Unraveling many-body effects in ZnO: Combined study using momentum-resolved electron energy-loss spectroscopy and first-principles calculations

We present a detailed study of the dielectric response of ZnO using a combination of low-loss momentum-resolved electron energy-loss spectroscopy (EELS) and first-principles calculations at several levels of theory, from the independent particle and the random phase approximation with different variants of density functional theory (DFT), including hybrid and $\mathrm{DFT}+U$ schemes; to the Bethe-Salpeter equation (BSE). We use a method based on the $f$-sum rule to obtain the momentum-resolved experimental loss function and absorption spectra from EELS measurements. We characterize the main features in the direct and inverse dielectric functions of ZnO and their dispersion, associating them to single-particle features in the electronic band structure, while highlighting the important role of many-body effects such as plasmons and excitons. We discuss different signatures of the high anisotropy in the response function of ZnO, including the symmetry of the excitonic wave functions.

Figure 1

Band structure (a) and projected density of state (PDOS) [(b)–(d)] of ZnO obtained with PBE, PBE0 and $\mathrm{PBE}+U$. Scissors corrections of 2.51 and 0.02 eV have been applied respectively to PBE and PBE0 results in order to match the $\mathrm{PBE}+U$ gap of 3.38 eV. The momentum $\mathbf{k}$ is given in grid units, i.e., the length of the $\mathrm{\Gamma }M, MK, KH, HA, A\mathrm{\Gamma }$, and $\mathrm{\Gamma }K$ high-symmetry lines in the band structure is proportional to the number of sampling points along them, which is 19 for $\mathrm{\Gamma }M$, 7 for $MK$, and 13 for the rest; with the $36\times 36\times 24\mathbf{k}$ grid. Three horizontal lines are drawn in each of the PDOS plots, the bottom of the conduction band determining the gap (dotted), the Fermi level at null energy (solid), and the energy from which the DOS become predominantly of $d$ character (dashed); $-2.96\mathrm{eV}$ for PBE, $-4.45\mathrm{eV}$ for PBE0, and $-5.66\mathrm{eV}$ for $\mathrm{PBE}+U$.

Figure 2

Loss function of ZnO in the $\mathbf{q}\to 0$ limit along the $\mathrm{\Gamma }M$ line computed at the IPA level with PBE, PBE0 and $\mathrm{PBE}+U$ and the RPA level on top of $\mathrm{PBE}+U$. The normalized experimental measurements are also plotted. The same scissors corrections from Fig. 1 are used. Eight features of the loss function are indicated in the plot.

Figure 3

Color-map representation of the energy and momentum-dependent inverse dielectric function, $\mathrm{Im}[-{\varepsilon }^{-1}(\mathbf{q},\omega )]$, along the $\mathrm{\Gamma }M$ line. Comparison between several flavors of theory: IPA@PBE (a), IPA@PBE0 (b), IPA@$\mathrm{PBE}+U$ (c) and RPA@$\mathrm{PBE}+U$ (d), and EELS experiments (e). The dispersion of the features F1–4 defined in Fig. 2 is appreciable, as indicated with labels in the experimental panel (e). The edge of the onsets is drawn with approximate lines to guide the eye, while a fit is provided in Table 1. The dotted gray line at $q_{\mathrm{\Gamma }M}=0.18{\AA }^{-1}$ in (e) indicates the change of experimental data set (see description in Sec. 3a).

Figure 4

Similar color maps to Fig. 3 but for the energy and momentum-dependent dielectric function, $\mathrm{Im}\left[\varepsilon \right(\mathbf{q},\omega \left)\right]$. In the case of the experiment we plot the results of the Kramers-Kronig analysis on the experimental loss function (see Appendix pp2). The same borderlines from Fig. 3 are plotted here to show the similarity of the dispersion of the onset. In (e), we also plot the same dotted gray line at $q_{\mathrm{\Gamma }M}=0.18{\AA }^{-1}$. In addition, we include labels indicating the position of the F1–4 features defined for the loss function in Fig. 2.

