A Theoretical Study on the Structural and Energy Spectral Properties

Sep 4, 2012 - Ce3+ are extracted from the calculated energies and wave functions. The calculated .... orbitals of main character Ce 4f, 5d, and 6s. Wi...
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A Theoretical Study on the Structural and Energy Spectral Properties of Ce3+ Ions Doped in Various Fluoride Compounds Jun Wen,† Lixin Ning,‡ Chang-Kui Duan,*,† Yonghu Chen,† Yongfan Zhang,§ and Min Yin† †

Department of Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China Department of Physics, Anhui Normal University, Wuhu 241000, People's Republic of China § Department of Chemistry, Fuzhou University, Fuzhou, Fujian 350002, People's Republic of China ‡

S Supporting Information *

ABSTRACT: Geometry optimization and wave function-based complete-activespace self-consistent field-embedded cluster calculations have been performed for a series of Ce3+-doped fluoride compounds (CaF2, YF3, LaF3, KMgF3, LiYF4, K2YF5, and KY3F10) to investigate local coordination structures, crystal field parameters, and 5d1 energy-level structures of doping Ce3+ ions. The crystal-field parameters of Ce3+ are extracted from the calculated energies and wave functions. The calculated crystal-field parameters and 5d1 energy-level structures show excellent consistency with the experimental results. Our calculations show that the onset of 4f → 5d absorption, which is important in phosphors and scintillators, can be wellpredicted. Apart from that, the distortion of local structure due to doping, the wave functions, and the crystal-field parameters of 4f1 and 5d1 states of Ce3+ in the hosts can be obtained. Those can seldom be obtained by fitting empirical crystal-field Hamiltonian to experimental data but are required by some detailed theoretical analysis, such as the calculation of transition intensities and hyperfine splittings. The obtained crystal-field parameters of Ce3+ may also be useful for other lanthanide ions in the same hosts.

1. INTRODUCTION Cerium-doped luminescent materials have attracted much attention due to their wide applications in laser and scintillator crystals and phosphors for white light-emitting diodes (LED) and displaying.1−6 Luminescence in Ce3+-doped materials corresponds to the electric dipole-allowed 5d → 4f optical transitions, where the information for the onset and crystal-field (CF) splitting of the 5d1 energy levels of Ce3+ ion is desired, and the wave functions for 5d1 states are also helpful for the detailed theoretical analysis. Dorenbos7−10 has analyzed a large number of the experimentally measured 5d1 energy-level data for Ce3+-doped oxide, fluoride, and other inorganic compounds and given an empirical model to study the variation trend of 5d 1 energy-level structures with respect to the local coordination surroundings. Although the onsets and CF splittings of 5d1 energy levels for the Ce3+-doped systems can be obtained experimentally, the wave functions and CF parameters can seldom be obtained from the experimental data. The first-principles calculations, which do not rely on prior experimental spectroscopic data, can in principle provide not only the local coordination structures but also the electronic structures of the doping ions in lanthanide-doped systems, which can aid in a deeper understanding of experimental results and also in the design of the new luminescent materials.11−13 With first-principles band-structure calculations, Stephan et al.14 have studied 5d → 4f transitions for a number of lanthanide-doped fluoride and oxide compounds. Watanabe et al.15 have investigated the 4f → 5d transition energy of Ce3+ in various fluoride hosts based on the discrete-variational Dirac− © 2012 American Chemical Society

Slater (DV-DS) calculations using Slater's transition-state theory. As is well-known, the band-structure calculations using density functional theory (DFT) are usually not suitable for the prediction of the position of localized 4fN states of lanthanide ions. In addition to this, the results of DV-DS calculation usually rely heavily on the choice of funnel potentials. Recently, Seijo16,17 and Ning18,19 have performed ab initio model potential (AIMP) embedded cluster calculations on Ce3+-doped solid-state compounds to investigate structural properties and 4f → 5d absorption of Ce3+ ion and obtained significant information on the energies and relative intensities of the 4f → 5d transitions. Still, it is believed that the parametric CF model of 4fN and 4fN−15d energy levels of lanthanide ions is at present the most practicable model to analyze and simulate the energy-level structures of lanthanide ions in crystal hosts.20,21 The energy levels and transition intensities of lanthanide ions in crystals can be effectively simulated with the CF Hamiltonian and the electric dipole operator, which involve many interaction parameters obtained from fitting to the experimental spectroscopic data. On the basis of the combination of parametric CF model with firstprinciples ab initio calculations, Reid et al.22−24 presented a method for constructing an effective Hamiltonian and extracting CF parameters from ab initio calculations for lanthanide-doped inorganic compounds. Received: June 27, 2012 Revised: August 26, 2012 Published: September 4, 2012 20513

