Neutron scattering from quantum materials Bernhard Keim er Max Planck Institute for Solid State Research Max Planck – UBC – UTokyo Center for Quantum Materials Detection of bosonic elem entary excitations in quantum m aterials by inelastic neutron scattering ● theory & instrumentation ● applications to cuprates & ruthenates understanding of unconventional metallicity & superconductivity
Quantum materials conventional Mott insulator m etal superconductor zero conductivity finite conductivity infinite conductivity quantum oscillations at low temperatures Fermi sphere energy gap subtle re-entanglem ent m assive re-entanglem ent well understood frontier of research after ~ 100 years of research
Quantum materials diverse electronic ordering phenom ena near Mott m etal-insulator transition cuprates m anganates ruthenates cobaltates Dagotto et al. Science 2005
Neutron scattering neutron E 1 q 1 excitation: E = E 1 − E 2 E 2 q 2 q = q 1 − q 2 interaction strong ( nuclear) interaction m agnetic ( dipole-dipole) interaction elastic lattice structure elastic magnetic structure inelastic lattice dynamics 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
Elastic neutron scattering
Elastic neutron scattering Q = k f – k i momentum transfer
Elastic nuclear neutron scattering scattering length b ~ size of nucleus ~ 10 -15 m Bragg peaks at reciprocal lattice vectors K
Elastic nuclear neutron scattering
Inelastic neutron scattering d σ d Ω # of neutrons scattered into = elastic cross section • d Ω (unit time) (incident flux) σ d Ω d 2 # of neutrons scattered into = inelastic cross section • • dEd Ω (unit time) (incident flux) (energy) inelastic nuclear neutron scattering initial, final state of sample energy of excitation created by neutron in sample partition function
Inelastic nuclear neutron scattering thermal average Debye-Waller factor due to thermal lattice vibrations K) K)} phonon creation phonon annihilaion neutron energy loss neutron energy gain
Neutron spectroscopy analyzer detector monochromator sample Bertram Brockhouse Nobel Prize 1994
TRISP @ FRM-II
Phonon dispersions in Pb Brockhouse, PRL 1962 well described by modern ab-initio lattice dynamics
Conventional superconductors pairing boson quantitative description based on electron electron pairing bosons bosonic spectrum ferm ionic spectrum from neutron scattering from tunneling Brockhouse, PRL 1962 Savrasov, PRB 1996
Quantum materials conventional Mott insulator m etal superconductor zero conductivity finite conductivity infinite conductivity Fermi sphere energy gap subtle re-entanglem ent m assive re-entanglem ent well understood frontier of research after ~ 100 years of research
Elastic magnetic neutron scattering
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
Elastic magnetic neutron scattering one atom approximated as magnetized sphere, magnetization density M(r)
Elastic magnetic neutron scattering generalization for collinear m agnets Bragg peaks magnetic structure factor polarization factor magnetic reciprocal lattice vectors
YBa 2 Cu 3 O 6+ x electronic structure lattice structure Cu d-orbitals YBa 2 Cu 3 O 7 , “YBCO” (T c ~ 90 K) x 2 -y 2 CuO 2 3z 2 -r 2 CuO 2 xz yz xy hole content in x 2 -y 2 orbital dopant O 2- ions arranged in chains controlled by oxygen concentration
YBa 2 Cu 3 O 6 spin structure spin orientation extracted from magnetic Bragg reflections J ⊥ 2 layer b J ⊥ 1 Tranquada et al., PRB 1989 layer a J || H = Σ ij (J || S i (a,b) ) + Σ i (J ⊥ 1 S i (a,b) • S j (a) • S i (b) + J ⊥ 2 S i (b) • S i (a) ) sign, but not strength of exchange parameters determined by elastic neutron scattering
YBa 2 Cu 3 O 6+ x phase diagram 400 300 temperature (K) 200 100 AFI SC 0.05 0.1 0.15 hole concentration
