KEY ENTITIES:

Localized excitations = = unconventional forms of lattice dynamics described by

classical degrees of freedom

Complex oxides = = multicomponent perovskites exhibiting slightly different ground states that can easily be converted upon small extrinsic perturbations

Polar nanoregions = = spatial regions that are to small to approach

the thermodynamic phase transition limit and still are large enough for cooperativity

of their atomic displacements. Unique counterpart of relaxor ferroelectrics.

KEY ENTITIES:

Localized excitations = = unconventional forms of lattice dynamics described by

classical degrees of freedom

Complex oxides = = multicomponent perovskites exhibiting slightly different ground states that can easily be converted upon small extrinsic perturbations

Polar nanoregions = = spatial regions that are to small to approach

the thermodynamic phase transition limit and still are large enough for cooperativity

of their atomic displacements. Unique counterpart of relaxor ferroelectrics.

LOCALIZED EXCITATIONS IN COMPLEX OXIDES: POLAR NANOREGIONS

E. Klotins

Institute of Solid State Physics, University of Latvia

LENCOS-09,Seville, July,13-17,(2009)

STATE OF ART

Discrete nonlinear systems contrasting expectations of canonical statisticsDiscrete nonlinear systems contrasting expectations of canonical statistics

Ferroelectric relaxors with polar nanoregions as the unique counterpartFerroelectric relaxors with polar nanoregions as the unique counterpart

First principle effective (phonon) HamiltoniansFirst principle effective (phonon) Hamiltonians

DNLSDNLS

Grand canonical statistics in action-angle approach

K.Ø.Rasmussen, T. Cretegny, P.G. Kevrekidis, N.Grønbech-Jenssen, PRL 84,3740(2000)

M. Johansson, Physica D 216,62 (2006)

Grand canonical statistics in action-angle approach

K.Ø.Rasmussen, T. Cretegny, P.G. Kevrekidis, N.Grønbech-Jenssen, PRL 84,3740(2000)

M. Johansson, Physica D 216,62 (2006)

Phase space separation (at temperatures above localization transition)

B.Rumpf, PRE 69, 016618 (2004)

Phase space separation (at temperatures above localization transition)

B.Rumpf, PRE 69, 016618 (2004)

Extremal entropy approach: interaction with phonons

B.Rumpf, EPL 78 (2007)26001 B. Rumpf, PLA 372 (2008) 1579

Extremal entropy approach: interaction with phonons

B.Rumpf, EPL 78 (2007)26001 B. Rumpf, PLA 372 (2008) 1579

2

MICROSCOPIC STARTING POINT : ELEMENTARY LATTICE

Ba

Ti

O

Structure of perovskite BaTiO3. Arrows indicate displacements for a local mode polarized along x axis ( )

Structure of perovskite BaTiO3. Arrows indicate displacements for a local mode polarized along x axis ( )

Microscopic theory gives:

Local modes assigned to elementary lattice cells

Dipole moment associated with local mode

First principles effective (phonon) Hamiltonian

Microscopic theory gives:

Local modes assigned to elementary lattice cells

Dipole moment associated with local mode

First principles effective (phonon) Hamiltonian

Challenges:

Finite temperature properties

Spatio-temporal behavior

Challenges:

Finite temperature properties

Spatio-temporal behavior

3

llelasshortdipselftot EEEEEE ,int uuuu

222222422 izizixiyiyixiii uuuuuuuukE u 222222422 izizixiyiyixiii uuuuuuuukE u

ji ij

jijiijjidip

R

ZE

3

2 ˆˆ3 uRuRuuu

ji ij

jijiijjidip

R

ZE

3

2 ˆˆ3 uRuRuuu

ji

jiijshort uuJE

,2

1u

ji

jiijshort uuJE

,2

1u

lHelas

HlIelas

Ilelas EEE ,, lH

elasHlI

elasIl

elas EEE ,, iiilll uuBE RRRu 2

1,int

iiilll uuBE RRRu 2

1,int

Dipole moment associated with local mode is di = Z*ui

Dipole moment associated with local mode is di = Z*ui

Total elastic energy expanded to quadratic order as a sum of homogeneous and inhomogeneous constituents

Total elastic energy expanded to quadratic order as a sum of homogeneous and inhomogeneous constituents

Energy contribution due the interactions between neighboring local modes.

