arXiv:1311.3502v1 [cond-mat.mtrl-sci] 14 Nov 2013 clmcaim noapeitv oe.Oeo h ma- the of One model. predictive a framework into phys- aforementioned mechanisms numerical the ical several combined established successfully of no has spite that currently situation, In this is resolve unclear. there to still attempts of is promising importance events recent relative understood these the level, of are qualitative each emission a on photon least final at to creation track s s oee,lmtdadde o etterequire- the meet identification. not isotope does radioactive and for and ments limited NaI however, that as is, resolution such scintillators energy CsI classical The using achievable source. identifica- is radiation allows the principle in of resolu- this tion energy high spec- the sufficiently energy Provided is the signal. tion deduce incoming radi- the to incident of possible an trum is of it energy during quantum, the with ation generated interval photons time photons. of short energy a number 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arneLivermor Lawrence Directorate, Sciences Life and Physical fsdu oieadsrnimidd rmmn-oypertur many-body from and iodide sodium of o w motn ytm,NIadSrI and NaI properties systems, excitonic and important spectra two absorption for as well as tion f26mVfrNIad195 and NaI binding for obtain meV 216 we Furthermore, of experiment. with agreement otoa epne epeitbn asfrNIadSrI and NaI for gaps band predict We response. has latter portional the while non-proportionality well-documented spectr quasiparticle of study principles first comparative iltr oicdn aito epnil o eue re reduced for responsible radiation incident to tillators xet ftehgl nstoi o-nryectn nSr in excitons low-energy we Furthermore, anisotropic highly similar. the are of energies extents binding their if even hw htteectn nNIdffrsrnl rmtoei Sr in those from p strongly electron-hole differ NaI coarse-grained in a excitons the of that shown means by extent spatial w otrei em ftefl it thl aiao h ele 78.70.Ps the 71.15.Qe of 71.20.-b maxima 71.35.-y half at numbers: width PACS full the of terms in three to two us-atceseta bopinseta n excitoni and spectra, absorption spectra, Quasi-particle eivsiaetebscqatmmcaia rcse behi processes mechanical quantum basic the investigate We 1 .INTRODUCTION I. hlesUiest fTcnlg,Dprmn fApidP Applied of Department Technology, of University Chalmers 1,2 alErhart, Paul 4 5–9 o hs plctos n fre- one applications, these For uigtels eae thas it decades last the During ,2 1, ± Andr´e Schleife, 5mVfrSrI for meV 25 3 2 10–14 h omri tnadsitlao aeilwith material scintillator standard a is former The . 2 2 eaayetedge fectnaiorp and anisotropy exciton of degree the analyze We . aa Sadigh, Babak ainlLbrtr,Lvroe aiona950 USA 94550, California Livermore, Laboratory, National e e.1)adSrI and 17) Ref. ntepoete fSrI of properties both the on extensively rather characterized experimentally LaBr been While already NaI. to im- has compared significantly resolution and energy luminosity proved high very exhibit which adgpaesilt edtrie ossetyadthe and consistently determined be to still are gap band sparse. is ture, n o cniltrapiain r LaBr are applications scintillator for methods ing significant the a because with part come in effects burden. these least computational access at to is excitonic interaction required specifically This the excitations, to and effects. relate carriers scin- charge that of of processes both number of and understanding and a quantities knowledge our for in mobili- gap studied nounced such carrier been properties materials. charge recently tillating electronic and has of gaps experimen- ties impact band guide models. fundamental The and such as complement for efforts. to data tal serve input also but They insight not physical providing electronic- for only valuable very parameter-free are difficult calculations experimentally, notoriously structure are measure quantities Since these to states. of excitonic some and least carriers recom- at free Auger of hole as well rates and as bination electron rates, trapping excitons, defect migra- self-trapped mobilities, function, of dielectric barriers as tion such parameters, physical ela hi epciecnrbtost h epneof response the to contributions as system. the respective excitons) their and holes, as excita- well electron, of (primarily transport carriers uncertainty and tion and thermalization complexity with enormous connected the is reasons jor w aeil htaekont evr promis- very be to known are that materials Two xsigmodels Existing I ouin o hsproe ecnuta conduct we purpose, this For solution. eetybe on oehbtavr pro- very a exhibit to found been recently 2 ntebsso the of basis the on a 2 hwta ut nxetdytespatial the unexpectedly quite that show nfc xedtoei a yafco of factor a by NaI in those exceed fact in ysligteBteSlee equation Bethe-Salpeter the solving by i-orlto ucin hrb,i is it Thereby, function. air-correlation f55ad52e,rsetvl,i good in respectively, eV, 5.2 and 5.5 of I 2 nrisfrtegonsaeexcitons groundstate the for energies dnnpootoa epneo scin- of response non-proportional nd to-oepi-orlto function. pair-correlation ctron-hole ntrso tutr n symmetry, and structure of terms in 2 n Daniel and 31 yis ohnug Sweden Gothenburg, hysics, 4,22–24 vnfnaetlqatte uha the as such quantities fundamental Even 2 E seeg,Rf.1–1,bt of both 18–21), Refs. e.g., (see :Eu 15,16 10–14 n theoretically, and 2 npriua t lcrncstruc- electronic its particular in , hr ean,hwvr pro- a however, remains, There yial eyo ubrof number a on rely typically Aberg ˚ G properties c 0 W aintheory bation 0 2 approxima- 24–30 3 C see.g., (see :Ce information 3 2 influence of excitonic and polaronic effects is entirely un- tional Tables of Crystallography, Ref. 35). Its unit cell known at this point. Yet, a deeper understanding of this contains 24 atoms with nine internal degrees of free- material is crucial since properties such as the band gap, dom. The experimentally determined lattice parame- exciton binding energies and dielectric functions consti- ters are a = 15.22 A,˚ b = 8.22 A,˚ and c = 7.90 A˚ with tute essential building blocks for any theoretical study of the following internal coordinates: Sr on Wyckoff site 8c e.g., free carrier transport, polaron formation and migra- (x = 0.1105, y = 0.4505, z = 0.2764), I(1) on Wyckoff tion as well as Auger recombination. In addition, they site 8c (x =0.2020, y =0.1077, z =0.1630), and I(2) on represent important