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 3:
Vanadates:
Competing nn and nnn interactions yield
Incommensurate order
Competing anisotropies yield complex field dependent
phase diagrams
Ni and Co have very different magnetic structures
Theoretical tools introduced in pervious lectures suffice
to 2explain most features
General outline:
Cuprates
Lecture 3
3
Vanadates
Buckled Kagome
Ni3V2O8
S=1
4
Co3V2O8
S=3/2
Ni3V2O8
5
Co3V2O8
Buckled Kagome
Crystal Structure of Ni3V2O8
Only magnetic (S=1) Ni ions are shown
b
?
a
Cross-tie is
c
6
Cross-tie
Spine
FRUSTRATED
Ni3V2O8
7
H || a
Magnetic Field (T)
5
0
H || b
5
0
5
H || c
0
0
2
4
6
8
10
Temperature (K)
8
9
Specific heat
Weak ferromagnetism in C phase
Neutron scattering intensities
in C, LTI and HTI
Incommensurate wave vector
10
MAGNETIC PHASE DIAGRAM
OF Ni3V2O8
Paramagnetic
HTI = High Temperature
Incommensurate Phase
LTI = Low Temperature
Incommensurate Phase
CAF = Antiferromagnetic
+ weakly ferromagnetic
CAF’ = Incommensurate?
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MAGNETIC PHASE DIAGRAM
OF Ni3V2O8
12
Theory
Step I: Main interactions along spines:
Superexchange, Ni—O—Ni and Ni—O—O--Ni
H  J1 (S1  S2 )  J 2 (S1  S3 )  H A
O
Ni
O
Ni
O
Ni
Ni
Explain HTI, LTI, CAF
13
Incommensurability? -- simplest model:
H  J1 (S1  S2 )  J 2 (S1  S3 )  H A
HTI
S n  ( S cos( qna   ),0,0)
LTI
S n  ( S1 cos(cos( qna   ), S 2 sin( qna   ),0)
(q locked in)
cos(qa)   J1 /( 4 J 2 )
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At low T, anisotropy wins again  CAF
Step II: Anisotropy comes from spin orbit interactions
Spin-orbit interaction
generates Antiferromagnetic
bond-dependent
spin anisotropy
Also Dzyaloshinskii-Moria antisymmetric exchange
O
Ni
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Ni
Oxygen tilted along z
Bilinear coupling between staggered
Moment along a and ferromagnetic
Moment along c
D along y, AFM along x
FM along z
Step III: spin on cross-tie NI?
Pseudodipolar interactions
y
x
y
x
II
I1
16
II
I2
I1
I2
More recent results: Multiferroic behavior
Ferroelectric moment along
b, only in LTI phase!
Can switch ferroelectric
moment with magnetic field!
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PHASE DIAGRAM
b
a
H || a
5
0
P( mC/m2)
Magnetic Field (T)
SPONTANEOUS POLARIZATION
5
T=4K
P || b
0
H || c
5
H || c
T=5K
0
0
2 4 6 8 10
Temperature (K)
0 0.5
1 1.5
2
2.5
Magnetic Field (T)
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3 3.5
LANDAU THEORY WITH
TWO ORDER PARAMETERS
2
4
2
F  a (T  TM ) M  b M  c (T  TP ) P  d P
4
THIS DOES NOT WORK!!
WE DO NOT BELIEVE IN ACCIDENTAL
DEGENERACY (TP = TM). ALSO BOTH
M AND P DEPEND STRONGLY ON H, SO
2
2
F  a (T  TM ) M  b P   P M
x
THEN, WHEN WE MINIMIZE WITH RESPECT
TO P, P APPEARS ONLY WHEN M IS NONZERO:
x
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P  const  M
MAGNETOELECTRIC COUPLING
H 
xy
a x, y ,  x ( q)  y (  q) P
where x,y are LTI or HTI and  = x,y,z
(q) = (-q)* is an order parameter
In the HTI phase we have a single order parameter
which has a node at some lattice site. About this
site there will be inversion symmetry. So
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I (q) = (-q) = (q)*
I = inversion operator
H   a |  ( q) |2 P
( IH = H)

=0
MANETOELECTRIC INTERACTION
Thus the trilinear magnetoelectric
interaction is of the form
H=
HTI LTI
P + d P2
So, after we minimize with respect to P:
P = const HTI
21
LTI
= const LTI
This qualitatively explains the
dependence of P on T and H
Can arise from DM and PD interactions
Confirm mean field
trilinear term from
microscopic Hamiltonian
22
B2u-phonons
Mode Number: 64
Mode Energy: 69.24 meV (experimental value
is about 80 meV!)
Dipole Moment: 0.4612 (One of the largest Connected to V
dipole moment!)
Mode Description: Two oxygen atoms connected
to cross-tie Ni moves along b-axis, significantly
effecting the Ni-O-Ni bond angle for the spine
spins (see the animations; side and top views).
Spine-spins (a-axis)
Cross-tie
b-axis
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24
25
Phys. Rev. B, in press
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(Spins along spine parallel to each other)
FM, d=0
AFM, d=1/2
27
Theory
x(J3)
28
Quartic terms
Higher harmonics
Lock-in
Lock-in
29
Dielecric constant
Ferroelectricity???
30
Conclusions:
Vanadates are almost frustrated;
interesting phase diagrams
Can explain incommensurate
phases by competing interactions
Multiferroics!
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THE END
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