J ij,αβ is the interaction matrix.

Energy contribution due the interactions between neighboring local modes.

J ij,αβ is the interaction matrix.

(Electro) elastic interaction

(Electro) elastic interaction

FIRST- PRINCIPLES EFFECTIVE (PHONON) HAMILTONIAN

Local and dipole – dipole terms share essential properties of Klein – Gordon lattices Local and dipole – dipole terms share essential properties of Klein – Gordon lattices

Local mode at cell Ri with amplitude {ui} relative to that of the perfect cubic structure

Local mode at cell Ri with amplitude {ui} relative to that of the perfect cubic structure

W.Zong, D. Vanderbilt, K.M. Rabe, PRL 73, 1861 (1994 ) W.Zong, D. Vanderbilt, K.M. Rabe, PRB 52, 6301 (1995)

4

Temperature development of supecell averaged soft-mode components. u1 (diamonds) , u2, u3 (triangles). Effective Hamiltonian for BaTiO3. Local + dipole-dipole terms.

Temperature development of supecell averaged soft-mode components. u1 (diamonds) , u2, u3 (triangles). Effective Hamiltonian for BaTiO3. Local + dipole-dipole terms.

STRUCTURAL PHASE TRANSITION

Supercell averaged soft-mode components. u1 (diamonds) ,u2,u3 (triangles).

Supercell averaged soft-mode components. u1 (diamonds) ,u2,u3 (triangles).

Conventional phase transition at critical temperature Tc

Conventional phase transition at critical temperature Tc

Emergence of polar nanoregions at appropriate chemical content

Too small to approach the thermodynamic phase transition limit

Large enough that the cooperativity of their atomic displacements is evident in the neutron data.

Polar nanoregions are at the heart of ultrahigh performance

Emergence of polar nanoregions at appropriate chemical content

Too small to approach the thermodynamic phase transition limit

Large enough that the cooperativity of their atomic displacements is evident in the neutron data.

Polar nanoregions are at the heart of ultrahigh performance

?

STATISTICS IN CANONICAL ENSEMBLESTATISTICS IN CANONICAL ENSEMBLE STATISTICS IN GRAND – CANONICAL ENSEMBLESTATISTICS IN GRAND – CANONICAL ENSEMBLE

W.Zong, D. Vanderbilt, K.M. Rabe, PRL 73, 1861 (1994 )S.Tinte,J. Inguez, K.M. Rabe, D. Vanderbilt, PRB 67, 064106 (2003)

5

DIRECT EXPERIMENTAL EVIDENCES OF POLAR NANOREGIONS

Formation of polar nanoregions 20-200 Å supported by local electric fields and chemical disorder

TBTBTfTf

BURNS TEMPERATURE (TB)

Below TB the intensity of the central peak ( ICP) as a function of temperature becomes measurable

BURNS TEMPERATURE (TB)

Below TB the intensity of the central peak ( ICP) as a function of temperature becomes measurable

High temperature

FREEZING TEMPERATURE (Tf)

In the neighborhood of Tf ICP rises sharply then plateaus

FREEZING TEMPERATURE (Tf)

In the neighborhood of Tf ICP rises sharply then plateaus

Dielectric dispersion of relaxation nature Dynamic motion in THz range slowing down at some freezing temperature

Low temperature

INDIRECT EXPERIMENTAL EVIDENCES OF POLAR NANOREGIONS

Neutron scattering measurements

6

STATISTICAL PHYSICS == IN GRAND – CANONICAL ENSEMBLE

DYNAMICS == NONLINEAR (mode frequency depends on amplitude)

SYSTEM == PERIODIC

∑==

+RESEMBLANCE BETWEEN PNR AND INTRINSIC LOCALIZED EXCITATIONS

WORKING HYPOTESIS

KEY PROBLEM:

To what degree the concepts and mathematical techniques developed for localized excitations are valid for PNR

KEY PROBLEM:

To what degree the concepts and mathematical techniques developed for localized excitations are valid for PNR

CONTENTS: DNLS representation of effective (phonon) Hamiltonian for complex oxides