parameters for the interpretation of Wyckoff site 8c (x = −0.0341, y =0.2682, z =0.0054).36 experimental data. All calculations were carried out using the projector- The objective of the present work is to alleviate this augmented wave method to describe the electron- situation by providing predictions for electronic and op- interaction.37,38 We used a plane-wave expansion for tical properties on the basis of modern theoretical spec- the wave functions with a cutoff energy of 228 eV for troscopy techniques. We specifically aim at accurately both materials. DFT and G0W0 electronic structures describing band structures and densities of states (includ- were generated using the Vienna Ab-initio Simulation ing fundamental band gaps), optical absorption spectra, Package.39–43 The corresponding BSE implementation exciton binding energies and exciton localization. In or- has been discussed in Refs. 44–46. der to assess the reliability of our parameter-free simu- For NaI and SrI2 we use the generalized-gradient lations and to put our results in context, we also carry approximation47 and the local-density approximation,48 out calculations for the classic scintillator NaI, for which respectively, to represent exchange-correlation effects at extensive experimental data are available. the DFT level. Brillouin-zone (BZ) integrations for The remainder of this paper is organized as follows. both DFT and G0W0 calculations were carried out by In Sect. II A we summarize the computational approach summing over Γ-centered 6 × 6 × 6 Monkhorst-Pack49 employed in this work. A convenient decomposition of (MP) grids in the case of NaI. For SrI2 the density of the six-dimensional exciton wave function into single- states (DOS) was computed on the DFT level using a particle densities and an envelope function is introduced 6 × 11 × 11 Γ-centered MP k-point grid while the band in Sect. II B. Results for band structure, dielectric prop- structure was obtained on the basis of a 4 × 7 × 7 mesh. erties and absorption spectra are presented in Sects. IIIA Calculations of G0W0 QP energies were carried out for and III B. Exciton binding energies as well as an analysis several k-point grids up to Γ-centered 3 × 4 × 4 and of their spatial extent are the subject of Sects. IIIC and up to 2880 bands were included in the calculations to III D. Finally, we summarize the results and discuss them achieve convergence of the dielectric function entering the in the context of the scintillation properties of NaI and screened interaction W . Spin-orbit coupling (SOC) was SrI2 in Sect. IV. taken into account using the projector-augmented wave implementation described in Ref. 30. Based on conver- gence tests we estimate that these computational param- II. THEORETICAL APPROACH eters yield QP shifts around the band edges that are con- verged to within 50 meV for both materials. Since this work aims at a precise description of elec- In order to describe optical properties, we calculated tronic structure as well as optical properties it requires dielectric functions using regular MP meshes of 16 × 16 × techniques beyond standard density functional theory 16 and 4 × 6 × 6 k-points for NaI and SrI2, respectively. (DFT). Specifically, it is essential to employ computa- For a more efficient sampling of the BZ, each grid was tional schemes that are capable of describing quasipar- displaced by a small random vector. For NaI and SrI2 ticle (QP) and excitonic effects. To this end, we use we used 32 and 48 conduction bands, respectively. In ad- 32 a combination of DFT and G0W0 calculations to de- dition, the number of Kohn-Sham states contributing to scribe single-particle excitations that govern band struc- the BSE Hamiltonian is limited by the BSE cutoff energy tures and densities of states. Two-particle (electron-hole) which specifies the maximum non-interacting electron- excitations have to be taken into account when comput- hole pair energy that is taken into account. Here we ing exciton binding energies, dielectric functions, and op- used a BSE cutoff energy of at least 13.0 and 5.0eV for tical absorption spectra. This is achieved by solving the NaI and SrI2 to set up the excitonic Hamiltonian from Bethe-Salpeter equation (BSE) for the optical polariza- independent electron-hole pairs. The screened Coulomb tion function.33,34 interaction W was constructed assuming the q-diagonal model function of Bechstedt et al.50 and static electronic dielectric constants of 3.69 and 4.58 as obtained for NaI A. Computational parameters and SrI2 on the DFT level. The Coulomb singularity present in the BSE Hamilto- In this work we use experimental values for the crys- nian effectively prevents accurate one-shot calculations tallographic geometries. adopts the rock- of exciton binding energies for a given k-point mesh.45,51 salt structure with a lattice constant of 6.48 A.˚ SrI2 be- This is somewhat alleviated by the introduction of so- longs to space-group Pbca (number 61 in the Interna- called singularity corrections. Still present implemen- 3 tations are left with error terms that are proportional following we will derive explicit expressions for the ex- to the inverse number of k-points. Therefore, the most citonic single-particle densities and an approximate cor- efficient scheme currently consists of extrapolating cal- relation function that turns out to be solely a function culated binding energies for a number of k-point grids of the electron-hole separation. Thereby, we will obtain to the continuum limit.45 We sample the BZ using both a partitioning of the pair distribution function into two regular and hybrid k-point meshes as defined in Ref. 45. single-particle cell-periodic densities, representing the lo- For NaI we employed regular Γ-centered grids up to cal variations of electron and hole densities, as well as 21 × 21 × 21 as well as 113 : 63 : x3 hybrid grids an associated pair distribution function that is coarse- with x={22, 25 2/3, 29 1/3, 33}. For SrI2 the finest regular grained over each periodic cell. meshes we used were 7×9×9, 5×11×11, and 3×13×13 The exciton wave function can be represented in terms and, in addition, we employed hybrid meshes of 5×5×5 : of electron and hole coordinates via the amplitude58,59 4 × 2 × 2 : x with x={5 × 15 × 15, 5 × 20 × 20, 5 × 25 × 25} † for finer sampling around the Γ-point in the ky and kz χj (re, rh)= N;0 ψ (re)ψ(rh) N; j , (3) directions. b b where N; j is the j-th excited state of the N-electron

