  A comparative theoretical study is presented for the rhombohedral R3 and R3m
phase HfO2, of two possible forms in its heavily Zr-doped ferroelectric thin
films found recently in experiments. Their structural stability and
polarization under the in-plane compressive strain are comprehensively
investigated. We discovered that there is a phase transition from R3 to R3m
phase under the biaxial compressive strain. Both the direction and amplitude of
their polarization can be tuned by the strain. By performing a symmetry mode
analysis, we are able to understand its improper nature of the
ferroelectricity. These results may help to shed light on the understanding of
the hafnia ferroelectric thin films.
