We present the electronic structure, lattice relaxation, and formation energies of iodine vacancy defects in SrI2 for the one-electron, two-electron, and ionized charge states. We use a local generalized gradient approximation as well as non-local hybrid functionals within the framework of density functional theory, as it is commonly accepted that the latter can improve accuracy of the band gap and hence relevant energy levels. Comparison is made to published results on chlorine vacancy defects in NaCl calculated with similar methods and functionals, and also to a recent first-principles study of one- and two-electron occupancy in MgO vacancy centers. Using the parameters that are calculable from first principles in SrI2 as a starting point, we incorporate available experimental data and adaptations of simple models to predict a range of results that can help guide or interpret future experiments such as absorption energy, configuration coordinate curves, vibrational lineshape, thermal trap depth, and Mollwo–Ivey comparison to alkaline-earth fluorides.
In 2008, SrI2:Eu2+ became the focus of intense interest in the search to develop higher resolution gamma-ray scintillation spectrometers 1, 2 for use in fields such as chemical and isotope security screening, medical molecular imaging, and high-energy physics experiments. For roughly six decades, monovalent alkali iodide scintillators had remained the stalwart choice for many applications in radiation detection. Their light yield and proportionality (both related to energy resolution 3) were only modest compared to theoretical limiting values and their response was slow. But even the modest values of the first two performance parameters along with ease of crystal growth were enough to keep them ahead of most competing scintillator materials for much of the six decades. Oxide hosts doped with Ce gave faster response and more rugged mechanical characteristics for medical applications beginning from about 1990 3–5, but still at generally lower light yield (until very recently 6) and comparable or lower resolution than the alkali halides 3. Under mounting needs for a breakthrough advance in sensitivity and gamma energy resolution, the discovery of the tri-valent metal halide scintillators LaCl3:Ce3+ and LaBr3:Ce3+7, 8 finally approached close to theoretical maximum light yield 3 and achieved resolution unprecedented in a scintillator to that time. Then the detailed experimental re-examination and development starting from 2008 of SrI2:Eu2+ scintillation (first discovered decades earlier by Hofstaedter 9) set a new record in scintillator performance with arguably the best combined light yield and proportionality achieved to date 1–3. It was soon followed with other discoveries of similar high performance in combined light yield and proportionality, e.g., BaBrI:Eu2+10, CsBa2I5:Eu2+10, Cs2LiLaBr6:Ce3+11, and Cs2LiYCl6:Ce3+12. Interestingly, those recent top performers so far are always in host crystals describable as multivalent or complex metal halides.
This distinctive dependence of scintillation performance on the host crystal structure, first noted by Payne et al. 13 in summarizing survey measurements of a large number of materials, is an interesting puzzle in its own right. We have recently proposed 14 that a main reason for the better performance of complex halides over simple alkali halides involves both hot electron transport within the dense ionization track and the concentration and properties of deep electron traps in the host. Together, the diffusion distance, trap concentration, and capture cross-sections determine a linear quenched fraction k1, which was shown in Refs. 15, 16 to be a controlling factor in both nonproportionality and the total light yield. Part of the motivation of this paper is to gain understanding of electron traps contributing to the size of k1 in what is arguably the pre-eminent representative of the high-performance new multivalent halides, SrI2:Eu2+. According to a numerical model of interacting defect traps and carrier diffusion in high concentration gradients of electrons and holes 15, 17, and also an analysis by the method of rate equations 18, k1 acts both to scale the “halide hump” in measurements of light yield versus initial electron energy and to limit the maximum light yield. Elimination of deep traps and other causes of linear quenching of electrons (k1), could in principle eliminate the halide hump, which is the main contributor to poor proportionality of alkali halide scintillators, and also increase the potential light yield to very high values 16. One immediately wonders if the main material advantage of SrI2:Eu2+ as a scintillator host over, e.g., alkali halides might just be a particularly low concentration or cross-section of defects serving as deep electron traps in SrI2. In any case, as a divalent halide, SrI2:Eu2+ is the next step up in chemical complexity from the alkali halides, bringing a very remarkable change in properties for a short step in chemical complexity but a large step in structural complexity.
With this motivation, we want to characterize the properties of some of the expected common lattice defects in SrI2. As it turns out, SrI2 is extremely hygroscopic, and until recently it was moderately difficult to grow good crystals. As a result of these complicating experimental factors and the absence of a strong driving interest until the recent need for improved gamma detectors, there is very little known about defects in SrI2. Thermoluminescence measurements have been performed 19. However, optical absorption and EPR spectroscopy of native or radiation induced lattice defects in SrI2 are basically absent. The first optical absorption spectroscopy on SrI2 that has been performed in our laboratory is on short-lived species induced by band-gap excitation 20. The first “defect” calculations on SrI2 up to now are of an intrinsic transient species, self-trapped excitons 21. Experiments to introduce and study conventional lattice defects in SrI2 are being planned, but meanwhile this seems a good occasion for first principles electronic structure theory to lead experiment.
