A. Atomic geometries

In this work we adopt experimental atomic geometries for NaI and SrI₂ as described in Ref. Reference Erhart, Schleife, Sadigh and Åberg29 and for LaBr₃ as described in Ref. Reference Åberg, Sadigh and Erhart28. NaI crystallizes in the rocksalt structure in equilibrium, SrI₂ belongs to the Pbca space group ^{Reference Erhart, Schleife, Sadigh and Åberg29} (no. 61 of the International Tables of Crystallography ^{Reference Hahn54} ), and LaBr₃ adopts a hexagonal lattice (P6₃/m, no. 176 in Ref. Reference Hahn54).

For orthorhombic BaI₂ (space group Pnma, no. 62 in Ref. Reference Hahn54), we fully relaxed the atomic positions until all Hellman–Feynman force components on any atoms were less than 0.02 eV/Å. This leads to a = 9.01 Å, b = 5.43 Å, and c = 10.87 Å. The Ba atom and both iodine atoms occupy the 4c Wyckoff site with x = 0.244, z = 0.112 (for Ba), x = 0.142, z = 0.425 (for the first I), and x = 0.020, z = 0.836 (for the second I). All unit cells for the materials studied in this work are visualized in Fig. 1.

FIG. 1. Unit cells of the materials studied in this work: (a) NaI, (b) LaBr₃, (c) BaI₂, and (d) SrI₂.

B. Single-particle electronic structure

DFT does not provide an accurate description of the electronic structure, since it neglects quasiparticle effects, which manifests itself for instance in the infamous band-gap underestimation. ^{Reference Onida, Reining and Rubio57} In the present work, we need to include quasiparticle effects to accurately describe both band gaps and electronic structure as the foundation of a reliable description of optical properties. Here, we use a scissor operator Δ that shifts all conduction bands to higher energies. The shift Δ is determined using band gaps available in the literature for NaI, LaBr₃, and SrI₂ (see Table I). For BaI₂ we carry out generalized-gradient approximation (GGA) + G ₀ W ₀ calculations of quasiparticle energies using a Γ-centered 3 × 6 × 3 Monkhorst–Pack ^{Reference Monkhorst and Pack58} (MP) k-point grid and 1024 bands (out of which 54 are occupied) to achieve convergence of the screened interaction. ^{Reference Shishkin and Kresse59} This yields a band gap of 4.98 eV for BaI₂ (see Table I).

TABLE I. Kohn–Sham results and literature values for the fundamental band gaps (in eV) are given as well as the scissor value (in eV) used in this work. In addition, DFT results and literature values are given for the static electronic dielectric constants.

In addition, local or semilocal approximations to exchange and correlation suffer from self-interaction errors. This error particularly affects localized orbitals such as d and f states. Hence, we compute the electronic structure of LaBr₃ within the rotationally invariant GGA + U approach, ^{Reference Liechtenstein, Anisimov and Zaanen60} to correct for the La 4f states that incorrectly appear within the band gap in DFT-PBE calculations. ^{Reference Åberg, Sadigh and Erhart28} Here we use U = 11.0 eV and J = 0.68 eV, acting on these La 4f electrons. ^{Reference Czyżyk and Sawatzky61}

C. Two-particle excitations: excitons

To achieve an accurate description of optical properties (including excitonic and local-field effects) from first principles, we solve the BSE for the optical polarization function. ^{Reference Onida, Reining and Rubio57} This allows us to include two-particle (electron–hole) excitations in the description of the dielectric function. The underlying DFT Kohn–Sham states are used to compute the optical transition matrix elements in the longitudinal approximation ^{Reference Gajdoš, Hummer, Kresse, Furthmüller and Bechstedt49} and the statically screened Coulomb attraction as well as the unscreened exchange terms that determine the excitonic Hamiltonian. ^{Reference Rödl, Fuchs, Furthmüller and Bechstedt52,Reference Fuchs, Rödl, Schleife and Bechstedt53,Reference Onida, Reining and Rubio57} After constructing this Hamiltonian, the dielectric function is computed from it using a time-propagation technique. ^{Reference Schmidt, Glutsch, Hahn and Bechstedt62} The model dielectric function of Bechstedt et al. ^{Reference Bechstedt, Del Sole, Cappellini and Reining63} is used to compute the screened Coulomb interaction W in the excitonic Hamiltonian, using the static electronic dielectric constants obtained on DFT level (see Table I).