Iron pnictide superconductors electron concentration (x) hole concentration (x) ● lattice structure different from cuprates ● phase diagram very similar to cuprates ● focus on m agnetic m echanism s of Cooper pairing
Understanding unconventional superconductors boson w orking hypothesis electron electron electronic and bosonic quasiparticles key experim ental 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. w orking hypothesis pairing bosons = spin fluctuations d-wave superconductivity
Inelastic magnetic neutron scattering polarization factor spin-spin correlation function fluctuation-dissipation theorem dynamical magnetic susceptibility response to time- and position-dependent H-field
Inelastic magnetic neutron scattering localized electrons Heisenberg antiferrom agnet, m agnon creation η ˆ - ˆ Q K m ) K m , a = 0, 1 q, K m magnon dispersions
YBa 2 Cu 3 O 6 magnons H = Σ ij (J || S i (a,b) ) + Σ i (J ⊥ 1 S i (a,b) • S j (a) • S i (b) + J ⊥ 2 S i (b) • S i (a) ) E 200 meV J ⊥ 2 optic 70 meV acoustic layer b J ⊥ 1 ( π , π ) q layer a exchange parameters from magnon dispersions J || J || ~ 100 meV Tranquada et al., PRB 1989 J ⊥ 1 ~ 10 meV Reznik et al., PRB 1996 J ⊥ 2 ~ 0.01 meV
Ca 2-x Sr x RuO 4 phase diagram Mott insulator-metal transition driven by electronic bandwidth through Ru-O-Ru bond angle
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 Jain et al., Nature Phys. 2017 Higgs mode well defined at q= (0,0) strongly damped at q= ( π , π ) Max Krautloher presentation
Ca 2-x Sr x RuO 4 phase diagram Mott insulator-metal transition driven by electronic bandwidth through Ru-O-Ru bond angle
Inelastic magnetic neutron scattering itinerant electrons electrons Lindhard function & RPA − f ( E ) f ( E ) ∑ + ↑ ↓ χ ω = k q k ( q , ) ω − − − ∆ + ε 0 k h ( E E ) i band dispersions + k q k E Fermi sphere q q-dependent enhancem ent of χ by correlations χ ω ( q , ) χ ω = 0 ( q , ) − χ ω RPA expression 1 J ( q ) ( q , ) 0
Sr 2 RuO 4 spin excitations χ ´´( q,ω ) from RPA calculation Ferm i surface from ARPES strongly nested Mazin et al., PRL 1999
Sr 2 RuO 4 spin excitations Iida et al., PRB 2011 from inelastic neutron scattering spin fluctuation m ediated superconductivity?
Magnetic exitations in cuprates superconductor antiferrom agnetic insulator 300 meV E E 400 40 meV 300 ( π , π ) temperature (K) ( π , π ) q q magnons paramagnons 200 100 AFI SC 0.05 0.1 0.15 hole concentration
Magnetic resonant mode Neutron intensity Energy (meV) Energy (meV) Inosov et al., Nature Phys. 2010 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
INS from superconductors coherence factor ε ε +∆ ∆ − χ ω = Σ + f E ( ) f E ( ) + + + scattering of 1 k k q k k q k q k ( , q ) { (1 ) ω − − + δ k 2 E E ( E E ) i + + thermally excited pairs k k q k q k ε ε +∆ ∆ − − + − 1 f E ( ) f E ( ) + + + 1 k k q k k q k q k (1 ) ω + + + δ pair annihilation 4 E E ( E E ) i + + k k q k q k ε ε +∆ ∆ + − + − f E ( ) f E ( ) 1 + + + 1 k k q k k q k q k (1 ) } ω − + + δ pair creation 4 E E ( E E ) i + + k k q k q k = ε + ∆ 2 2 E 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
Magnetic resonant mode RPA reproduces lower branch of spin excitations of a hour-glass dispersion d-w ave superconductor Im χ direct excitonic collective mode Umklapp incoherent spin flips ω ( π , π ) superconducting energy gap 2 ∆ q Eremin et al. + PRL 2005 dispersion of resonant m ode _ _ momentum-space signature of Cooper-pair wave function +
Paramagnon-mediated superconductivity antiferrom agnetic param agnons electronic band dispersions from neutron scattering from photoemission q 2 q 1 q 1 quantitative cross-correlation q 2 paramagnon electron electron Dahm et al., Nature Phys. 2009
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