Strain gradients in epitaxial ferroelectrics

X-ray analysis of ferroelectric thin layers of ${\mathrm{Ba}}_{1∕2}{\mathrm{Sr}}_{1∕2}\mathrm{Ti}{\mathrm{O}}_3$ with different thicknesses reveals the presence of strain gradients across the films and allows us to propose a functional form for the internal strain profile. We use this to calculate the influence of strain gradient, through flexoelectric coupling, on the degradation of the ferroelectric properties of films with decreasing thickness, in excellent agreement with the observed behavior. This paper shows that strain relaxation can lead to smooth, continuous gradients across hundreds of nanometers, and it highlights the pressing need to avoid such strain gradients in order to obtain ferroelectric films with bulklike properties.

Figure 1

(Color online) Williamson-Hall plots, showing maximum slope for films of intermediate thickness. Inset: 004 diffraction peaks; peak shapes are well fitted (solid lines) assuming an exponential strain profile. The peaks at $\sim 103°$ correspond to the SRO electrode.

Figure 2

Inhomogeneous strain as a function of film thickness. The solid line is a visual guide. Inset: average out-of-plane strain.

Figure 3

Calculated and measured dielectric constant as a function of temperature for films of thickness 950, 660, 340, 280, 220, and $145\mathrm{nm}$. Inset: The temperature of maximum permittivity, experimental (dots) and calculated (solid line).