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 : Tb Mn O 3 , Ni 3 V 2 O 8 , Ba 0.5

Sr 1.5

Zn 2 Fe 12 O 22 , CoCr 2 O 4 , Mn WO 4 ,

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-to helical 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 Nagaosa 6 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: (RMn 2 O 5 ) Dzyaloshinskii-Moriya type: ( RMnO 3 and many others )

Model Hamiltonian

The exchange integral J ij , which is a functional of the distance between nearest neighbor lattice sites, is Taylor expanded to first order.



Classical Analysis

Nearest neighbor spins make a 120  angle with respect to each other

y x 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 1 st order gives out

Diagonalization of the Hamiltonian

Divide the Hamiltonian into two parts, H 0 and H SP , 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)

-6 Original spinwave/phonon dispersions k z ,0 1 0.8

0.6

0.4

0.2

Spinwave dispersion Phonon dispersion

Coupled spinwave/phonon dispersions k z ,0 1 0.8

0.6

0.4

E 1k , J 1 =0.1

E 1k , J 1 =0.3

E 2k , J 1 =0.1

E 2k , J 1 =0.3

0.2

-4 -2 2 4 6 k z k z -3 -2 -1 1 2 3

Increase of ~0.12J

0 in E 1k Decrease of ~0.31J

0 in E 2k at (k z , k y )=(2

,0

)

Weakly(J 1 =0.1) and strongly(J 1 =0.3) coupled spinwave/phonon dispersions k z ,0 1 0.8

0.6

0.4

0.2

E 1k dispersion E 2k dispersion

k z ,0 1 0.8

0.6

0.4

0.2

-3 -2 -1 J 1 =0.1

1 2 3 k z -3 -2 -1 J 1 =0.3

1 2 3 k z

Energy Spectrum (2)

Uncoupled spinwave/phonon dispersion Weakly coupled (J 1 =0.1) spinwave/phonon dispersions 1 0.8

0.6

0.4

0.2

0 k z Strongly coupled (J 1 =0.3) spinwave/phonon dispersions -5 0 k y 5 1 0.8

0.6

0.4

0.2

0 k z 1 0.8

0.6

0.4

0.2

0 -5 0 k y 5 k z -5 0 k y 5

Emergence of roton-like minima points

New Spin Configuration

New Spin configuration Critical Value of J 1 =0.321

k z ,0 1 0.8

0.6

0.4

0.2

-3 -2 -1 1 Roton-like minima points 2 3 k z

Magnetization

Surprisingly little change due to spin-phonon coupling!

Phonon Correlation Function

Spin-Spin Correlation Function

-2 -1 0 1 2 We calculate the spin-spin correlation function, , from which we 2.

-2 -1 0 1 2 can extract the electromagnon spectral function, directly observable by neutron scattering experiments. 1.5

1.5

1.5

2.

1.5

1.

1.

1.

1.

0.5

2.

-2 0 -2 1.5

-1 -1 0 0 1 1 1.

0.5

0 -2 -1 0 1 0.5

2 2 0 2.

1.5

1.

0.5

2 0 0.5

2.

-2 0 -2 1.5

-1 -1 1.

0.5

0 -2 -1 0 0 0 1 1 1 0.5

2 2 2.

0 1.5

1.

0.5

2 0

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