Chen et al. 22 reported results on chlorine vacancy defects in NaCl calculated with similar methods and functionals to those we employ here. Their work provides a useful validation of the accuracy and appropriateness of the computational methods in a similar ionic material where experimental data are abundant. In discussing the data, we will be interested in finding what may be viewed as extensions of behavior seen in the simpler alkali halides, but also new features or characteristics due to the divalent cations and more complex crystal structure. It is anticipated that spectroscopic defect data on SrI2 will soon be coming from a number of laboratories. The present calculated predictions should be of help in planning experiments and interpreting the data. Finally, comparison of fully interpreted data with the calculations will provide a retrospective validation or route to improvement of the calculation methods as applied in this material class.
Another useful comparison will be drawn between three types of crystals exhibiting one- and two-electron vacancy centers. In SrI2, the one- and two-electron centers are termed F and F−, respectively in the common nomenclature, where lattice-neutral trapped electron defects are designated F centers. One important case for comparison is the one- and two-electron vacancy centers in MgO, termed F+ and F, respectively. These were the subject of a recent first-principles calculations 23 of optical spectra using the GW approach and the Bethe–Salpeter equation (BSE) aimed partly at elucidating the experimental observation 24 that the one- and two-electron centers in MgO have almost identical first optical absorption transitions, i.e., the optical binding energy of the second electron in the vacancy is almost the same as that of the first one. This might seem at first counter-intuitive in a static-lattice Coulomb potential picture. We encounter a similar result on examining the optical binding energies of one- and two-electrons in the iodine vacancy of SrI2. This affects how F and F− centers both behave as deep electron traps in SrI2 and so feeds back to the practical consequences for scintillator performance.
2 Material parameters and calculation methods
2.1 Crystal structure and experimental parameters
The crystal structure of SrI2 is orthorhombic, space group Pbca (No. 61 of the International Tables of Crystallography), with lattice constants 15.22, 8.22, and 7.90 Å, respectively 25. The experimental band gap of SrI2 is still being refined, and seems to be converging to about 5.5 eV. Experimental and theoretical values of band gap since 2008 have been quoted as 3.7 eV estimated from absorption and luminescence spectra in thick samples 2, 4.5 eV calculated in DFT with Engel–Vosko GGA 26, 5.7 eV synchrotron luminescence excitation 27, ≥5.1 eV measured in transmission of a 100 µm crystal 28, and ≈5.5 eV deduced from the 1s exciton dip in synchrotron radiation luminescence excitation with estimated 0.26 eV exciton binding energy from dielectric constant 29.
Another experimental parameter that will be used in later analysis and discussion is the LO phonon frequency. Cui et al. 30 measured Raman spectra and reported the highest Ag mode to be 124.5 cm−1, i.e., the highest-frequency zone-center phonon ωLO = 2.6 × 1013 s−1 in SrI2.
2.2 Computational methods
Our ab initio calculations are carried out in the projector augmented wave framework in the Vienna ab initio simulation package (VASP) 31, 32. We employ both screened and unscreened hybrid functionals (HSE06 33 and PBE0 34) and compare the results to the Perdew–Burke–Ernzerhof (PBE) parametrization of the generalized gradient approximation (GGA) 35. The mixing fraction is 0.25 for both hybrid functionals and the screening parameter used for HSE06 is 0.2 Å−1 following Ref. 33. The kinetic cut-off energy is 300 eV. A self-consistency convergence criterion of 1 × 10−6 eV is used for all calculations and the structures are relaxed until all force components are less than 0.01 eV Å−1. The bulk properties of ideal SrI2 are calculated with the primitive unit cell of 24 atoms. A Γ-centered 2 × 4 × 4 Monkhorst–Pack k-point mesh is applied for all three exchange-correlation (xc) functionals. To simulate the iodine vacancy, we choose a 1 × 2 × 2 supercell which is roughly cubic in overall shape containing 96 atoms, and remove one iodine atom from a site as specified below. For the defect calculations we use a Γ-centered 2 × 2 × 2 Monkhorst–Pack k-point mesh for GGA-PBE, and only Γ point calculations for hybrid functionals due to the computational complexity. To test the effect of this economization, we calculated with HSE06 the perfect crystal energy with a 2 × 4 × 4 sampling in a unit cell and with 1 × 1 × 1 sampling in the supercell specified above. The energies differed by 1.7 meV atom−1. There are two distinguishable iodine sites at the 8c Wyckoff positions: site 1 (−0.202, −0.108, −0.163) and site 2 (0.202, 0.108, 0.163). We have calculated the formation energy for both of them and they differ by ∼0.2 eV. In the current paper, all the results correspond to the lower energy iodine vacancy unless specified otherwise.
