Influence of epitaxial strain on multiple-mode compounds: The case of ${\mathrm{SrBi}}_2{\mathrm{Nb}}_2{\mathrm{O}}_9$

We study the effect of a tetragonal, symmetry-conserving epitaxial strain on the phase transition sequence and the onset of three symmetry-breaking modes in ${\mathrm{SrBi}}_2{\mathrm{Nb}}_2{\mathrm{O}}_9$ by first-principles calculations. The interplay of the strain variation of energy expansion coefficients results in an interesting phase diagram which includes proper, hybrid improper, and a different kind of improper ferroelectric phase transitions. The symmetries of the epitaxially stabilized phases and the possible phase-transition sequences are compared with the unstrained case.

Figure 1

Three views of the parent structure (left column) and of the distortions present in the ferroelectric phase (right column). The axes are labeled according to the tetragonal structure and the black lines correspond to the ferroelectric unit cell.

Figure 2

Amplitudes of the modes allowed in the $A{2}_{1}am$ structure, $Q_1, Q_2$, and $Q_3$, as a function of misfit strain at $T=0$. The amplitudes are given for modes normalized within the primitive unit cell of the ferroelectric structure.

Figure 3

Phase diagrams of epitaxially strained SBN for different temperature renormalization of the stiffness constants. In (a) the three modes renormalize homogeneously ($a_1=a_2=a_3$). The other three figures are calculated by assuming that one of the $a_i$ coefficients is 20% smaller than the other two: (b) for $a_1$, (c) for $a_2$, and (d) for $a_3$. The shaded regions represent the $Amam$ (dots/red), $F2mm$ (horizontal lines), and $Abam$ (vertical lines/blue) phases.

Figure 4

Depths per formula unit of the energy wells associated with the pure distortions with respect to the relaxed tetragonal parent configuration as a function of misfit strain, at $T=0$. The plot is divided into four regions: I contains the new avalanche improper ferroelectric regime, II meets the hybrid improper ferroelectric conditions, III represents the improper ferroelectric behavior,and IV corresponds to proper ferroelectricity.