Effects of biaxial strain on the improper multiferroicity in $h-\mathrm{LuFe}{\mathrm{O}}_3$ films studied using the restrained thermal expansion method

Elastic strain is potentially an important approach in tuning the properties of the improperly multiferroic hexagonal ferrites, the details of which, however, have been elusive due to experimental difficulties. Employing the method of restrained thermal expansion, we have studied the effect of isothermal biaxial strain in the basal plane of $h-\mathrm{LuFe}{\mathrm{O}}_3$ (001) films. The results indicate that a compressive biaxial strain significantly enhances the $K_3$ structural distortion (the order parameter of the improper ferroelectricity), and the effect is larger at higher temperatures. The compressive biaxial strain and the enhanced $K_3$ structural distortion together cause an increase in the electric polarization and a reduction in the canting of the weak ferromagnetic moments in $h-\mathrm{LuFe}{\mathrm{O}}_3$, according to our first principles calculations. These findings are important for understanding the strain effect as well as the coupling between the lattice and the improper multiferroicity in $h-\mathrm{LuFe}{\mathrm{O}}_3$. The experimental elucidation of the strain effect in $h-\mathrm{LuFe}{\mathrm{O}}_3$ films also suggests that the restrained thermal expansion can be a viable method to unravel the strain effect in many other thin film materials.

Figure 1

(a) Schematics of the strain generated by the restrained thermal expansion in comparison with that by the isobaric thermal expansion. The arrows indicate the directions of thermal expansion. (b) Illustration of the restrained thermal expansion and isobaric thermal expansion in the ($a, T$) space, where $a$ is the in-plane lattice constants, $T$ is temperature, $f$ is a general physical property. (c) Trimer model of the $K_3$ structural distortion in $h-\mathrm{LuFe}{\mathrm{O}}_3$, which corresponds to the rotation of $\mathrm{Fe}{\mathrm{O}}_5$ trigonal bipyramids, the buckling of the Lu layers, and the trimerization. The arrows indicate the atomic displacements when $K_3$ is enhanced under the compressive biaxial strain. (d) Illustration of the experimental setup for the pump (laser) and probe (x-ray) measurements.

Figure 2

Thermal strain and the intensity change of the (106) peak in the restrained and isobaric thermal expansions. (a) The diffraction profile of the (106) peak before and after the laser $\left(1.0\mathrm{mJ}/\mathrm{m}{\mathrm{m}}^2\right)$ pulse. (b) The decay of the thermal strain $\frac{\mathrm{\Delta }c}{c}$ and the (106) peak intensity in the restrained thermal expansion, as well as the fit using the thermal conduction model. Inset: the thermal strain $\frac{\mathrm{\Delta }c}{c}$ and the (106) peak intensity at $t_{\mathrm{delay}}=0.1\mathrm{ns}$ as a function of laser fluence. (c) The thermal strain and (106) peak intensity, as a function of temperature in the isobaric thermal expansion. (d) The relation between the (106) peak intensity and the thermal strain $\frac{\mathrm{\Delta }c}{c}$ in the restrained and isobaric thermal expansions. The (106) peak intensities are normalized using the values at room temperature.

Figure 3

(a) Temperature dependence of the amplitude of the $K_3$ distortion ($Q_{K_3}$) in both isobaric and restrained thermal expansion as a function of temperature. Representative error bars are displayed. (b) The effect of isothermal biaxial compressive strain on $Q_{K_3}$. Each data point represents a change of $Q_{K_3}$ caused by a strain $\frac{\mathrm{\Delta }a}{a}$ (bottom axis) at the corresponding temperature (top axis). The line is a guide to the eyes. (c) and (d) are the effect of the biaxial strain $\frac{\mathrm{\Delta }a}{a}$ on the $K_3$ distortion and on the electric polarization and the weak ferromagnetic moment, respectively, calculated using the density functional theory.