2.3 Finite supercell size corrections
In the supercell approximation there are spurious interactions between the defects 22, 36, 37. For charge neutral defects, the strain energy is the leading error and scales roughly with L−338, 39, where L is distance between the periodic defects. Makov and Payne considered the convergence of the energy of charged species in periodic systems and established a correction on the basis of a multipole expansion as follows 40:
where q is the charge of the defect, and Q is the quadrupole moment. The leading term corresponds to the monopole–monopole interaction and can be analytically determined from the Madelung constant αMd of the Bravais lattice of the supercell and the static dielectric constant of the material. Because the F center is lattice-neutral, only small inward displacement of the nearest Sr2+ ions occurs on relaxation, so no correction is needed. For the charged iodine vacancies, we choose five different supercells with different sizes (containing 48, 96, 144, 288, and 432 atoms, respectively) and relax the structures for both F− and F+ center using GGA-PBE. We calculate the monopole–monopole interaction terms for each of them explicitly using the calculated static dielectric constant tensor 41 and extrapolate the corrected data assuming an L−3 dependence. Here L is defined as the cubic root of the supercell volume. Note that this term contains both the quadrupole term of the Makov–Payne scheme and the strain energy 36. The resulting total correction terms for the F− and F+ center of 0.22 and 0.06 eV, respectively are applied to the hybrid functional results when specifying defect formation energies.
2.4 Chemical potentials, formation energies, and thermodynamic transitions
We assume that the crystal is in equilibrium with a reservoir of strontium metal (fcc crystal) and molecular iodine (orthorhombic crystal). The formation energy of an iodine vacancy at charge state q is 22, 36, 37
where Ed is the total energy of a supercell containing one iodine vacancy in charge state q, Elat is the total energy of the perfect supercell. EVBM and EF are valence band maximum (VBM) and electron Fermi energy, respectively. To simulate the energy cost of removing one electron from VBM, one needs a sufficiently large supercell to reach the dilute limit.
The chemical potential of the iodine reservoir crystal is (solid). is the change in chemical potential of iodine from the reservoir upon incorporation in SrI2. is the analogous chemical potential change from the strontium reservoir. The formation enthalpy of the SrI2 crystal is thus
can vary from in the Sr-rich limit up to 0 in the iodine-rich limit. The thermodynamic transition energy, defined by the value of the electron chemical potential at which the charge state of the vacancy changes from q to q′, is given by the following expression 22, 37:
3.1 Crystal structure and experimental parameters
The bulk properties of the ideal crystal are listed in Table 1. All three xc functionals overestimate the lattice constants. PBE0 predicts the smallest deviation of the unit cell volume from experiment. Use of the PBE0 functional produces a close match with the experimental bandgap discussed in Section 2.1.
Table 1. Lattice constant, deviation of the unit cell volume from experiment, and bandgap (Eg) calculated using three different xc functionals.
3.2 Defect formation energies and thermodynamic transition energies
The formation energies at the Sr-rich limit for three different charge states as a function of Fermi energy calculated from PBE0 are plotted in Fig. 1. Note that there are two distinguishable iodine sites in the crystal – site 1 has four nearest Sr ion neighbors and site 2 has three nearest Sr ion neighbors. The F center at site 1 has ∼0.2 eV lower formation energy than at site 2, so we focus on the type 1 site in the rest of the paper. We can see from Fig. 1 that the thermodynamic transition energies ε(+/0) and ε(0/−) are both within the band gap, which suggests the stability of all three charge states of iodine vacancy when the Fermi energy is varied within the band gap. We can also see that the finite size correction widens the region of stability for the neutral F center.
In Fig. 2, the thermodynamic transition energies calculated using the three different density functionals are compared. The ε(+/0) level increases from 2.8 to 3.74 eV as the band gap widens from GGA-PBE to PBE0, however the energy window between ε(+/0) and ε(0/−) remains similar.
The calculated formation energies for the iodine vacancy in different charge states are listed in Table 2, for the Fermi energy at the VBM. The dependence of formation energies on choice of functional is more significant for the charged centers. Similar trends have been found in NaCl in Ref. 22.
Table 2. Formation energies (eV) of iodine vacancies in different charge states calculated with different functionals at Sr rich and I rich limits. The Fermi energy is set at EVBM. All results are corrected for finite size effect.
3.3 Lattice relaxation and electron density contours around the iodine vacancy
It can be seen in Table 3 that the nearest-neighbor Sr2+ ions around the F center remain almost at the perfect lattice distance, since the F center with one unpaired electron is lattice neutral in SrI2. In the F+ center, the electron bound to the iodine vacancy has been removed, leaving an effective positive charge at the vacancy. The nearest neighbor Sr2+ ions relax outward due to the net repelling potential, and the unoccupied defect level moves closer to the CBM and delocalizes more. When the vacancy is doubly occupied as in the F− center, the nearest neighbor Sr2+ ions see a negative charged potential at the vacancy and relax inward accordingly, as seen in Table 3. Their positive charge and the shrinking confinement cage they represent keeps the doubly occupied defect level well localized despite its negative charge and brings the energy down.
