KIAS Emergent Materials 2006 Bond Polarization induced by Magnetic order Jung Hoon Han Sung Kyun Kwan U. Reference: cond-mat/0607 Collaboration Chenglong

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)


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.


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:


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.



2006


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


2006

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


2006

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


2006

CoCr2O4

Tokura group PRL 06

Co spins have ferromagnetic + spiral (conical) componentsEmergence of spiral component accompanied by P


2006

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


2006

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 HH S L

H U m S

H E p p

H V d p d p d p d p h c


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


2006

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.


2006

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


2006

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


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)


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


2006

Numerical Results for one holeRotate 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


2006

Numerical results for two holes

Transverse and longitudinal components exist which we were able to fit using very simple empirical formulas:

KNB


2006

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


2006

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?


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).


2006

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.


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.

Documents

KIAS Emergent Materials 2006 Bond Polarization induced by Magnetic order Jung Hoon Han Sung Kyun Kwan U. Reference: cond-mat/0607 Collaboration Chenglong