Table 3.

The dipolar interactions in hafnia

${\tilde{J}}_{12}^{\prime }$ (eV Å−2)	${\tilde{J}}_{13}^{\prime }$ (eV Å−2)	${\tilde{J}}_{12}^{″}$ (eV Å−2)	${\tilde{J}}_{13}^{″}$ (eV Å−2)
$\left(\begin{array}{ccc}\hfill 3.30\hfill & \hfill 0.00\hfill & \hfill 0.00\hfill \\ \hfill 0.00\hfill & \hfill -2.24\hfill & \hfill -2.67\hfill \\ \hfill 0.00\hfill & \hfill -2.67\hfill & \hfill -2.24\hfill \end{array}\right)$	$\left(\begin{array}{ccc}\hfill -2.24\hfill & \hfill 0.00\hfill & \hfill -2.67\hfill \\ \hfill 0.00\hfill & \hfill 3.30\hfill & \hfill 0.00\hfill \\ \hfill -2.67\hfill & \hfill 0.00\hfill & \hfill -2.24\hfill \end{array}\right)$	$\left(\begin{array}{ccc}\hfill 3.07\hfill & \hfill 0.88\hfill & \hfill 0.73\hfill \\ \hfill -0.88\hfill & \hfill -2.48\hfill & \hfill -2.92\hfill \\ \hfill -0.73\hfill & \hfill -2.92\hfill & \hfill -2.45\hfill \end{array}\right)$	$\left(\begin{array}{ccc}\hfill -2.12\hfill & \hfill 0.25\hfill & \hfill -2.67\hfill \\ \hfill 0.25\hfill & \hfill 3.24\hfill & \hfill 0.06\hfill \\ \hfill -2.67\hfill & \hfill 0.06\hfill & \hfill -2.26\hfill \end{array}\right)$

The ${\tilde{J}}_{\tau \kappa }^{\prime }$ [(τ, κ) = (1, 2) or (1, 3)] matrix characterizes the interactions (between electric dipoles centered on the Hf_τ and Hf_κ sites) for the cubic $Fm\overline{3}m$ phase, while ${\tilde{J}}_{\tau \kappa }^{″}$ describes those for $Fm\overline{3}m$ phase with a $O_x^{A_y}$ distortion. Here, the magnitude of the $O_x^{A_y}$ distortion is fixed to 0.05 a_fc, a typical value occurred in the structural phases of HfO₂ (see “Methods” for details).