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In this work, the results of first-principles density-functional-theorycalculations are used to construct the energy landscapes of HfO$_2$ and its Yand Zr substituted derivatives as a function of symmetry-adapted lattice-modeamplitudes. These complex energy landscapes possess multiple local minima,corresponding to the tetragonal, oIII ($Pca2_1$), and oIV ($Pmn2_1$) phases. Wefind that the energy barrier between the non-polar tetragonal phase and theferroelectric oIII phase can be lowered by Y and Zr substitution. InHf$_{0.5}$Zr$_{0.5}$O$_2$ with an ordered cation arrangement, Zr substitutionmakes the oIV phase unstable, and it become an intermediate state in thetetragonal to oIII phase transition. Using these energy landscapes, weinterpret the structural transformations and hysteresis loops computed forelectric-field cycles with various choices of field direction. The implicationsof these results for interpreting experimental observations, such as thewake-up and split-up effects, are also discussed. These results and analysisdeepen our understanding of the origin of ferroelectricity and field cyclingbehaviors in HfO$_2$-based films, and allow us to propose strategies forimproving their functional properties.

Phase Competition in HfO$_2$ with Applied Electric Field from First Principles