Order parameters for symmetry-breaking structural transitions: The tetragonal-monoclinic transition in ZrO2

Abstract

Group/subgroup structural phase transitions are exploited in a wide variety of technologies, including those that rely on shape-memory behavior and on transformation toughening. Here, we introduce an approach to identify symmetry-adapted strain and shuffle order parameters for any group/subgroup structural transition between a high-symmetry parent phase and its symmetrically equivalent low-symmetry product phases. We show that symmetry-adapted atomic shuffle order parameters can be determined by the diagonalization of an orbital covariance matrix, formed by taking the covariance among the atomic displacement vectors of all symmetrically equivalent product phase variants. We use this approach to analyze the technologically important tetragonal to monoclinic structural phase transformation of ${\mathrm{ZrO}}_{2}$. We explore the energy landscapes, as calculated with density functional theory, along distinct paths that connect $m{\mathrm{ZrO}}_{2}$ to $t{\mathrm{ZrO}}_{2}$ and to other $m{\mathrm{ZrO}}_{2}$ variants. The calculations indicate favorable pairs of variants and reveal intermediate structures likely to exist at coherent twin boundaries and having relatively low deformation energy. We identify crystallographic features of the monoclinic ${\mathrm{ZrO}}_{2}$ variant that make it very sensitive to shape changing strains.