spin waves in stripe ordered systems e. w. carlson d. x. yao d. k. campbell
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Spin Waves in Stripe Ordered Systems
E. W. Carlson
D. X. Yao
D. K. Campbell
Strong Correlations
nickelates manganites cuprate superconductors organic superconductors
All show some evidence
of real space order
Strong CorrelationFermi Liquid
• Interaction energy is important
• Real space structure– spin
– charge
• Kinetic energy is
minimized
• k-space structure
• Real space homogeneity
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)
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)
Cuprates and Nickelates
Cu-O or Ni-O Planes
Other Layers
Layered structure quasi-2D system
Cuprates and Nickelates
Dope with holes(remove spins)
Topological Doping
Ni: S=1Cu: S=1/2
Oxygen Cu or Ni
Cuprates
Dope with holes
Superconducts at certain dopings
Oxygen Cu or Ni
T
x
AF SC
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
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
Cuprates
from Almason and Maple (1991)
stripes: interleaved charge and spin density
(Kivelson, Emery)
(Zaanen)
(Castro Neto, Morais-Smith)
bond-ordered charge density
(Sachdev)
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
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)
Model and MethodJb
Bond-centered, p=3Ja > 0 (AFM) Jb < 0
(FM)
Ja Heisenberg model
Elastic Response
Magnetic Reciprocal Lattice VectorsSite-centered p=3
Bond-centered p=3
π
π
Spacing p=3
Magnetic Reciprocal Lattice Vectors
Bond-centered p=4
π
π
Spacing p=4
Site-centered p=4
Elastic Neutron Scattering
g(m)
f(n)
Elastic Neutron Scattering p=3Site-centered
π
π
g(m)
f(n)
Elastic Neutron Scattering p=3Site-centered
g(m)
f(n)
π
π
Elastic Neutron Scattering p=3Bond-centered
g(m)
f(n)
π
π
Elastic Neutron Scattering p=3Bond-centered
g(m)
f(n)
π
π
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
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
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
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)
Inelastic Response: Spin Waves
Model and MethodJb
Bond-centered, p=3Ja > 0 (AFM) Jb < 0
(FM)
Ja Heisenberg model
Model and Method
Heisenberg model
Up Spins: Down Spins:
Holstein-Primakoff Bosons
Model and Method
Heisenberg model
Fourier transformation + symplectic transformationyield spectrum and eigenstates
Spin Structure Factor
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
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)
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
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
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=
Site-centered velocities
v velocity along the stripe direction
v velocity perpendicular to the stripe direction
v velocity of pure 2D antiferromagnet
||
AF
Bond-centered velocities
v velocity along the stripe direction
v velocity perpendicular to the stripe direction
v velocity of pure 2D antiferromagnet
||
AF
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