Neutron scattering from quantum materials

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1 Neutron scattering from quantum materials Bernhard Keimer Max Planck Institute for Solid State Research Max Planck UBC UTokyo Center for Quantum Materials Detection of bosonic elementary excitations in quantum materials by inelastic neutron scattering theory & instrumentation applications to cuprates & ruthenates understanding of unconventional metallicity & superconductivity

2 Quantum materials conventional superconductor infinite conductivity metal finite conductivity quantum oscillations at low temperatures Mott insulator zero conductivity energy gap Fermi sphere subtle re-entanglement well understood after ~100 years of research massive re-entanglement frontier of research

3 Quantum materials diverse electronic ordering phenomena near Mott metal-insulator transition manganates cuprates ruthenates cobaltates Dagotto et al. Science 2005

4 Neutron scattering neutron E 1 q 1 excitation: E = E 1 E 2 E 2 q 2 interaction q = q 1 q 2 strong (nuclear) interaction elastic lattice structure inelastic lattice dynamics magnetic (dipole-dipole) interaction elastic magnetic structure inelastic magnetic excitations additional features of neutron scattering five-dimensional data sets: E, q, intensity + temperature, magnetic field, pressure etc. scattering cross section precisely understood

5 Elastic neutron scattering

6 Elastic neutron scattering Q = k f k i momentum transfer

7 Elastic nuclear neutron scattering scattering length b ~ size of nucleus ~ m Bragg peaks at reciprocal lattice vectors K

8 Elastic nuclear neutron scattering

9 Inelastic neutron scattering elastic cross section dσ dω # of neutrons scattered into dω = (unit time) (incident flux) inelastic cross section d 2 σ dedω # of neutrons scattered into dω = (unit time) (incident flux) (energy) inelastic nuclear neutron scattering initial, final state of sample energy of excitation created by neutron in sample partition function

10 Inelastic nuclear neutron scattering thermal average Debye-Waller factor due to thermal lattice vibrations K) K)} phonon creation neutron energy loss phonon annihilaion neutron energy gain

11 Neutron spectroscopy analyzer monochromator sample detector Bertram Brockhouse Nobel Prize 1994

12 FRM-II

13 Phonon dispersions in Pb Brockhouse, PRL 1962 well described by modern ab-initio lattice dynamics

14 Conventional superconductors quantitative description based on pairing bosons electron pairing boson electron bosonic spectrum from neutron scattering fermionic spectrum from tunneling Brockhouse, PRL 1962 Savrasov, PRB 1996

15 Quantum materials conventional superconductor infinite conductivity metal finite conductivity Mott insulator zero conductivity energy gap Fermi sphere subtle re-entanglement well understood after ~100 years of research massive re-entanglement frontier of research

16 Elastic magnetic neutron scattering

17 Elastic magnetic neutron scattering one electron classical electron radius non-spin-flip σ z σ x, σ y spin-flip (not possible for nuclear scattering) separate nuclear and magnetic neutron scattering by spin polarization analysis unpolarized beam average spin-flip and non-spin-slip channels

18 Elastic magnetic neutron scattering one atom approximated as magnetized sphere, magnetization density M(r)

19 Elastic magnetic neutron scattering generalization for collinear magnets Bragg peaks polarization factor magnetic structure factor magnetic reciprocal lattice vectors

20 YBa 2 Cu 3 O 6+x lattice structure YBa 2 Cu 3 O 7, YBCO (T c ~ 90 K) electronic structure Cu d-orbitals x 2 -y 2 CuO 2 3z 2 -r 2 CuO 2 yz xz xy dopant O 2- ions arranged in chains hole content in x 2 -y 2 orbital controlled by oxygen concentration

21 YBa 2 Cu 3 O 6 spin structure spin orientation extracted from magnetic Bragg reflections J 2 Tranquada et al., PRB 1989 layer b J 1 layer a J H = Σ ij (J S i (a,b) S j (a,b) ) + Σ i (J 1 S i (a) S i (b) + J 2 S i (b) S i (a) ) sign, but not strength of exchange parameters determined by elastic neutron scattering

22 YBa 2 Cu 3 O 6+x phase diagram 400 temperature (K) AFI SC hole concentration

23 Iron pnictide superconductors electron concentration (x) hole concentration (x) lattice structure different from cuprates phase diagram very similar to cuprates focus on magnetic mechanisms of Cooper pairing

24 Understanding unconventional superconductors working hypothesis electronic and bosonic quasiparticles electron boson electron key experimental challenges detect collective excitations in high-t c superconductors by inelastic scattering detect feedback mechanims of superconductivity on bosonic spectra quantify strength of pairing interaction calculate T c, energy gap, Eliashberg theory is the only method that is currently available. working hypothesis pairing bosons = spin fluctuations d-wave superconductivity

25 Inelastic magnetic neutron scattering polarization factor fluctuation-dissipation theorem spin-spin correlation function dynamical magnetic susceptibility response to time- and position-dependent H-field

