C. W. Thiel a,b and R. L. Cone a

1. C. W. Thiela,b and R. L. Conea a Physics Department, Montana State University, Bozeman, MT, USA 59717 b Spectrum Lab, Montana State University, Bozeman, MT, USA 59717 Email: thiel@physics.montana.edu Investigating Electron Binding Energies of Impurity Ion States and Host Crystal Bands in Rare-Earth-Doped Optical Materials Research was supported in part by the Air Force Office of Scientific Research, Scientific Materials Corporation, and the National Science Foundation 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

2. Electronic Structure of Rare-earth Materials Rare-earth Ion Host Crystal Conduction Band States 4f N15d ??? Valence Band States 4f N The atomic-like electronic structure of localized rare-earth ion states is well understood The electronic band structure of de-localized crystal states is well understood To predict and explain many optical properties and electron transfer processes, it is essential to understand how these two classes of states are related and interact in rare-earth-activated optical materials 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

3. Importance of Crystal Band States Broad Impact on Rare-Earth-Activated Optical Materials: • Optical Memories and Processors—Photorefractive and photon-gated hole burning techniques may use photoionization for non-volatile operation • Laser Materials—Excited-state absorption to conduction band can limit gain and tuning range and cause optical damage • Phosphors and Solid-State Lighting—Ionization provides a non-radiative relaxation pathway while charge transfer provides an optical pumping mechanism • Scintillators— Ionization reduces light yield while efficiency of energy transfer from electron hole pairs is influenced by relative energies of ion and band states • Electroluminescence—Field-induced ionization and thermal ionization may limit performance in rare-earth-doped semiconductor materials Studying the Relationships Between Rare Earth and Band States: • Need a broad picture for the electronic structure of the host-impurity system to understand optical properties of materials • Motivates fundamental theoretical understanding • Helps explain and predict optical properties of materials • Guides the logical design of new materials with optimum properties 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

4. Methods for Studying Electron Transfer There are many Methods for Probing Broad Energy Level Structure • Optical Spectroscopy—Absorption or reflectivity spectra reveal charge transfer and photoionization transition energies, as well as fundamental host absorption • Electron Spectroscopy—Photoemission and inverse photoemission directly measure electron binding energies of occupied and unoccupied electronic states • Photoconductivity and Photocapacitance—Electron transfer detected from mobility of generated electron or hole charge carriers • Thermally Stimulated Luminescence Excitation—Electron transfer detected in fluorescing materials from charge recombination and relaxation • Microwave-detected Electron Transfer—Electron transfer detected by transient changes in the material’s dielectric constants and the effect on a resonant microwave cavity • Photo-EPR—Electron transfer detected by change in ground state spin of ionized or reduced centers in the material, or EPR signature of trapped charges ...and others Each method has unique advantages and disadvantages, and generally a combination of methods is required to fully investigate the electronic structure 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

5. Photoemission Directly Measures Electron Binding Energies Relative to a Common Energy Reference Electron Photoemission Spectroscopy KE Spectrum EVacuum Host Conduction Band (CB) VBM Binding Energy hp 4f Binding Energy RE3+ (4fN) Ground State Valence Band Maximum (VBM) Host Valence Band (VB) • Incident photons eject electrons from the occupied states in the material • Difference between Photon Energy (hnp) and ejected electrons’ Kinetic Energy (KE) gives the Binding Energy (BE) of the electrons in the sample • The energy distribution of photoelectrons gives the binding energies of all occupied electronic states—provides relative energies of the 4f electrons and the host valence band Extract Host and Ion Features: • Resonant Photoemission (RPES) exploits resonances in the rare-earth PES cross-sections to identify and extract 4f electron PES • May also compare spectra of samples with different rare-earth ion concentration 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

6. The 4f photoemission exhibits structure extending over a range of up to 10 eV that corresponds to the tetravalent rare-earth ion final electronic states Photoemission Final-state Structure • We are interested in threshold energies—a method is required to estimate the minimum energy required to remove a 4f electron from the trivalent ion • The 4f photoemission “Final-state Structure” may be predicted from the electronic states of the tetravalent ions—related to “Coefficients of Fractional Parentage” • The theoretical final-state structure is fit to the observed photoemission to accurately determine 4f binding energies • This final-state projection theory describes general trends in free-ion inter-configurational transition probabilities (4f-5d) 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

7. We are interested in comparing the energies of the 4f electrons to the energies of host band states—an estimate for the valence band maximum (VBM) is required for each material Locating the Valence Band Maximum • We estimate valence band photoemission cross-sections from theoretical atom-resolved partial density of states (PDOS) and atomic cross-sections • Fit theoretical cross-section to spectrum to locate VBM • The top of the valence band is very flat throughout the Brillouin zone for rare-earth oxides and fluorides • Other approximations for VB structure may be used if PDOS not known (e.g. a simple “Top Hat” shape often gives good VBM estimates in ionic materials with Egap > 5eV) Orbital PES Cross Sections from Yeh & Lindau 1985 YAG PDOS from Xu & Ching 1999 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