Modulation instability in presence of dipole-dipole interaction

Localization transition: from action-angle approach to extremal entropy

Entropy/energy balance: localization transition and emergence of polar

nanoregions paralleled

7

EFFECTIVE HAMILONIAN - DNLS APPROACH

Intersite (dipole-dipole) interactions

ji ij

jijiijjiiiiii R

uRuRuuZuuuuuH

3

242

202

ˆˆ3

422

1

ji ij

jijiijjiiiiii R

uRuRuuZuuuuuH

3

242

202

ˆˆ3

422

1

#1 Hamiltonian#1 HamiltonianInsite (anharmonic) interactions

#3 Fourier transform (mean value ->symmetry breaking)#3 Fourier transform (mean value ->symmetry breaking)

0)()(62*

3)0()0()1()1(20

nnnnn dipoledipole

Zaatata

0)()(6

2*3)0()0()1()1(2

0

nnnnn dipoledipole

Zaatata

#2 Fourier transform (fundamental frequency -> conventional modulation instability conserving symmetry#2 Fourier transform (fundamental frequency -> conventional modulation instability conserving symmetry

0233 )1(2*

2)1()1()1(2)0(220

ibiiinb aidipoledipole

Zaaaa

0233 )1(

2*2)1()1()1(2)0(22

0

ibiiinb aidipoledipole

Zaaaa

Fundamental frequencyFundamental frequency

8

MODULATION INSTABILITY IN PRESENCE OF DIPOLE-DIPOLE INTERACTION: NUMERICAL EVIDENCES

Lattice with unit spacing

Time scale = 5 periods of nonlinear plane wave

Initial conditions:

Plane wave amplitude = 0.006

Uniformly distributed fluctuations = 10-7

Initial conditions:

Plane wave amplitude = 0.006

Uniformly distributed fluctuations = 10-7

Plane wave amplitude

M.Öster, M. Johansson, Phys. Rev. E 71, 025601(R)(2005)

Y.S. Kivshar, Phys.Lett. A 173 (1993) (172-178)

Dipole-dipole interaction favuors the spatial size of excitationsDipole-dipole interaction favuors the spatial size of excitations

9

MUTUAL EFFECT OF LOCALIZED EXCITATIONS AND RANDOM FIELDS - DNLS APPROACH #1

LOCALIZED EXCITATIONS POLAR NANOREGIONS

DIPOLE – DIPOLE INTERACTION FACTOR 0.01 a.u. (SMALL)

Nonlinearity factor β= 6

Plane wave: wave number q = 0.000001, amplitude = 0.00018

Time scale: 1 period of plane wave

Random field

10

MUTUAL EFFECT OF LOCALIZED EXCITATIONS AND RANDOM FIELDS - DNLS APPROACH #2

LOCALIZED EXCITATIONS POLAR NANOREGIONS

DIPOLE – DIPOLE INTERACTION FACTOR 0.4 a.u. (MEDIUM)

Nonlinearity factor β= 6

Plane wave: wave number q = 0.000001, amplitude = 0.00018

Time scale: 1/2 period of plane wave

NO LOCALIZED EXCITATIONS

11

MUTUAL EFFECT OF LOCALIZED EXCITATIONS AND RANDOM FIELDS - DNLS APPROACH #3

LOCALIZED EXCITATIONS POLAR NANOREGIONS

DIPOLE – DIPOLE INTERACTION FACTOR 0.5 a.u. (LARGE)

Nonlinearity factor β= 6

Plane wave: wave number q = 0.000001, amplitude = 0.00018

Time scale: 1/2 period of plane wave

NO LOCALIZED EXCITATIONS

12

LOCALIZED EXCITATIONS: OVERCRITICAL AMPLITUDE DNLS APPROACH #4

LOCALIZED EXCITATIONSLOCALIZED EXCITATIONS

DIPOLE – DIPOLE INTERACTION FACTOR 0.01 a.u. (SMALL)

Nonlinearity factor β= 6

Plane wave: wave number q = 0.000001, amplitude = 0.0009

Time scale: 1/2 period of plane wave

12.1

AMPLITUDE 0.0009 (OVERCRITICAL)

MODULUS SQUARE AMPLITUDE

MODULUS SQUARE AMPLITUDE

STATISTICAL MECHANICS (ACTION – ANGLE APPROACH)

