Investigating Electron Binding
Energies of Impurity Ion States
and Host Crystal Bands in
Rare-Earth-Doped Optical Materials
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
Research was supported in part by the Air Force Office of Scientific Research,
Scientific Materials Corporation, and the National Science Foundation
Email: thiel@physics.montana.edu
17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010
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
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
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
Electron Photoemission Spectroscopy
 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
KE  Spectrum
EVacuum
Photoemission Directly
Measures Electron
Binding Energies Relative
to a Common Energy
Reference
Host Conduction Band (CB)
Extract Host and Ion Features:
VBM
Binding
Energy
Valence
Band
Maximum
(VBM)
hnp
4f Binding
Energy
RE3+ (4fN)
Ground State
Host Valence Band (VB)
 Resonant Photoemission
(RPES) exploits resonances
in the rare-earth PES crosssections to identify and
extract 4f electron PES
 May also compare spectra of
samples with different rareearth ion concentration
17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010
Photoemission Final-state Structure
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
30% Er:YAG
 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
4f Spectrum
Final-state Structure
Fit of FSS
Photoelectron Counts
 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
24
22
20 18 16 14 12 10
Electron Binding Energy (eV)
8
6
4
 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
Locating the Valence Band Maximum
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
Photoelectron Counts
YAG
hn = 125 eV
PES Data
Theory
Fit of Theory
 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
VBM at 8.7 eV
20.0
17.5
15.0
12.5
10.0
7.5
Electron Binding Energy (eV)
Orbital PES Cross Sections from Yeh & Lindau 1985
YAG PDOS from Xu & Ching 1999
5.0
 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)
17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010
Systematic Trends of 4f Binding Energies
The measured binding energies of the rare-earth ion 4f electrons display a
characteristic trend across the 4fN series
-4
YAG
-3
VBM
BE Relative to VBM (eV)
Experimental Spectra
Fit of Theory to Data
Final States
7%Gd:YAG
TbAG
DyAG
HoAG
-2
-1
1
2
3
4
30%Er:YAG
5
LuAG
20
PES Data
Optical Data
 This “zig-zag” trend is related to variation in
effective nuclear charge, inter-electronic
repulsion, 4f spin-pairing energy, and spinorbit coupling
50%Yb:YAG
22
YAG
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
TmAG
24
Band Gap
Valence Band
0
18
16
14
12
10
Binding Energy (eV)
8
6
4
 First quantitatively described by Jørgensen’s
Refined Spin-Pairing Energy Theory in 1962
17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010
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
An Empirical Model for 4f Binding Energies
A simple two-parameter semi-empirical form of the electrostatic model
accurately describes relative 4fN energies of all rare-earth ions in a material
0
Conduction Band
Electron Binding Energy (eV)
2
 The chemical shift consists of a large
constant shift (EL) and a smaller shift
that depends on ionic radius (aR)
[Pedrini, McClure, & Anderson 1979]
4
6
Egap
3+
RE :LaF3
8
4f
10
N
12
14
 Model the 4f binding energies as freeion values shifted by interactions with
the lattice—“Chemical Shift”
Valence Band
16
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
 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
Using Host PES to Predict 4f Energies
A consequence of the electrostatic model is that all core electrons of any
chemically similar ions should 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
Material
Peak
EBE (exp.)
EL (est.)
EL (exp.)