8. The measured binding energies of the rare-earth ion 4f electrons display a characteristic trend across the 4fN series Systematic Trends of 4f Binding Energies • This “zig-zag” trend is related to variation in effective nuclear charge, inter-electronic repulsion, 4f spin-pairing energy, and spin-orbit coupling • First quantitatively described by Jørgensen’sRefined Spin-Pairing Energy Theory in 1962 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

9. Electron Binding Energies in Ionic Crystals Model the 4f electron binding energy as the free-ion value shifted by (mostly) electrostatic interactions with each host lattice • Free-ion 4f electron energy (~40 eV) modified by electrostatic potential, or Madelung potential (~30 eV), of crystalline environment [Pauling 1929] • Covalency modifies effective ionic charges and the Madelung potential (~1 eV to ~15 eV) [Fadley, Hagstrom, Klein, & Shirley 1968] • Lattice polarizability screens charges and stabilizes ionized final state (~5 eV) [Mott & Littleton 1938] • Change in inter-atomic Born repulsive energy (~0.5 eV to 1 eV) [Citrin & Thomas 1972] • Change in van der Waals interaction and vibrational zero-point energies (~0.5 eV) [Poole, Szajman, Leckey, Jenkin, & Liesegang 1975] • Distortion of wavefunctions, central-field covalency, nephelauxetic effect, … (few eV?) [Jørgensen 1962] • For doped materials, the impurity-induced distortion of the lattice site affects all of these energy terms (< 5 eV) [Pedrini, McClure, & Anderson 1979] 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

10. A simple two-parameter semi-empirical form of the electrostatic model accurately describes relative 4fN energies of all rare-earth ions in a material An Empirical Model for 4f Binding Energies • Model the 4f binding energies as free-ion values shifted by interactions with the lattice—“Chemical Shift” • The chemical shift consists of a large constant shift (EL) and a smaller shift that depends on ionic radius (aR) [Pedrini, McClure, & Anderson 1979] • If we treat the model parameters EL and aR as empirical values that must be measured, they then predict 4f binding energies for all fourteen rare earths in a host [Thiel et. al 2001] • We found that this simple approach is very successful across a broad range of materials [Thiel et. al 2002, 2003] This model is successful for rare earths due to their chemical similarity, small variation in ionic radii, and the shielded, non-bonding character of 4f electrons 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

11. Using Host PES to Predict 4f Energies A consequence of the electrostatic model is that all core electrons of any chemically similar ionsshould experience the same chemical shift • When we substitute rare-earth impurity ions for similar host cations such as Y3+ or La3+, the measured chemical shift of the host cation’s core electrons gives us an estimate of EL • We find that using a fixed value of aR ~ 10 eV/Å in the empirical model gives a sufficient degree of accuracy for many optical materials • Analysis of photoemission measurements on undoped host crystals indicate that this approach predicts the 4f electron binding energies for rare-earth dopants to within the experimental accuracy of ~0.5 eV This simple estimation method works well! 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

12. Non-ionic Materials The model may be tested using 4f electron binding energies in the elemental rare-earth metals, the opposite extreme from ionic insulators • The 4f binding energies are known very accurately in the elemental metals—no charging effects and negligible vibrational broadening [Lang et. al 1981] • The PES structure establishes that metals have same 4fN configurations as trivalent ions, except Eu and Yb • The model is remarkably successful in describing the relative 4f energies of the rare-earth metals • Considerations of effective charge and electronic screening in covalent or metallic materials leads to a similar form for the binding energy variations [Thiel 2002] The simple two parameter model is successful for materials ranging from metals to ionic insulators 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

13. The 4fN-15d Binding Energies Combining 4fN to 4fN-15d transition energies with 4fN binding energies gives the binding energies of the 4fN-15d states • The lowest 4fN to 4fN-15d transitions in a material may be accurately described using a one-parameter empirical model [Dorenbos 2001] • The model for the 4fN-15d transition energies may be combined with the 4fN binding energy model to give a simple three-parameter empirical model that describes both the 4fN and 4fN-15d binding energies [Thiel et. al 2002] • Similar behavior is expected for other mixed configurations such as 4fN-16s and 4fN-16p These results show that the 4fN-15d binding energies are similar for all rare-earth ions in a host material with maximum variations of 0.5 eV 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

14. Inverse Photoemission and the 4fN+1 States Inverse Photoemission Spectroscopy (IPES) is the time reverse of PES and measures binding energies of unoccupied electron acceptor states • The same empirical model used for 4fN states may be applied to 4fN+1 • Energy difference between 3+ and 2+ states is not as large for ions in solids (~5-9eV) as for free ions (~15-20eV) • Polarization of lattice decreases the 4fN binding energy and increases the 4fN+1 binding energy • In metals DEL = -12.8 eV, and this estimate predicts DEL = -13.3 eV • In ionic materials, this rough estimate has errors up to 10-20% due to neglect of other terms in model IPES and PES values from Lang, Baer, & Cox 1981 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