Such calculations of converged results for optical properties across a large energy range are numerically very challenging, since different competing requirements have to be fulfilled: the sampling of the Brillouin zone needs to be fine enough to converge the onset of the optical absorption spectrum, but at the same time a large number of conduction bands is required to compute optical properties for high-energy optical transitions. Due to the large computational cost of the BSE approach, we employ the BSE cutoff energies (maximum noninteracting electron–hole pair energy taken into account) and MP k-point grids as listed in Ref. 64. Convergence is improved by displacing each grid by a small random vector to lift degeneracies that occur for unshifted MP meshes.

For computing the electron-energy loss function, we also need to converge the real part of the dielectric function (i.e., related to the imaginary part via a Kramers–Kronig relation) at low photon energies. This is achieved by including optical transitions with transition energies between the BSE cutoff and 200 eV on the level of DFT (as described in Refs. Reference Schleife, Rödl, Fuchs, Furthmüller and Bechstedt35 and Reference Schleife65).

A. Dielectric functions

Using the computational framework described in Sec. II, we computed the optical properties (dielectric functions) of ideal crystals of NaI (see Fig. 2), LaBr₃ (see Fig. 3), BaI₂ (see Fig. 4), and SrI₂ (see Fig. 5). Due to the absence of experimentally determined (for instance via ellipsometry) dielectric functions, these results represent highly accurate predictions. In these figures, we compare the results within the independent-quasiparticle approximation, i.e., where quasiparticle effects on the Kohn–Sham eigenvalues are taken into account using the DFT + Δ scheme, to the solution of the BSE, that, in addition, includes excitonic and local-field effects. This allows us to draw conclusions about the influence of excitonic effects in the following.

FIG. 2. Imaginary part of the dielectric function ε(ω) computed using the DFT + Δ approximation (top) as well as the BSE approach (bottom) for NaI.

FIG. 3. Imaginary part of the dielectric function ε(ω) computed using the DFT + Δ approximation (top) as well as the BSE approach (bottom) for LaBr₃. The optical anisotropy is shown for both.

FIG. 4. Imaginary part of the dielectric function ε(ω) computed using the DFT + Δ approximation (top) as well as the BSE approach (bottom) for BaI₂. The optical anisotropy is shown for both.

FIG. 5. Imaginary part of the dielectric function ε(ω) computed using the DFT + Δ approximation (top) as well as the BSE approach (bottom) for SrI₂. The optical anisotropy is shown for both.

Excitonic effects are particularly dramatic for NaI (see Fig. 2), where the onset of optical absorption is dominated by a bound-excitonic state that occurs as a pronounced peak in the spectrum. It can be seen that this is an additional peak at about 5.5 eV in the lower panel of Fig. 2, that is absent in the upper panel. We visualize part of the two-particle electron–hole wave function of this bound-exciton state in Fig. 6 by fixing the position of the hole at an iodine atom and plotting the resulting electron distribution. The spherical shape confirms its similarity to a 1 s state within a hydrogen-like Wannier–Mott-exciton picture. In Fig. 6, it can be clearly seen that this electron density is centered around the iodine atom at which the hole is fixed. We previously studied this bound-excitonic state in more detail and determined the exciton-binding energy to be 216 meV and the Bohr radius to be 9 Å. ^{Reference Erhart, Schleife, Sadigh and Åberg29}

FIG. 6. The two-particle electron–hole wave function is visualized for the lowest bound-exciton state in NaI. The position of the hole is fixed on a I atom in the center of the spherical structure that represents the electronic part.

Furthermore, comparing the upper and lower panels in Fig. 2 shows that peaks that appear in the DFT + Δ spectrum at higher energies (labeled A, B, C in the figure) are shifted to lower photon energies and show slightly redistributed spectral weight when excitonic effects are taken into account. This red shift as well as the spectral redistribution has been traced back to excitonic effects for other materials (e.g., GaN, MgO, or ZnO) that show a similar structure of the uppermost valence and the lowest conduction band. ^{Reference Schleife, Rödl, Fuchs, Furthmüller and Bechstedt35,Reference Benedict and Shirley66} In our calculations, we found that the peaks A, B, and C are attributed to transitions from the I 5p electrons into the conduction bands. The largest contribution to A is transitions into empty Na s states and to B and C transitions into empty I d states. Empty I s states contribute significantly only to A, while B and C also include important contributions from transitions into empty Na and I p states.

