Magnetic Interactions and Order-out-of-disorder in Insulating Oxides

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Magnetic Interactions andOrder-out-of-disorderi
n Insulating Oxides
Amnon Aharony

• Ora Entin-Wohlman, A. Brooks Harris, Taner
  Yildirim
• Robert J. Birgeneau, Marc A. Kastner, Koichi
  Katsumata
• R. Ramirez, C. Broholm, J. W. Lynn
• TAU, BGU, U Penn, NIST, MIT, RIKEN, Lucent, JHU

Les Houches summer school on Quantum Magnetism,
June 2006
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Lecture 2 Generalized superexchange add
spin-orbit and involve all 10 d orbitals
Tetragonal lattice bond dependent
anisotropies, Remove frustration by spin wave
zero-point energy Order out of
disorder Orthorhombic lattice
Dzyaloshinskii-Moriya, dependence on CuOCu
bond angle Interplanar coupling Shender
mechanism, Pseudo-dipolar anisotropies
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LCO, 214
YBCO, 123
What determines the easy axes for the spins (in
plane and between planes)?
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Simple theory Super-exchange
Hubbard Hamiltonian
Perturb in t, keep low energy states
Heisenberg Hamiltonian
ordered moment
2
1
(the manifold, each site has
only a single electron)
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Superexchange
No phase transition in the 2D isotropic
Heisenberg model??
Order arises due to small anisotropies plus weak
interplane coupling Plus quantum fluctuations!
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Theory
On Cu
Cu---O---Cu
On Cu
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cuprates full Hamiltonian
full Coulomb interaction
spin-orbit interaction
hopping among all 5 d-orbitals
further hopping processes
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Tetragonal symmetry
Spin-orbit anisotropy
y
3
z
x
2
1
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Orthorhombic symmetry Also
Dzyaloshinskii-Moriya antisymmetric exchange
O
Cu
Cu
Oxygen tilted along z D along y, AFM
along x FM along z
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Moriya's anisotropic superexchange interaction,
frustration, and Dzyaloshinsky's weak
ferromagnetism, L. Shekhtman, O. Entin-Wohlman
and AA, Phys. Rev. Lett. 69, 000836 (1992)
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For a 90 degrees CuOCu bond, electrons must
hop from each Cu ion to two orthogonal orbitals
on the oxygen, and then Hunds rule turns the
superexchange ferromagnetic. Superexchange then
turns antiferromagnetic as the bond angle
increases towards 180 degrees.
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NNN
NN
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Tetragonal symmetry
y
f
3
z
x
2
1
Mean field theory
Frustration!
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f
3
z
x
2
1
Order out of disorder!
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2342
The Shender mechanism
3
2
1
3
1
2
2122
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Pseudodipolar interactions
x
y
x
y
II
II
I2
I1
I1
I2
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Interplane interactions
1
2
Also explain
need Nd-Cu and Pr-Cu interactions
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Order determined by delicate competition With
quartic spin anisotropies on the Pr or Nd
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2342
Fits to magnetic moment versus Direction of
magnetic field yield Many of the interesting
parameters
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Spin wave theory for 2342
Shender
4-fold
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ESR in plane anisotropy gap in 2122 and 2342, In
and out of plane gaps for CuII at low T
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Spin chains, ladders same interactions as in
2342?
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Conclusions Cuprates taylored 2D and 1D S1/2
AFMs Hubbard spin-orbit
effective magnetic anisotropies
Pseudodipolar, Dzyaloshinskii-Moriya Fluctuations
Shender, in-plane 4-fold Explain structures
of many cuprates 2342 measure many
anisotropies, useful for chains and
ladders Relevance to ladders, chains,
nickelates, cobaltates, titanates
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THE END (More tomorrow)