Neutron scattering from quantum materials Bernhard Keim er Max - - PowerPoint PPT Presentation

Bernhard Keim er Max Planck Institute for Solid State Research Max Planck – UBC – UTokyo Center for Quantum Materials

Neutron scattering from 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

Fermi sphere energy gap

m etal finite conductivity quantum oscillations at low temperatures conventional superconductor infinite conductivity Mott insulator zero conductivity subtle re-entanglem ent well understood after ~ 100 years of research m assive re-entanglem ent frontier of research

Quantum materials

diverse electronic ordering phenom ena near Mott m etal-insulator transition m anganates ruthenates cuprates cobaltates

Dagotto et al. Science 2005

Neutron scattering

strong ( nuclear) interaction elastic lattice structure inelastic lattice dynamics neutron excitation: E = E1 − E2

q = q1 − q2

interaction E1 q1 E2 q2 m agnetic ( 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

Elastic neutron scattering

Elastic neutron scattering

Q = kf – ki momentum transfer

Elastic nuclear neutron scattering

Bragg peaks at reciprocal lattice vectors K

scattering length b ~ size of nucleus ~ 10-15 m

Elastic nuclear neutron scattering

Inelastic neutron scattering

elastic cross section

flux) (incident time) (unit into scattered neutrons

• f

#

• =

dΩ dΩ dσ

inelastic cross section

(energy) flux) (incident time) (unit into scattered neutrons

• f

#

• =

dΩ dEdΩ σ d 2

inelastic nuclear neutron scattering initial, final state of sample partition function energy of excitation created by neutron in sample

Inelastic nuclear neutron scattering

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

Neutron spectroscopy

monochromator sample detector analyzer

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

quantitative description based on pairing bosons electron electron pairing boson ferm ionic spectrum from tunneling

Savrasov, PRB 1996

bosonic spectrum from neutron scattering

Brockhouse, PRL 1962

Quantum materials

Fermi sphere energy gap

m etal finite conductivity conventional superconductor infinite conductivity Mott insulator zero conductivity subtle re-entanglem ent well understood after ~ 100 years of research m assive re-entanglem ent frontier of research

Elastic magnetic neutron scattering

Elastic magnetic neutron scattering

non-spin-flip “classical electron radius”

• ne electron

σz → σx , σy

spin-flip (not possible for nuclear scattering) unpolarized beam average spin-flip and non-spin-slip channels separate nuclear and magnetic neutron scattering by spin polarization analysis

Elastic magnetic neutron scattering

• ne atom

approximated as magnetized sphere, magnetization density M(r)

Elastic magnetic neutron scattering

polarization factor magnetic structure factor magnetic reciprocal lattice vectors

generalization for collinear m agnets

Bragg peaks

YBa2Cu3O6+ x

lattice structure YBa2Cu3O7, “YBCO” (T

c ~ 90 K)

CuO2 CuO2

x2-y2 yz xz 3z2-r2 xy electronic structure Cu d-orbitals hole content in x2-y2 orbital controlled by oxygen concentration dopant O2- ions arranged in chains

YBa2Cu3O6 spin structure

H = Σij (J|| Si

(a,b)• Sj (a,b)) + Σi (J⊥1 Si (a)• Si (b) + J⊥2 Si (b)• Si (a))

J⊥1 J⊥2 J||

layer a layer b sign, but not strength of exchange parameters determined by elastic neutron scattering Tranquada et al., PRB 1989 spin orientation extracted from magnetic Bragg reflections

YBa2Cu3O6+ x phase diagram

hole concentration temperature (K) AFI SC 0.05 0.1 0.15 400 300 200 100

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

w orking hypothesis electronic and bosonic quasiparticles electron electron boson 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

Km) Km,

q, Km a = 0, 1

η ˆ ˆ Q

magnon dispersions

YBa2Cu3O6 magnons

H = Σij (J|| Si

(a,b)• Sj (a,b)) + Σi (J⊥1 Si (a)• Si (b) + J⊥2 Si (b)• Si (a))

q

(π,π)

