Electronic Structure and Transition Intensities in Rare-Earth Materials

Transcription

1 Electronic Structure and Transition Intensities in Rare-Earth Materials Michael F Reid School of Physical and Chemical Sciences Te Kura Matū University of Canterbury Christchurch, New Zealand January 2018 v03 1

2 Outline Background on rare-earth (lanthanide) ions. States and transitions Effective Hamiltonian for 4f N Some simple calculations of energy levels Transition intensities 4f N-1 5d Ab-initio calculations 2

3 Gerhard Dieke Johns Hopkins

4 Bill Carnall 4

5 1960 s Theory: Judd, Ofelt, Wybourne 5

6 s Filling of orbitals p f d s 6 6

7 Collapse of 4f orbitals JP Connerade,

8 Lanthanide 3+ ground state: 5s 2 5p 6 4f N 5d 0 5d 4f 5s 5p 8

9 4f N Lanthanides: 4f N, 4f N-1 5d, Excitons Sharp lines Long lifetimes So ideal for laser and phosphor applications 4f N-1 5d Broad absorption bands from 4f N Useful for absorbing energy Short lifetimes useful in some applications, such as scintillators Excitons Excited electron can become delocalized, giving an excitonic state. e - Charge Transfer Transitions Ligand to lanthanide electron transfer 9

10 Transitions 4f N 4f N No configuration shift Sharp lines 4f N 4f N 5d Configuration shift Broad bands Excitonic states 4f N Large configuration shift Very broad bands 10

11 How do we proceed? Ab-initio (first principles) calculations Well established in atoms Hartree-Fock + perturbations Now viable for lanthanide complexes but slow Effective Hamiltonians ( crystal field ) Requires parameter fitting Relatively quick and easy, allowing rapid interpretation of spectra. Can be related to ab-initio calculations. 11

12 Tutorial Mike Reid, Personal Home Page Electronic Structure Calculations Handout. Tutorial Presentation (Asia, January 2018). 12

13 Effective Hamiltonian Calculations H Ψ i > = E i Ψ i > (H is Hamiltonian) H eff φ i > = E i φ i > (H eff is Effective Hamiltonian) E Theory E Expt ( H) (H eff ) (H eff ) Σ α P α (M α ) 13

14 z Crystal Field ( B k ( k ) q C ) q k,q - - f z x,y L Electrostatic Overlap Covalency All increase energy of z orbital more than x,y Orbital energies crystal-field parameters 14

15 Understanding the energy levels: 4f N - Coulomb Spin-orbit Crystal-field 15

16 Effective Hamiltonian for 4f N H=E avg + k= 2,4,6 F k f k +ζ f A so + k,q B q k C q ( k ) Coulomb Spin-Orbit Crystal Field Correlation and other corrections +αl( L+1 )+βg (G 2 )+γg (R 7 )+ + h=0,2,4 M h m h + k= 2,4,6 P k p k i= 2,3,4,6,7,8 T i t i 16

17 Parameter trends across the lanthanide series. C-K Duan and P A Tanner, J. Phys. Chem. A, 2010, 114, pp Coulomb F 2 F 4 F 6 ζ Spin-orbit F 4 /F 2 Ln 3+ in Cs 2 NaLnCl 6 B 4 F 6 /F 2 B 6 Crystal-field

18 Dieke Diagram 1960s Carnall, Goodman, Rajnak, Rana, ,000 cm nm 2.5 ev 18

19 Calulating Matrix elements Wigner-Eckart Theorem matrix element geometrical factors 3j symbol or Clebsh-Gordan Coeffieient reduced matrix element Selection Rules M +q = M J k J form a triangle: J-J k J+J SLJM> states More complex versions of WET. Triangle rules for S S L L J J and operator labels. 19

20 Ce 3+ : 4f 1 5d 1 2 F J=7/2 E avg ζ = 2253 cm -1 S=1/2 L=3 J=5/2 E avg ζ = 0 cm -1 Free-ion splitting is 2253 cm -1 so ζ = 644 cm -1 Similarly for 5d 1 20