system, and ψ(r) and ψ†(r) are the standard annihila- B. Electron-hole densities tion and creationb operatorsb acting on coordinate r, re- spectively. Translated into the basis of Bloch orbitals The electron-hole separation of a free exciton is read- this becomes ily obtained within the effective mass approximation for j ∗ χj (re, rh)= X Ak φkv(rh)φkc(re), (4) a two-band model.52 In cases where this approximation is cv kcv not applicable, the BSE wave functions must be analyzed instead. To this end, Dvorak et al. have computed the rel- j where the coefficients Akcv are the eigenvectors of the ative distribution of the electron around the hole as well BSE matrix. Therefore we can write the associated as spread and localization lengths by a straightforward electron-hole density as integration of the full excitonic wave function.53 The BSE wave functions and electron hole distribution function j j∗ j ρeh(re, rh)= X X AkcvAk′c′v′ are, however, six-dimensional functions of the electron kcv k′c′v′ and hole coordinates which renders the problem, at least ∗ ∗ × φk (r )φ (r )φ ′ ′ (r )φk′ ′ (r ). (5) computationally, more involved. Furthermore, when dis- v h kc e k v h c e cussing the spatial distribution of electrons or holes, a By integrating over either the electron or hole coordi- charge density (or wave function) is usually constructed nate we obtain the cell-periodic hole and electron charge by fixing the electron (hole) at the position of an atom densities, respectively, belonging to the conduction (valence) band edge.54–57 j∗ j This choice is though somewhat arbitrary since the Bloch j ∗ ′ ′ ρe(re)= X X φkc(re)φk c (re) X AkcvAkc′v, (6a) states may be strongly hybridized. To alleviate this prob- k cc′ v lem and to avoid the representation of a density over the j ∗ j∗ j r k r ′ ′ r ′ entire Born-von-K´arm´an cell on a dense Cartesian mesh ρh( h)= X X φ v( h)φk v ( h) X AkcvAkcv . (6b) ′ in the case of calculating the electron-hole separation, we k v v c follow another route. We note that the electron-hole density of the entire N- j The electron-hole pair distribution function ρ(re, rh) j 0 j electron system can be written as ρ = ρ + ρe − ρh, specifies the probability of simultaneously finding an elec- where ρ0 is the ground state charge density. This can be tron at re and a hole at rh. In the single-particle band seen by expanding the N-electron exciton wave function picture we thus can write χj (r1,..., rN ) = r1,..., rN N; j into Slater determi- nants. ρ(re, rh)= ρe(re)ρh(rh). (1) Returning to the question of electron-hole separation we again consider the two-particle density of Eq. (5) and In this approximation both electron and hole can be com- ′ make the variable substitution re/h = re/h +Re/h, where pletely delocalized over the entire crystal. However, since now r′ is constrained to one unit cell and R denotes electron and hole are coupled through the Coulomb inter- e/h e/h a lattice vector. By integrating the resulting two-particle action the distribution function assumes a more complex density over r′ and r′ we obtain a coarse-grained pair form e h distribution function

