Trilinear-coupling-driven ferroelectricity in ${\mathrm{HfO}}_2$

Ferroelectricity in hafnia is often regarded as a breakthrough discovery in ferroelectrics, potentially able to revolutionize the whole field. Despite increasing interests, a deep and comprehensive understanding of the many factors driving ferroelectric stabilization is still lacking. We here address the phase transition in terms of a Landau-theory-based approach, by analyzing symmetry-allowed distortions connecting the high-symmetry paraelectric tetragonal phase to the low-symmetry polar orthorhombic phase. By means of first-principles simulations, we find that the ${\mathrm{\Gamma }}_{3-}$ polar mode is only weakly unstable, whereas the other two symmetry-allowed distortions, ${\mathrm{Y}}_{2+}$ and ${\mathrm{Y}}_{4-}$ (showing a nonpolar and antipolar behavior, respectively), are hard modes. While none of the modes, taken alone or combined with one other mode, is able to drive the transition, the key factor in stabilizing the ferroelectric phase is identified as the strong trilinear coupling among the three modes. Furthermore, the experimentally acknowledged importance of substrate-induced effects in the growth of ${\mathrm{HfO}}_2$ ferroelectric thin films, along with the lack of a clear order parameter in the transition, suggested the extension of our analysis to strain effects. Our findings suggest a complex behavior of the ${\mathrm{Y}}_{2+}$ mode, which can become unstable under certain conditions (i.e., a tensile strain applied along the $a$ direction), and an overall weakly unstable behavior for the ${\mathrm{\Gamma }}_{3-}$ polar mode for all the strain conditions. In any case, a robust result emerges from our analysis: independently of the different applied strain (be it compressive or tensile, applied along the $a, b$, or $c$ orthorhombic axis), the need for a simultaneous excitation of the three coupled modes remains unaltered. Finally, when applied to mimic experimental growth conditions under strain, our analysis shows a further stabilization of the ferroelectric phase with respect to the unstrained case, in agreement with experimental findings.

Figure 1

The supergroup tree of the orthorhombic $Pca{2}_1$ structure. The parent structure $P{4}_2/nmc$ distorted into the $Pca{2}_1$ primitive cell is connected to the $Ccce$ phase through the action of the symmetry-lowering strain mode ${\mathrm{\Gamma }}_{4+}$. The transformation to the $Ccce$ structure also involves a totally symmetric ${\mathrm{\Gamma }}_{1+}$ mode comprising both a strain and an atomic displacement. The space-group number of each structure is reported in parentheses. For each distortion, the amplitude of the maximum atomic displacement is reported in angstroms below the corresponding irreps. The displacements are intended from the $Ccce$ phase. The distortion modes are labeled according to the Ccce (${P4}_2/nmc$) irreps.

Figure 2

Top: Ab initio energies (in eV/f.u.) of the single symmetry-distortion modes (a), along with their couplings (b), as a function of the generalized distortion coordinate $\lambda$. Each path consists of ten intermediate structures. The energy trend relative to the simultaneous excitation of the three modes is reported with a bold black line. Bottom: Atomic displacements of each single mode. Hafnium is colored in gold, and oxygen is colored in red. The arrows point in the direction of the displacement with a length proportional to its magnitude. For every mode the arrows relative to hafnium are magnified by a factor of 5.

Figure 3

Fit of ab initio energies, represented by colored dots, with the free energy expressed in Eq. (1) for the zero-strain case. The path connects the high-symmetry structures from the $Ccce$ to the $Pca{2}_1$. The figure shows only part of the energy landscape, i.e., it reports only some portions of the fit: two-mode and three-mode combinations are not shown. Each high-symmetry structure is labeled with the values of the vector $({\mathrm{Q}}_{{\mathrm{\Gamma }}_{3-}},{\mathrm{Q}}_{{\mathrm{Y}}_{2+}},{\mathrm{Q}}_{{\mathrm{Y}}_{4-}})$, thus $(0,0,0)$ corresponds to the $Ccce$ and $(1,1,1)$ corresponds to the $Pca{2}_1$.

Figure 4

(a), (b) Ab initio energies (in eV/f.u.) of the single symmetry-distortion modes (a), along with their couplings (b), as a function of the generalized distortion coordinate $\lambda$, when a 3% tensile strain is applied along $a$. The inset in panel (b) reports a zoom of the energies for $\lambda$ in [0.0,0.5]. The colors choice follows the one in Fig. 2: ${\mathrm{\Gamma }}_{3-}$ in black, ${\mathrm{Y}}_{2+}$ in blue, ${\mathrm{Y}}_{4-}$ in red (a); ${\mathrm{\Gamma }}_{3-}\cup {\mathrm{Y}}_{2+}$ in light blue, ${\mathrm{\Gamma }}_{3-}\cup {\mathrm{Y}}_{2+}$ and ${\mathrm{\Gamma }}_{3-}\cup {\mathrm{Y}}_{4-}$ in purple, and $Y_{2+}\cup {\mathrm{Y}}_{4-}$ in green (b). The excitation of the three modes is shown in black in panel (b). Each path consists of ten intermediate structures. (c) The dependence of energy minima of the modes ${\mathrm{\Gamma }}_{3-}$, in black, and ${\mathrm{Y}}_{2+}$, in blue, on the applied strain. In each panel, the application direction is represented by a different line style.

Figure 5

Difference in ab initio energies between the $Ccce$ and the $Pca{2}_1$ as a function of the total polarization along the distortion. Only the trends for the most favorable strain states reported in Table 2 are shown.

Figure 6

Strain dependence of the three leading coefficients in Eq. (1). Top, middle, and lower panels show the ${\epsilon }_{113},{\epsilon }_{131},\text{and}{\gamma }_{111}$ coefficients, respectively. The application directions of the strain are represented by different line styles (see legend).