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Symmetry Elements in a Crystal

Symmetry Elements in a Crystal

Crystal symmetry operations are isometric transformations that map objects onto indistinguishable copies while preserving distances, angles, and volumes. The simplest symmetry operation is translation, which shifts the entire infinite crystal lattice parallelly by a translation vector.Crystallographic rotations involve rotations by an angle of 2π/n around an axis without changing the positions of points on the axis. It is called the rotational axis of the symmetry, denoted by n. The combination...

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Related Experiment Video

Updated: Jun 18, 2026

Atomically Defined Templates for Epitaxial Growth of Complex Oxide Thin Films

Atomically Defined Templates for Epitaxial Growth of Complex Oxide Thin Films

Published on: December 4, 2014

Boosting Inversion Symmetry Breaking in Epitaxial Tetragonal ZrO₂ Via Atomic Layer Deposition.

Jung Woo Cho¹, Dongmin Kim^2,3,4, In Hyeok Choi⁵

¹Department of Physics Education, Seoul National University, Seoul 08826, Republic of Korea.

|July 18, 2025

Summary

We stabilized a new polar tetragonal phase of zirconium dioxide (ZrO2) in thin films. This discovery advances ferroelectric materials by revealing unexpected properties in fluorite-structured oxides.

Keywords:

4D-STEM Atomic layer deposition Epitaxial growth HfO2 ZrO2

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Last Updated: Jun 18, 2026

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Fabrication of Schottky Diodes on Zn-polar BeMgZnO/ZnO Heterostructure Grown by Plasma-assisted Molecular Beam Epitaxy

Published on: October 23, 2018

Area of Science:

Materials Science
Solid-State Physics
Crystallography

Background:

Stabilizing intermediate polar phases in fluorite-structured oxides is crucial for ferroelectric and antiferroelectric applications.
Existing research outlines expected phase transitions in hafnium zirconate (Hf_xZr_1-xO₂) systems.

Purpose of the Study:

To stabilize and characterize the epitaxial polar tetragonal (T) phase of zirconium dioxide (ZrO₂).
To investigate the phase transitions and electrical properties of Hf_xZr_1-xO₂ thin films.

Main Methods:

Epitaxial Hf_xZr_1-xO₂ thin films (x = 0, 0.25, 0.5, 0.75, 1) were synthesized on yttria-stabilized zirconia substrates using atomic layer deposition.
Second harmonic generation measurements were employed to detect symmetry breaking.
High-resolution 4D-scanning transmission electron microscopy was used to confirm electric dipoles and atomic structure.

Main Results:

The unprecedented polar T-ZrO₂ phase was successfully stabilized, deviating from predicted phase transitions.
Second harmonic generation confirmed inversion symmetry breaking in the T-phase ZrO₂.
4D-STEM revealed electric dipoles due to off-centered oxygen displacements in T-phase ZrO₂.

Conclusions:

This study establishes a low-temperature epitaxial synthesis route for HfO₂-ZrO₂-based materials.
The findings provide critical insights into the polar nature and potential applications of T-phase ZrO₂.