Thermal Fluctuations of Ferroelectric Nanodomains in a Ferroelectric-Dielectric ${\mathrm{PbTiO}}_3/{\mathrm{SrTiO}}_3$ Superlattice

Ferroelectric-dielectric superlattices consisting of alternating layers of ferroelectric ${\mathrm{PbTiO}}_3$ and dielectric ${\mathrm{SrTiO}}_3$ exhibit a disordered striped nanodomain pattern, with characteristic length scales of 6 nm for the domain periodicity and 30 nm for the in-plane coherence of the domain pattern. Spatial disorder in the domain pattern gives rise to coherent hard $x$-ray scattering patterns exhibiting intensity speckles. We show here using variable-temperature Bragg-geometry $x$-ray photon correlation spectroscopy that $x$-ray scattering patterns from the disordered domains exhibit a continuous temporal decorrelation due to spontaneous domain fluctuations. The temporal decorrelation can be described using a compressed exponential function, consistent with what has been observed in other systems with arrested dynamics. The fluctuation speeds up at higher temperatures and the thermal activation energy estimated from the Arrhenius model is $0.35\pm 0.21\mathrm{eV}$. The magnitude of the energy barrier implies that the complicated energy landscape of the domain structures is induced by pinning mechanisms and domain patterns fluctuate via the generation and annihilation of topological defects similar to soft materials such as block copolymers.

Figure 1

(a) The geometry of the experiment in real space. $k_{\mathrm{in}}$ and $k_{\mathrm{out}}$ represent the directions of the incident and scattered $x$-ray beams, respectively. The red and blue stripes in the serpentine pattern represent unit cells with up and down polarization in the ${\mathrm{PbTiO}}_3/{\mathrm{SrTiO}}_3$ superlattice. (b) The reciprocal-space scattering geometry. The sample and detector are arranged so that the Ewald sphere [translucent red (spherical) surface] intersects the ring of domain diffuse scattering (green ring), but does not pass through the 002 superlattice Bragg reflection (red sphere). (c) Representative coherent scattering speckle pattern from the superlattice sample. (d) Ensemble-averaged scattering pattern obtained from Fig. 1 via digital smoothing and used as an approximation of the incoherent scattering pattern.

Figure 2

(a) Spatial profile of diffuse domain scattering intensity in reciprocal space measured from rocking curve scans at different temperatures. The dashed lines are fits to Gaussian profiles. (b) Peak intensity of the domain scattering as a function of temperature. (c) Peak position of the domain scattering (in $Q_x$) as a function of temperature.

Figure 3

(a) Two-time correlation map for coherent domain diffuse scattering patterns at 393 K. The color scale highlights the nondiagonal values. Correlations on the exact diagonal are much higher due to self-correlation of the noise from counting statistics [18]. (b) ISFs acquired from the superlattice sample at four different temperatures. The solid lines are fits with the compressed exponential $\exp (-{[\tau /{\tau }_0]}^p)$ with $p=1.32$.

Figure 4

Arrhenius plot of the characteristic decorrelation time ${\tau }_0$ as a function of temperature. The solid line is a fit to the Arrhenius temperature dependence.