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In the present work, we have combined the first-principles calculations (geometry optimization calculations and AIMP embedded cluster calculations) with the parametric CF model to investigate local coordination structures, CF parameters, and 5d1 energy levels of Ce3+ ions in inorganic compounds. A series of Ce3+-doped fluoride compounds (CaF2, YF3, LaF3, KMgF3, LiYF4, K2YF5, and KY3F10) have been chosen for our investigation. In our calculations, the local coordination structures of doping Ce3+ ions were first optimized with DFT using the supercell model. The effect of Ce3+ doping and charge compensation on the cation−anion bond lengths is then analyzed in detail. On the basis of the optimized supercell structures, Ce3+ center embedded clusters were constructed, on which the state-averaged complete active-space self-consistent field (SA-CASSCF) calculations were performed to obtain the energies and wave functions of the 4f1 and 5d1 levels. Then, the CF parameters of Ce3+ in these fluoride compounds were extracted from the previously calculated energies and wave functions and compared with the values obtained from the least-squares fitting to the experimental spectroscopic data. With the derived CF parameters and the empirically determined quasi-free ion parameters for the spin-orbit interaction, the energies for the 4f1 and 5d1 levels were refined for Ce3+ in the fluoride hosts under study. The energy-level results were compared to the experimental results in terms of the onset, the centroid energy, and the CF splitting for the 4f→ 5d absorption. With the 5d energy-level structures of Ce3+ in compounds provided by our method, the first transition energies of 4fN → 4fN−15d of the other lanthanide ions in the same host then can be predicted from the empirical model of Dorenbos.10 Furthermore, the CF parameters of the other lanthanide ions also may be predicted according to the variation tendency of CF parameters with respect to the lanthanide series in crystals,20 once the CF parameters of Ce3+ in the same host are derived. On the basis of the parametric CF model, the energy-level structures and spectra for other lanthanide ions in crystals then can be simulated and analyzed.

application to lanthanide ions in crystals can be found in refs 29−31. For each of the chosen Ce3+-doped systems, a defect cluster with doping Ce3+ ion at the center was embedded in the AIMP representations of the pure fluoride hosts. All of the ions located within a sphere of radius 10.0 Å surrounding the embedded cluster were modeled with the AIMP embedding potentials, which were made of electrostatic, exchange, and Pauli repulsion interactions; point charges have been used for the rest of ions located within a sphere of radius 30.0 Å. For the embedded cluster, we performed SA-CASSCF32−35 calculations with a scalar relativistic many-electron Hamiltonian without spin−orbit terms. These wave function-based calculations were performed by using the program MOLCAS.36 In the SACASSCF calculations, a [4f, 5d, 6s] complete active space was chosen so that it comprises all configurations in which the single unpaired electron occupies one of the 13 molecular orbitals of main character Ce 4f, 5d, and 6s. Within the chosen defect clusters, a relativistic effective core potential ([Kr] core) and a (14s10p10d8f3g)/[6s5p6d4f1g] Gaussian valence basis set from ref 37 were used for Ce; a [He] core effective core potential and a (5s6p1d)/[2s4p1d] Gaussian valence basis set from ref 38 were used for F; a [Mg] core effective core potential and a (9s7p5d)/[2s3p3d] Gaussian valence basis set from ref 38 were used for Ca; a [Zn] core effective core potential and a (11s8p7d3f)/[3s3p4d1f] Gaussian valence basis set from ref 39 were used for Y; a [Cd] core effective core potential and a (13s10p8d)/[3s3p3d] Gaussian valence basis set from ref 40 were used for La; a [Be] core effective core potential and a (7s6p1d)/[2s3p1d] Gaussian valence basis set from ref 38 were used for Mg; a [Mg] core effective core potential and a (9s7p)/[2s3p] Gaussian valence basis set from ref 38 were used for K; and a [He] core effective core potential and a (5s1p)/[1s1p] Gaussian valence basis set from ref 38 were used for Li. The minimum Gaussian valence basis set was used for Y and K in the larger embedded clusters for Ce3+doped LiYF4 and KY3F10 to constrain the calculation costs, and the impact of this on the results for the energy levels of Ce3+ is expected to be minor due to the localization of 4f and 5d orbitals. The detailed embedding AIMPs for the simulation of the hosts can be found in refs 31 and 36. The method for constructing the single-electron effective Hamiltonian Heff in a model space from the quantum-chemical ab initio calculation was given by Reid et al.22−24 Here is a brief outline of constructing the Heff and extracting CF parameters from the embedded cluster SA-CASSCF calculations in this work. The CASSCF wave function Ψ of the system is formed as a linear combination of configuration state functions ΦK:

2. METHODOLOGY The geometry optimization for Ce3+-doped fluoride compounds was carried out by DFT calculations within the generalized gradient approximation (GGA), using the Vienna ab initio Simulation Package (VASP).25,26 The configurations of Ce 5s25p64f15d16s2, F 2s22p5, Ca 3s23p64s2, Y 4s24p64d15s2, La 5s25p65d16s2, Mg 2p63s2, K 3p64s1, and Li 2s1 were treated as the valence electrons, and their interactions with the cores were described by the projected augmented wave (PAW) method.27 The Perdew−Burke−Ernzerhof (PBE) exchangecorrelation functional28 has been employed. The structural relaxation was performed by using the conjugate gradient technique. The equilibrium structures were obtained by optimizing atomic positions until the energy change was less than 10−4 eV and the Hellmann−Feynman forces on atoms were less than 0.01 eV/Å. Also, we used a plane-wave cutoff energy of 500 eV and one k-point Γ for the sampling of the irreducible part of the Brillouin zone for the structural optimization of the Ce3+-doped cases. On the basis of the optimized crystal structures, the 4f and 5d energies without spin−orbit coupling effects as well as the corresponding wave functions of Ce3+ ions in these fluoride compounds were calculated with the AIMP embedded cluster approach. The detailed descriptions of the method and its

