Phase Stability and Phase Transition Pathways in the Rhombohedral Phase of HfO₂

Determining the stability of complex phases in HfO₂ is fundamental to advancing its development and application as ferroelectric material. However, there is ongoing debate regarding whether the ferroelectric phase of HfO₂ originates from the orthorhombic phase or the rhombohedral one. Using first-principles calculations with symmetry group and phonon structure analysis, we have derived multiple phase transitions and ferroelectric switching pathways for the rhombohedral phase, and analyzed their static and dynamic stability. The results indicate that the <i>R</i>3<i>m</i> phase, characterized by imaginary frequencies, is a metastable structure that spontaneously transitions to the tetragonal one through multiple pathways. Although the <i>R</i>3 phase lacks imaginary frequencies, the high formation energy due to internal stress leads to all its ferroelectric switching pathways decaying to lower energy orthorhombic or tetragonal phases. Additionally, different Zr distributions in Hf_0.5Zr_0.5O₂ disrupt the spatial group symmetry of the <i>R</i>3 phase, causing it to spontaneously transition to orthorhombic or monoclinic phases. Consequently, both the phase stability and ferroelectric cycling endurance of the rhombohedral phase are inferior to those of the orthorhombic phase for practical applications. These findings are crucial for experimentally determining the phase structure of HfO₂ and further developing its ferroelectric mechanisms and potential.