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Neutron and x-ray scattering
studies of superconductors
B. Keimer
Max-Planck-Institute for Solid State Research
lecture 1
conventional superconductors
inelastic nuclear neutron scattering from phonons
unconventional superconductors
magnetic structure determination by elastic magnetic neutron scattering
inelastic magnetic neutron scattering from magnons and paramagnons
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Neutron and x-ray scattering
studies of superconductors
lecture 2: unconventional superconductors
magnetic neutron scattering continued
resonant inelastic x-ray scattering from magnons and paramagnons
resonant elastic x-ray scattering from charge density waves
lecture 3: cuprate and nickelate superlattices
orbital occupation
magnetic order
charge density waves
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Neutron scattering
strong (nuclear) interaction elastic lattice structure inelastic lattice dynamics magnetic (dipole-dipole) interaction elastic magnetic structure inelastic magnetic excitations
neutron
excitation: E= E2-E1
q=q2-q1
interaction
E1 q1
E2 q2
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Neutron sources
FRM-II Garching, Germany
research reacor
neutron flux
Maxwellian profile
energy ~ 30 meV
spectrum
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Neutron sources
SNS Oak Ridge, TN
1. Source
2. Linac
3. Beamlines
4. Accumulator ring
5. Target area
spallation source
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Elastic neutron scattering
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Elastic neutron scattering
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Elastic nuclear neutron scattering
Bragg peaks at reciprocal lattice vectors K
scattering length b ~ size of nucleus ~ 10-15 m
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Neutron scattering lengths
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Neutron radiography
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Inelastic neutron scattering
elastic cross section flux) (incidenttime) (unit into scattered neutrons of #
•=
dΩdΩdσ
inelastic cross section (energy)flux) (incidenttime) (unit into scattered neutrons of #
••=
dΩdEdΩ
σd 2
inelastic nuclear neutron scattering
initial, final state of sample
partition function
energy of excitation created by neutron in sample
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Inelastic nuclear neutron scattering
thermal average
characterized by population ns of phonons of energy in branch s )(ks
ωλ
K) K)
Debye-Waller factor due to thermal lattice vibrations
phonon creation neutron energy loss
phonon annihilaion neutron energy gain
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Triple-axis spectrometer
monochromator sample detector
analyzer
if
if
kkq
kkm−=
−= )(2
222
ω
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Phonon dispersions in Pb
excellent agreement with
ab-initio lattice dynamics Munnikes, Boeri et al.
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Electron-phonon interaction
example MgB2
Kong et al., PRB 2001
strong coupling short phonon lifetime
typical phonon linewidth: 1-100 μeV
electron-phonon interaction in simple metals predicted by ab-initio LDA
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Conventional superconductors
understanding based on quasiparticles
fermion fermion
pairing boson
fermionic spectrum from tunneling
experimental tunnel spectrum calculated spectrum based on phonon dispersions from neutrons
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Resonant mode in conventional superconductors
N. Munnikes after Allen et al., PRB 1997
phonon dispersion
T > Tc T < Tc
feedback of pairing interaction on intermediate boson
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Resonant mode in conventional superconductors
first observed in borocarbides
Stassis et al., PRB 1997 Weber et al., PRL 2008
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Neutron spin echo spectroscopy
monochromator sample detector
analyzer
triple axis spectrometer: excitation energy ~ 1-100 meV energy resolution ~ 0.1-10 meV
coil 1 coil 2
monochromator sample detector
analyzer
triple axis spectrometer: excitation energy ~ 1-100 meV energy resolution ~ 0.1-10 meV triple axis – spin echo spectrometer: excitation energy ~ 1-100 meV energy resolution ~ 1 – 100 µeV 3 orders of magnitude gain in energy resolution
→ possible to resolve excitation lifetimes in solids
if
if
kkq
kkm−=
−= )(2
222
ω
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TRISP Spectrometer at FRM-II
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Electrn-phonon interaction
electron-phonon linewidths
in good agreement with
ab-initio lattice dynamics Munnikes, Boeri et al.