Ψ=

∑ CK ΦK

(1)

K

The configuration state functions (i.e., antisymmetric multielectron Slater-determinant functions) are generated from molecular spin orbitals: ΦK = Â {ϕK (x1), ϕK (x 2), ..., ϕK (xN )} 1

2

N

(2)

where x = r, s, and  is an antisymmetrizer. The molecular spin orbitals ϕi(x) (i = K1, ..., KN) are ϕi(x) = ϕi(r, s) = φi(r)θi(s)

(3)

where φi(r) is the molecular orbital and θi(s) is a spin function. Furthermore, these molecular orbitals are obtained as expansions in a set of atomic-center basis functions χp: 20514

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Table 1. Crystal Structures for the Hosts, the Supercells for the Structural Optimization Calculations, and the Clusters Adopted for the Embedded Cluster Calculations of Energy Levels space group

site symmetry

CNa

unit cell sizeb

supercells

embedded clusters

CaF2 YF3 LaF3

Fm3m Pnma P-3c1

2+

Ca :Oh Y3+:Cs La3+:C2

2+

Ca :8 Y3+:9 La3+:11

4 4 6

2×2×2 2×2×3 2×3×2

KMgF3

Pm3m

K+:Oh

K+:12

1

3×3×3

LiYF4 K2YF5 KY3F10

I41/a Pna21 Fm3m

Y3+:S4 Y3+:Cs Y3+:C4v

Y3+:8 Y3+:7 Y3+:8

4 4 8

3×3×2 2×2×2 1×1×2

(CeF8Ca12)19+ (CeF13Y4)2+ (CeF13La4)2+ (CeF12Mg8K6)13+ (Oh) (CeF12Mg8K4)11+ (C4v) (CeF12Li8Y4)11+ (CeF8Y2K6)7+ (CeF16Y4K4)3+

host

a

CN, coordination number. bDefined as the number of chemical formula in a unit cell.

φi =

Ca2+ ion farthest from the doping Ce3+ ion is substituted by a Na+ ion in the supercell. Experimentally, to avoid the cluster sites of Ce3+ ions, the dopant concentration of Ce3+ and Na+ ions is very low.42 So, it is reasonable that the Ca2+ site substituted by a Na+ ion is far away from the Ce3+ ion. Such a local coordination structure of Ce3+ ion keeps Oh symmetry. For Ce3+-doped KMgF3, according to the analysis in ref 43, we consider two possible sites for the doping Ce3+ ions. One site is the K+ substitutional site with a Mg2+ vacancy in the supercell, and the other one is the same site with two nearest K+ vacancies, which lower the local symmetry of doping Ce3+ ion to C4v. For the first substitution case, the compensation vacancy is set at the farthest Mg2+ site away from the doping Ce3+ ion in the supercell so that the local coordination structures of Ce3+ ion remains Oh symmetry. The optimized lattice parameters of pure fluoride compounds are listed in Table 2 together with the experimentally

∑ Cipχp p

(4)

The seven Ce center 4f basis functions (five Ce center 5d basis functions) were used to form the “4f” (“5d”) model space. The calculated matrices for both the seven 4f (five 5d) eigenvalues and the corresponding eigenvectors projected into the “4f” (“5d”) model space were then chosen to construct the Heff. Meanwhile, from the point of view of the parametric CF model, the Heff can be expanded in terms of a complete set of parameters and operators describing the effective interactions in the “4f” (“5d”) model space.41 Consequently, the interaction parameters (CF parameters in this work) can be extracted in a straightforward way. The SA-CASSCF embedded cluster ab initio calculations do not automatically include the spin−orbit coupling effects, which are important for the 4f levels of lanthanide ions. The strength of spin−orbit coupling for a given lanthanide ion barely changes from one host to another host and has been known for many hosts.20,42 The spin−orbit parameters together with the CF parameters obtained from the SA-CASSCF calculations can produce more accurate 4f1 and 5d1 energy levels for Ce3+ in crystals. For fluoride hosts, the parameter values (ζ4f = 614.9 cm−1, and ζ5d = 1082 cm−1) from ref 42 are used for all of the investigated fluoride compounds. As for the energy differences between the centroids of 4f1 and 5d1 energy levels, the corresponding values from the previous SA-CASSCF calculations are used.

