ultrashort lifetime expansion for resonant inelastic x-ray scattering luuk ament in collaboration...
TRANSCRIPT
![Page 1: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/1.jpg)
Ultrashort Lifetime Expansionfor
Resonant Inelastic X-ray Scattering
Luuk AmentIn collaboration with
Jeroen van den Brink and Fiona Forte
![Page 2: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/2.jpg)
What is RIXS?
Synchrotron radiation
Interestingsample
1s
4p
res (5-10 keV)
ResonantInelasticX-ray Scattering
Momentumtransfer q
Energy loss Core-hole
![Page 3: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/3.jpg)
Why RIXS?
Complementary to Raman and neutrons
• Momentum dependence• Tune X-rays to different
resonances• Small samples• Why not?
![Page 4: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/4.jpg)
Outline
1. Framework: the UCL expansionWhat can one measure?
2. Charge excitations3. Magnetic RIXS on La2CuO4
Coupling of core-hole to spins• Cu K-edge: indirect RIXS• Cu L-edge: direct RIXS
4. Orbital RIXS on LaMnO3
![Page 5: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/5.jpg)
Typical RIXS spectra
M.Z. Hasan et al. Science 288, 1811 (2000)
![Page 6: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/6.jpg)
What happens in detail
initial finalintermediate
1s core
conduction
4p empty
valence
Ucore Ucore
![Page 7: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/7.jpg)
What happens in detail
initial final
1s core
conduction
4p empty
valence
Ultrashort Core-holeLifetime (UCL):Integrate out
intermediate states!
intermediate
Ucore Ucore
![Page 8: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/8.jpg)
Theory
• Kramers-Heisenberg relation
with
Intermediate statesDetuning fromresonance
Inverse core-holelifetime
![Page 9: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/9.jpg)
Theory - UCL expansion
Using the Ultrashort Core-hole Lifetime gives
Scattering amplitude
![Page 10: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/10.jpg)
What does RIXS measure?
We obtain
Intermediate stateHamiltonian
Valenceelectrons
Interaction ofvalence electrons
with core-hole
J. van den Brink & M. van Veenendaal, EPL 73, 121 (2006)L. Ament, F. Forte & J. van den Brink, PRB 75, 115118 (2007)
At T = 0, approximate for both strong and weak potentials
Arbitrary range hopping & Coulomb interactions
Assume immobile core-hole,and local core-hole potential
![Page 11: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/11.jpg)
What does RIXS measure?Spinless fermions:
J. van den Brink & M. van Veenendaal, EPL 73, 121 (2006)L. Ament, F. Forte & J. van den Brink, PRB 75, 115118 (2007)
Fermions with spin:
Longitudinal spin densityCharge density
with
Chargedensity
Resonantprefactor
![Page 12: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/12.jpg)
Can we probe other excitations than charge ones?
What does RIXS measure?
Conclusion: core-hole couples to charge.
RIXS measures charge density andlongitudinal spin density (= higher order chargedensity correlation function).
Data on Cu K-edge of La2CuO4 below Mott gap (~ 2 eV)
Phonons?d-d excitation?Magnons?
Nd2CuO4: no shift500 meV peak!
J. van den Brink & M. van Veenendaal, EPL 73, 121 (2006)L. Ament, F. Forte & J. van den Brink, PRB 75, 115118 (2007)J. Hill et al., arXiv:0709.3274
![Page 13: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/13.jpg)
Magnetic RIXS on La2CuO4
Perovskite layers of CuO2:
Cu 3d eg holes at low energy:
Single band Hubbard model
Ground state is antiferromagnetically orderedCollective excitations: magnons
![Page 14: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/14.jpg)
Magnetic RIXS on La2CuO4
How does the core-hole couple to the spins?Go back to Hubbard model!
Hopping amplitude t Coulomb repulsion U
K-edge:Core-hole locally modifies superexchange!