Table 3. Average nearest neighbor Sr2+ distance from the vacancy center in the relaxed structures of iodine vacancy in SrI2 in different charge states calculated in a 96 atom supercell. The last column lists configuration coordinate force constants deduced in Section 3.4.
average nearest Sr2+ distance (Å)
γ (eV Å−2)
Figure 3 compares the energies of the valence band maximum (VBM), each defect level of specified charge (−1, 0, +1), and the conduction minimum (CBM) for each of the three functional choices. The defect level of the F− center is doubly occupied, and for the F+ center it is unoccupied. In spin-polarized calculations, the F center has one spin level occupied (Fs1) and the other unoccupied (Fs2).
It can be seen in Fig. 3 that both electrons of the F− center in SrI2 are almost as deeply bound as the single electron of the F center. Furthermore, the isosurface plots in Fig. 4 show that the spatial confinement of the two electrons in the central vacancy of the F− center is almost the same as in the single F electron. In fact, the Bader analysis below shows that the two electrons in the F− center are confined in a smaller central distribution than the F center electron. This can be explained by the strong role of the inward relaxing divalent Sr2+ ions responding to and stabilizing the net negative charge in the F−.
The similarity of the F and F− energy levels in Fig. 3 despite the net charge difference is suggestive of the case of F+ and F centers in MgO addressed in a very recent first principles study 23 and earlier experiments 24. In MgO (with divalent anion and cation), the absorption bands of the F+ center (one electron) and lattice-neutral F center (two electrons) are almost superimposed.
Figure 4 shows the charge density contours of the occupied (gold) and unoccupied (red) defect states. Figures on the left show the contour representing 2% of the maximum density; on the right, 10% of the maximum.
Henkelman et al. have established a method using Bader analysis to separate the atoms in a crystal according to their electronic charge density 42–44. In Table 4, we show the integrated electron density at the vacancy and the minimum distance from the vacancy to the Bader surface for the iodine vacancy electron density distribution. For the F− center, the hybrid functionals predict smaller volume of the iodine vacancy relative even to the F center, which is consistent with the results of the vacancy – nearest neighbor Sr2+ ion distance shown in Table 3. More electron density is confined within the smaller vacancy volume predicted by hybrid functionals. This suggests stronger tendency for localizing electronic states compared to semilocal functionals in which self-interactions are more dominant.
Table 4. Results of the Bader analysis, listing integrated electron density within the Bader surface, DBader, in units of e, and the minimum distance from the vacancy to the Bader surface, dmin (Å).
3.4 Configuration coordinate diagram
For first-principles input toward determining approximate optical transitions and vibrational lineshapes (rather than a full BSE approach as done recently for the F and F+ centers in MgO 23) we calculated approximate configuration coordinate curves from the first-principles energies of each of the three defect charge states in the following way. The finite-size corrected formation energies in each different charge state were first calculated at the energy-minimized lattice configuration using PBE0. To describe the procedure farther, we focus specifically on the F center as the ground state and the F+ center as its ionization limit. The complete set of coordinates of the ground state F (1s) and the ionization limit (F+) are denoted as Q0 and Q+, respectively. We linearly interpolate these two lattice configurations (for all ions) by using 15 intermediate interpolations along the lattice configuration gradient defined by Q0 and Q+ as two endpoints. The F and F+ energies are then calculated at each of the fixed intermediate lattice configurations. The energy values as a function of interpolated Q are plotted as points superimposed on the fitted F and F+ parabolas in Fig. 5 and can be seen to match the parabolas very closely. From this fit, we can extract the effective force constant γ, which is listed for each charge state in Table 3 and will be used for calculating vibrational wave functions in Section 4.3.
The configuration coordinate curves for the F center as ground state and the F+ center as its ionized state are plotted in Fig. 5. The true horizontal axis is the interpolated configuration coordinate Q involving all ion positions. In the lower axis label, this is referred to as configuration coordinate even though it is not representing a specific normal mode. To give a numerical feel, we also plot in Fig. 5 the average nearest-neighbor Sr2+ distance from the vacancy (Rnn) appearing in each Q configuration of the F center ground state. But this does not imply that the energy change is a function of changing only the nearest-neighbor distance. The Q for the F− ground state will be along a somewhat different line in configuration space than for the F center ground state.