ρ(re, rh)= ρe(re)ρh(rh)geh(re, rh), (2) j j∗ j geh(Re, Rh)= X X AkcvAk′c′v′ e kcv k′c′v′ where geh represents the explicit pair correlation between ′ ∗ ′ ′ × exp i k − k · (Re − Rh) Ikv,k′v′ Ikc,k c , electron and hole. Thus, given an electron at re the prob-    (7) ability of finding a hole at rh is ρh(rh)geh(re, rh). In the 4

15 0.3 eV upon inclusion of SOC due to a shift of the va- (a) (b) I d lence band maximum (VBM). Otherwise the structure of 10 I p both valence and lower conduction band states is largely I s preserved. Figure 2(c) demonstrates that there are no Na d 5 qualitative changes in going from DFT to G0W0 within Na p a band manifold. The valence band width increases from Na s 2.86 eV to 3.00 eV and the band gap from 3.7 to 5.5 eV. 0 The band characters and their ordering are only weakly Energy (eV) affected and, as a result, a rigid upward shift (a scis- −5 sor correction, referred to as DFT+∆ from here on) of the DFT conduction band yields reasonable agreement −10 between DFT and DFT+G0W0 QP energies. This is Γ X K Γ L 0 1 2 3 4 demonstrated in Fig. 2(d), which shows QP energies from Density of states G0W0 superimposed on a DFT band structure. Due to the prohibitive computational cost, SOC effects FIG. 1. (a) NaI non-relativistic QP energies from G0W0 were not taken into account on the G0W0 level for SrI2. (empty circles) superimposed on DFT+∆ band structure (col- Our calculations for NaI as well as LaBr3 (see Ref. 30), ored lines). The spin-orbit split valence bands are shown by however, show that DFT and GW yield similar results black lines and filled squares. (b) Partial density of states for for SOC induced shifts. We therefore can employ the NaI as obtained from DFT+∆ calculations. band gap reduction calculated on the DFT level [compare Fig. 2(a,b)] to estimate the SOC corrected band gap of SrI2 as 5.2 eV. ′ ′ ∗ ′ ′ where Ikn,k n = R drφkn(r)φk n (r) is an integral over one unit cell only. Also note that geh of Eq. (7) is only a function of the electron-hole separatione R − R and e h B. Dielectric functions and absorption spectra thus resembles an envelope function. As a result, we can approximately write the electron-hole pair distribution function as We now address two-particle excitations and optical properties based on the dielectric function. As discussed j j j j ρeh(re + Rr, rh + Rh)= ρe(re)ρh(rh)geh(Re − Rh). above, the good agreement between DFT+G0W0 and e (8) rigidly shifted DFT band structures (DFT+∆) enables us to use the latter as the starting point for the BSE In this simplified picture, we can therefore write the exci- calculation. In this work we use values of 2.03 eV and ton pair distribution function as an envelope function geh 1.82 eV for the rigid shift ∆ for NaI and SrI2, respec- modulated by the product of two periodic single-particlee tively. j j charge densities, ρe and ρh. We first study the spectroscopic properties of NaI by comparing the imaginary part of the dielectric function computed using DFT+∆ to the one obtained from BSE III. RESULTS calculations. Figure 3 reveals a pronounced peak with large oscillator strength near the absorption onset related A. QP band structures to an excitonic bound state, which is not captured by the single-particle DFT picture approximation. Overall In the case of NaI DFT calculations yield a band gap the BSE result exhibits a pronounced redistribution of of 3.7 eV. This value reduces to 3.4 eV when SOC is peak weights leading to structural changes in the dielec- taken into account, which is considerably less than ex- tric function. One also notices a red shift of the entire perimental values that lie in the range between 5.8 and spectrum in going from DFT+∆ to BSE. These features 60–62 6.3 eV. The G0W0 band gap is 5.8 eV without and are attributed to excitonic effects and are also apparent 5.4 eV with SOC in much better agreement with experi- in the predicted absorption spectrum, which is shown ment, see Fig. 1. The major effect of SOC is to cause a in Fig. 4(b) along with experimental data recorded at splitting of the I 5p band by about 0.9 eV at the Γ-point. 10 K.60 The BSE spectrum is overall in good agreement The partial DOS of SrI2 is shown in Fig. 2 calculated with the experimental data, in particular if compared both without and with SOC taken into account. The to the DFT+∆ result. The most pronounced difference uppermost valence band is composed almost exclusively occurs between 6.5 and 7.5eV where the experimental of I5p states whereas the conduction band is dominated spectrum exhibits two peaks whereas there is only one by Sr 4d states with a contribution from Sr 5s states at distinct feature in the BSE data. the bottom of the conduction band. The largest change On the basis of the similarity between the absorption due to SOC is a splitting of 1.15 eV observed for the spectra of gaseous xenon and the iodide ion Teegarden Sr 4p states that lie between −15.8 and −17.0 eV below and Baldini60 proposed that the two lowest peaks, which the valence band maximum. The band gap decreases by are separated by approximately 1eV, originate from the 5

2 (a) (c) 60 I d

3.7 eV (eV) ×0.5 I p

I s DFT

Sr d ε 1 30 − Sr p CBM QP ∆ε = +0.9 eV Sr 4d Sr s ε 0 0 (b) ∆εVBM= −0.8 eV 60 3.4 eV ×0.5 −1 I 5s 30 I 5p Sr 4p Density of states (states/eV/cell)

0 Quasi−particle shifts −2 −10 −5 0 5 −20 −10 0 10 Energy (eV) DFT eigen values (eV)

3 (d)

(eV) 2 CBM ε 1 − ε 0 0

−1 (eV)

VBM LDA

ε −2 G0W0 (1) − ε G W (2) −3 0 0 R Z Γ X Z Y Γ R X Y

FIG. 2. Partial density of states for SrI2 obtained from DFT calculations (a) without and (b) with spin-orbit coupling. The major change upon the inclusion of SOC effects is an upward shift of the valence band edge by 0.3 eV, which is indicated by the gray bar in panel (b). (c) QP energy shifts from G0W0 calculations with respect to initial DFT eigenenergies. (d) QP energies from G0W0 superimposed onto DFT+∆ band structure. The two different sets of G0W0 QP energies correspond to Γ-centered k-point grids with (1) 2 × 2 × 2 and (2) 3 × 4 × 4 divisions, respectively. spin-orbit split 5p6s atomic state. In fact the sepa- C. Exciton binding energies ration of the two features is comparable to the spin- orbit splitting of 0.9 eV calculated for the I 4p band on As mentioned in Sect. IIA the slow convergence of ex- a DFT/G0W0 level [see Fig. 2(a)]. Unfortunately SOC citon binding energies with k-point sampling requires an is currently not included in our BSE calculations and extrapolation scheme45 the applicability of which is con- thus we cannot directly assess this assignment. It should, tingent upon reaching the linear regime. We illustrate however, be noted that the separation of the second and this approach for NaI in Fig. 5, where it can be seen third peak in the experimental spectrum, which overlap that linear behavior is not quite accomplished for regu- with a single peak in the BSE spectrum, also exhibit a lar meshes containing as many as 21 × 21 × 21 k-points. separation of about 0.9 eV. However, denser sampling was possible by using hybrid k-point meshes45 that are well suited in the case of a ma- terial with approximately parabolic bands. This denser Figure 3 displays the predicted dielectric function sampling justifies a linear fit of the exciton binding en- as well as absorption and transmission spectra at the ergy (see Fig. 5) and extrapolation yields a value for the DFT+∆ and BSE levels. As in the case of NaI the BSE lowest excitonic bound state of 216 meV. This number is spectra show a strong red shift with respect to DFT+∆ in good agreement with experimental measurements at and a redistribution of spectral weight due to excitonic 80 K, which yield a binding energy of 240 meV.64 effects. The absorption spectra are rather featureless but As illustrated in Fig. 2(d), the band structure of SrI2 is clearly the onset of absorption is below the fundamental much more complicated than the one for NaI. The min- QP band gap and a shoulder is apparent around 5.5eV. imum of the lowest conduction band is less pronounced The latter feature is corroborated by experimental emis- and, as a consequence, a large number of very flat con- sion spectra.63 duction bands lie within a small energy range and are 6