Figure 5

Color-map representation of the energy and momentum-dependent loss function, $\mathrm{Im}[-{\varepsilon }^{-1}(\mathbf{q},\omega )]$, (left) and dielectric function, $\mathrm{Im}\left[\varepsilon \right(\mathbf{q},\omega \left)\right]$, (right) along several high-symmetry lines in the Brillouin zone, computed at the RPA@$\mathrm{PBE}+U$ level. Similar borderlines to Fig. 3 are plotted, including the experimental edge when available. The same labels indicating the position of the experimental F1–4 features are shown.

Figure 6

Color-map representation of the momentum-dependent static dielectric constant, $\mathrm{Re}\left[\varepsilon \right(\mathbf{q},\omega =0\left)\right]$ of ZnO, computed at the IPA@$\mathrm{PBE}+U$ level of theory along in-plane and out-of-plane projections of the Brillouin zone. Contour isolines (gray shades) show anisotropy between the in-plane and out-of-plane directions at finite momentum. The dashed blue circumference, which is tangent to the second smaller isoline in the $M\mathrm{\Gamma }K$ lines, represents an ideal isotropic contour.

Figure 7

Color-map representation of the energy and momentum-dependent loss function, $\mathrm{Im}[-{\varepsilon }^{-1}(\omega ,\mathbf{q})]$ [(a) and (b)], and dielectric function, $\mathrm{Im}\left[\varepsilon \right]$ [(c) and (d)], computed at the BSE@$\mathrm{PBE}+U$ level and compared to the experiment along the $M\mathrm{\Gamma }A$ lines. Two points beside the high-symmetry ones are marked in the experimental panels, $m$ and $a$, which correspond to $q_{\mathrm{\Gamma }M}=0.18{\AA }^{-1}$ and $q_{\mathrm{\Gamma }A}=0.30{\AA }^{-1}$, respectively, the dotted lines at these points separate different experimental data sets (see description in Sec. 3a). Similar borderlines from Fig. 3 are plotted, and the same labels indicating the position of the experimental F1–4 features.

Figure 8

Comparison of the experimental and theoretical absorption spectra, $\mathrm{Im}\left[\varepsilon \right(\mathbf{q},\omega \left)\right]$, in the $\mathrm{\Gamma }M$ [(a) and (b)] and $\mathrm{\Gamma }A$ [(c) and (d)] lines in the $\mathbf{q}\to 0$ limit [(a) and (c)] and for finite momentum [(b) and (d)], $q_{\mathrm{\Gamma }M}\approx 0.4{\AA }^{-1}$ and $q_{\mathrm{\Gamma }A}\approx 0.3{\AA }^{-1}$. Black circles correspond to the reference experimental data extracted from Tables D2-1 and D2-2 of Ref. [81], solid red lines correspond to the dielectric function obtained by Kramers-Kronig analysis on the EELS data (see Appendix pp2) and dashed orange and dashed-dotted blue lines correspond respectively to BSE and RPA calculations on top of $\mathrm{DFT}+U$.

Figure 9

Symmetries of the excitonic wave functions manifested by the A, B, and C excitons of ZnO (see Table 2). The contours (yellow) correspond to the real-space probability distribution of finding a hole when an electron is placed at the dot in the center (blue) and vice versa. View along the direction perpendicular to the hexagonal plane of the crystalline structure of Zn (gray) and O (red) atoms.

Figure 10

Partial sum rule relative to its value for $q_{\mathrm{\Gamma }M}=0$ at a maximum frequency of 45 eV, corresponding to RPA@$\mathrm{PBE}+U$ results. (a) frequency dependence for four values of momentum, $q_{\mathrm{\Gamma }M}$. (b) Momentum dependence in the $\mathrm{\Gamma }M$ interval at the maximum frequency, fitted to a parabola with the form $S\left(q_{\mathrm{\Gamma }M}\right)/S\left(q_{\mathrm{\Gamma }M}=0\right)=1-a{q}_{\mathrm{\Gamma }M}^2$, where $a=0.13{\AA }^2$.