Junghoon Kim and Jung Hoon Han
Department of Physics
Sungkyunkwan University
Introduction
 Multiferroic materials :
Materials which posses two or more coupled ferroic properties, e.g.
ferroelectricity, (anti)ferromagnetism, etc., are called multiferroic materials, or
“multiferroics”. This strong coupling between magnetic and electric degrees of
freedom give rise to interesting physics. There has been a recent resurgence of
interest in multiferroic materials due to advances in experimental techniques,
potential technological applications (e.g. electrically accessible magnetic
memory), and discovery of new multiferroic materials. Some examples of
multiferroic materials are :
TbMnO3, Ni3V2O8, Ba0.5Sr1.5Zn2Fe12O22, CoCr2O4, MnWO4, etc.
(The elements indicated in red are magnetic ions)
 Electromagnons :
Small oscillations of the electric and magnetic moments with respect to the
equilibrium positions propagate as spin waves and polarization waves. This
coupling leads to collective modes called electromagnons.
Introduction
 Some recent experimental progress in multiferroics :
•
•
•
Observations of the development of dipole moments accompanying the collinear-tohelical spin ordering.1,2
Displacement of magnetic ions at the onset temperature of magnetic order in the
triangular net.3
Adiabatic control of dipole moments through sweeping of applied magnetic field.4,5
 Recent theoretical progress in multiferroics :
Katsura, Balatsky, and Nagaosa6 recently studied the low-energy dynamics of a one
dimensional system of coupled spins and electric polarization originating from
Dzyaloshinskii-Moriya type interactions. They found new collective modes of spin
and polarization waves that lead to a novel coupling between the dielectric and
magnetic properties.
Schematic ground state configuration of spins
(black arrows) and lattice displacements (blue arrows)
 Our approach :
We study the low energy dynamics of the spin-phonon coupled model in the
exchange-striction picture on a two dimensional triangular net.
Two types of spin-polarization coupling
Exchange-striction type:
(RMn2O5)
Dzyaloshinskii-Moriya type:
(RMnO3 and many others)
Model Hamiltonian
The exchange integral Jij, which is a functional of the distance
between nearest neighbor lattice sites, is Taylor expanded to first order.
 Classical Analysis
x
y
Nearest neighbor spins make a
120 angle with respect to each
other
z
Holstein-Primakoff (HP) Theory
 Analysis of small fluctuations using HP bosons :
 For low-lying states of the system such that the fractional
spin reversal is small;
 1/S expansion of the HP operators to 1st order gives out
Diagonalization of the Hamiltonian
Divide the Hamiltonian into two parts, H0 and HSP, and diagonalize each respectively.
Spinwave energy spectrum :
Truncate terms higher than the quadratic
Diagonalize through a series of
Bogoliubov transformations
Two branches of coupled spinwave/phonon dispersions
Energy Spectrum (1)
Original spinwave/phonon dispersions
k z,0
1
Coupled spinwave/phonon dispersions
k z,0
1
Spinwave
dispersion
0.8
Phonon
dispersion
0.4
E 2k, J 1 =0.1
0.4
E 2k, J 1 =0.3
0.2
-4
E 1k, J 1 =0.3
0.6
0.6
-6
E 1k, J 1 =0.1
0.8
0.2
-2
2
4
6
kz
-3
-2
-1
1
2
3
kz
Increase of ~0.12J0 in E1k
Decrease of ~0.31J0 in E2k at (kz, ky)=(2,0)
Weakly(J1=0.1) and strongly(J1=0.3) coupled spinwave/phonon dispersions
k z,0
1
k z,0
1
0.8
0.8
E1k dispersion
0.6
0.6
E2k dispersion
0.4
0.4
0.2
-3
-2
-1
0.2
1
J1=0.1
2
3
kz
-3
-2
-1
1
J1=0.3
2
3
kz
Energy Spectrum (2)
Weakly coupled (J1=0.1)
spinwave/phonon dispersions
Uncoupled spinwave/phonon dispersion
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
5
0
-5
ky
5
0
-5
ky
0
kz
0
-5
kz
5
-5
5
Strongly coupled (J1=0.3)
spinwave/phonon dispersions
Emergence of roton-like
minima points
1
0.8
0.6
0.4
0.2
0
5
0
-5
0
kz
-5
5
ky
New Spin Configuration
New Spin configuration
Critical Value of J1=0.321
k z ,0
1
0.8
0.6
0.4
0.2
-3
-2
-1
1
Roton-like minima points
2
3
kz
Magnetization
Surprisingly little change due to spin-phonon coupling!
Phonon Correlation Function
Spin-Spin Correlation Function
-2
-1
0
1
2
-2
-1
0
1
We calculate
the
spin-spin
correlation
function,
,
from
which
we
2.
2.
2.
can extract the electromagnon spectral function, directly observable by
neutron scattering experiments.
1.5
1.5
1.
1.
0.5
-2
-1
0
1
0
-1
0
1
0.5
2.
2.
1.
1.
0.5
0.5
0
0
0
1
0.5
2
-1
0
1
2
2.
0
0
-2
1.5
-1
1.
-2
2
1.5
-2
1.5
1.
0.5
0
-2
2.
1.5
2
2.
2
-1
0
1
2
1.5
1.5
1.
1.
0.5
0.5
0
0
-2
-1
0
1
2
Summary
 Spin-phonon coupled Hamiltonian is exactly
solved to quadratic order for the triangular lattice.
 The dispersion shows a roton-like minima at
k=(2,0) in the presence of critical coupling.
 Various measurable quantities, (magnetization,
phonon correlation function, spin-spin correlation
function) have been calculated.
References