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