A framework of formation energies, lattice relaxation, charge contour, optical and thermal ionization limits, and curvature (effective force constant) of the configuration coordinate diagram were given by the first principles calculations discussed above. In order to compare to available experiments, or in the case of SrI2 mostly to prepare for future experiments, we will now go sometimes outside the first principles DFT methods to finish deducing values of transition energies to bound excited states, vibrational lineshape of the transitions, comparison to a particle-in-a-box model by so-called Mollwo–Ivey plots, and activation energies of thermoluminescence. These predictions follow from first principles calculations as the first step, then are supplemented by model extensions and some additional experimental data. At present, they are mostly ahead of experiment in SrI2. However, some of the same first-principles computational methods have been used for chlorine vacancy centers in NaCl where experimental data do exist. Comparison to experiment in NaCl can provide guidance on which of the xc functional choices are most successful for ionic vacancy defects, and outline the approximate error achieved between theory and experiment. On this basis we will carry over the best NaCl defect calculation methods after confirmation against experiment to the (ground-breaking) predictions about iodine vacancy centers in SrI2.
4.1 Photo-ionization limit and optical absorption transitions of the F center
Figure 6 shows again the calculated F center configuration coordinate curves based on results with PBE0 hybrid functionals. It now includes additional labels and markings to be referenced in the present discussion. The vertical transition from A to B represents the optical ionization limit of the F center in the calculated results for SrI2. In the literature on alkali halide F centers 45, 46, the “F band” optical transition is associated with the transition labeled 1s → 2p in a hydrogenic model analog of the F center. There is a higher energy “K band” in the optical absorption spectrum that is interpreted as the unresolved envelope of all 1s → np transitions for n = 3, 4, …, ∞. The transition A → B in Fig. 5 represents 1s → ∞p in this terminology, and is the essential input provided here from the first principles calculation. For comparison to the F band transition in optical absorption, we need to deduce the 1s–2p energy. The effective mass hydrogenic model for shallow trapped electrons predicts transition energies to p-states of principle quantum number n according to 45
This is useful for shallow trapped-electron centers in solids and for conceptual discussions of F centers, but it is not accurate on its own for the F center. A shortfall of Eq. (5) for F centers is that their radial extent is too small for effective mass theory to apply in the ground state and the effective dielectric constant is between the optical limit for the ground state and tending toward static ε0 for the excited states. One is then left with m*/ε2 in Eq. (5) being an undetermined parameter. In the present treatment, we rely on the first principles calculation to provide the 1s → ∞p limit of Eq. (5), thus in an approximate sense determining the undetermined ε parameter. We interpolate Eq. (5) to estimate the 1s → 2p transition energy consistent with the calculated ionization limit. The fact that m*/ε2 in Eq. (5) still changes somewhat for different transitions is becoming a smaller correction in a smaller quantity than if ε were required to specify the scaling from a full hydrogen Rydberg of 13.6 eV, as in earlier attempts to apply Eq. (5) without first principles input.
Following the recipe summarized above, the predicted F band absorption transition (1s → 2p) in SrI2 is estimated to be 75% of the ionization limit calculated from first principles, i.e., ΔEF(1s→2p) = 1.67 eV. This transition of the F center in SrI2 is sketched in Fig. 5, where the interpolated F(2p) potential curve is approximated with a broken line.
Reference 22 presented a configuration coordinate diagram for NaCl analogous to Fig. 5. As we have also done, they evaluated the transition energy A → B from the F center vertically to the unrelaxed F+ center using three different choices of DFT xc functionals, GGA, HSE06, and PBE0. However, they compared the ionization transition A → B (1s → n∞) directly to the F band absorption transition in NaCl, ΔEF = 2.77 eV. Comparing this experimental transition energy in the same column of their Table 5 with the calculated ionization limits appeared to make the HSE06 prediction of 2.56 eV look better than the PBE0 prediction of 3.50 eV. Similarly, the HSE06 prediction of 1.07 eV recombination emission from the NaCl F+ state to the F(1s) defect ground state was compared to experimental 0.98 eV emission, which is actually from the relaxed F(2p) state. Comparison to recombination from the ionized state rather than the relaxed excited state made HSE06 look much more successful in matching experiment than PBE0, which predicted 1.88 eV for recombination from ionization. The emission energies will be discussed below and are listed with absorption energies in Table 5.
Table 5. Transitions Ea and Ee from Ref. 22 multiplied by the 0.75 factor discussed above before comparison to experiment. The last column notes that ZPL from Ref. 22 is the thermal trap depth of the F center, ΔEf,th.
What we want to point out, with some importance for applications of the various xc functionals to SrI2 in the present paper, is that PBE0 gives much better predictions for the experimental absorption transition energy in NaCl if the deduction of 1s → 2p F band transition energy is done by the procedure outlined above before comparing to the experimental transition. The experimental emission energy falls between the HSE06 and PBE0 predictions in NaCl. The results for F band (1s → 2p) absorption energies deduced from the Chen et al. 22 calculation of the 1s → ∞p ionization limit for NaCl F centers are shown in Table 5.