The spectra of the other three materials show either much weaker additional peaks/spectral features attributed to bound-excitonic states (see BaI₂ and SrI₂ in Figs. 4 and 5) or none is visible at all (see LaBr₃ in Fig. 3). Partly, this can be attributed to the larger dielectric screening in these materials (see Table I): NaI has the lowest electronic dielectric constant of all four cases investigated here. Consequently the electron–hole interaction is strongest in this material, which is confirmed by the strong excitonic effects observed in the dielectric function. In addition, also the band structures of the materials differ: especially in the case of LaBr₃, we attribute the lack of a bound-exciton peak at the absorption onset to the more complicated band structure (see Ref. Reference Åberg, Sadigh and Erhart28) that does not have a pronounced valence-band maximum and conduction-band minimum but rather dispersionless bands. The spectral red shift of higher-energy peaks due to excitonic effects is clearly visible for all materials.

In Figs. 3–5, it can be seen that the spectra of LaBr₃, BaI₂, and SrI₂ feature a fairly broad (more than 2 eV of full width at half maximum) single peak structure starting right above the respective absorption onsets. This clearly distinguishes the spectra of these three materials from the one of NaI (see Fig. 2). Using our data, we are able to trace this feature back to transitions into empty d states that do not occur in NaI but in the other three materials. For LaBr₃ this feature arises mostly due to transitions from Br 4p electrons into empty La d states and for BaI₂ it is mostly transitions from I 5p into empty Ba d states. In the case of SrI₂, it is transitions from I 5p electrons into empty Sr d states that cause the appearance of this spectral feature.

The four different scintillator materials also differ regarding their optical anisotropy: cubic NaI is optically isotropic due to its lattice symmetry. However, also the spectra of the noncubic materials BaI₂ and SrI₂ show almost no dependence on the light polarization (see Figs. 4 and 5). This agrees well with the dielectric functions computed by Singh using the random phase approximation. ^{Reference Singh67} In the case of LaBr₃, the overall anisotropy is still not very strong; however, the energy position and the spectral shape of the absorption edge depend on the polarization of the light (see Fig. 3). This is due to the energy splitting of the uppermost Br 4p derived valence bands by about 88 meV.

B. Electron-energy loss function

To understand the energy loss processes of electrons in the scintillator materials, we use the complex dielectric function to compute the electron-energy loss function for zero momentum transfer according to

(1) $$- {\mathop{\rm Im}\nolimits} \,{\varepsilon ^{ - 1}}\left( \omega \right) = {{{\mathop{\rm Im}\nolimits} \,\varepsilon \left( \omega \right)} \over {{{\left( {{\mathop{\rm Re}\nolimits} \,\varepsilon \left( \omega \right)} \right)}^2} + {{\left( {{\mathop{\rm Im}\nolimits} \,\varepsilon \left( \omega \right)} \right)}^2}}}\quad .$$

For all four materials studied in this work, the results are shown in Fig. 7. In this figure, we compare results computed within the independent-particle approximation, i.e., neglecting excitonic effects, to electron-loss functions derived from the BSE data (i.e., taking excitonic effects into account).

FIG. 7. Electron-energy loss function −Im ε⁻¹(ω) computed from the dielectric function including (solid lines) and excluding (dashed lines) excitonic effects, according to Eq. (1) for NaI (a), LaBr₃ (b), BaI₂ (c), and SrI₂ (d). For NaI, a measured curve ^{Reference Creuzburg68} is included for comparison.

Figure 7 clearly illustrates the impact of excitonic effects on −Im ε⁻¹ across the entire energy range studied here. For all four materials, we observe a remarkable red shift when the electron–hole interaction is included, along with significant spectral redistribution. In particular for NaI, which shows the strongest excitonic effects due the weak screening of the electron–hole interaction, the difference between both computational results are large: a peak that occurs at around 16 eV in independent-particle approximation occurs at ≈13.5 eV due to excitonic effects and also its height is significantly reduced. In the case of NaI, we also compare to an experimental result ^{Reference Creuzburg68} and find very good agreement with our theoretical data that takes excitonic effects into account. Even though the y axis scaling of the experiment is arbitrary, it is immediately clear that both absolute peak positions and relative peak heights agree very well with our computational result. The data for all four materials illustrate that both quasiparticle and excitonic effects need to be treated accurately, to achieve reliable predictions of electron-energy loss function that can be used, for instance, in multiscale modeling approaches. ^{Reference Gao, Xie, Wang, Kerisit, Wu, Campbell, Van Ginhoven and Prange22}