(a) NaI BSE 0.25 (a) dipole forbidden ∆ 8 DFT+ ε 5.8 eV 0.20 ±

Im E = 195 25 meV 4 b

0.15 nx = 3 0 k x 20 nk = 5

(b) experiment Binding energy (eV) 0.10 nx = 7 15 k /cm) 5 10 x 0.25 (b) dipole allowed, polarized along (10

α 5

0 0.20 4 6 8 10 12 ± Eb = 180 20 meV Energy (eV) x 0.15 nk = 3 x FIG. 3. (a) Imaginary part of the dielectric function and nk = 5 Binding energy (eV) 0.10 x (b) absorption spectrum for NaI. The vertical arrow indicates nk = 7 the fundamental QP band gap (excluding SOC effects). The experimental absorption data is taken from Ref. 60. z 0.25 (c) dipole allowed, polarized along

(a) SrI 0.20 ± 2 Eb = 190 25 meV 8 x ε 5.5 eV 0.15 nk = 3

Im x 4 BSE nk = 5 Binding energy (eV) DFT+∆ 0.10 nx = 7 SrI2 k 0 0.0 0.1 0.2 0.3 20 (b) y z excitation (430 nm) Linear k−point density 1/nk = 1/nk 15 excitation (570 nm) /cm) 5 10 FIG. 6. Convergence of exciton binding energies in SrI2 from (10

α 5 BSE with respect to k-point sampling for (a) dipole forbidden and (b,c) dipole allowed transitions. Empty and filled sym- 0 4 6 8 bols denote binding energies calculated using standard and Energy (eV) hybrid meshes, respectively. Arrows and gray bars indicate extrapolated binding energies and associated error estimates. FIG. 4. (a) Imaginary part of the dielectric function and (b) absorption spectrum for SrI2. Panel (b) also shows excitation spectra recorded by Pankratov et al. (Ref. 63). The verti- likely to contribute to the lowest excitonic states. This cal arrow indicates the fundamental QP band gap (excluding SOC effects). situation is further exacerbated by the large number of very shallow valence bands with several extrema that are

x energetically close [see Fig. 2(d)] and, hence, are also Number of k−points nk expected to contribute to the formation of the lowest ex- 33 25 19 15 13 11 0.24 citonic states. In addition, unlike NaI, SrI2 is not of regular k−point grid cubic symmetry, hence, the exciton localization is gener- hybrid k−point grid 0.22 ally anisotropic, an aspect that will be explored in more Eb = 216 meV detail below. This situation motivates the application of anisotropic k-point meshes when converging exciton 0.20 binding energies using both regular (empty symbols in Fig. 6) and hybrid grids (filled symbols in Fig. 6). 0.18