The experimental emission band is from the F(2p) relaxed excited state to the F(1s) unrelaxed ground state. The relaxed excited state equilibrium configuration is between Q0 and Q+. The factor applied in the table above is 0.75 to take account of the excited state being 2p rather than ∞p (ionized, F+). The “≥” notation is used on the corrected emission energies because the lattice configuration of the F(2p) excited state is displaced from the F+ equilibrium configuration. The approximate F(2p) potential curve is suggested schematically in Fig. 6 by the dashed qualitative excited state curve.
4.2 Thermal trap depth of the F center and thermoluminescence data
Referring again to Fig. 5, we review the distinction between the optical trap depth (A → B) and the thermal trap depth (A → C). The optical trap depth can be measured experimentally by photoconductivity spectroscopy or by analysis of the optical absorption series limit. Its final state is on the unrelaxed F+ potential curve, meaning the crystal with one vacancy per supercell, at the local lattice configuration Q0 with an electron at the conduction band minimum. This can be equivalently denoted CBM(Q0) as in Fig. 6. The thermal trap depth from the minimum of the F center ground state (point A in Fig. 5) to the minimum of the F+ potential curve (point C) can in principle be measured experimentally by thermoluminescence or thermally stimulated current spectroscopy. In the motivating context of this paper, thermoluminescence is important because of its well-established utility for diagnosing scintillator defect properties 47. In the case of SrI2 discussed in Section 1, its good scintillation performance seems to imply a small linear quenched fraction k115, and one circumstance giving small k1 could be if the dominant electron traps such as F centers have small thermal depths. Thus evaluating the F and F− thermal depths in SrI2 is one of the practical goals of this paper. The thermal trap depth of the F center can be directly obtained from the first principles calculation. It is the energy of relaxed F+ minus the energy of relaxed F. For SrI2 using PBE0, it is ΔEF,th = 1.56 eV. The F− center has .
The values of thermal trap depth that can be deduced from the calculations by Chen et al. 22 are listed in the last column of Table 5. They labeled this value as ZPL for zero-phonon line. However, our calculated vibrational ground state of the F center in SrI2 (Section 4.3) shows that there will not be a zero-phonon line associated with optical transitions to the ionization limit nor with the F band (1s → 2p) spectrum, and experiments in NaCl have established that there is no ZPL observable from its F center either. These are both cases of strong linear coupling to the lattice. In any case, the energy values of the last column in Table 5 also correspond to thermal trap depth as defined above, and from the calculated results of Ref. 22, we label them as ΔEF,th in the second-line column title.
In SrI2, thermoluminescence is one of the first defect spectroscopies other than luminescence that has been published 19. Yang et al. found 9 thermoluminescence peaks in the temperature range 50–259 K, and none in the range 260–550 K. The highest activation energy (thermal depth) in the measured range was 0.431 eV for the 255 K peak. The others were lower than 0.28 eV. None of these are a good match for the calculated F center thermal trap depth in SrI2. This suggests that the F center and F− center likely function as deep electron traps (effectively quenchers) in SrI2 after all. The reason for small k1 in SrI2 and other complex halides may lie elsewhere 14, 48. It has often been found in thermoluminescence of alkali halides that the F centers are destroyed by more mobile species including halogen interstitial atoms or other hole species before releasing their trapped electrons. Thermoluminescence trap depths for release of electrons from F− centers were measured in NaF, NaCl, and LiF as 0.72, 0.62, and 1.06 eV, respectively 49.
4.3 Vibrational wavefunctions and modeled optical absorption bands of F and F− centers in SrI2
The configuration coordinate diagram in Fig. 6 has a lower potential curve (F center ground state) that is customarily approximated as quadratic in a configuration coordinate Q representing a single most important interacting vibrational mode (usually the symmetric breathing mode)
In the present case of first-principles energies minimized for full lattice relaxation at the (e.g., F and F+) endpoints, Q is a configuration coordinate in the comprehensive sense of a single parameter labeling configurations of all ions in the lattice. We have fit Eq. (6) to the calculated lower CC curve to determine the effective force constants as listed in Table 3. For example, γ = 8.58 eV Å−2 in the F center ground state. In Fowler's compilation of data on F centers in alkali halides, the local mode frequency that fits the F-band width in each alkali halide is about 1/2 of the LO phonon frequency in that crystal 45. The basic reason is that the ions neighboring a vacancy with an electron partly in it and partly out see much softer restoring force in breathing mode vibration than the ions surrounding a normal lattice site with a hard rare-gas configuration halide ion in the center. Cui et al. 30 have measured Raman spectroscopy from which the highest LO phonon frequency in SrI2, ωLO = 2.34 × 1013 s−1, is determined. For comparison, ωLO = 2.0 × 1013 s−1 in RbI, where Sr mass is adjacent to Rb. The F center local mode frequency in RbI is ωA = 1.0 × 1013 s−145. Scaling in the same way for SrI2, we arrive at the estimate ωA = 1.17 × 1013 s−1 for that case. From ωA and γ, the effective mass M of the mode is found from .
