Connections between resonant inelastic x-ray scattering and complementary x-ray spectroscopies: Probing excitons at the $\mathrm{Al} K \text{and} L_1$ edges of $\alpha \text{−}{\mathrm{Al}}_2{\mathrm{O}}_3$

We present an ab initio study of neutral core and valence electronic excitations in $\alpha \text{−}{\mathrm{Al}}_2{\mathrm{O}}_3$ by solving the Bethe-Salpeter equation (BSE) of many-body perturbation theory within an all-electron framework. Calculated spectra at the Al K and $L_1$ edges are in remarkable agreement with available experiments from x-ray absorption (XAS) and x-ray Raman spectroscopy once excitonic effects are taken into account. The combination of the BSE spectra for the two techniques confirms the dipole-forbidden nature of the exciton prepeak as suggested by recent calculations based on density-functional theory. Moreover, we make predictions for resonant inelastic x-ray scattering (RIXS) spectra at K and $L_1$ edges, which strikingly fully overlap also beyond an independent-particle picture. The RIXS calculations reveal two distinct regimes as a function of incoming photon energy. Below and at the XAS threshold, we observe Raman-like features, characterized by strong excitonic effects, which we directly compare to peaks in the loss function. Above the XAS threshold, instead, fluorescence features become predominant: RIXS spectra can be well described and analyzed within an independent-particle approximation showing similarity with the x-ray emission spectrum.

Figure 1

Al K edge absorption spectra calculated within the IPA, RPA and BSE for two polarization directions: (top panel) parallel ($xy$) and (bottom panel) perpendicular ($z$) to the $\alpha \text{−}{\mathrm{Al}}_2{\mathrm{O}}_3$ layers. The IPA and RPA spectra are on top of each other, highlighting the negligible contribution of local fields. For both polarizations the calculated spectra are compared to experimental and simulated XAS spectra from Manuel et al. [37], which have been normalized to match the maximum of the BSE intensity. The experiment has been done at T = 300 K and the simulation has been carried out in a DFT core-hole approach (where the coupling with atomic vibrations is excluded).

Figure 2

Loss function calculated within the BSE for momentum transfers $\mathbf{q}$ along the $x$ (top) and $z$ (bottom) Cartesian directions, with a broadening of 0.7 eV. The experimental XRS spectrum from Ref. [38] was measured for momentum transfer $\left|\mathbf{q}\right|=10$ Å${}^{-1}$. Here, it has been scaled to match the maximum intensity of the calculated spectrum at $\left|\mathbf{q}\right|=10$ Å${}^{-1}$. We note that the peak positions of the XRS spectrum [38] have a redshift of $\sim 0.7$ eV with respect to the XAS spectra in Fig. 1.

Figure 3

Left: XAS spectra calculated in the BSE for the Al K and $L_1$ edges (orange and blue curves, respectively) for polarization along the $xy$ direction. Note the different energy scales for the two edges. The spectrum for the K edge has been multiplied by 35. Both spectra have been convoluted with a 0.1 eV Lorentzian broadening. Right: RIXS spectra at the K and $L_1$ edges (orange and blue curves, respectively) as a function of the energy loss $\omega$, evaluated for excitation energies ${\omega }_1$ (separated by 0.2 eV steps) across the corresponding XAS edge in the left panel. The RIXS spectra have been calculated within the BSE for photon polarizations vectors ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ along the $x$ direction. Each RIXS spectrum has been normalized to its maximum. The thin black lines connecting the different RIXS spectra are guides for the eye to identify Raman losses and fluorescence features.

Figure 4

BSE RIXS spectra (blue curves) as a function of the energy loss $\omega$ at the $L_1$ edge for excitation energies ${\omega }_1<126$ eV. They are compared to the BSE loss function $-\text{Im}{\epsilon }_M^{-1}\left(\omega \right)$ (dark yellow curve) for $\mathbf{q}\to 0$ along the $x$ direction. The black dotted lines connect Raman peak in RIXS spectra with corresponding inelastic losses in $-\text{Im}{\epsilon }_M^{-1}\left(\omega \right)$. Each spectrum has been normalized to its maximum and offset for clarity.

Figure 5

Absorption oscillator strengths $|t_{{\lambda }_{\mu }}^{\left(1\right)}|$ (top) and excitation pathways $|t_{{\lambda }_o,{\lambda }_{\mu }}^{\left(2\right)}|$ (center) as a function of core excitation energy $E_{{\lambda }_{\mu }}$ compared to the cumulative RIXS oscillator strengths $|t_{{\lambda }_o}^{\left(3\right)}\left(E\right)|$ (bottom) for different exciton energies $E_{{\lambda }_o}$: 8.45, 8.99, and 9.46 eV, as specified in the legends. In the four cases, the incoming photon energy ${\omega }_1$ is set to 125.6 eV.

Figure 6

Comparison between BSE (blue curves) and IPA (dark yellow curve) RIXS spectra as a function of the emission energy ${\omega }_2$ for different excitation energies ${\omega }_1$ across the XAS Al $L_1$ threshold and above it. The fluorescence features are also compared with the XES spectrum (blue curve at the top) at the same Al edge. The RIXS spectrum has been calculated for ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ along the $x$ direction, and the XES spectrum for $\mathbf{q}\to 0$ in the $x$ direction. In the plot, each spectrum has been normalized to its maximum and offset for clarity.

Figure 7

Comparison between experimental x-ray emission spectrum at Al K edge extracted from Ref. [90] and calculated XES within IPA for the two polarization directions $xy$ and $z$. The calculated XES can be well described by the Al $3p$ component of the PDOS. In all cases, the top of the valence band in the PDOS and the XES have been set to 0 eV.

Figure 8

Cumulative function $|t_{{\lambda }_o}^{\left(3\right)}\left(E\right)|$ for different incoming photon energies within the Raman regime. The two panels represent different optical exciton energies $E_{{\lambda }_o}$, associated to RIXS peaks at energy loss of (a) 8.99 eV and (b) 9.46 eV, as seen in Fig. 4.

Figure 9

Absolute value of the absorption oscillator strength $|t_{{\lambda }_{\mu }}^{\left(1\right)}|$ and the excitation pathways $|t_{{\lambda }_o,{\lambda }_{\mu }}^{\left(2\right)}|$ as a function of the core excitation energy $E_{{\lambda }_{\mu }}$ for valence excitations $E_{{\lambda }_o}=10.4$ eV, 10.8 eV and 11.2 eV, as specified in the legends, for (left) BSE and (right) IPA RIXS.

Figure 10

Cumulative function $|t_{{\lambda }_o}^{\left(3\right)}\left(E\right)|$ for different incoming photon energies ${\omega }_1$ in the fluorescence regime obtained with the BSE for optical excitation energies $E_{{\lambda }_o}$ of (a) 10.42 eV and (b) 10.80 eV. Similarly, in the right panels the $|t_{{\lambda }_o}^{\left(3\right)}\left(E\right)|$ has been plotted, calculated within IPA for $E_{{\lambda }_o}$ equal to (c) 10.39 eV and (d) 10.81 eV.