Coulomb repulsion U - Uc
![Page 15: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/15.jpg)
4-spin correlation function
Magnetic RIXS on La2CuO4
How does the core-hole couple to the spins?Indirect RIXS locally modifies superexchange!To first order in UCL:
At T = 0, 2-magnon excitations are probed:
J. van den Brink, cond-mat/0510140F. Forte, L. Ament & J. van den Brink, arXiv:0705.0263
![Page 16: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/16.jpg)
Magnetic RIXS on La2CuO4
Selection rule: vanishing intensity atq = (0,0) and q = (,)
J. Hill et al., arXiv:0709.3274
F. Forte, L. Ament & J. van den Brink, arXiv:0705.0263
![Page 17: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/17.jpg)
J
Magnetic RIXS on La2CuO4
J. Hill et al., arXiv:0709.3274
Theory
Improve theorywith longer rangeinteractions:
Values from neutron scattering.Coldea et al., PRL 86, 5377 (2001)
Theory + experimentalresolution
F. Forte, L. Ament & J. van den Brink, arXiv:0705.0263
![Page 18: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/18.jpg)
Magnetic RIXS on La2CuO4
F. Forte, L. Ament & J. van den Brink, arXiv:0705.0263J. Hill et al., arXiv:0709.3274
![Page 19: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/19.jpg)
Magnetic RIXS on La2CuO4
L-edge: direct RIXS - photoexcited electron in 3d-shell. G. Ghiringhelli et al. are measuring this right now!
Sneak preview:• Dispersive feature similar to K-edge RIXS,
becoming elastic in the low-q limit.• Another feature appearing at q = 0 around = 3 J.• This feature disappears off-resonance.
What about the theory?
![Page 20: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/20.jpg)
Magnetic RIXS on La2CuO4
L-edge: direct RIXS - core-electron ends up in 3d-stateBack to Hubbard model!
The UCL results for the K-edge and L-edge are the same!
The extra 3d-electron locally blocks superexchange
![Page 21: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/21.jpg)
Magnetic RIXS on La2CuO4
Currently, q = 0 is being measured by G. Ghiringhelli et al.
To be continued!
L-edge has more intensity than K-edge.Can we go beyond the UCL approximation?
Off-resonance: bigger, and becomes more important!
Reminder
![Page 22: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/22.jpg)
• LaMnO3 has a cubic structure:
Outlook - Orbital RIXS
LaMnO3: A-type AFM order:spins align FM in each layer.
• MnO6 octahedron induces crystal field:
3deg
t2g
eg
t2g
![Page 23: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/23.jpg)
Kugel-Khomskii model without Hund’s rule coupling:
To first order, orbitals of different layers decouple!
Orbital RIXS
eg
t2g
Order by superexchangeand/or Jahn-Tellerdistortion of octahedra
• eg orbitals order ‘antiferro-orbitally’:
• Excitations: eg orbital waves (analogous to spin waves)
![Page 24: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/24.jpg)
Orbital RIXS
Looks like Heisenberg, but no conservation of pseudo-Sz. This leads to single orbiton excitations.
S. Ishihara & S. Maekawa, PRB 62, 2338 (2000)J. van den Brink, P. Horsch, F. Mack & A. M. Oles, PRB 59, 6795
(1999)
eg
Initial FinalIntermediate
![Page 25: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/25.jpg)
Orbital RIXS
U’=U+Uc
U’=U-Uc
1s
4p
Initial
3d egIntermediate
4p
Final
1s
3d eg
Again, core-hole locallymodifies superexchange!
How does the core-hole couple to the eg orbitals?
![Page 26: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/26.jpg)
Orbital RIXS
One-orbiton peak
Two-orbiton continuum
Experimental limitations:Resolution = 100 meV, inelastic peak
J = 25 meV
Orbital RIXS spectrum for LaMnO3
![Page 27: Ultrashort Lifetime Expansion for Resonant Inelastic X-ray Scattering Luuk Ament In collaboration with Jeroen van den Brink and Fiona Forte](https://reader037.vdocuments.us/reader037/viewer/2022110320/56649c9e5503460f9495de23/html5/thumbnails/27.jpg)
Conclusions
• Experimental progress is enormous
• Theoretical spectra can be easily obtained with the UCL expansion using e.g. a model Hamiltonian
• We now know which charge correlation functions we measure in indirect RIXS
• Both direct and indirect RIXS are a new probe for magnons (La2CuO4)
• We can possibly detect orbitons in the manganites!