The ground state vibrational wavefunction is, in one dimension appropriate to a single normal mode,
The probability based on γ from first principles and ωA scaled empirically from the experimental highest ωLO in SrI2 is plotted in Fig. 6.
By comparing the width of the vibrational wavefunction to the displacement ΔQ between the F(1s) and F(2p) minima, we can see that there will be no zero phonon line. The ground state vibrational wavefunction projects up onto the highly excited upper-state vibrational wavefunctions with strong peaks at their classical turning points. We have not done the full vibrational overlap calculation, but regard each classical turning point at high n to be a delta function at that point on the F(2p) curve. In this way, we produce the modeled approximate F(1s → 2p) absorption lineshape in SrI2 at low temperature shown in Fig. 7.
Figure 7 also plots the lineshape of the F− (1s → 2p) transition (red dashed) obtained in the same way from the F− configuration coordinate curve and the F curve as its ionization limit. Although Table 3 shows that there is considerable softening of the configuration coordinate curves in the sequence F+, F, F−, the degree of softening (fractional change in force constant γ) is about the same at each stage of the sequence, so the band width is predicted to be similar for both the F and F− first absorption transitions. This is similar to what is found experimentally for the one- and two-electron transitions (F+ and F) in MgO, and different from alkali halides like NaCl, where the F− transition is significantly wider than the F.
The predicted F and F− transition energies in Fig. 7 differ by 0.4 eV. In NaCl 22, using a similar DFT-PBE0 method and finite cell-size correction, the predicted F and F− transitions differed by 1.47 eV, whereas the experimental difference in NaCl is 0.34 eV. In MgO 23, DFT-LDA calculations with finite size corrections predicted a difference of 1.1 eV between absorption band peaks of one-electron (F+) and two-electron (F) centers, whereas GW and BSE methods duplicated the experimental finding 24 of nearly coincident F+ and F center absorption bands. Based on the trend exhibited in these two comparison cases of using DFT results to predict the splitting of absorption transitions of one- and two-electron charge states on a given vacancy, we suggest that when experimental results become available in SrI2, the F and F− transitions may be nearly coincident as in MgO.
We suggest further that part of the difficulty in trying to deduce optical transitions with respect to ionization limits posed by differences between the three charge states F+, F, and F− has to do with finite cell-size correction. It causes shifts in the two charged defect states and not in the neutral one. This is entirely appropriate if one is comparing the calculations to properties of the ground state of each charge state. However, it seems that consideration of the final-state electron, which is properly done in GW and BSE methods, will be needed in at least an approximate way in order to deduce features of the optical spectrum such as F–F− splittings from DFT results.
4.4 Comparison to a particle-in-a-box model and other halide crystals in Mollwo–Ivey plots
It has been well-known in the F center literature that a particle-in-a-box model works reasonably well, maybe even surprisingly well, for predicting variation of the 1s → 2p F center transition from crystal to crystal. The point ion potential in the vicinity of the anion vacancy is a flat-bottom well (at the Madelung energy) inside the radial region bounded by the nearest-neighbor cations, and at larger radii it oscillates up and down at each alternate shell of anion and cation neighbors, respectively, with diminishing amplitude. But most of the electron density (70% in SrI2) lies within the nearest-neighbor cation bounds, as we have already seen, and so the early developers of the particle-in-a-box model 45, 50, 51 tried the simple case of an infinite three-dimensional square well of radius a defined as the distance from the vacancy center to the nearest-neighbor cations. In that model 45, 50, 51 the 1s → 2p transition energy is
What counts is the 1/a2 dependence, meaning that the F center transition energy should, in this simple model, scale from crystal to crystal as approximately the inverse square of the nearest-neighbor distance. The log–log plot to test whether a power law relation is seen is known as a Mollwo–Ivey plot. In the cubic rock-salt alkali halides, a good linear relation is obtained with the exponent −1.84 45, 51. Williams et al. 52 made a Mollwo–Ivey plot for both F center and STE absorption transitions in the alkaline-earth fluorides BaF2, SrF2, CaF2, and MgF2, where the fluorite structures of the first three have a unique nearest-neighbor distance, but MgF2 has a noncubic structure and three cation neighbors of the vacancy at two slightly different distances. In that case, the average nearest-neighbor cation distance was used, and the F centers in all four crystals formed a good linear Mollwo–Ivey plot, however with exponent −3. To see how SrI2 fits with the four alkaline earth fluorides, we simply plotted our 1.52 and 1.73 eV calculated F center transition energies predicted from HSE06 and PBE0 results, respectively, at the calculated 3.38 eV nearest-neighbor Sr2+ distance on the same graph as the alkaline-earth fluoride crystals 52. The resulting plot is shown in Fig. 8. Considering that the alkaline-earth fluoride F band energies are experimental and SrI2 calculated, the plot is not bad. If we keep the solid line with slope −3 that fit the alkaline-earth fluoride crystals, the extended fluoride F band line actually intersects the SrI2 nearest-neighbor distance at an energy prediction of about 1 eV rather than our calculated 1.73 eV PBE0 result. On the other hand if we enforce the slope of −2 dictated by the particle-in-a-box model, the dashed line fits CaF2, SrF2, and calculated SrI2 rather well, while MgF2 and BaF2 lie farther off. When experimental optical absorption data on SrI2 F centers finally emerge, it will be interesting to see whether DFT hybrid functional theory or particle-in-a-box hits it closer.
