate some cancellation of errors between self-energy and excitonic effects, even for excitations from the Hf $4f$ shell. We also simulate the dispersion with increasing momentum transfer for plasmon and collective excitation peaks.

Figure 1

Cross-sectional HRTEM image of $\mathrm{m}-{\mathrm{HfO}}_2$ showing a polycrystalline area (left) and a single crystalline domain (right).

Figure 2

VEELS spectra for $\mathrm{m}-{\mathrm{HfO}}_2$. Thin solid green line: HRTEM-VEELS averaged on a polycrystalline sample; dashed red line: DFT without LF effects; and solid black line: TDDFT RPA with LF effects. Theoretical curves have been convoluted with a Gaussian broadening of 1.5 eV.

Figure 3

The density of states (DOS) for $\mathrm{m}-{\mathrm{HfO}}_2$. The inset shows the projected DOS near the band gap in greater detail.

Figure 4

Energy-loss function (top), imaginary part of the dielectric function (middle), and the real part of the dielectric function (bottom) for $\mathrm{m}-{\mathrm{HfO}}_2$. Dashed red line: RPA without LF effects; solid black line: RPA with LF effects; and dot-dashed green line: HRTEM-VEELS derived dielectric functions. Theoretical curves are calculated at a reduced Lorentzian broadening of 0.1 eV.

Figure 5

Energy-loss function (top), imaginary part of the dielectric function (middle), and the real part of the dielectric function (bottom) for $\mathrm{m}-{\mathrm{HfO}}_2$. Solid blue line: RPA NLF without semicore (Hf $4f 5p$) electrons; dashed red line: RPA NLF explicitly including semicore electrons; and dot-dashed green line: HRTEM-VEELS derived dielectric functions. Theoretical curves are calculated at a reduced Lorentzian broadening of 0.1 eV.

Figure 6

VEELS spectra for $\mathrm{m}-{\mathrm{HfO}}_2$. Thin solid green line: HRTEM measured VEELS on a polycrystalline sample; solid black line: TDDFT RPA with LF effects; dot-dashed blue line: TDLDA; and dashed magenta line: RPA with the SO-corrected electronic structure. Theoretical curves have been convoluted with a Gaussian broadening of 1.5 eV.

Figure 7

Energy-loss function (top), imaginary part of the dielectric function (middle), and the real part of the dielectric function (bottom) for $\mathrm{m}-{\mathrm{HfO}}_2$. Dashed red line: RPA without LF effects; solid black line: RPA without LF effects on top of a SO-corrected electronic structure; and dot-dashed green line: HRTEM-VEELS derived dielectric functions. In the inset: RPA SO real part of the dielectric function along the main reciprocal lattice directions. Theoretical curves are calculated at a reduced Lorentzian broadening of 0.1 eV.

Figure 8

Energy-loss function dispersion along [001]. Open circles are a guide to the eye to indicate the dispersion of the plasmon and the collective excitation. Spectra have been convoluted with a Gaussian broadening of 1.5 eV.