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KEY ENTITIES: Localized excitations = = unconventional forms of lattice dynamics described by

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

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KEY ENTITIES: Localized excitations = = unconventional forms of lattice dynamics described by

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  1. 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 • 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. LENCOS-09,Seville, July,13-17,(2009)

  2. 2 STATE OF ART Discrete nonlinear systems contrasting expectations of canonical statistics Ferroelectric relaxors with polar nanoregions as the unique counterpart First principle effective (phonon) Hamiltonians DNLS 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) Extremal entropy approach: interaction with phonons B.Rumpf, EPL 78 (2007)26001 B. Rumpf, PLA 372 (2008) 1579

  3. 3 Ba Ti O MICROSCOPIC STARTING POINT : ELEMENTARY LATTICE 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 • Challenges: • Finite temperature properties • Spatio-temporal behavior

  4. 4 FIRST- PRINCIPLES EFFECTIVE (PHONON) HAMILTONIAN Local and dipole – dipole terms share essential properties of Klein – Gordon lattices Energy contribution due the interactions between neighboring local modes. J ij,αβis the interaction matrix. Dipole moment associated with local mode is di = Z*ui (Electro) elastic interaction Total elastic energy expanded to quadratic order as a sum of homogeneous and inhomogeneous constituents 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)

  5. 5 STRUCTURAL PHASE TRANSITION STATISTICS IN CANONICAL ENSEMBLE STATISTICS IN GRAND – CANONICAL ENSEMBLE Supercell averaged soft-mode components. u1 (diamonds) ,u2,u3 (triangles). 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 Conventional phase transition at critical temperature Tc ? Temperature development of supecell averaged soft-mode components. u1 (diamonds) , u2, u3 (triangles). Effective Hamiltonian for BaTiO3. Local + dipole-dipole terms. W.Zong, D. Vanderbilt, K.M. Rabe, PRL 73, 1861 (1994 ) S.Tinte,J. Inguez, K.M. Rabe, D. Vanderbilt, PRB 67, 064106 (2003)

  6. 6 DIRECT EXPERIMENTAL EVIDENCES OF POLAR NANOREGIONS • Neutron scattering measurements BURNS TEMPERATURE (TB) Below TB the intensity of the central peak ( ICP) as a function of temperature becomes measurable FREEZING TEMPERATURE (Tf) In the neighborhood of Tf ICP rises sharply then plateaus Tf TB Formation of polar nanoregions 20-200 Å supported by local electric fields and chemical disorder Low temperature High temperature INDIRECT EXPERIMENTAL EVIDENCES OF POLAR NANOREGIONS • Dielectric dispersion of relaxation nature • Dynamic motion in THz range slowing down at some freezing temperature

  7. 7 WORKING HYPOTESIS + SYSTEM == PERIODIC DYNAMICS == NONLINEAR (mode frequency depends on amplitude) STATISTICAL PHYSICS ==IN GRAND – CANONICAL ENSEMBLE ∑== RESEMBLANCE BETWEEN PNR AND INTRINSIC LOCALIZED EXCITATIONS • 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

  8. 8 EFFECTIVE HAMILONIAN - DNLS APPROACH Insite (anharmonic) interactions Intersite (dipole-dipole) interactions #1 Hamiltonian #2 Fourier transform (fundamental frequency -> conventional modulation instability conserving symmetry #3 Fourier transform (mean value ->symmetry breaking) Fundamental frequency

  9. 9 MODULATION INSTABILITY IN PRESENCE OF DIPOLE-DIPOLE INTERACTION: NUMERICAL EVIDENCES Dipole-dipole interaction favuors the spatial size of excitations Plane wave amplitude Time scale = 5 periods of nonlinear plane wave Lattice with unit spacing Initial conditions: Plane wave amplitude = 0.006 Uniformly distributed fluctuations = 10-7 M.Öster, M. Johansson, Phys. Rev. E 71, 025601(R)(2005) Y.S. Kivshar, Phys.Lett. A 173 (1993) (172-178)

  10. 10 MUTUAL EFFECT OF LOCALIZED EXCITATIONS AND RANDOM FIELDS - DNLS APPROACH #1 DIPOLE – DIPOLE INTERACTION FACTOR 0.01 a.u. (SMALL) LOCALIZED EXCITATIONS POLAR NANOREGIONS Random field Nonlinearity factor β= 6 Plane wave: wave number q = 0.000001, amplitude = 0.00018 Time scale: 1 period of plane wave