15. The 4fN+1 States and Charge Transfer Understanding the location of 4fN+1 acceptor states relative to the host valence band is critical to understand charge transfer transitions [Happek et. al 2001] • We may compare IPES results to measured charge transfer energies to determine the regions of the valence band density of states (DOS) that have largest transition probability to the RE 4f orbitals • In ionic materials, the anion ligands have the greatest DOS near the VBM • This type of model was developed and applied over a wide range of materials to describe and predict relative RE charge transfer energies [Dorenbos 2003] • It has also been found that the aR parameter in the 4fN model has the same value for the 4fN+1 states if same set of radii are used for both [van der Kolk & Dorenbos 2006] IPES data from Park & Oh 1993 CT Absorption Data from Heaps, Elias, & Yen 1976, Yang & DeLuca 1978, Krupa, Gerard, & Martin 1993 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

16. To understand electron transfer processes, it is essential to understand how the lattice relaxes for the different states involved [de Boer & van Geel 1935] Effect of Lattice Relaxation • Relaxation of the total adiabatic energy of the ion & lattice may be approximately described for linear electron-lattice coupling using configurational coordinate diagrams • After a change in electronic state, the equilibrium position of ligands shifts on the timescale of the lattice vibrational frequencies • In the few cases studied in detail, the 4fN-1+e- ionized state relaxation energy is ~2-4 times larger than for 4fN-15d with the same sign of DQ • For ionized states, the optical and DC dielectric constants can be used to estimate the relaxation energy [Mott & Littleton 1938] Different ionization thresholds are measured by photoemission (PES), excited-state absorption (ESA), and photoconductivity (PC) techniques 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

17. Example Analysis for Pr3+:YAG Experimental Energy Diagram for Pr3+:Y3Al5O12 • Configurational coordinate energy curves are shown for 4fN (blue), 4fN-15d (red), and host band gap (crosshatched region) • The ionized Pr4+ state is indicated by the shaded region • The vertical “frozen lattice” energy of ionized Pr4+ was determined from our PES results • The measured relaxation energy of ionized states in RE3+:YAG is ~1.4 eV [Mayolet et. al 1995], which compares well with the calculated value of ~1.6 eV • The observed ESA threshold of photoionization from the lowest 4fN-15d state [Cheung & Gayen 1994] is plotted as an arrow • The dashed line is the energy where photoconductivity has been observed [Wittmann & Macfarlane 1996] This picture successfully explains all the observed processes in Pr3+:YAG 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

18. Material trends may be identified from analysis of 4f electron binding energies over a wide range of materials Material Trends for 4f Electron Energies • Rare-earth impurity concentration has no observable effect on the 4f binding energies when substituting for Y3+ or other RE • Crystal structure weakly affects 4f energies • Changing cations with the same valence has a weak effect on 4f binding energies • Changing anions has a significant effect • Binding energies decrease as covalency increases Variation in the relative energies of 4f electron and host band states between materials is mostly due to shifts in the host bands, with weaker shifts in the 4f electron energies observed 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

19. The electrostatic model predicts that material trends are dominated by initial-state energy effects (e.g. Madelung potential), but this generally does not agree well with observed trends in optical materials A Final-state Model for Material Trends • We find that electrostatic model calculations overestimate energy differences between materials and even predict the wrong sign for some material trends (RE-halides) • This is partly explained by changes in bonding covalency and ligand distances tending to compensate initial-state energy variations • From these considerations, we compare 4f binding energies to a simple empirical model that only includes lattice polarization effects • Using the simple Mott-Littleton approximation for Epol that only requires the index of refraction, we find a good linear correlation with the observed material trends This simple model is surprisingly successful  suggests that 4f electron energies may be accurately predicted using only the host crystal’s index of refraction 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

20. Conclusions • Photoemission and Inverse Photoemission spectroscopy are powerful tools for determining 4fN and 4fN+1 electron energies relative to host band structure • The 4fN-15d energies relative to host bands can be found from the 4fN binding energies and the 4fN to 4fN-15d transition energies • Simple models may be used to describe and predict all 4fN, 4fN-15d, and 4fN+1 energies in a host from measurements on one or two ions • Understanding lattice relaxation for ionized and reduced states is critical for predicting electron transfer processes and the stability of these states • ESA energies, photoionization thresholds, thermal activation energies, etc. may be obtained by measurement of 4fN and 4fN-15d energies relative to host states • A simple empirical model predicted the material dependent trends in 4fN binding energies of rare-earth-ions over a wide range of materials, but further testing is required to confirm the success of this model for additional material systems Acknowledgements This material is based on work supported by the Air Force Office of Scientific Research under Grants F49620-97-1-0411, F49620-98-1-0171, and F49620-00-1-0314, Scientific Materials Corporation, and the National Science Foundation under Grants 0903937 and DGE-4189-9553556. 17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010