Electronic excitations in monoclinic hafnia as studied by spatially and momentum-resolved electron energy-loss spectroscopy and ab initio density functional theory calculations

Monoclinic hafnia ($m\text{−}{\mathrm{HfO}}_2$) with its high dielectric permittivity ($\kappa$) and larger band gap of $\sim 5.5\text{−}5.9\mathrm{eV}$ deposited as a few nanometers thick $m\text{−}{\mathrm{HfO}}_2$ thin film on a Si substrate was introduced as an effective gate oxide in complementary metal-oxide-semiconductor devices by Intel in 2007. However, the existence and complex anisotropic excitation characters of this centrosymmetric monoclinic crystal structure—which involve both single-particle and collective electron excitations such as plasmons, and include electron-hole (excitonic) and electron-electron (self-energy) interactions—remain elusive. Therefore, the electronic nature of the material needs to be explored in depth for applications in semiconductor technology. In this study, spatially and momentum-resolved electron energy-loss spectroscopy (EELS) in conjunction with first-principles calculations of the electronic band structure and dielectric function have been employed to investigate electronic excitations in $m\text{−}{\mathrm{HfO}}_2$. The phase purity and crystallinity of $m\text{−}{\mathrm{HfO}}_2$ were confirmed by x-ray diffraction and EELS. Low-loss EELS performed using $aloof$ electron beam setups and energy-filtered transmission electron microscopy spectrum imaging (EFTEM-SI) revealed spectral features at 13.5 and 16 eV energy loss assigned to surface plasmons and volume plasmons (VPs), respectively. Surface exciton polaritons (SEPs) with surface resonances associated with excitonic onsets above the band gap were also observed at $\sim 7.5$ and $\sim 28\mathrm{eV}$ energy loss. The surface excitation character of these features was confirmed by EFTEM-SI and relativistic calculations of energy versus momentum ($E\text{−}k$) maps. Using collection-angle (β) and momentum ($q$)-resolved EELS, it was found that the SEP intensity at $\sim 7.5\mathrm{eV}$ energy loss is a function of β and $q$, and no anisotropic shape for the VP is observed along the [100], [010], and [001] directions. Furthermore, the peak at $\sim 48\mathrm{eV}$ energy loss was assigned to a semicore $\mathrm{Hf}5p$ plasmon involving multiplet resonant processes. All the VPs, the SEPs at 28 eV energy loss, and the $\mathrm{Hf}5p$ plasmons at 48 eV energy loss display parabolic dispersion behavior with an energy shift of $1–3\mathrm{eV}$.

Figure 1

(a) XRD pattern of $m\text{−}{\mathrm{HfO}}_2$ recorded at room temperature and Rietveld refinement results. The inset shows the unit cell of $m\text{−}{\mathrm{HfO}}_2$. (b) The ELNES of the O $K$ edge of $m\text{−}{\mathrm{HfO}}_2$. The Hf-O bonding distances calculated by Rietveld refinement from XRD pattern(a) are listed in Table 2.

Figure 2

(a) Aloof beam low-loss STEM-EEL spectra acquired from $m\text{−}{\mathrm{HfO}}_2$ with the electron probe positioned at the locations marked in the STEM image in (c). The color circles denote probe positions in the inset. Corresponding EEL spectra are shown in the same colors. (b) The real (black) and imaginary (red) parts of the dielectric permittivity of $m\text{−}{\mathrm{HfO}}_2$ derived from the orange spectrum in (a). (c) shows the STEM-HAADF image with locations marked for aloof low-loss STEM-EEL spectra. (d) The calculated relativistic $E\text{−}k$ intensity maps for 50 nm thick $m\text{−}{\mathrm{HfO}}_2$ slab with impact parameters $b=-14$, 0, 4, and 12 nm. The electron probe position exploited for the calculations is depicted in each corresponding inset. (e) The calculated relativistic loss probabilities of 50 nm thick $m\text{−}{\mathrm{HfO}}_2$ slab per unit electron path length integrated over $k$ in the $0\text{−}0.03{\AA }^{-1}$ range with impact parameters $b=-14$, 0, 4, 8, and 12 nm.

