Phase Stability\nand Phase Transition Pathways in the\nRhombohedral Phase of HfO<sub>2</sub>

Determining the stability of complex phases in HfO<sub>2</sub> is\nfundamental to advancing its development and application as ferroelectric\nmaterial. However, there is ongoing debate regarding whether the ferroelectric\nphase of HfO<sub>2</sub> originates from the orthorhombic phase or\nthe rhombohedral one. Using first-principles calculations with symmetry\ngroup and phonon structure analysis, we have derived multiple phase\ntransitions and ferroelectric switching pathways for the rhombohedral\nphase, and analyzed their static and dynamic stability. The results\nindicate that the <i>R</i>3<i>m</i> phase, characterized\nby imaginary frequencies, is a metastable structure that spontaneously\ntransitions to the tetragonal one through multiple pathways. Although\nthe <i>R</i>3 phase lacks imaginary frequencies, the high\nformation energy due to internal stress leads to all its ferroelectric\nswitching pathways decaying to lower energy orthorhombic or tetragonal\nphases. Additionally, different Zr distributions in Hf<sub>0.5</sub>Zr<sub>0.5</sub>O<sub>2</sub> disrupt the spatial group symmetry\nof the <i>R</i>3 phase, causing it to spontaneously transition\nto orthorhombic or monoclinic phases. Consequently, both the phase\nstability and ferroelectric cycling endurance of the rhombohedral\nphase are inferior to those of the orthorhombic phase for practical\napplications. These findings are crucial for experimentally determining\nthe phase structure of HfO<sub>2</sub> and further developing its\nferroelectric mechanisms and potential.