Table 2. Calculated and Measured Lattice Parameters (Collected from Refs 44−50) for the Investigated Fluoride Hosts exptl

3. RESULTS AND DISCUSSION 3.1. Structural Properties. Table 1 presents a summary of the structural properties of the pure fluoride compounds studied here. As is well-known, for LiYF4, K2YF5, and KY3F10, the most probable cation sites substituted by Ce3+ are Y3+, rather than Li+ and K+ sites. For KMgF3, we consider the K+ substitutional sites. The supercells chosen for the geometry optimization calculations and the defect clusters chosen for the AIMP embedded cluster calculations are also listed in Table 1. Each Ce3+-doped fluoride compound was modeled by an appropriate supercell chosen by considering both the size of the unit cell and the computational cost. Within the AIMP embedded cluster calculations, a few layers of the coordination ions of Ce3+ are employed to construct the embedded cluster. Because of the charge difference between doping Ce3+ ion and host cations in CaF2 and KMgF3, the effect of the charge compensation should be taken into account. For Ce3+-doped CaF2, the charge compensation can be provided by a local interstitial fluoride ion or a Na+ ion replacing a Ca2+ ion. In this work, we consider the charge compensation case in which the

calcd

host

a (Å)

b (Å)

c (Å)

a (Å)

b (Å)

c (Å)

CaF2 YF3 LaF3 KMgF3 LiYF4 K2YF5 KY3F10

5.464 6.353 7.185 3.990 5.171 10.791 11.540

5.464 6.850 7.185 3.990 5.171 6.607 11.540

5.464 4.393 7.351 3.990 10.748 7.263 11.540

5.512 6.377 7.244 4.053 5.225 11.013 11.689

5.512 6.949 7.244 4.053 5.225 6.716 11.689

5.512 4.512 7.386 4.053 10.855 7.358 11.689

measured lattice parameters. The optimized lattice parameters are all slightly larger than the measured ones, which is typical in the GGA-DFT calculations. All of the errors are about 2% or less. The good agreement between the calculated and the experimentally measured lattice parameters of the pure fluoride compounds provides a solid foundation for the geometry optimization of Ce3+-doped cases and even the subsequent embedded cluster calculations. Table 3 lists the optimized bond lengths of the cations and fluoride ions for both the pure and the Ce3+-doped fluoride compounds. The optimized bond lengths of the host cations and fluoride ions are slightly larger than the experimentally measured ones, and most of the deviations are about 2% or less. For Ce3+-doped LaF3 and KMgF3, due to the smaller ionic radius of Ce3+ than that of the substituted cation, the bond lengths of Ce3+−F− are shorter than those of the substituted cation and fluoride ions in pure crystal. The situations are opposite for the rest of the chosen 20515

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Table 3. Bond Lengths (in Å) of Cations (Host Cations and Ce3+ Ion) and Fluoride Ions for Pure and Ce3+-Doped Fluoride Compounds (the Host Cations Substituted by Ce3+ in the Fluoride Compounds Are Listed in Table 1) pure exptla

host

Ce3+-dopedb (calcd)

calcd

CaF2

8 × 2.366

8 × 2.387

YF3

1 3 2 2 1

× × × × ×

2.259 2.260 2.300 2.328 2.594

1 1 2 2 1

× × × × ×

2.284 2.286, 2 × 2.321 2.341 2.318 2.663

1 1 2 2 1

× × × × ×

2.373 2.412, 2 × 2.394 2.401 2.429 2.691

LaF3

2 1 2 2 2 2

× × × × × ×

2.417 2.442 2.459 2.487 2.640 3.002

2 1 2 2 2 2

× × × × × ×

2.430 2.482 2.468 2.488 2.648 3.077

1 1 1 1 1 1

× × × × × ×

2.415, 2.474 2.446, 2.456, 2.633, 3.088,

12 × 2.821

12 × 2.866

LiYF4

4 × 2.247 4 × 2.300

4 × 2.269 4 × 2.328

K2YF5

2 1 2 2

KY3F10

4 × 2.195 4 × 2.331

KMgF3

× × × ×

2.184 2.208 2.256 2.281

2 1 1 1

× × × ×

2.211 2.222 2.286, 1 × 2.287 2.310, 1 × 2.311

4 × 2.225 4 × 2.383

8 × 2.389

1 × 2.423 1 1 1 1

× × × ×

2.452 2.468 2.634 3.106

12 × 2.629 (Oh) 8 × 2.608, 4 × 2.712 (C4v) 4 × 2.371 4 × 2.415 1 1 1 1

× × × ×

2.304, 1 × 2.325 2.319 2.376, 1 × 2.378 2.408, 1 × 2.420

4 × 2.314 4 × 2.479

Experimental values from refs 44−50. bThe optimized spherical coordinates of the coordination fluoride ions around Ce3+ in the chosen fluoride compounds are listed in Tables S1 and S2 in the Supporting Information. A cutoff value of 3.2 Å is used for the Ce3+−F− bond length. a