Binding energy (eV) Due to the symmetry of the states that are involved, NaI the lowest bound exciton state corresponds to an opti- 0.16 0.00 0.02 0.04 0.06 0.08 0.10 cally (dipole) forbidden transition. In Fig. 6 we show the x k Linear k−point density 1/nk dependence of the binding energy on -point sampling for this state as well as for the first states that are dipole allowed for x-polarized (E || x) and z-polarized (E || z) FIG. 5. Convergence of binding energy of lowest exciton light, respectively. The most highly converged data point state for NaI. in Fig. 6 corresponds to 685 k-points in the full BZ and 7 requires setting up and (iteratively) diagonalizing a BSE terms of the effective mass equation. The hole and elec- matrix with a rank of over 140,000. We extract bind- tron densities of the three states discussed in the previ- ing energies by linear extrapolation45 based on the two ous section are displayed in Fig. 9. In the dipole forbid- most highly converged data points for the data sets cor- den and x-polarized states the electron densities exhibit x x responding to nk = 5 and nk = 7 as indicated in Fig. 6. s-like maxima centered on all atoms with equal ampli- In this fashion we obtain a binding energy of 195 ± tude, whereas the Sr 4d-character dominates in the z- 25 meV for the dipole forbidden state from separate fits polarized case. Note that the exciton single particle den- to regular and hybrid meshes, where the error estimate is sities shown in Fig. 9(a-c) does not simply coincide with based on the deviation between the fits. For the dipole al- the single-particle density of the lowest lying conduction lowed excitons we obtain binding energies of 180±20 meV band state at Γ, which primarily exhibits Sr 5s-character. (x-polarized) and 190±25 meV (z-polarized). These val- The hole densities shown in Fig. 9(d-f) display I p- ues are in rough agreement with the estimate of 260meV character for all three considered states. It is noteworthy obtained by Pankratov et al. based on the effective mass that the dipole forbidden state only has appreciable hole approximation.63 Note, however, that this agreement density around atoms belonging to the I(2) set of should be considered rather fortuitous due to the strong Wyckoff sites (compare Sect. II A) as shown in Fig. 9(d). anisotropy of the hole effective mass tensor. The analy- The same behavior is observed for other dipole forbid- sis of the excitonic wave functions in the following sec- den states. In the dipole allowed excitons we find hole tion will provide further evidence for anisotropic exci- densities of equal amplitude on all iodide atoms. tonic properties. The envelope functions for the SrI2 excitons discussed here differ from those of NaI in several aspects: (i) They do not display hydrogenic (ρ(r) ∝ exp[−2r/a]) D. Exciton densities and electron-hole separation density dependence, at least not in the vicinity of the center-of-mass. Rather, by fitting several different func- The cell-periodic electron and hole densities [see tional types, an ellipsoidal Gaussian function was judged Eqs. (6a) and (6b)] for the three degenerate exciton to most closely reproduce the BSE result. In prin- ground state states in NaI are spherically centered around ciple, it might be possible to benchmark our results against the eigenvalues of the anisotropic effective mass iodine atoms, reflecting the fact that the heavy hole 65,66 bands are almost exclusively formed from iodine p-states. equation. This is, however, beyond the scope of the The conduction bands of NaI are strongly hybridized and present work. (ii) Because of the crystal symmetry the thus it is not entirely obvious that also the electron will envelope functions in SrI2 are anisotropic as illustrated in localize on iodine atoms. This picture is, however, cor- Fig. 7(c,d). For the dipole forbidden state the best fit re- roborated by visualizing the electron charge density of sulted in anisotropic full width at half maxima (FWHM) ˚ the exciton with the hole placed at an iodine atom as of 21, 17, and 21 A in the x, y, and z-directions, respec- ˚ shown in Fig. 7(a). The exciton ground state envelope tively. (iii) The effective Bohr radius of 9.3 A for NaI ˚ function shown in Fig. 7(a) is spherically symmetric as corresponds to a FWHM of 6.4 A. As a result, despite well. The deviations from spherical symmetry at large the apparent dominance of the localized Sr 4d states of electron-hole separation that are apparent in Fig. 7(a) the conduction bands, the lowest lying SrI2 excitons have are an artifact of finite k-point sampling and the pe- a spread which is about two to three times larger than the riodic boundary conditions applied in our calculations. 1s exciton in NaI. This is solely due to the single s-like Fitting the envelope function to the charge density of a minimum at the Γ-point [see top panel of Fig. 2(d)]. 1s hydrogen-like wave function ρ(r) ∝ exp[−2r/a] yields a Bohr radius (defined as the most probable electron- hole distance) of a = 9.3 A˚ for a 143 Γ-centered MP IV. SUMMARY AND CONCLUSION grid. This corresponds to an exciton binding energy of 210meV in very good agreement with the extrapolated In this work we have studied from first principles, sin- value of 216 meV obtained above. Thus it is no surprise gle and two-particle excitations in two prototypical scin- that for NaI also the effective mass approximation is rea- tillator materials, NaI and SrI2. This was motivated by sonably accurate yielding a binding energy of 256 meV. the need to understand the role of free excitons in scin- Figure 8 illustrates the relation between the envelope tillator non-proportionality. The two systems were ju- function geh(r) introduced in Eqs. (7–8) and the electron- diciously chosen to represent two significantly different hole densitye ρeh(re, rh) in the case of NaI. While the lat- responses to incident radiation. NaI is a standard scin- ter exhibits fluctuations that obey the lattice periodicity, tillator material with strongly non-linear dependence of the envelope function is smooth and decays monotoni- light-yield as a function of incoming photon/electron en- cally and exponentially. The lattice periodicity enters in ergy, while SrI2 has recently been discovered to have ex- Eq. (8) via the excitonic single-particle densities ρe(re) cellent proportionality. and ρh(rh). On the basis of DFT and G0W0 calculations we ob- SrI2 again proves itself much more complex and ap- tained rather similar band gaps of 5.5 and 5.2 eV for NaI pears to be not well suited for a simple description in and SrI2, respectively. The dielectric functions of NaI 8

30 30 (Å) (Å) (a) NaI (b) NaI (c) SrI2 (d) SrI2 y y 6.480 Å 20 20 10 10

0 0 ¥

10 ¢ 10 ¤

20 20 £ 30 ¡ 30 Distance along [010] Distance along [010] −20 0 20 −20 0 20 −40 −20 0 20 40 −20 0 20 − Distance along [101] (Å) Distance along [100] x (Å) Distance along [100] x (Å) Distance along [001] z (Å)

FIG. 7. (a) The logarithm of the electron-hole density ρeh(re, rh) defined in Eq. (5) for the exciton ground state state of NaI in a {110} plane assuming the hole is located at an iodine site, i.e. rh = 0. The black bar indicates the lattice constant. Also shown are logarithmic contour maps of the envelope function geeh(r) defined in Eq. (7) for (b) the ground state exciton of NaI and (c,d) the lowest lying dipole forbidden exciton state of SrI2 in projected onto (001) and (100).

Excitonic single-particle electron densities 1.2 ~ geh(r) (a) dipole forbidden(b) x -polarized (c) z -polarized 1.0 ρ (r, 0) eh 6.48 Å

0.8

0.6

0.4

NaI 0.2 Electron−hole distribution function 0.0 −30 −20 −10 0 10 20 30 Distance along 〈100〉 r (Å)