Upon comparing results to available lattice constant and band-gap data for SrI2 and upon analyzing the results of corresponding calculation methods in NaCl for comparison to experimental F center optical transitions, we conclude at the first step that DFT with PBE0 hybrid functionals gives the best predictions of available experimental data for these ionic crystals and their vacancy defects relative to the other approaches tried with GGA-PBE and HSE06. Then continuing with the DFT-PBE0 method, we calculated iodine vacancy defect formation energies in the charge states q = +1, 0, and −1 relative to lattice neutrality, and thermodynamic transition energies between them, predicting stability versus Fermi level. We used an interpolation scheme to construct configuration coordinate diagrams for the F and F− centers based on the first-principles defect energies at lattice configurations along the linearized configuration path from the potential minimum to the ionized equilibrium configuration for each of the defect charge states. Thermal trap depth and optical trap depth are directly obtained from first principles. Furthermore, construction of the configuration coordinate diagram permits determination of the effective ground and ionized state force constants from fitting the first-principles potential curves. This allows calculating the ground-state and ionized-state defect vibrational wave functions for prediction of optical lineshapes. Departing from first principles methods, but using the results noted above to remove large uncertainties surrounding appropriate ε and m* parameters in the simple hydrogenic model of F center optical transitions, we were able to use that model to make interpolations of excited state energies based on the 2.31 eV energy interval from ground to ionized state of the defect rather than the 13.6 eV Rydberg basis of the full hydrogenic model. In this way the uncertainties in excited state energies of the F center become of a tolerable size to make meaningful comparisons and predictions with experiments. For example, the vibrationally broadened 1s → 2p optical absorption of the F band in SrI2 was predicted, as a simpler alternative to the GW and BSE predictions of F+ and F bands as has been used in MgO 23, another crystal with divalent alkaline earth cations. The prediction of similar optical binding energies for the one-electron F and two-electron F− centers in SrI2 corresponds interestingly to the calculation 23 and experiments 24 for the one-electron F+ and two-electron (lattice neutral) F centers in MgO. There are far fewer existing experimental data on SrI2 than on MgO or NaCl, so many of our detailed predictions are just that – predictions waiting for the experiments which should be coming soon given the recently realized importance of SrI2:Eu2+.
This work was supported by the Office of Nonproliferation Research and Development (NA-22) of the U.S. Department of Energy under contracts DE-NA0001012 (Fisk-WFU), DE-AC02-05CH11231 (LBNL-WFU), and DE-AC52-07NA27344 (LLNL). Calculations were performed on the Wake Forest University DEAC Cluster, a centrally managed facility with support in part by the University. The calculations were performed using the ab initio total-energy program VASP (Vienna ab initio simulation program) developed at the Institüt für Materialphysik of the Universität Wien. We thank Natalie Holzwarth, Babak Sadigh, Timo Thonhauser, and Miguel Moreno for helpful discussions.
Qi Li received his B.S. in Optics Information Science and Technology from the Special Class for Gifted Young at the University of Science and Technology of China in 2009. Currently he is pursuing Ph.D. research on carrier transport modeling and first principles electronic structure calculations in radiation detection materials at Wake Forest University under the supervision of Prof. Richard Williams and in collaboration with Dr. Daniel Aberg of Lawrence Livermore National Laboratory. He has also conducted first principles calculations on hydrogen storage in clathrates and metal organic frameworks with Prof. Timo Thonhauser at Wake Forest. He has 15 papers published or accepted in 3 years of research.
Richard T. Williams received his Ph.D. in physics from Princeton University in 1973 and worked at the US Naval Research Laboratory until joining Wake Forest University as Reynolds Professor of physics from 1985 to present. His research interests include self-trapping of excitons and ultrafast spectroscopy of carrier relaxation and defect formation dynamics with current emphasis on materials for radiation detection. He has 10 patents and 170 scientific publications cited over 4280 times.
Daniel Åberg received his Ph.D. in physics from Uppsala University, Sweden, in 2004 and joined Lawrence Livermore National Laboratory, USA, in 2006 where he is now a Staff Scientist. His area of expertise includes first principles modeling of the electronic structures of atomic and condensed matter systems. His current research is directed toward the material science of semiconductor and scintillator radiation detectors.