E acoustic

• ptic

70 meV 200 meV J⊥1 J⊥2 J||

layer a layer b

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

Ca2-xSrxRuO4 phase diagram

Mott insulator-metal transition driven by electronic bandwidth through Ru-O-Ru bond angle

Longitudinal “Higgs” mode in Ca2RuO4

spin waves Higgs mode

spin-polarized triple-axis neutron scattering

Higgs mode well defined at q= (0,0) strongly damped at q= (π,π) damping at q= (π,π) due to decay into transverse modes

Jain et al., Nature Phys. 2017  Max Krautloher presentation

Ca2-xSrxRuO4 phase diagram

Mott insulator-metal transition driven by electronic bandwidth through Ru-O-Ru bond angle

Inelastic magnetic neutron scattering

) , ( ) ( 1 ) , ( ) , ( 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 ) ( ) ( ) ( ) , ( h

itinerant electrons electrons  Lindhard function & RPA band dispersions RPA expression

Fermi sphere

E q

q-dependent enhancem ent of χ by correlations

Sr2RuO4 spin excitations

Ferm i surface from ARPES strongly nested

χ´´(q,ω) from RPA calculation

Mazin et al., PRL 1999

Sr2RuO4

Iida et al., PRB 2011

spin excitations from inelastic neutron scattering spin fluctuation m ediated superconductivity?

Magnetic exitations in cuprates

magnons (π, π) q 300 meV E antiferrom agnetic insulator hole concentration temperature (K) AFI SC 0.05 0.1 0.15 400 300 200 100 paramagnons superconductor 40 meV (π, π) q E

Magnetic resonant mode

Energy (meV) Energy (meV) Neutron intensity 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

( ) ( ) 1 2 ( ) 1 ( ) ( ) 1 4 ( ) ( ) ( ) 1 1 4 ( ) 2 2

( , ) { (1 ) (1 ) (1 ) }

k k q k k q k q k k k q k q k k k q k k q k q k k k q k q k k k q k k q k q k k k q k q k

f E f E k E E E E i f E f E E E E E i f E f E E E E E i k k k

q E

ε ε ω δ ε ε ω δ ε ε ω δ

χ ω ε

+ + + + + + + + + + + + + + +

+∆ ∆ − − − + +∆ ∆ − − + + + +∆ ∆ + − − + +

= Σ + + − + − = + ∆

coherence factor scattering of thermally excited pairs pair annihilation pair creation

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

Fong et al., PRL 1995 Monthoux & Scalapino, PRL 1994

Magnetic resonant mode

incoherent spin flips Imχ

ω

superconducting energy gap 2∆ excitonic collective mode

spin excitations of a d-w ave superconductor q (π,π)

Eremin et al. PRL 2005

Umklapp direct

+ + _ _

dispersion of resonant m ode momentum-space signature of Cooper-pair wave function RPA reproduces lower branch of hour-glass dispersion

Paramagnon-mediated superconductivity

antiferrom agnetic param agnons from neutron scattering

q1 q2 q1 q2

electronic band dispersions from photoemission quantitative cross-correlation electron electron paramagnon

Dahm et al., Nature Phys. 2009

Spin dynamics from neutron scattering

spin waves (π, π) q 300 meV E antiferrom agnetic insulator hole concentration temperature (K) AFI SC 0.05 0.1 0.15 400 300 200 100 magnetic resonant mode superconductor 40 meV (π, π) q E neutron blind spots

• high energies
• high doping levels

 RI XS

Resonant inelastic x-ray scattering (RIXS)

Energy loss (eV)

• 4
• 3
• 2
• 1

2 1

2003 2008 2000 2007

• rder-of-magnitude increase in in energy resolution
• L. Braicovich, G. Ghiringhelli (Politecnico Milano)

incoming photon scattered photon magnon La 2CuO4 Cu 2p  3d photon energy ~ 931 eV triple-axis spectrom etry w ith soft x-rays

Resonant inelastic x-ray scattering (RIXS)

total length ~ 12 m RI XS spectrom eter e.g. ERIXS @ ESRF

ERIXS Spectrometer @ ESRF