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Electron-phonon interaction in Pb
Pb
Keller et al., PRL 2006
lifetime renormalization below superconducting Tc = 7.2 K
phonon energy
energy gap E
Tc T
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Electron-phonon interaction in Pb
∆ (0.8 Tc)
∆ (0.07 Tc)
phonon energy
energy gap ∆
ξ
E
Aynajian et al. Science 2008
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Electron-phonon interaction in Pb
superconducting energy gap
merges with second linewidth
maximum at low T
origin: Kohn anomaly
due to Fermi surface nesting
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Accident ?
no! same effect observed in Nb
niobium
Aynajian et al. Science 2008
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Electron-phonon interaction in Pb
Aynajian et al. Science 2008
Kohn anomalies not predicted in TA branch by ab-initio LDA calculations many-body correlations beyond LDA charge density wave fluctations?
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Electron-phonon interaction in Pb and Nb
scenario • many-body effects beyond LDA: charge density wave fluctuations
• dynamical nesting → Kohn anomalies
• interference between CDW and superconducting fluctuations limits growth of superconducting energy gap • not explain by BCS/Eliashberg theory remains open problem
Johnston et al., PRB 2011
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High temperature superconductivity
year
boiling point of nitrogen
tem
pera
ture
(K)
mechanism phonon-mediated Cooper pairing BCS 1956, Eliashberg … 1960’s
LaFeAsO1-xFx
BaFe2-x(Co,Ni)xAs2
magnetic mechanism ?
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Copper oxide superconductors
lattice structure e.g. YBa2Cu3O6+x Tc ≤ 93 K
CuO2 CuO2
phase diagram
hole concentration electron concentration
temperature
SC SC
AF
TN
Tc Tc
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YBa2Cu3O6+x
NMR high homogeneity, low disorder
Bobroff et al. PRL 2002
untwinned crystals scattering & transport probes can discriminate between uniaxial and biaxial modulations
This image cannot currently be displayed.
quantum oscillations fermionic quasiparticles
Doiron-Leyraud et al. Nature 2007
Sebastian et al. Nature 2008
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Iron pnictide superconductors
electron concentration (x) hole concentration (x)
lattice structure different from cuprates
phase diagram similar to cuprates
focus on magnetic mechanisms of Cooper pairing
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Elastic magnetic neutron scattering
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Elastic magnetic neutron scattering
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Elastic magnetic neutron scattering
non-spin-flip
“classical electron radius”
one electron
σz → σx , σy spin-flip (not possible for nuclear scattering)
average for unpolarized beam
separate nuclear and magnetic neutron scattering by spin polarization analysis
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Elastic magnetic neutron scattering
one atom
approximated as magnetized sphere, magnetization density M(r)
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Elastic magnetic neutron scattering
polarization factor magnetic structure factor
magnetic reciprocal lattice vectors
generalization for collinear magnets
Bragg peaks
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Example one-dimensional ferromagnet
use interference between nuclear and magnetic scattering to create spin-polarized neutrons
ηησ ˆˆ~ 22 bbdd
++Ω
(up to prefactors)
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Example one-dimensional antiferromagnet
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Neutron diffractometers
powder single crystal
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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
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Phase diagram of YBa2Cu3O6+x
hole concentration
tem
pera
ture
(K)
AFI SC
0.05 0.1 0.15
400
300
200
100
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Spin-polarized neutrons
manipulate relative orientation of vectors σ, η, Q
accurate determination of complex spin structures
neutron spin operator
neutron spin states defined by spin polarizers
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Spin-polarized neutrons
Polarized neutron spectrometer
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Spin density wave
Haug et al., PRL 2009, NJP 2012 Porras, Loew et al.
undoped YBa2Cu3O6
commensurate antiferromagnetism
lightly doped YBa2Cu3O6+x
noncollinear incommensurate structure,
facilitates propagation of doped holes
spin structures from spin-polarized neutron scattering
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Competing order in YBa2Cu3O6+x
uniaxial incommensurate spin density wave
hole concentration
tem
pera
ture
(K)
AFI SC
SDW
0.05 0.1 0.15
400
300
200
100
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Inelastic magnetic neutron scattering
polarization factor
spin-spin correlation function
fluctuation-dissipation theorem
dynamical magnetic susceptibility response to time- and position-dependent H-field
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Inelastic magnetic neutron scattering
localized electrons Heisenberg antiferromagnet, magnon creation
Km) Km, q, Km a = 0, 1
ηQ
magnon dispersions
-
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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 optic 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