vacancies, the neighboring F− ions around Ce3+ are divided into two groups (2.608 and 2.712 Å). 3.2. 4f1 and 5d1 Energy Levels. The energies of 4f1 and 1 5d levels without spin−orbit coupling effects for Ce3+ in CaF2, LiYF4, and LaF3 are listed in Table 4, where the corresponding energy levels of Ce3+ in the other fluoride compounds are not given for simplicity. The overall distribution and splittings of the calculated 5d1 energy levels agree reasonably well with the experimentally measured values. The largest deviation between the calculated and the corresponding measured 5d1 energy levels is less than 3000 cm−1. The centroids (E̅5d) of the calculated 5d1 energy levels relative to the lowest 4f energy levels for all of the chosen Ce3+-doped fluoride compounds are located at about 44000 cm−1, which is consistent with the experimental values. The splittings (ΔE5d) of 5d1 energy levels are also consistent with the experimental values, and the largest deviation is about 2000 cm−1. On the basis of the energies (without spin−orbit coupling effects) and wave functions obtained from the SA-CASSCF calculations, the 4f and 5d CF parameters are extracted. The detailed discussion of CF parameters will be presented in section 3.3. These CF parameters are then used together with the quasi-free ion parameters for spin−orbit interaction42 to calculate the refined 4f and 5d energy levels using the parametric CF model. These energy levels are also listed in Table 4 together with the available experimentally measured values. The inclusion of spin−orbit interaction increases the

fluoride compounds. Judging from the deviation of the calculated from the measured bond lengths for the hosts, we estimate that the deviation between the optimized and the actual Ce 3+ −F − bond lengths in Ce 3+ -doped fluoride compounds is also about 2% or less. The effect of Ce3+ doping on the cation−anion bond lengths can be approximately described by the following relation: d(Ce3 +−F−) r(Ce3 +) + r(F−) = × η d(Mn +−F−) r(Mn +) + r(F−)

(5)

where d(Ce3+−F−) is the calculated average value of the bond lengths of Ce3+ ion and neighboring F− ions, d(Mn+−F−) is the average value of the experimentally measured bond lengths of Mn+ ion and neighboring F− ions in pure fluoride compounds, Mn+ is the cation substituted by Ce3+, and r(X) is the Shannon's ionic radii of X ion corresponding to the coordination number and the valence state in crystals.51 The scaling factor η should be close to 1. Using the average bond lengths from our ab initio geometry optimization calculations, we obtain η values in the range of 0.995−1.013, except for η = 1.04 for Ce3+-doped KMgF3. This indicates that the ionic radii of cations and coordinated anion usually determine the bond length variation of Ce3+−F− relative to Mn+−F−, while the abnormal η for Ce3+doped KMgF3 is attributed to the larger difference in electric charge between Ce3+ and K+. Besides, for the C4v case of Ce3+doped KMgF3, due to the existence of charge-compensating K+ 20516

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32300 32300 51600 53300 55200 44940 22900

0

exptlb

Oh 0 0 168 1946 2329 2453 2453 42147 42147 47426 47426 48692 45568 6545

CF + SO

C4v 0 186 275 2034 2404 2474 2639 41866 42461 45255 48392 49377 45470 7511

CF + SO

42735 44053 absence 47619 49261 45917 6526

0

exptlb 0 196 196 297 504 504 1321 32389 40274 48640 48640 52213 44431 19824

CF

LiYF4 0 247 481 2214 2255 2409 3016 33378 41142 49404 50144 53520 45518 20142

CF + SO

33433 41101 48564 50499 52790 45277 19357

0

exptlb 0 189 262 303 394 743 1165 39217 41907 44498 46083 51491 44639 12274

CF

LaF3 0 191 361 2122 2233 2491 2808 39959 42681 45326 47672 52668 45661 12709

CF + SO

40161 42735 45872 48077 51546 45678 11385

0

exptlb

YF3 0 312 479 2148 2393 2583 2770 38068 41813 47077 49124 51655 45547 13587

CF + SO

39063 41841 46296 49261 51546 45601 12483

0

exptlb

K2YF5 0 413 822 2129 2524 2707 3337 33819 34534 46072 49891 59198 44703 25379

CF + SO

KY3F10 0 526 774 2133 2522 2857 3051 31781 41463 50528 50848 52202 45364 20421

CF + SO

a

Units are cm−1. CF and CF + SO denote the CF energy levels without and with the spin-orbit interaction included, respectively; E̅ 5d and ΔE5d denote the centroid and splitting of the 5d1 energy levels of Ce3+. bExperimental values for CaF2 and LiYF4 from ref 42 and for YF3, LaF3, and KMgF3 from ref 7.