  11. 11 MUTUAL EFFECT OF LOCALIZED EXCITATIONS AND RANDOM FIELDS - DNLS APPROACH #2 DIPOLE – DIPOLE INTERACTION FACTOR 0.4 a.u. (MEDIUM) LOCALIZED EXCITATIONS POLAR NANOREGIONS NO LOCALIZED EXCITATIONS Nonlinearity factor β= 6 Plane wave: wave number q = 0.000001, amplitude = 0.00018 Time scale: 1/2 period of plane wave

  12. 12 MUTUAL EFFECT OF LOCALIZED EXCITATIONS AND RANDOM FIELDS - DNLS APPROACH #3 DIPOLE – DIPOLE INTERACTION FACTOR 0.5 a.u. (LARGE) LOCALIZED EXCITATIONS POLAR NANOREGIONS NO LOCALIZED EXCITATIONS Nonlinearity factor β= 6 Plane wave: wave number q = 0.000001, amplitude = 0.00018 Time scale: 1/2 period of plane wave

  13. 12.1 LOCALIZED EXCITATIONS: OVERCRITICAL AMPLITUDE DNLS APPROACH #4 DIPOLE – DIPOLE INTERACTION FACTOR 0.01 a.u. (SMALL) AMPLITUDE 0.0009 (OVERCRITICAL) LOCALIZED EXCITATIONS MODULUS SQUARE AMPLITUDE Nonlinearity factor β= 6 Plane wave: wave number q = 0.000001, amplitude = 0.0009 Time scale: 1/2 period of plane wave

  14. 13 STATISTICAL MECHANICS (ACTION – ANGLE APPROACH) Grand – canonical partition function Chemical potential Action – angle variables Action – angle Hamiltonian Action - angle excitation norm K.Ø. Rasmussen, T. Cretegny, P.G.Kevrekidis, PRL 84,3740 (2000) M. Johansson, Physica D, 62 (2006)]

  15. 14 STATISTICAL MECHANICS : LOCALIZATION TRANSITION The state of a system is distinguished by two types of initial conditions. LOCALIZATION TRANSITION (β->0) EXCITATIONS GIBBSIAN THERMALIZATION

  16. 15 STATISTICAL MECHANICS : PHASE SPACE SPLITING Exchange between peaks and fluctuations vanishes when temperature and chemical potential is the same Contributes little in total entropy High amplitude domain K<<(N-K) peaks Max entropy is reached if the fluctuations contain appropriate amount of energy Low amplitude domain (N-K) fluctuations Mathematical objective: Find extremal entropy as a function of the conserved quantities and B.Rumpf,PRE 69,016618 (2004)

  17. 16 EXTREMAL ENTROPY APPROACH • 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 PHONONS B.Rumpf,EPL, 78 (2007)26001

  18. 17 LOCALIZED EXCITATION IN PHONON BATH EXCITATION (GROWTH) PHONONS Entropy gain Decreases entropy of phonons EXCITATION (STATIONARY STATE) PHONONS

  19. 18 STATE OF AFFAIRS IN APPLICATION OF DNLS TO POLAR NANO - REGIONS INTRINSIC LOCALIZED EXCITATIONS ADVANCEMENTS/DRAWBACKS FOR PNR DYNAMICS Nonlinear Hamiltonian lattices with nearest neighbor interaction Effective lattice Hamiltonians with d-d, random fields and phonon bath Action – angle approach for grand – canonical statistics Action – angle approach is invalid in case of d-d interaction STATISTICS Phase space splitting Details of ordering transition missed Extremal entropy Details of ordering transition as well as relaxation of PNR captured CHALLENGES Nonconservative Hamiltonians Field response and time propagation of PNR Theory of dipolar glasses

  20. 19 CONCLUSIONS • DNLS representation of effective (phonon) Hamiltonian is promising for PNR in complex oxides • Heuristic interpretation of long-living PNR is that they constitute the state of maximum entropy for certain values of (conserved) initial conditions • Growth of PNR corresponds to the relaxation toward the state of maximum entropy • Highly motivated developments are addressed to nonconservative Hamiltonians with applications to field response and time development of PNR and the theory of dipolar glasses

  21. THANKS FOR YOUR ATTENTION!

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