Figure 3

(a) The EEL spectrum in Fig. 2 redrawn to illustrate the band gap measurements. Inset is Tauc's plot for the band gap analysis of $m\text{−}{\mathrm{HfO}}_2$ from the optical absorption data measured at room temperature. The calculated relativistic $E\text{−}k$ maps (E is energy loss and $k$ is momentum transfer) for $m\text{−}{\mathrm{HfO}}_2$ slabs with thicknesses of 20 nm (b), 50 nm (c), and 100 nm (d) at accelerating voltage of 200 kV. (e) The calculated relativistic loss probabilities of $m\text{−}{\mathrm{HfO}}_2$ slab per unit electron path length integrated over $k$ in the range of $0–00.3{\AA }^{-1}$ for the thickness of 20, 30, 40, 50, 70, and 100 nm.

Figure 4

(a) Zero-loss EFTEM image. (b)–(d) The EFTEM-SI intensity maps acquired at energy losses of $8\pm 1\mathrm{eV}$ (b), $13\pm 1\mathrm{eV}$ (c), and $16\pm 1\mathrm{eV}$ (d). The color scale bar represents the linearly normalized image intensity. (e) SAED pattern of $m\text{−}{\mathrm{HfO}}_2$ recorded at the zone axis of [001] complementary to the HRTEM image in (a). The color circles indicate the different collection β angles used for recording the EEL spectra, and corresponding EEL spectra are shown by the same colors in (f). The extracted energy-loss spectrum from the EFTEM-SI datasets was plotted with a yellow-green dotted line for comparison. The intensities of the spectra were normalized to zero-loss peak and then vertical shifted to easily visualize.

Figure 5

(a) SAED pattern of $m\text{−}{\mathrm{HfO}}_2$ recorded at the zone axis of [010]. (b) The $\omega \text{−}q$ map recorded along the [100] direction to the BZ boundary ($\sim 0.6{\AA }^{-1}$). Momentum ($q$)-resolved EEL spectra collected with the $q$ parallel to the [100] [top panel in (c)], [010] [middle panel in (c)], and [001] [bottom panel in (c)] directions. The spectra are all normalized by the intensity of the peak at $\sim 15\mathrm{eV}$ and then displaced vertically to improve readability. (d) The energy of the VP at $\sim 16\mathrm{eV}$ energy loss and the SEP at $\sim 28.5\mathrm{eV}$ energy loss along the [100] (black square), [010] (red circle), and [001] (blue triangle) directions as a function of $q$.

Figure 6

(a) The energy-loss function, (b) the imaginary part of the dielectric function, and (c) the real part of the dielectric function for $m\text{−}{\mathrm{HfO}}_2$ calculated along [100] (black curve and gray dashed line for calculated without and with LFE, respectively), [010] (red curve and pink dashed line for calculated without and with LFE, respectively), and [001] (green curve and dark yellow dashed line for calculated without and with LFE, respectively) directions. Experimental EEL spectrum [inset, the right panel in (a)], corresponding to the imaginary part and real part of the dielectric function, and calculated energy-loss function plotted with a blue dashed line for comparison. The inset, right panel in (b), is the enlarged real (black) and imaginary (red) parts of the dielectric permittivity of $m\text{−}{\mathrm{HfO}}_2$ from Fig. 2.

Figure 7

The $\omega \text{−}q$ map for the energy-loss range $30\text{−}55\mathrm{eV}$ energy loss acquired along the [100] direction to the BZ boundary ($\sim 0.6{\AA }^{-1}$). (b) The EEL spectra extracted from the map (a) for different $q$ values. (c) The energy of the $\mathrm{Hf}{O}_{2,3}$ edge at $\sim 35\mathrm{eV}$ (red circle) and $\sim 39\mathrm{eV}$ (blue triangle), and the $5p$ semicore plasmon at $\sim 48\mathrm{eV}$ (black square) along the [100] direction as a function of $q$.