26 Inelastic magnetic neutron scattering localized electrons Heisenberg antiferromagnet, magnon creation - ˆQηˆ K m ) K m, a = 0, 1 q, K m magnon dispersions

27 YBa 2 Cu 3 O 6 magnons H = Σ ij (J S i (a,b) S j (a,b) ) + Σ i (J 1 S i (a) S i (b) + J 2 S i (b) S i (a) ) E 200 mev layer b layer a J 2 J 1 70 mev (π,π) acoustic q optic J exchange parameters from magnon dispersions J ~ 100 mev J 1 ~ 10 mev J 2 ~ 0.01 mev Tranquada et al., PRB 1989 Reznik et al., PRB 1996

28 Ca 2-x Sr x RuO 4 phase diagram Mott insulator-metal transition driven by electronic bandwidth through Ru-O-Ru bond angle

29 Longitudinal Higgs mode in Ca 2 RuO 4 spin-polarized triple-axis neutron scattering spin waves Higgs mode damping at q=(π,π) due to decay into transverse modes Higgs mode well defined at q=(0,0) strongly damped at q=(π,π) Jain et al., Nature Phys Max Krautloher presentation

30 Ca 2-x Sr x RuO 4 phase diagram Mott insulator-metal transition driven by electronic bandwidth through Ru-O-Ru bond angle

31 Inelastic magnetic neutron scattering ), ( ) ( 1 ), ( ), ( 0 0 ω χ ω χ ω χ q q J q q = + = + + k k q k k q k i E E E f E f q ε ω ω χ ) ( ) ( ) ( ), 0( h itinerant electrons electrons Lindhard function & RPA band dispersions RPA expression Fermi sphere E q q-dependent enhancement of χ by correlations

32 Sr 2 RuO 4 spin excitations Fermi surface from ARPES χ (q,ω) from RPA calculation strongly nested Mazin et al., PRL 1999

33 Sr 2 RuO 4 spin excitations from inelastic neutron scattering Iida et al., PRB 2011 spin fluctuation mediated superconductivity?

34 Magnetic exitations in cuprates antiferromagnetic insulator superconductor E 300 mev E mev temperature (K) (π, π) magnons q (π, π) q paramagnons AFI SC hole concentration

35 Magnetic resonant mode Neutron intensity Energy (mev) Energy (mev) Inosov et al., Nature Phys Suchaneck et al., PRL 2010 paramagnons in normal state magnetic short-range order feedback effect of superconductivity on paramagnon spectrum similar amplitude, T-dependence in two families of high-t c superconductors

36 INS from superconductors coherence factor 1 εε k k+ q+ k k+ q f ( Ek+ q) f ( Ek) k 2 EE ω ( E E) + iδ χ( q, ω) =Σ { (1 + ) k k+ q k+ q k 1 εε k k+ q+ k k+ q 1 f ( Ek+ q) f ( Ek) 4 (1 EE ) ω+ ( E + E) + iδ + k k+ q k+ q k 1 εε k k+ q+ k k+ q f ( Ek+ q) + f ( Ek) 1 4 EE ω ( E + E) + iδ + (1 ) } k k+ q k+ q k scattering of thermally excited pairs pair annihilation pair creation E = ε k k k Fong et al., PRL 1995 Monthoux & Scalapino, PRL 1994 χ 0 at q = (π,π) in s-wave superconductor resonant mode implies sign change in superconducting gap function d-wave in cuprates, s ± in iron pnictides

37 Magnetic resonant mode spin excitations of a d-wave superconductor RPA reproduces lower branch of hour-glass dispersion Imχ excitonic collective mode superconducting energy gap 2 _ + + _ incoherent spin flips ω dispersion of resonant mode q (π,π) direct Umklapp momentum-space signature of Cooper-pair wave function Eremin et al. PRL 2005

38 Paramagnon-mediated superconductivity antiferromagnetic paramagnons from neutron scattering electronic band dispersions from photoemission q 2 q 1 q 1 quantitative cross-correlation q 2 paramagnon electron electron Dahm et al., Nature Phys. 2009

39 Spin dynamics from neutron scattering antiferromagnetic insulator superconductor E 300 mev E mev temperature (K) AFI (π, π) spin waves q SC (π, π) q magnetic resonant mode neutron blind spots - high energies - high doping levels RIXS hole concentration

40 Resonant inelastic x-ray scattering (RIXS) triple-axis spectrometry with soft x-rays order-of-magnitude increase in in energy resolution L. Braicovich, G. Ghiringhelli (Politecnico Milano) La 2 CuO 4 Cu 2p 3d photon energy ~ 931 ev incoming photon magnon scattered photon Energy loss (ev) 2008

41 Resonant inelastic x-ray scattering (RIXS) RIXS spectrometer e.g. ESRF total length ~ 12 m

42 ERIXS ESRF