Ferroelectric soft and central modes in perovskite ceramics and thin films

Jan PetzeltInstitute of Physics Acad. Sci. Czech Republic

Prague, Czech Republicpetzelt@fzu.cz

Normal vibrational modes

Normal modes (whose quanta are phonons) are plane-wave excitations consisting of specific linear combinations of small atomic displacements (vibrations), which in the case of harmonic lattice (vibrations of atoms in parabolic potentials) do not interact with each other.Number of phonon branches is equal to the number of vibrationaldegrees of freedom in one primitive unit cell, i.e. 3s, where s is the number of atoms in the unit cell. 3 of them are acoustic, remaining ones are optic.Symmetry classification of all phonons is according to irreducible representations of the corresponding crystal space group; Modes transforming according to one irreducible representation belong to one symmetry type (species). By symmetry arguments, they are linearly uncoupled from all other symmetry-types modes and obey the same selection rules (same activity in IR and Raman spectra).

Some useful keywords and concepts:

Mode degeneracy – given by the dimensionality of the corresponding ir. rep.: 3-dim – only in cubic structures (F (sometimes T) symbols)

2-dim – in optically uniaxial and cubic structures (E symbols)1-dim – everywhere (A and B symbols)

Transverse optical (TO) modes – IR active (polar) modesLongitudinal optical (LO) modes – do not couple with the el-mag

radiation, but can be evaluated from IR (or Raman) experimentSilent modes – in a given structure are inactive in any of the IR and

Raman spectraExclusion rule - in centro-symmetric structures no mode is both IR

and Raman activeTotally symmetric mode – the vibrational displacements do not break

the crystal symmetry (always 1-dim. ir. rep. with A symbol).Structural phase transitions – mostly change the symmetry ⇒ change

in selection rules and possibly in degeneracy (e.g. splitting if the symmetry reduces).

More general is factorized form (generalized 4-parameter oscillators, longitudinal damping γLO added), which reduces to Lyddanne-Sachs-Teller (L-S-T) relations for ω = 0. It is frequently used to fit the IR reflectivity.

Dielectric function of lattice vibrations (evaluated from the IR experiment)

The simplest model is sum of classical damped harmonic oscillators:

Both models display n complex poles and zeros whose frequency define the (complex) transverse and longitudinal normal modes, respectively. ωTOj andωLOj are moduli of these complex frequencies, respectively.

Simple perovskites

(a) Slater mode (b) Last mode (c) Axe modeFerroelectric soft mode Ferroelectric soft mode Highest frequencyin non-Pb perovskites in Pb-containing perovskites

IR active normal modes in simple cubic perovskites ABO3

A O B

SrTiO3 (ST)Incipient ferroelectric, at 105 K antiferrodistortive transition from simple cubic Pm3m structure to tetragonal phase I4/mcm.

Ferroelectric SM

Structural SM (doublet)

Mode frequencies in ST ceramics

(Petzelt et al., PRB 64, 184111 (2001))

x

y

z

TO1 (FE SM)(9-90) cm-1

TO2175 cm-1

TO4545 cm-1

Polar modes ( IR-active, R-inactive)

O2-

Sr2+

Ti4+

Below Ta ≈ 105 K - tetragonal symmetry

Cubic symmetry AFD SM –

non-polar Raman active

SrTiO3 – incipient ferroelectric

Antiferrodistortive (AFD) transition in SrTiO3 in R-point of the Brillouin zone

Eg

A1g Splitting of the AFD soft mode (Raman active below Ta)

Ta

Anisotropy of permittivity and splitting of the ferroelectric soft-mode in ST below the AFD transition

BaTiO3 (BT)

Factor-group analysis of the vibrational modes in different phases

Strong dielectric anisotropy in the tetragonal phase of BT single-domain crystal (Wemple et al., J. Phys. Chem. Solids 29, 1797 (1968), Camlibel et al., ibid. 31, 1417 (1969)).

Soft-mode in BaTiO3 crystal

0 200 400 600 800 1000 1200 14000

20406080

100120140160180200220240260280

CTR O

E

E

Freq

uenc

y (c

m-1)

Luspin_80 (IR refl.) Vogt_82 (HR) Laabidi_90 (R) Perry_65 (R) Burns_78 (R) Dougherty_94 (femt. ISRS)

Temperature (K)

B2

B1?

A1

F1u

TO1 mode in BaTiO3 single crystal

Temperature dependence of the overdamped soft-mode component in BaTiO3 .

Plotted is the equivalent Debye-relaxation frequency ω02/γ which corresponds

approximately to the maxima in ε”(ω) spectra.

87

BaTiO3 - rotational dispersion of long-wavelength IR phonon modes frequencies at room temperature

TO1(E)

TO2(E)

LO2(A) TO2(A)LO2(E)

TO3(E)TO1(A)LO3(E)

LO1(A)

TO4(E)LO1(E)TO4(E)TO4(A)

LO4(E)LO4(A)

k II c k ⊥ c

Extremely large A-E TO1 mode splitting ⇒ large dielectric anisotropy

Order-disorder model for BaTiO3 phase transitions

Ordered Ba

Dynamically disordered Ti

Chain-cluster structure proposed by Comes et al, Sol. State Commun. 6, 715 (1968):

Dipole-moment correlation of about 10-20 unit cells (4-8 nm) along the cubic crystal axes.