FIG. 8. Envelope function geeh(r) and exciton electron Excitonic single-particle hole densities density ρeh(r, 0) for NaI as defined in Eqs. (7) and (5) pro- (d) dipole forbidden(e) x -polarized (f) z -polarized jected onto h100i direction with the hole placed on an iodine (rh + Rh = 0). The vertical dotted lines indicate the lattice spacing (a0 = 6.480 A).˚ I Sr and SrI2 displayed significant red shift of the oscillator strengths due to excitonic effects. As a result the op- tical spectra calculated from BSE deviate substantially both in intensity and structure from the single-particle RPA spectra calculated with DFT+∆ over the energy range considered in this study, i.e. up to approximately 6 eV above the conduction band edge. Although the dif- ference is expected to diminish at high energies, these results highlight the need for incorporating excitonic ef- r fects in the dielectric models used in the study of carrier FIG. 9. Excitonic single-particle (a-c) electron ρe( e) and (d- f) hole densities ρh(rh) according to Eqs. (6a) and (6b) for the and exciton generation by the photoelectrons during the lowest energy dipole forbidden, x-polarized, and z-polarized cascade. excitons in SrI2. Small purple and large green spheres indicate Almost all models for scintillation require knowledge I and Sr , respectively. of the binding energy of excitons, which determines their population relative to free carriers at very early times < 1 ps after the impact of the ionizing radiation. In this regard the main result of this work is that the calculated correlated with the population of the free excitons. groundstate exciton binding energies fall in the range be- To study localization and spatial structure of excitons tween 200 and 220meV for both NaI and SrI2. Hence, we introduced a decomposition of the six-dimensional ex- the superior energy resolution of SrI2 cannot be directly citon wave function int a product of single-particle den- 9 sities and an envelope function. In the case of NaI we of carrier density during early stages of scintillation from obtained a perfectly spherical envelope function, which first principles, and considering the relative importance can be fit very well assuming a hydrogen-like wave func- of free versus self-trapped excitons in scintillator non- tion. The binding energy obtained in this way is in good proportionality. agreement with the values obtained directly from BSE calculations and via the effective mass approximation, which is expected given the hydrogen-like character of the excitonic groundstate. ACKNOWLEDGMENTS The SrI2 low-energy excitons revealed more complex character, which could not be fitted using simple hydro- We acknowledge fruitful discussions with C. R¨odl. This genic wave functions. The envelope function was found work was performed under the auspices of the U.S. De- to possess anisotropic first moments in accord with its partment of Energy by Lawrence Livermore National anisotropic effective mass tensor. More surprising is the Laboratory under Contract DE-AC52-07NA27344 with finding that excitonic wave functions in SrI2 are more support from the National Nuclear Security Adminis- extended than in NaI, which suggests that non-radiative tration Office of Nonproliferation Research and Devel- exciton-exciton annihilation mediated by exchange is opment (NA-22). P.E. acknowledges support through stronger in SrI2. This is unexpected as such processes the “Areas of Advance – Materials Science” at Chalmers contribute to stronger non-proportional response in con- and computer time allocations by the Swedish National ventional models of scintillation. This study thus calls Infrastructure for Computing at NSC (Link¨oping) and for more detailed investigation of the temporal evolution C3SE (Gothenburg).

1 P. A. Rodnyi, Physical processes in inorganic scintillators Moses, Appl. Phys. Lett. 92, 083508 (2008). (CRC Press, Boca Raton, 1997). 19 C. M. Wilson, E. V. van Loef, J. Glodo, N. Cherepy, 2 G. F. Knoll, Radiation detection and measurement; 4th ed. G. Hull, S. Payne, W. S. Choong, W. Moses, and K. S. (Wiley, New York, NY, 2010). Shah, Proc. SPIE 7079 (2008), 10.1117/12.806291. 3 K. E. Nelson, T. B. Gosnell, and D. A. Knapp, 20 R. Hawrami, M. Groza, Y. Cui, A. Burger, Nucl. Instrum. Meth. A 659, 207 (2011). M. D. Aggarwal, N. Cherepy, and S. A. Payne, 4 P. Dorenbos, IEEE Trans. Nucl. Sci. 57, 1162 (2010). Proc. SPIE 7079, Y790 (2008). 5 P. Dorenbos, J. T. M. de Haas, and C. W. E. Van Eijk, 21 N. J. Cherepy, S. A. Payne, S. J. Asztalos, G. Hull, IEEE Trans. Nucl. Sci. 42, 2190 (1995). J. D. Kuntz, T. Niedermayr, S. Pimputkar, J. J. Roberts, 6 B. Rooney and J. Valentine, R. D. Sanner, T. M. Tillotson, E. van Loef, C. M. Wil- IEEE Trans. Nucl. Sci. 43, 1271 (1996). son, K. S. Shah, U. N. Roy, R. Hawrami, A. Burger, 7 A. V. Vasil’ev, IEEE Trans. Nucl. Sci. 55, 1054 (2008). L. A. Boatner, W. S. Choong, and W. W. Moses, 8 W. W. Moses, S. A. Payne, W.-S. Choong, G. Hull, and IEEE Trans. Nucl. Sci. 56, 873 (2009). B. W. Reutter, IEEE Trans. Nucl. Sci. 55, 1049 (2008). 22 P. Dorenbos, E. V. D. van Loef, A. P. Vink, 9 W. Moses, G. Bizarri, R. Williams, S. Payne, A. Vasil’ev, E. van der Kolk, C. W. E. van Eijk, K. W. Kr¨amer, J. Singh, Q. Li, J. Q. Grim, and W. Choong, H. U. G¨udel, W. M. Higgins, and K. S. Shah, IEEE Transactions on Nuclear Science 59, 2038 (2012). J. Luminescence 117, 147 (2006). 10 S. A. Payne, N. J. Cherepy, G. Hull, J. D. 23 F. P. Doty, D. McGregor, M. Harrison, K. Findley, and Valentine, W. W. Moses, and W. Choong, R. Polichar, Proc. SPIE 6707, 670705 (2007). IEEE Trans. Nucl. Sci. 56, 2506 (2009). 24 G. Bizarri and P. Dorenbos, 11 S. Kerisit, K. M. Rosso, B. D. Cannon, F. Gao, and Y. Xie, Phys. Rev. B 75, 184302 (2007). J. Appl. Phys. 105, 114915 (2009). 25 R. M. Van Ginhoven, J. E. Jaffe, S. Kerisit, and K. M. 12 G. Bizarri, W. Moses, J. Singh, A. Vasilev, and Rosso, IEEE Trans. Nucl. Sci. 57, 2303 (2010). R. Williams, J. Lumin. 129, 1790 (2009). 26 A. Canning, A. Chaudhry, R. Boutchko, and N. Grønbech- 13 R. T. Williams, J. Q. Grim, Q. Li, K. B. Ucer, and W. W. Jensen, Phys. Rev. B 83, 125115 (2011). Moses, Phys. Status Solidi A 248, 426 (2011). 27 J. Andriessen, E. van der Kolk, and P. Dorenbos, 14 Z. Wang, R. T. Williams, J. Q. Grim, F. Gao, and Phys. Rev. B 76, 075124 (2007). S. Kerisit, Phys. Status Solidi B 250, 1532 (2013). 28 D. J. Singh, Phys. Rev. B 82, 155145 (2010). 15 W. Setyawan, R. M. Gaume, R. Feigelson, and S. Cur- 29 M. E. McIlwain, D. Gao, and N. Thompson, IEEE Nucl. tarolo, IEEE Trans. Nucl. Sci. 56, 2989 (2009). Sci. Symposium Conference Record 1-11, 2460 (2007). 16 Q. Li, J. Q. Grim, R. T. Williams, G. A. Bizarri, and 30 D. Aberg,˚ B. Sadigh, and P. Erhart, W. W. Moses, J. Appl. Phys. 109, 123716 (2011). Phys. Rev. B 85, 125134 (2012). 17 E. V. D. van Loef, P. Dorenbos, C. W. E. 31 D. J. Singh, Appl. Phys. Lett. 92, 201908 (2008). van Eijk, K. Kr¨amer, and H. U. G¨udel, 32 L. Hedin, Phys. Rev. 139, A796 (1965). Appl. Phys. Lett. 79, 1573 (2001). 33 E. E. Salpeter and H. A. Bethe, 18 N. J. Cherepy, G. Hull, A. D. Drobshoff, S. A. Payne, Phys. Rev. 84, 1232 (1951). E. van Loef, C. M. Wilson, K. S. Shah, U. N. Roy, 34 G. Onida, L. Reining, and A. Rubio, A. Burger, L. A. Boatner, W.-S. Choong, and W. W. Rev. Mod. Phys. 74, 601 (2002). 10