E̅5d ΔE5d

5d1

4f1

0 0 480 2139 2139 2170 3650 32702 32702 53226 53226 54763 45324 22061

0 0 0 49 49 49 2040 31590 31590 52483 52483 52483 44126 20893

CaF2

CF + SO

CF

KMgF3

Table 4. Calculated and Measured Energies of 4f1 and 5d1 Levels of Ce3+ Doped in the Fluoride Compoundsa

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splitting of 4f levels by about 2000 cm−1 but has a much smaller effect on 5d levels. As a result of this additional 4f splitting, the energy of the lowest 5d level (relative to the 4f ground level) increased by around 1000 cm−1. It is shown that the inclusion of spin−orbit interactions is extremely important for the 4f splititngs and improves the prediction of the onset of 5d absorption. The largest deviation between the calculated and the corresponding measured 5d1 energy levels is about 1600 cm−1. The following is a detailed analysis of the 5d energy-level structures of Ce3+ in the fluoride compounds. For Ce3+-doped CaF2, the calculated 5d1 energy levels without spin−orbit coupling effects fall into two groups (2E and 2 T2 states) separated by 20893 cm−1, with 2T2 being higher in energy. When considering the spin−orbit coupling effects, the 2 T2 state further splits into two subgroups separated by 1537 cm−1. It should be noted that according to the analysis in ref 42, the presence of the lowest energy level of 2T2 state (i.e., the 5d1 energy level located at 51600 cm−1) in the experimental excitation spectrum is not explained by the theory. We attribute it to the distortion of the local coordination structures around the Ce3+ ion due to the doping Na+ ions, which lower the site symmetry of Ce3+ ion. The degeneracy of energy level is then removed. For Ce3+-doped KMgF3, the embedded cluster calculations are performed on the clusters (CeF 12 Mg 8 K 6 ) 13+ and (CeF12Mg8K4)11+ embedded in pure KMgF3 host, which correspond to the local coordination structures of Ce3+ ions at Oh and C4v sites, respectively. The additional splitting of the 5d levels when the symmetry is lowered from Oh to the chargecompensated C4v results into the larger 5d splitting of Ce3+ at C4v site (7511 cm−1) than that of Ce3+ at Oh site (6545 cm−1). The calculated 5d splittings of Ce3+ for the two sites in KMgF3 are much smaller than those for the other fluoride compounds. It can be attributed to the relatively large average Ce3+−F− bond length and coordination number (12) of Ce3+ ion in KMgF3. For Ce3+-doped LiYF4, it should be noted that the middle 5d1 energy level and its upper energy level are not very well produced by our calculations. The calculated energy difference (740 cm−1) between these two levels is much smaller than the corresponding experimental value (1935 cm−1). This problem is thought to be a result of relaxation of the ligands in the excited state, which lowers the symmetry.42 For Ce3+-doped YF3 and LaF3, the difference in both the site symmetries of Ce3+ and Ce3+−F− bond lengths is quite small. Therefore, the calculated 5d1 energy levels for the two systems are similar, as expected. Generally speaking, the calculated 5d1 energy levels with spin−orbit coupling effects for Ce3+ in the chosen fluoride compounds show excellent consistency with the corresponding experimental results, which is also illustrated in Figure 1. It shows that all of the calculated lowest 5d1 energy levels are smaller than the corresponding experimental values by only hundreds of wavenumbers. Figure 2 plots the centroids and splittings of the calculated and experimentally measured 5d1 energy levels. The centroids of the calculated 5d1 energy levels change slightly with the local coordination structures and are located at about 45000 cm−1, which is excellently consistent with the experimental values. The biggest deviation between the calculated and the corresponding experimental 5d splittings is less than 1500 cm−1. Moreover, the calculated 5d splittings tend to increase with decreasing coordination number of Ce3+, which is consistent with the analysis of Dorenbos in ref 7.

Figure 1. Energies of the calculated (black bars, left) and measured (red bars, right) 5d1 levels for Ce3+ in various fluoride hosts. The number in brackets is the coordination number of Ce3+ in the host.

Figure 2. Centroids (E̅ 5d) and splittings (ΔE5d) of the calculated and experimental 5d1 energy levels for Ce3+ in various fluoride compounds (calculated E̅ 5d, □; experimentally measured E̅ 5d, ○; calculated ΔE5d, △; and experimentally measured ΔE5d, ▽). The number in brackets is the coordination number of Ce3+ in the host.

Because of the excellent consistency between the calculated and the experimentally measured energy levels, our calculations can be used to calculate the energy-level structures for the Ce3+doped systems without experimental energy-level data, such as the calculations for Ce3+-doped K2YF5 and KY3F10, whose experimental data are not available. According to our calculations, the centroids of the 5d1 energy levels of Ce3+ in K2YF5 and KY3F10 are in accordance with the other fluoride compounds. Furthermore, the calculated 5d splittings for Ce3+doped K2YF5 and KY3F10 satisfy the overall variation tendency of the 5d splittings with respect to the coordination number. Herein, we deduce that the calculated 5d1 energy levels for Ce3+ in K2YF5 and KY3F10 are reasonable. On the basis of the excellent consistency between the calculated and the experimentally measured results, the approach presented here on the energy-level structures of Ce3+ can be expanded to the Ce3+-doped inorganic compounds. According to the empirical model of Dorenbos,10 the first 4f → 5d transition energies of lanthanide ions in compound A can be predicted by applying the following expression: E(Ln, A) = E(Ce, A) + ΔELn,Ce

(6)

where E(Ln, A) and E(Ce, A) are the lowest 5d energies of Ln3+ and Ce3+ ions in compound A, respectively. ΔELn,Ce is defined as the difference of 4f → 5d energies between Ln3+ and Ce3+ and can be regarded as an intrinsic property of the 20518

