b ond p olarization induced by m agnetic order
DESCRIPTION
B ond P olarization induced by M agnetic order. Jung Hoon Han Sung Kyun Kwan U. Reference: cond-mat/0607 Collaboration Chenglong Jia (SKKU, KIAS) Naoto Nagaosa (U. Tokyo) Shigeki Onoda (U. Tokyo). B ond P olarization induced by M agnetic order. - PowerPoint PPT PresentationTRANSCRIPT
KIAS Emergent Materials 2006
Bond Polarization induced by Magnetic order
Jung Hoon Han Sung Kyun Kwan U.
Reference: cond-mat/0607
Collaboration
Chenglong Jia (SKKU, KIAS)Naoto Nagaosa (U. Tokyo)Shigeki Onoda (U. Tokyo)
KIAS Emergent Materials 2006
Bond Polarization induced by Magnetic order
Electric polarization, like polarization of spin, is responsible for loss of symmetry in the system,
in this case, inversion symmetry. Its phenomenological description bears natural
similarity to that of magnetic ordering. Normally, however, we do not think of the two ordering
tendencies as “coupled”.
Here we discuss experimental instances and theoretical models where the onset of electric polarization is “driven” by a particular type of
spin ordering.
KIAS Emergent Materials 2006
Two order parameters X, Y are coupled in a GL theory
Introduction to spin-polarization coupling via GL theory
If X condenses (aX < 0) and Y does not (aY > 0), but a linear coupling ~ XY exists, simultaneous condensation of Y occurs:
KIAS Emergent Materials 2006
Spin <S> and polarization <R> break different symmetries:
<S> breaks time-inversion symmetry
<R> breaks space-inversion symmetry
Naively, lowest-order coupling occurs at <S>2 <R>2 .
If the system already has broken inversion symmetry lower-order coupling
<S>2 <R> is possible.
Even without inversion symmetry breaking,
<S>2 <grad R> or <S><grad S><R> is possible.
Introduction to spin-polarization coupling via GL theory
KIAS Emergent Materials 2006
Introduction to spin-polarization coupling via GL theory
Generally one can write down
that result in the induced polarization
For spiral spins
induced polarization has a uniform component given by
Mostovoy PRL 06
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Experimental Evidence of spin-lattice coupling
Uniform induced polarization depends product M1 M2
- Collinear spin cannot induce polarization - Only non-collinear, spiral spins have a chance
Recent examples (partial)
Ni3V2O8 – PRL 05
TbMnO3 – PRL 05
CoCr2O4 – PRL 06
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Ni3V2O8
Lawes et al PRL 05
TbMnO3
Kenzelman et al PRL 05
Collinear to non-collinear spin transition accompanied by onset of polarization with P direction consistent with theory
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CoCr2O4
Tokura group PRL 06
Co spins have ferromagnetic + spiral (conical) componentsEmergence of spiral component accompanied by P
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Microscopic Theory of Katsura, Nagaosa, Balatsky (KNB)
A simple three-atom model consisting of M(agnetic)-O(xygen)-M ions is proposed to “derive” spin-induced polarization from microscopic Hamiltonian
KNB PRL 05
Dagotto PRB 06(different perspective)
Polarization orthogonal to the spin rotation axis and modulation wavevector develops; consistent with phenomenological theories
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Elements of KNB Theory
The cluster Hamiltonian assuming t2g levels for magnetic sites
KNB Hamiltonian is solved assuming SO > U
,, , ,
, ,
, , , ,
. .
SO M O V
SO
M a a la r l l xy yz zx
O p b bb x y z
V l xy y l zx z r xy y r zx z
H H H H H
H S L
H U m S
H E p p
H V d p d p d p d p h c
KIAS Emergent Materials 2006
Why spin-orbit is important
Conceptual view: spin orientations leave imprints on the wave functions, leading to non-zero polarization
Technical view: Spin-orbit Hamiltonian mixes oxygen pz with magnetic dyz, px with dxy within the same eigenstate, non-zero <dyz|y|pz>, <dxy|y|px> is responsible for polarization
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Motivation for our work
(0) KNB result seems so nice it must be general.