0

2

0 1

N

m

AHmm

AAAAedAdZ

Grand – canonical partition function Chemical potential

Action – angle variables

Action – angle Hamiltonian

Action - angle excitation norm

nn

nnnnn

AA AAAH 211 2

1cos2

nn

nnnnn

AA AAAH 211 2

1cos2

n

nAA AN

nn

AA AN

K.Ø. Rasmussen, T. Cretegny, P.G.Kevrekidis, PRL 84,3740 (2000) M. Johansson, Physica D, 62 (2006)]

13

0 1 2 3 4 5 6AVERAGE NORM DENSITYa0

5

10

15

20

25

30

35EGAREVA

YGRENE

YTISNEDh a,hPARAMETER SPACE

STATISTICAL MECHANICS : LOCALIZATION TRANSITION

GIBBSIAN THERMALIZATION

EX

CIT

AT

ION

S

The state of a system is distinguished by two types of initial conditions.

N

Aa AA

N

Hh AA

2ah

2ah

LOCALIZATION TRANSITION (β->0)

14

STATISTICAL MECHANICS : PHASE SPACE SPLITING

High amplitude domain

K<<(N-K) peaks

High amplitude domain

K<<(N-K) peaks

Low amplitude domain

(N-K) fluctuations

Low amplitude domain

(N-K) fluctuations

Contributes little in total entropyContributes little in total entropy

Max entropy is reached if the fluctuations contain appropriate amount of energy

Max entropy is reached if the fluctuations contain appropriate amount of energy

S

S

Exchange between peaks and fluctuations vanishes when temperature and chemical potential is the same

Exchange between peaks and fluctuations vanishes when temperature and chemical potential is the same

B.Rumpf,PRE 69,016618 (2004)

Mathematical objective:

Find extremal entropy as a function of the conserved quantities and

Mathematical objective:

Find extremal entropy as a function of the conserved quantities andH A

15

EXTREMAL ENTROPY APPROACH

B.Rumpf,EPL, 78 (2007)26001

In more realistic models the zone boundary modes becomes essential

Statistical problem is growth/decay of excitations is caused by interaction with phonons

Their distribution over all wavenumbers in the BZ may be captured by Rayleigh-Jeans distribution

In more realistic models the zone boundary modes becomes essential

Statistical problem is growth/decay of excitations is caused by interaction with phonons

Their distribution over all wavenumbers in the BZ may be captured by Rayleigh-Jeans distribution

EXCITATIONS

)(excE

)(excN

)( phE

)( phN

PHONONS

(int)E

(int)Nk

phk

ph NS )()( ln

16

EXCITATION

(GROWTH)

)(excE

)(excN

)( phE

)( phN

PHONONS(int)E

(int)N

k

phk

ph NS )()( ln

EXCITATION

(STATIONARY STATE)

)(excE

)(excN

)( phE

)( phN

PHONONS(int)E

(int)N k

phk

ph NS )()( ln

Decreases entropy of phonons

Entropy gain

LOCALIZED EXCITATION IN PHONON BATH17

Nonlinear Hamiltonian lattices with nearest neighbor interaction

Effective lattice Hamiltonians with d-d, random fields and phonon bath

DY

NA

MIC

S

Phase space splitting Details of ordering transition missed

Extremal entropy Details of ordering transition as well as relaxation of PNR captured

ST

AT

IST

ICS

Field response and time propagation of PNR

CH

AL

LEN

GE

S

INTRINSIC LOCALIZED EXCITATIONS ADVANCEMENTS/DRAWBACKS FOR PNR

Action – angle approach for grand – canonical statistics

Action – angle approach is invalid in case of d-d interaction

Nonconservative Hamiltonians

STATE OF AFFAIRS IN APPLICATION OF DNLS TO POLAR NANO - REGIONS

Theory of dipolar glasses

18

CONCLUSIONS

Highly motivated developments are addressed to nonconservative Hamiltonians with applications to field response and time development of PNR and the theory of dipolar glasses

Growth of PNR corresponds to the relaxation toward the state of maximum entropy

Heuristic interpretation of long-living PNR is that they constitute the state of maximum entropy for certain values of (conserved) initial conditions

DNLS representation of effective (phonon) Hamiltonian is promising for PNR in complex oxides

19

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