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Table 5. Calculated CF Parameters (in cm−1) for Ce3+ Ions in CaF2, KMgF3, LiYF4, and KY3F10 in Comparison with the Fitted ones CaF2 CF parameters 4f

5d

B2 B4 B4 B6 B6 B2 B4 B4

0(ff) 0(ff) 4(ff) 0(ff) 4(ff) 0(dd) 0(dd) 4(dd)

KMgF3

calcd

fitteda

−2949 −1762 1171 −2190

−2185 −1306 734 −1373

−601 −359 −868 1625

−43876 −26221

−44016 −26305

−10723 −6408

calcd (Oh)

LiYF4

KY3F10

calcd (C4v)

calcdb

fittedc

calcd

fittedd

414 −788 −78 −592 1614 2900 −14529 −1941

310 −1104 −1418 −70 −1140 + 237i 4312 −18862 −24878

481 −1150 −1228 −89 −1213 4673 −18649 −23871

−1217 −1911 495 683 −239 −10147 −37066 12161

−644 −1543 343 891 −30

a Fitted values from refs 52 (4f CF parameters of Dy3+ in CaF2) and 42 (5d CF parameters of Ce3+ in CaF2). bThe calculated B4 4(ff) and B4 4(dd) have been transformed to be real by rotating the coordinate system about the z-axis by 10.98° and 10.75° clockwise, respectively. cFitted values from refs 41 (4f CF parameters of Pr3+ in LiYF4) and 42 (5d CF parameters of Ce3+ in LiYF4). dFitted values from ref 53 (4f CF parameters of Pr3+ in KY3F10).

Table 6. Scalar CF Strengths (S4f and S5d, Defined in Terms of the CF Parameters in Wybourne Normalization) and the Splittings (ΔE4f and ΔE5d) of 4f1 and 5d1 Energy Levels for Ce3+-Doped Fluoride Compoundsa K2YF5 ΔE4f S4f ΔE5d S5d CNf

CaF2

LiYF4

KY3F10

YF3

calcd

calcd

exptlb

calcd

exptlb

calcd

calcd

3337 689 25379 12577 7

3650 913 22061 13541 8

643c 22900 13584

3016 520 20142 9508 8

502d 19357 9210

3051 517 20421 10152 8

2770 403 13587 6304 9

LaF3 exptlb

calcd

12483

2808 452 12709 5483 11

KMgF3 exptlb

399e 11385

calcd (Oh)

calcd (C4v)

2453 421 6545 3309 12

2639 421 7511 3604

exptlb

6526

Units are cm−1. bExperimental splittings of 5d1 energy levels from ref 42 for CaF2 and LiYF4; experimental splittings of 5d1 energy levels from ref 7 for LaF3, YF3, and KMgF3. cFitted values (Dy3+ in CaF2) from ref 52. dFitted values (Pr3+ in LiYF4) from ref 41. eFitted values from ref 55. fCN, coordination number of Ce3+. a

where Sk is the rotational invariants57 defined in terms of the CF parameters (in Wybourne normalization).

lanthanide ions. As previously mentioned, the E(Ce, A) can be accurately provided by the method presented in this work, together with the values of ΔELn,Ce listed by Dorenbos in ref 10, and the first 4f → 5d transition energies of the other lanthanide ions in compound A can be straightforwardly predicted. The predicted results are presented in Tables S3 and S4 and Figures S1−S4 in the Supporting Information, showing good agreement with the experimentally measured 5d onset energies. 3.3. Crystal Field Parameters. The calculated 4f and 5d CF parameters for Ce3+ in CaF2, KMgF3, LiYF4, and KY3F10 are listed in Table 5 together with the fitted results if available. For Ce3+-doped YF3, LaF3, and K2YF5, the site symmetries of Ce3+ are so low that the coordinate axis for the fitted case is unknown and rules out a direction comparison of CF parameters with refs 54 and 55. The calculated CF parameters for Ce3+ at low symmetry sites (listed in Table S5 in the Supporting Information) should be reasonable, judging from not only the agreement of the calculated to the measured 5d1 energy levels but also the variation tendency of CF strength with the local coordination surroundings around Ce3+. The scalar CF strength (S), which reflects the overall CF interaction, can be calculated as follows:56 ⎛ 1 S4f = ⎜⎜ ⎝3

⎞1/2 ∑ Sk⎟⎟ ⎠ k = 2,4,6

(7)

⎛ 1 S5d = ⎜⎜ ⎝2

⎞1/2 ∑ Sk⎟⎟ ⎠ k = 2,4

(8)

Sk =

1 2k + 1

k

∑ q =−k

|Bqk |2 (9)

