Temperature dependence of polar modes in hybrid improper ferroelectrics

An atomistic effective Hamiltonian, along with a presently developed analytical model, are employed to investigate and analyze low-frequency polar, antipolar and antiferrodistortive phonons at finite temperature in a prototypical hybrid improper ferroelectric, that is, the $\left({\mathrm{BiFeO}}_3\right)/\left({\mathrm{NdFeO}}_3\right)$ [abbreviated as ${\left(\mathrm{BFO}\right)}_1/{\left(\mathrm{NFO}\right)}_1$] 1:1 superlattice. In the high-temperature paraelectric phase, phonons having both polar and antipolar characters are found to exist, as a result of a bilinear coupling between different cation motions, with these phonons having frequencies that are basically independent of temperature. In contrast, phonons having fluctuations of either in-phase or antiphase octahedral tiltings are soft in this high-temperature phase (with these two fluctuations being uncoupled), which results in their condensation below some critical temperature and the emergence of a low-temperature phase. In this latter low-temperature phase, trilinear energetic couplings between these two types of octahedral tiltings and Bi and Nd cations' motions lead to the appearance of a spontaneous polarization, consistent with the nature of hybrid improper ferroelectricity. These trilinear energetic couplings also yield an increase in the number of phonons possessing both polar and antipolar characters in the low-temperature phase, with most of these phonons softening when approaching the ferroelectric-to-paraelectric transition from below, as a result of the fact that they also exhibit antiferrodistortive features. The different temperature behaviors of polar modes at high versus low temperatures emphasize the uniqueness of hybrid improper ferroelectrics.

Figure 1

Temperature dependence of (a) the supercell averaged ${\mathbf{u}}_{\mathrm{\Gamma }}$ and ${\mathbf{u}}_X$ vectors characterizing the electrical polarization and antiferroelectric vector at the $X$ point of the cubic first Brillouin zone, respectively, (b) the local mode centered on Nd and Bi cations (${\mathbf{u}}_{\mathrm{Nd}}$ and ${\mathbf{u}}_{\mathrm{Bi}}$, respectively), and (c) ${\mathit{\omega }}_R$ and ${\mathit{\omega }}_M$ pseudovectors quantifying antiphase and in-phase tiltings, respectively, of the oxygen octahedra in the ${\left(\mathrm{BFO}\right)}_1/{\left(\mathrm{NFO}\right)}_1$ superlattice.

Figure 2

Frequency dependence of the imaginary part of the ${\chi }_{\alpha \alpha }^{A{A}^{\prime }}\left(\nu \right)$ susceptibilities in our ${\left(\mathrm{BFO}\right)}_1/{\left(\mathrm{NFO}\right)}_1$ superlattice for the following order parameters: (a) ${\mathbf{u}}_{\mathrm{\Gamma }}$, (b) ${\mathbf{u}}_X$, (c) ${\mathbf{u}}_{\mathrm{Nd}}$, (d) ${\mathbf{u}}_{\mathrm{Bi}}$, (e) ${\mathit{\omega }}_R$, and (f) ${\mathit{\omega }}_M$ at a temperature of 1780 K, that is, for the $P4/mmm$ state. For (a)–(e), $\alpha =x^{\prime }$, where $x^{\prime }$ is along the pseudocubic [110] direction, while $\alpha =z$ (which is along [001]) for (f). The black line displays the calculated data, while the red line represents their fit by DHOs. Insets zoom the peaks that are difficult to see.

Figure 3

Same as Fig. 2, but for a temperature of 750 K, that is, for the $Pmc{2}_1$ state.

Figure 4

Temperature dependence of natural frequencies of phonon modes having ferroelectric, antiferroelectric, and antiferrodistortive characters. See the text for the notations of these resonant frequencies. The vertical line delimits the two different phases obtained in our calculations. Error bars for ${\nu }_6^{Pmc{2}_1}$ arise from the slight difference in frequency that different responses (see Fig. 3) can have around this frequency.