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22 < 2 F 5/2,M A so 2 F 5/2,M > =

23 Splitting of J=5/2 multiplet J=5/2 J=5/2 M=+5/2 M=+3/2 M=+1/2 M=-1/2 M=-3/2 M=-5/2 M=±1/2 E=258 M=±3/2 E=172 M=±5/2 E= 0 Magnetic field Even spacing: 0.4 cm -1 for 1T field Crystal Field Uneven spacing Example: B 2 0 = 500 cm -1 More complex cases mix up M labels. 23

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25 Splitting: ( ) * = 0.4 cm -1 25

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27 ±1/ *500 = ±3/ *500 = ±5/ *500 =

28 Superposition Model 28

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30 z y x x: 3* = -1 y: 3* = -1 z: 3* = +2 Sum = 0. 30

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32 Pr 3+ : 4f 2 Coulomb + Spin-orbit 1 G 3 F 3 H Coulomb operators: S=0, L=0, J=0 J= Spin-orbit operator: S=1, L=1, J=

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35 Tm 3+ : 4f 12 Spin-orbit much larger and matrix elements change sign. J=4 1 G 3 F H

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37 Transition Intensities Electric Dipole, Magnetic Dipole, ED between 4f N and 4f N-1 5d can be calculated directly But require modelling of vibronic bands. ED within 4f N are parity forbidden. Construct Effective ED operator that accounts for mixing of configurations of opposite parity on ion or ligand. First detailed treatment: Judd, Ofelt,

38 Effective Electric Dipole Operator Can derive a parametrization. λ=2,4,6, t=λ±1, λ Dipole strength Oscillator strength Einstein A coefficients (1/τ) 38

39 Simulation Burdick, et al. Phys. Rev. B: 50: (1994) nm 39 39

40 Multiplet-Multiplet transitions Judd 1962 For solutions and glasses at room temperature. Sum over all states in a multiplet and all polarizations. Reduces to three-parameter linear fit. Ω λ parameters with λ=2,4,6 Over 3000 citations! 40

41 Eu 3+ : 5 D 0 7 F J Emission MD ED Ω 2 Ω 4 41

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45 Eliminate options: Pr-Yb downconversion 1 G 4 3 H 4 can never be strong 45

46 Understanding the energy levels: 4f N-1 5d T 2 Cubic: higher energy E Cubic: lower energy Crystal field Coulomb, 46 etc 40 x 4f (20,000 vs 500) 46

47 Conduction Band, Free Electrons, Excitons Conduction Band 5d 4f Valence Band 47 47

48 T 2 E CaF 2 (cubic sites) Ce 3+ : 4f 1 5d 1 Energy Pr 3+ : 4f 2 4f 1 5d 1 Nd 3+ : 4f 3 4f 2 5d 1 48

49 First-Principles Calculations 49 Relativistic ab-initio calculations are now possible for these systems. Ogasawara et al. J. Solid State Chem. 178, 412 (2005). Calculations for entire series. Some inaccuracies. Seijo et al. J. Chem. Phys. 125, (2006) Very accurate and detailed calculations for particular ions, including potential surfaces. Pr 3+ :LiYF 4 49

50 Excited-state geometry: CaF 2 :Ce 3+ Pascual, Schamps, Barandiaran, Seijo, PRB 74, (2006) BaF 2 :Ce 3+ cubic sites. Potential surfaces: 5d E is contracted 5d T 2 is expanded As bond length contracts 6s orbital becomes delocalized. Ce 3+ : 4f 1 5d 1 T 2 E Energy Ce 3+ :CaF 2 4f 1 5d

51 SrCl 2 :Yb 2+ Sánchez-Sanz et al. J. Chem. Phys. 133, Some of our recent work is on extracting parameters from these calculations. Absorption 51

52 Conclusion Effective Hamiltonian for 4f N Examples of energy level calculations Transition intensities 4f N-1 5d Ab-initio calculations Further information and exercises: mike.reid@canterbury.ac.nz 52

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