(1) Effective Zeeman energy U is derived from Hund coupling (as well as superexchange), which is much larger than SO interaction. The opposite limit U >> SO must be considered also.
(2) What about eg levels?
(3) From GL theory one expects some non-uniform component too.
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Our strategy for large U limit
Large-U offers a natural separation of spin-up and spin-down states for each magnetic site. All the spin-down states (antiparallel to local field) can be truncated out. This reduces the dimension of the Hamiltonian which we were able to diagonalize exactly.
Truncated Truncated HSHS
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Our Model (I): eg levels
The model we consider mimics eg levels with one (3x2-r2)-orbital for the magnetic sites, and px, py, pz orbitals for the oxygen. Within eg manifold SO is ineffective. Real multiferroic materials have filled t2g and partially filled eg!
IDEA(S. Onoda): Consider oxygen SO interaction. It will be weak, but better than nothing!
Our calculation for large-U gives
KIAS Emergent Materials 2006
Our Model (II): t2g levels
Going back to t2g, we considered strong-U limit, truncating +U subspace leaving only the –U Hilbert space.
Spontaneous polarization exists ALONG the bond direction.No transverse polarization of KNB type was found. (NB: KNB’s theory in powers of U/, our theory in powers of /U)
KIAS Emergent Materials 2006
Numerical approach
Surprised by, and skeptical of our own conclusion, we decided to compute polarization numerically without ANY APPROXIMATION
Exact diagonalization of the KNB Hamiltonian (only 16 dimensional!) for arbitrary parameters (/V,U/V)
For each of the eigenstates compute P = <r>
The results differ somewhat for even/odd number of holes; In this talk we mainly presents results for one and two holes. Other even numbers give similar results.
,, , ,
, ,
, , , ,
. .
M SO O V
M a a la r l l xy yz zx
SO O p b bb x y z
V l xy y l zx z r xy y r zx z
H H H H H
H U m S
H S L H E p p
H V d p d p d p d p h c
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Numerical Results for one hole
Rotate two spins within XY plane: Sl=(cos l,sinr,0) Sr=(cos l,sinr,0) and compute resulting polarization.
Numerical results for one hole is in excellent qualitative agreement with analytical calculation
Not only longitudinal but also transverse components were found in P
KIAS Emergent Materials 2006
Numerical results for two holes
Transverse and longitudinal components exist which we were able to fit using very simple empirical formulas:
KNB
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Uniform vs. non-uniform
When extended to spiral spin configuration, Px gives oscillating polarization with period half that of spin. Py has oscillating (not shown) as well as uniform (shown) component
KNB
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Uniform vs. non-uniform
What people normally detect is macroscopic (uniform) polarization but that may not be the whole story. Non-uniform polarization, if it exists, is likely to lead to some modulation of atomic position which one can pick up with X-rays.
How big is the non-uniform component locally?
KIAS Emergent Materials 2006
KNB
Coefficients
22
21
200 /
80 /
A B nC cm
B nC cm
The uniform transverse component B1 is significant
for small U (KNB limit).
A and B2 (non-uniform) are dominant for large U (our limit).
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Comparison to GL theory
Within GL theory non-uniform polarization is also anticipated. On comparing Mostovoy’s prediction with ours, a lot of details differ.
A large non-uniform component could not have been predicted on GL theory alone.
Bear in mind that t2g break full rotational symmetry down to cubic; corresponding GL theory need not have that symmetry built in. A new kind of GL theory is called for.
KIAS Emergent Materials 2006
Summary
Motivated by recent experimental findings of non-collinear-spin-induced polarization, we examined microscopic model of Katsura, Nagaosa, Balatsky in detail.
Induced polarization has longitudinal and transverse, uniform and non-uniform components with non-trivial dependence on spin orientations.
Detecting such local ordering of polarization will be interesting.