17 th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010 C. W. Thiel a,b and R. L. Cone

17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010

C. W. Thiela,b and R. L. Conea

a Physics Department, Montana State University, Bozeman, MT, USA 59717b Spectrum Lab, Montana State University, Bozeman, MT, USA 59717

Email: [email protected]

Investigating Electron BindingInvestigating Electron Binding

Energies of Impurity Ion StatesEnergies of Impurity Ion States

and Host Crystal Bands inand Host Crystal Bands in

Rare-Earth-Doped Optical Materials 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


Electronic Structure of Rare-earth MaterialsElectronic Structure of Rare-earth Materials

4f N15d

4f N

Rare-earth Ion

The atomic-like electronic structure of localized rare-earth

ion states is well understood

Host Crystal

Conduction Band States

Valence Band States

The electronic band structure ofde-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

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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

Importance of Crystal Band StatesImportance of Crystal Band States

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


Methods for Studying Electron Transfer 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


Photoemission Directly Measures Electron

Binding Energies Relative to a Common Energy

Reference

Incident photons eject electrons from the occupied states in the material

Difference between Photon Energy (hp) 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

Electron Photoemission SpectroscopyElectron Photoemission Spectroscopy

RE3+ (4fN) Ground State

EVacuum

KE Spectrum

hp 4f Binding Energy

ValenceBand

Maximum(VBM)

VBM Binding Energy

Host Valence Band (VB)

Host Conduction Band (CB)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


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

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)

Photoemission Final-state Structure Photoemission Final-state Structure


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

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)

Locating the Valence Band Maximum Locating the Valence Band Maximum

Orbital PES Cross Sections from Yeh & Lindau 1985YAG PDOS from Xu & Ching 1999


The measured binding energies of the rare-earth ion 4f electrons display a characteristic trend across the 4fN series

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’s Refined Spin-Pairing Energy Theory in 1962

Systematic Trends of 4f Binding Energies Systematic Trends of 4f Binding Energies


Electron Binding Energies in Ionic Crystals 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]


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 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 (R) [Pedrini, McClure, & Anderson 1979]

If we treat the model parameters EL and R 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


A consequence of the electrostatic model is that all core electrons of any chemically similar ions should experience the same chemical shift

Using Host PES to Predict 4f EnergiesUsing Host PES to Predict 4f Energies

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 R ~ 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!


The model may be tested using 4f electron binding energies in the elemental rare-earth metals, the opposite extreme from ionic insulators

Non-ionic MaterialsNon-ionic Materials

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]

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu11

10

9

8

7

6

5

4 Elemental Rare-earth Metals

EF=3.3 eV

EL=34.2 eV

R=17.9 eV/Å

Bin

ding

Ene

rgy

(eV

)

The simple two parameter model is successful for materials ranging from metals to ionic insulators


Combining 4fN to 4fN-15d transition energies with 4fN binding energies gives the binding energies of the 4fN-15d states

The 4fThe 4fN-1N-15d Binding Energies5d Binding Energies

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

La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

16

14

12

10

8

6

4

2

0

Measured 4fN-1

5d Energies

Model For Lowest 4fN-1

5d Energies

Measured 4fN Energies

Model for 4fN Ground State Energies

4fN

4fN-1

5d

LaF3

Conduction Band

Valence Band

Ele

ctro

n B

ind

ing

En

erg

y (e

V)

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


Inverse Photoemission Spectroscopy (IPES) is the time reverse of PES and measures binding energies of unoccupied electron acceptor states

Inverse Photoemission and the 4fInverse Photoemission and the 4fN+1N+1 States 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 EL = -12.8 eV, and this estimate predicts EL = -13.3 eV

In ionic materials, this rough estimate has errors up to 10-20% due to neglect of other terms in model

La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu8

6

4

2

0

-2

-4

-6

"Divalent"

4fN Configuration

"Trivalent"

4fN Configuration

4fN+1

Valence Band

4fN+1

EL=21.4 eV

=8.3 eV/Å

4fN

EL=34.2 eV

=17.7 eV/Å

Elemental Rare Earth Metals

Bin

ding

Ene

rgy

Rel

ativ

e to

Fer

mi E

nerg

y (e

V)

4fN

IPES and PES valuesfrom Lang, Baer, & Cox 1981


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]

The 4fThe 4fN+1N+1 States and Charge Transfer States and Charge Transfer

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 R 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]


16

14

12

10

8

6

4

2

0

IPES Data

4fN+1 Model

LaF3

4fN+1

Conduction Band

Valence Band

Ele

ctro

n B

ind

ing

En

erg

y (e

V)


16

14

12

10

8

6

4

2

0

ChargeTransfer

Transitions

Ele

ctro

n B

ind

ing

En

erg

y (e

V)

Optical Data

IPES data from Park & Oh 1993CT Absorption Data from Heaps, Elias, & Yen 1976,

Yang & DeLuca 1978, Krupa, Gerard, & Martin 1993


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 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 Q

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

2 210 2( )U Q E AQ M Qw= - + 0 2

AQ

Mw= 2 21

02RE M Qw= D


Example Analysis for PrExample Analysis for Pr3+3+:YAG :YAG 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]

0

0

1

2

3

4

5

6

7

8Pr

3+(3H4) + h

+ + e

-

Pr3+

(4f15d)

Pr4+

(2F5/2) + e

-

Pr3+

(3H4)

Pr3+:YAG

Ene

rgy

(eV

)

Configurational Coordinate

This picture successfully explains all the observed processes in Pr3+:YAG

Experimental Energy Diagram for Pr3+:Y3Al5O12


Material trends may be identified from analysis of 4f electron binding energies over a wide range of materials

Material Trends for 4f Electron Energies 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

13

12

11

10

9

8

7

6

5

4

3

2

1

0

LuGd

CB

4fN

4fN-15d

VB

Ce LuGdCe LuGdCe

Bin

din

g E

ne

rgy

(eV

)

YIGYGGYAG

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


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 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


Conclusions 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 4f N 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

AcknowledgementsThis 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.

Documents

17 th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010 C. W. Thiel a,b and R. L. Cone