LiYF4
LiYF4
YF3
YF3
LaCl3
LaBr3
YAG
YAG
Y3+ 4s
Y3+ 3d5/2
Y3+ 4s
Y3+ 3d5/2
La3+ 4d5/2
La3+ 4d5/2
Y3+ 4p3/2
Y3+ 3d5/2
52.4 eV
164.7 eV
51.6 eV
164.3 eV
108.8 eV
108.8 eV
30.4 eV
162.5 eV
30.1 eV
29.8 eV
30.9 eV
30.2 eV
30.8 eV
30.8 eV
31.4 eV
32.0 eV
29.9 eV
29.9 eV
30.2 eV
30.2 eV
31.2 eV
31.3 eV
31.5 eV
31.5 eV
0.2 eV
-0.1 eV
0.7 eV
0.0 eV
-0.4 eV
-0.5 eV
-0.1 eV
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
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
4
Elemental Rare-earth Metals
Binding Energy (eV)
5
6
 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
7
8
9
10
EF=3.3 eV
EL=34.2 eV
aR=17.9 eV/Å
11
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
The simple two parameter model is
successful for materials ranging from
metals to ionic insulators
 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]
17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010
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
0
Conduction Band
Electron Binding Energy (eV)
2
4
N-1
4f 5d
N-1
Measured 4f 5d Energies
6
LaF3
8
10
N-1
Model For Lowest 4f 5d Energies
N
Measured 4f Energies
N
Model for 4f Ground State Energies
N
4f
12
Valence Band
14
16
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
 The lowest 4fN to 4fN-15d
transitions in a material may be
accurately described using a oneparameter 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
Inverse Photoemission and the 4fN+1 States
Binding Energy Relative to Fermi Energy (eV)
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
-6
Elemental Rare Earth Metals
-4
N+1
4f
EL=21.4 eV
N+1
4f
a=8.3 eV/Å
-2
0
Valence Band
"Divalent"
4f Configuration
N
2
4
 Polarization of lattice decreases the
4fN binding energy and increases the
4fN+1 binding energy
N
4f
N
6
 Energy difference between 3+ and 2+
states is not as large for ions in solids
(~5-9eV) as for free ions (~15-20eV)
4f
EL=34.2 eV
a=17.7 eV/Å
"Trivalent"
4f Configuration
N
8
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
IPES and PES values
from Lang, Baer, & Cox 1981
 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
17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010
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]
N+1
Electron Binding Energy (eV)
0
4f
Conduction Band
2
4
Optical Data
6
8
10
LaF3
Charge
Transfer
Transitions
IPES Data
N+1
4f Model
12
14
Valence Band
 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]
16
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
IPES data from Park & Oh 1993
CT Absorption Data from Heaps, Elias, & Yen 1976,
Yang & DeLuca 1978, Krupa, Gerard, & Martin 1993
 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]
17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010
Effect of Lattice Relaxation
To understand electron transfer processes, it is essential to understand how
the lattice relaxes for the different states involved [de Boer & van Geel 1935]
U (Q) = E0 - AQ + 12 M w2Q2
Q0 =
A
M w2
ER =
1
2
M w2D Q02
Ionization
Threshold
Excited
State
Energy
ESA
PES
PC
Ground
State
Configurational Coordinate Q
Different ionization thresholds are measured by
photoemission (PES), excited-state absorption
(ESA), and photoconductivity (PC) techniques
 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]
17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010
Example Analysis for Pr3+:YAG
Experimental Energy Diagram for Pr3+:Y3Al5O12
3+ 3
+
-
Pr ( H4) + h + e
8
7
4+ 2
Energy (eV)
-
Pr ( F5/2) + e
6
5
3+
1
Pr (4f 5d)
4
3
2
3+ 3
1
Pr ( H4)
Pr3+:YAG
0
0
Configurational Coordinate
This picture successfully explains all the
observed processes in Pr3+: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]
17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010
Material Trends for 4f Electron Energies
Material trends may be identified from analysis of 4f electron binding
energies over a wide range of materials
6
3+
Ce 4f electrons
8
Valence
Band
Maximum
9
10
0
11
CeF3
CeCl3
CeBr3
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
3
4f
N-1
5d
4
5
6
7
8
9
VB
10
11
4f
N
12
13
Ce
17th
YIG
CB
2
12
YGG
YAG
1
Binding Energy (eV)
Binding Energy (eV)
7
 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
Gd
Lu Ce
Gd
Lu Ce
Gd
International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010
Lu
A Final-state Model for Material Trends
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
 We find that electrostatic model
calculations overestimate energy
differences between materials and
even predict the wrong sign for
some material trends (RE-halides)
Measured Chemical Shift (eV)
35
Metals
EL = 24.8 eV – 1.31·Epol
34
33
 This is partly explained by changes
in bonding covalency and ligand
distances tending to compensate
initial-state energy variations
Oxides
32
Chlorides
 From these considerations, we
compare 4f binding energies to a
simple empirical model that only
includes lattice polarization effects
31
Fluorides
30
–3.5
–4.0
–4.5
–5.0
–5.5
–6.0
–6.5
–7.0
Calculated Polarization Energy (eV)
This simple model is surprisingly successful
 suggests that 4f electron energies may be
accurately predicted using only the host
crystal’s index of refraction
–7.5
 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
17th International Conference on Dynamical Processes in Excited States of Solids, Argonne National Laboratory, June 2010
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