The calculated CF strengths for all of the chosen Ce3+-doped fluoride compounds are listed in Table 6 together with the corresponding CF splitting values. It can be found that the variation tendency of the calculated S4f and S5d is generally in accord with that of CF splittings. The S5d strongly relies on local coordination surroundings around the doping Ce3+ ion. For Ce3+-doped YF3 and LaF3, due to the similar coordination structures of Ce3+, the scalar CF strengths are approximately equal. For the two substitution sites of Ce3+-doped KMgF3, S4f does not change. However, S5d of C4v site is larger than that of Oh site, which is consistent with the previous analysis of the calculated 5d energy-level splittings of Ce3+ ions in KMgF3. Also, we can find that S5d of Ce3+-doped KMgF3 is much smaller than those of other Ce3+-doped fluoride compounds. The detailed discussion of the CF parameters is then presented. Fundamentally, Table 5 shows that the calculated 4f and 5d CF parameters satisfy the symmetry requirement. For high symmetric cases (Ce3+-doped CaF2, LiYF4, and KY3F10), the signs of the calculated CF parameters are exactly the same as those of the fitted ones except for the calculated B6 4(ff) parameter of LiYF4, which is complex due to the symmetry requirement of the S4 site. The values of the calculated CF parameters generally show good agreement with those of the fitted ones. For Ce3+-doped CaF2 and LiYF4, the calculated 5d CF parameters are consistent with the fitted ones, which are 20519

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obtained from fits of the corresponding Ce3+ spectra.42 There are no fitted values available for the 4f CF parameters of Ce3+ in CaF2, LiYF4, and KY3F10, so we adopt the fitted values of Dy3+ in CaF252 and Pr3+ in LiYF441 and KY3F10.53 Then, it is reasonable that the calculated B4 0(ff) and B6 0(ff) magnitudes of Ce3+ in CaF2 are much larger than the fitted ones (corresponding to Dy3+ in CaF2) by about 764 and 437 cm−1, respectively. Moreover, the calculated 4f CF parameters of Ce3+ in LiYF4 and KY3F10 show good consistency with the fitted values (corresponding to Pr3+ in LiYF4 and KY3F10). A model has been presented to account for the variation of the CF parameters across the lanthanide series in ref 20. For example, according to the quantitative variation relationship, the 4f CF parameter B4 0 of Yb3+ in Cs2NaYbF6 is reduced by about 30% relative to that of Ce3+ in Cs2NaCeF6. Similarly, on the basis of the CF parameters of Ce3+ obtained from our calculations, the CF parameters of the other lanthanide ions in the same host can be derived. Consequently, our calculations on the CF parameters of Ce3+ may be used as a starting point to predict the complicated CF energy-level structures of the other lanthanide ions, where a first-principles calculation is less practical.

Article

ASSOCIATED CONTENT

S Supporting Information *

Optimized spherical coordinates of the coordination fluoride ions around Ce3+ in the chosen fluoride compounds, the predicted 5d onsets for all of the trivalent lanthanide ions in the fluoride compounds (CaF2, LaF3, YF3, and LiYF4) from our calculated 5d onsets of Ce3+ by adopting the empirical model of Dorenbos (eq 6), the calculated CF parameters for Ce3+ in the low-site symmetry systems (LaF3, YF3, and K2YF5), and the trends of the 5d CF parameters for trivalent lanthanide ions in CaF2 and LiYF4. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 11074315, 11274299, 11174005, and 90922022) and the Fundamental Research Funds for the Central Universities (Grant No. 2340000034). The numerical calculations in this paper have been partially done on the supercomputing system in the Supercomputing Center of University of Science and Technology of China.

4. CONCLUSIONS A series of Ce3+-doped fluoride compounds are chosen to perform the geometry optimization calculations and AIMP embedded cluster calculations to obtain the local coordination structures, CF parameters, and energy-level structures of Ce3+. The effect of the Ce3+ doping and the charge compensation on the cation−anion bond lengths are analyzed. The derived CF parameters are in excellent agreement with the values from the least-squares fitting to the experimental energy level data. The 4f1 and 5d1 energy levels of Ce3+ taking the spin−orbit interaction into account are calculated and compared with the experimentally measured values. Excellent consistencies are obtained as follows: (1) the calculated onsets of the 5d1 energy levels are smaller than the corresponding experimental values by only a few hundred wavenumbers; (2) the centroids of the calculated 5d1 energy levels of Ce3+ change slightly with respect to the coordination structures of Ce3+ and are located at about 45000 cm−1; and (3) the biggest deviation between the calculated and the experimentally measured splittings of the 5d1 energy levels of Ce3+ is less than 1500 cm−1, and also, there is a clear correlation in the trend of smaller 5d splitting for larger coordination number. Adopting the approach presented in this paper, we can provide the following important information on the study of lanthanide-doped phosphors and scintillators: (1) the distortion of the local coordination structures around lanthanide dopant ions and the effect of charge compensation can be obtained; (2) the onset of 4f → 5d absorption of Ce3+ can be well-predicted by ab initio calculations and with which the first 4f → 5d transition energies of the whole lanthanide series in the same compounds can be predicted via the empirical model of Dorenbos;10 and (3) the CF splittings and CF parameters of Ce3+ can be well-predicted via ab initio calculations, from which the CF splittings and parameters of other lanthanide ions can be estimated by using the variation trend of CF parameters across the lanthanide series where an accurate first-principles calculation is still unpractical.

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