35 T. Hahn, International Tables for Crystallography, Space- Phys. Rev. B 66, 165105 (2002). Group Symmetry, International Tables for Crystallography 52 R. J. Elliott, Phys. Rev. 108, 1384 (1957). (Wiley, 2005). 53 M. Dvorak, S.-H. Wei, and Z. Wu, 36 H. B¨arnighausen, H. Beck, H. W. Grueninger, E. T. Ri- Phys. Rev. Lett. 110, 016402 (2013). etschel, and N. Schultz, Z. Krist. 128, 430 (1969). 54 M. Rohlfing and S. G. Louie, 37 P. E. Bl¨ochl, Phys. Rev. B 50, 17953 (1994). Phys. Rev. Lett. 81, 2312 (1998). 38 G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). 55 S. Ismail-Beigi and S. G. Louie, 39 G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993). Phys. Rev. Lett. 95, 156401 (2005). 40 G. Kresse and J. Hafner, Phys. Rev. B 49, 14251 (1994). 56 K. Hummer, P. Puschnig, and C. Ambrosch-Draxl, 41 G. Kresse and J. Furthm¨uller, Phys. Rev. Lett. 92, 147402 (2004). Phys. Rev. B 54, 11169 (1996). 57 P. Cudazzo, C. Attaccalite, I. V. Tokatly, and A. Rubio, 42 G. Kresse and J. Furthm¨uller, Phys. Rev. Lett. 104, 226804 (2010). Comput. Mater. Sci. 6, 15 (1996). 58 G. Strinati, La Rivista del Nuovo Cimento 11, 1 (1988). 43 M. Shishkin and G. Kresse, 59 M. Rohlfing and S. G. Louie, Phys. Rev. B 74, 035101 (2006). Phys. Rev. B 62, 4927 (2000). 44 W. G. Schmidt, S. Glutsch, P. H. Hahn, and F. Bechstedt, 60 K. Teegarden and G. Baldini, Phys. Rev. 155, 896 (1967). Phys. Rev. B 67, 085307 (2003). 61 F. C. Brown, C. G¨ahwiller, H. Fujita, 45 F. Fuchs, C. R¨odl, A. Schleife, and F. Bechstedt, A. B. Kunz, W. Scheifley, and N. Carrera, Phys. Rev. B 78, 085103 (2008). Phys. Rev. B 2, 2126 (1970). 46 C. R¨odl, F. Fuchs, J. Furthm¨uller, and F. Bechstedt, 62 R. Poole, J. Jenkin, R. Leckey, and J. Liesegang, Phys. Rev. B 77, 184408 (2008). Chem. Phys. Lett. 26, 514 (1974). 47 J. P. Perdew, K. Burke, and M. Ernzerhof, 63 V. Pankratov, A. Popov, L. Shirmane, A. Kotlov, Phys. Rev. Lett. 77, 3865 (1996), erratum, ibid. 78, G. Bizarri, A. Burger, P. Bhattacharya, E. Tupit- 1396(E) (1997). syn, E. Rowe, V. Buliga, and R. Williams, 48 D. M. Ceperley and B. J. Alder, Radiat. Meas. 56, 13 (2013). Phys. Rev. Lett. 45, 566 (1980). 64 J. E. Eby, K. J. Teegarden, and D. B. Dutton, 49 H. J. Monkhorst and J. D. Pack, Phys. Rev. 116, 1099 (1959). Phys. Rev. B 13, 5188 (1976). 65 J.-A. D´everin, Nuovo Cimento B 63, 1 (1969). 50 F. Bechstedt, R. Del Sole, G. Cappellini, and L. Reining, 66 A. Schindlmayr, Eur. J. Phys. 18, 374 (1997). Solid State Communications 84, 765 (1992). 51 P. Puschnig and C. Ambrosch-Draxl,