### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Material parameters and calculation methods
- 3 Results
- 4 Discussion
- 5 Conclusions
- Acknowledgements
- Biographical Information
- Biographical Information
- Biographical Information

In 2008, SrI_{2}:Eu^{2+} 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 LaCl_{3}:Ce^{3+} and LaBr_{3}:Ce^{3+} 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 SrI_{2}:Eu^{2+} 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:Eu^{2+} 10, CsBa_{2}I_{5}:Eu^{2+} 10, Cs_{2}LiLaBr_{6}:Ce^{3+} 11, and Cs_{2}LiYCl_{6}:Ce^{3+} 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 k*_{1}, 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 *k*_{1} in what is arguably the pre-eminent representative of the high-performance new multivalent halides, SrI_{2}:Eu^{2+}. 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, *k*_{1} 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 (*k*_{1}), 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 SrI_{2}:Eu^{2+} 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 SrI_{2}. In any case, as a divalent halide, SrI_{2}:Eu^{2+} 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 SrI_{2}. As it turns out, SrI_{2} 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 SrI_{2}. Thermoluminescence measurements have been performed 19. However, optical absorption and EPR spectroscopy of native or radiation induced lattice defects in SrI_{2} are basically absent. The first optical absorption spectroscopy on SrI_{2} that has been performed in our laboratory is on short-lived species induced by band-gap excitation 20. The first “defect” calculations on SrI_{2} up to now are of an intrinsic transient species, self-trapped excitons 21. Experiments to introduce and study conventional lattice defects in SrI_{2} 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 SrI_{2} 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 SrI_{2}, 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 SrI_{2}. This affects how F and F^{−} centers both behave as deep electron traps in SrI_{2} and so feeds back to the practical consequences for scintillator performance.

### 4 Discussion

- Top of page
- Abstract
- 1 Introduction
- 2 Material parameters and calculation methods
- 3 Results
- 4 Discussion
- 5 Conclusions
- Acknowledgements
- Biographical Information
- Biographical Information
- Biographical Information

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 SrI_{2} 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 SrI_{2}. 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 SrI_{2}.

#### 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 SrI_{2}. 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 *n*p 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

- ((5))

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 *ε*_{0} for the excited states. One is then left with *m**/*ε*^{2} 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**/*ε*^{2} 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 SrI_{2} is estimated to be 75% of the ionization limit calculated from first principles, i.e., Δ*E*_{F(1s2p)} = 1.67 eV. This transition of the F center in SrI_{2} is sketched in Fig. 5, where the interpolated F(2p) potential curve is approximated with a broken line.

Table 5. Transitions *E*_{a} and *E*_{e} 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, Δ*E*_{f,th}. | *E*_{a} | 0.75*E*_{a} | *E*_{e} | 0.75*E*_{e} | ZPL |
---|

1s ∞p | 1s 2p | ∞p 1s | ∼2p 1s | Δ*E*_{F,th} |
---|

GGA | 2.03 | 1.52 | 0.65 | ≥0.49 | 1.19 |

HSE06 | 2.56 | 1.92 | 1.07 | ≥0.80 | 1.70 |

PBE0 | 3.50 | 2.63 | 1.88 | ≥1.41 | 2.66 |

Expt. | | 2.77 | | 0.98 | |

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 Q_{0} 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

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 SrI_{2} (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 Δ*E*_{F,th} in the second-line column title.

In SrI_{2}, 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 SrI_{2}. This suggests that the F center and F^{−} center likely function as deep electron traps (effectively quenchers) in SrI_{2} after all. The reason for small *k*_{1} in SrI_{2} 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 SrI_{2}

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)

- ((6))

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 SrI_{2}, *ω*_{LO} = 2.34 × 10^{13} s^{−1}, is determined. For comparison, *ω*_{LO} = 2.0 × 10^{13} s^{−1} in RbI, where Sr mass is adjacent to Rb. The F center local mode frequency in RbI is *ω*_{A} = 1.0 × 10^{13} s^{−1} 45. Scaling in the same way for SrI_{2}, we arrive at the estimate *ω*_{A} = 1.17 × 10^{13} s^{−1} for that case. From *ω*_{A} 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,

- ((7))

The probability based on *γ* from first principles and *ω*_{A} scaled empirically from the experimental highest *ω*_{LO} in SrI_{2} is plotted in Fig. 6.

By comparing the width of the vibrational wavefunction to the displacement Δ*Q* 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 SrI_{2} 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 SrI_{2}, 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 SrI_{2}) 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

- ((8))

What counts is the 1/*a*^{2} 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 BaF_{2}, SrF_{2}, CaF_{2}, and MgF_{2}, where the fluorite structures of the first three have a unique nearest-neighbor distance, but MgF_{2} 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 SrI_{2} 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 Sr^{2+} 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 SrI_{2} 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 SrI_{2} 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 CaF_{2}, SrF_{2}, and calculated SrI_{2} rather well, while MgF_{2} and BaF_{2} lie farther off. When experimental optical absorption data on SrI_{2} F centers finally emerge, it will be interesting to see whether DFT hybrid functional theory or particle-in-a-box hits it closer.

### 5 Conclusions

- Top of page
- Abstract
- 1 Introduction
- 2 Material parameters and calculation methods
- 3 Results
- 4 Discussion
- 5 Conclusions
- Acknowledgements
- Biographical Information
- Biographical Information
- Biographical Information

Upon comparing results to available lattice constant and band-gap data for SrI_{2} 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 SrI_{2} 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 SrI_{2} 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 SrI_{2} 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 SrI_{2}:Eu^{2+}.