Transcript Document

Magnetic Interactions and
Order-out-of-disorder
in 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
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 Cu—O—Cu bond angle
Interplanar coupling: Shender mechanism,
Pseudo-dipolar
anisotropies
2
LCO, 214
YBCO, 123
What determines the easy axes for the spins
(in plane and between planes)?
3
Simple theory: Super-exchange
Hubbard Hamiltonian:
H   tij ci c j  U  ni ni
 ij  
t  U
i
Perturb in t, keep
low energy states
Heisenberg Hamiltonian:
 
H  J  Si  S j
 ij 
2
t
J
U
ordered moment:
2
1
4
(the
U 
manifold, each site
has only a single electron)
 B
Superexchange:
J  t 2 /U
No phase transition in the 2D
isotropic Heisenberg model??
5
Order arises due to small anisotropies
plus weak interplane coupling
Plus quantum fluctuations!
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;
H 
ij
7
      
( J ij Si  S j  Dij  Si  S j  Si ˆ ij S j )
Tetragonal symmetry:
Spin-orbit
anisotropy
y
3
z
8
1
2
x
Orthorhombic symmetry:
Also Dzyaloshinskii-Moriya antisymmetric exchange
O
Cu
Oxygen tilted along z
9
Cu
D along y, AFM along x
FM along z
Moriya's anisotropic superexchange interaction, frustration, and
Dzyaloshinsky's
weak ferromagnetism, L. Shekhtman, O. Entin-Wohlman and AA,
10
Phys. Rev. Lett. 69, 000836 (1992)
For a 90 degrees Cu—O—Cu bond, electrons
must hop from each Cu ion to two orthogonal
orbitals on the oxygen, and then Hund’s rule
turns the superexchange ferromagnetic.
Superexchange then turns antiferromagnetic
11 the bond angle increases towards 180 degrees.
as
NN
12
NNN
y
f
Tetragonal symmetry:
3
z
1
2
x
Mean field theory:
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Frustration!
f
3
z
1
Order out of disorder!
14
2
x
2342
The “Shender” mechanism:
3
1
3
1
15
2
2122
2
Pseudodipolar interactions
y
x
y
x
II
I1
16
II
I2
I1
I2
Interplane interactions
1
17
Also explain
2
: need Nd-Cu and Pr-Cu interactions
18
Order determined by delicate competition
With quartic spin anisotropies on the Pr or Nd
2342
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Fits to magnetic moment versus
Direction of magnetic field yield
Many of the interesting parameters
Spin wave theory for 2342
Shender
4-fold
20
ESR: in plane anisotropy gap in 2122 and 2342,
In and out of plane gaps for CuII at low T
21
Spin chains, ladders: same interactions as in 2342?
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Conclusions:
Cuprates: taylored 2D and 1D S=1/2 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)
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