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LOCALIZED EXCITATIONS IN COMPLEX OXIDES: POLAR NANOREGIONS. E. Klotins Institute of Solid State Physics, University of Latvia. KEY ENTITIES: Localized excitations = = unconventional forms of lattice dynamics described by classical degrees of freedom - PowerPoint PPT Presentation
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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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