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Spin Waves in Stripe Ordered Systems
E. W. Carlson
D. X. Yao
D. K. Campbell
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Strong Correlations
nickelates manganites cuprate superconductors organic superconductors
All show some evidence
of real space order
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Strong CorrelationFermi Liquid
• Interaction energy is important
• Real space structure– spin
– charge
• Kinetic energy is
minimized
• k-space structure
• Real space homogeneity
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Organic Superconductors
-(ET)2 X
(TMTST)2 PF6
From E. Dagotto, cond-mat/0302550
From S. Lefebvre et al., Physica B 312: 578-583 (2002)
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Organic superconductors
CDW, SDW
Bond Order BCSDW (Campbell)
(Mazumdar, Clay, and Campbell, Synth. Met. 137, 1317 (2003)
D. Chow et al., Phys Rev Lett 85 1698 (2000)
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Cuprates and Nickelates
Cu-O or Ni-O Planes
Other Layers
Layered structure quasi-2D system
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Cuprates and Nickelates
Dope with holes(remove spins)
Topological Doping
Ni: S=1Cu: S=1/2
Oxygen Cu or Ni
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Cuprates
Dope with holes
Superconducts at certain dopings
Oxygen Cu or Ni
T
x
AF SC
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Neutron Scattering in Cuprates and Nickelates
Disappearance of (π,π) peak
with doping
Appearance of satellite peaks
AFM signal averages to zero
antiphase domain walls
π
π
δ=0
π
π
δ=0
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Issues:
nature – static vs. dynamic
orientation – vertical vs. diagonal
spacing – commensurate vs. incommensurate
width – one atom vs. two ...
location of holes – site-centered vs.
bond-centered
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Cuprates
from Almason and Maple (1991)
stripes: interleaved charge and spin density
(Kivelson, Emery)
(Zaanen)
(Castro Neto, Morais-Smith)
bond-ordered charge density
(Sachdev)
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Scattering Probes
Energy, MomentumPhase Information?
Yes in certain cases
Goals: Phase-sensitive information from diffraction
probe Guidance for microscopic theories of
superconductivity in cuprates, organics
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Site or Bond-Centered
Ja > 0 (AFM) Jb > 0 (AFM)
Jb Ja Jb
Ja > 0 (AFM) Jb < 0 (FM)
Ja Site-centered p=3 Bond-centered p=3
π
π Both produce weightat (π+ π/p, π)
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Model and MethodJb
Bond-centered, p=3Ja > 0 (AFM) Jb < 0
(FM)
Ja Heisenberg model
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Elastic Response
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Magnetic Reciprocal Lattice VectorsSite-centered p=3
Bond-centered p=3
π
π
Spacing p=3
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Magnetic Reciprocal Lattice Vectors
Bond-centered p=4
π
π
Spacing p=4
Site-centered p=4
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Elastic Neutron Scattering
g(m)
f(n)
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Elastic Neutron Scattering p=3Site-centered
π
π
g(m)
f(n)
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Elastic Neutron Scattering p=3Site-centered
g(m)
f(n)
π
π
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Elastic Neutron Scattering p=3Bond-centered
g(m)
f(n)
π
π
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Elastic Neutron Scattering p=3Bond-centered
g(m)
f(n)
π
π
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Site vs. Bond-Centered p=3
Bond-centered p=3
g(m)
f(n) π
π
Site-centered p=3
g(m)
f(n) π
π
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Site vs. Bond-Centered p=4
Bond-centered p=4
g(m)
f(n)
Site-centered p=4
g(m)
f(n) π
π
π
π
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Elastic Peaks
2D Antiphase Domain Walls
Site-centered: never weight at
Bond-centered: no weight at for p=EVEN
generic weight at for p=ODD
The presence of weight at with incommensurate peaks at is positive evidence of a bond-centered configuration
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Elastic Peaks3D Antiphase Domain Walls
Site-centered Bond-centered
p=EVEN
Vertical/Vertical - -
Diagonal/Vertical - -
Diagonal/Diagonal - -
Bond-centered
p=ODD
(0,,)
(0,,0)
(0,0,0)
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Inelastic Response: Spin Waves
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Model and MethodJb
Bond-centered, p=3Ja > 0 (AFM) Jb < 0
(FM)
Ja Heisenberg model
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Model and Method
Heisenberg model
Up Spins: Down Spins:
Holstein-Primakoff Bosons
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Model and Method
Heisenberg model
Fourier transformation + symplectic transformationyield spectrum and eigenstates
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Spin Structure Factor
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Number of Bands
Bond-centered p=4
Site-centered p=4p-1 spins per unit cellSpin up/Spin down degeneracy
) (p-1)/2 bands
3 bands for p=4
p spins per unit cellSpin up/Spin down degeneracy
) p/2 bands
4 bands for p=4
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Site-Centered: S(k,
Jb=0.4 Ja Jb=1.0 Ja Jb=2.5 Ja
kx
p=3
p=4
π
π
N.B. Site-centered consistent with F.Kruger and S. Scheidl, PRB 67, 134512 (2003)
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Jb= - 0.1 Ja Jb=-0.56 Ja Jb=-1.0 Ja
π
π
Bond-Centered: S(k, )
p=3
p=4
p=2
kx
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1 2 3 4 5
1
2
3
4
5
6
7
S3 k=(0,0)
1 2 3 4 5
2
4
6
k=(π, π)
1 2 3 4 5
1
2
3
4
5
6
7
S4
Energy dependence on λ=
1 2 3 4 5
1
2
3
4
5
6
7
Ja
Jb
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1 2 3 4 5
2
4
6
8
10
12
1 2 3 4 5
2
4
6
8
10
12
1 2 3 4 5
2
4
6
8
10
12
1 2 3 4 5
2
4
6
8
10
12
1 2 3 4 5
2
4
6
8
10
12
1 2 3 4 5
2
4
6
8
10
12
k=(0,0) k=(π, π)B2
B3
B4
Ja
JbEnergy dependence on λ=
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Site-centered velocities
v velocity along the stripe direction
v velocity perpendicular to the stripe direction
v velocity of pure 2D antiferromagnet
||
AF
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Bond-centered velocities
v velocity along the stripe direction
v velocity perpendicular to the stripe direction
v velocity of pure 2D antiferromagnet
||
AF
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ConclusionsElastic:
For both 2D and 3D antiphase domain walls, bond-centered p=ODD stripes show new peaks, forbidden for site-centered
Inelastic: – Number of bands distinguishes site- or bond-centered
Site: (p-1) bands Bond: (p) bands
– Qualitatively different spin wave spectraSite: all bands increase with J_bBond: lower bands independent of J_b
top band ~ 2 J_b
– Velocity anisotropyBond-centered is rather isotropic over a large range of parameters
Extensions:– Diagonal spin waves– Other spin textures