Structural Chirality and Natural Optical Activity across the $α$-to-$β$ Phase Transition in SiO$_2$ and AlPO$_4$ from first-principles

TL;DR

Using first-principles density-functional perturbation theory, the authors investigate natural optical activity in $SiO_2$ and $AlPO_4$ across their enantiomorphic high-symmetry to low-symmetry phase transition driven by a zone-center $\Gamma_3$ phonon. They show that the sign of optical rotation is controlled by the atomic-scale helicity of the most polarizable atoms and is preserved even as the space-group screw axis reverses from $P6_422$ (or $P6_222$) to $P3_121$ (or $P3_221$). The rotation magnitude decreases roughly linearly along the distortion path, with opposite NOA signs for the two materials arising from differences in oxygen site helicity. Overall, the work clarifies the nuanced relationship between structural chirality and NOA and establishes a robust first-principles framework for predicting optical activity during phase transitions in chiral crystals.

Abstract

Natural optical activity (NOA), the ability of a material to rotate the plane of polarized light, has traditionally been associated with structural chirality. However, this relationship has often been oversimplified, leading to conceptual misunderstandings, particularly when attempts are made to directly correlate structural handedness with optical rotatory power. In reality, the relationship between chirality and NOA is more nuanced: optical activity can arise in both chiral and achiral crystal structures, and the sign of the rotation cannot necessarily be inferred from the handedness of the space group. % In this work, we conduct a first-principles investigation of natural optical activity in SiO$_2$ and AlPO$_4$ crystals, focusing on their enantiomorphic structural phase transition from high-symmetry hexagonal ($P6_422$ or $P6_222$) to low-symmetry trigonal ($P3_121$ or $P3_221$) space groups. This transition, driven by the condensation of a zone-center $Γ_3$ phonon mode, reverses the screw axis type given by the space group symbol while leaving the sign of the optical activity unchanged. By following the evolution of the structure and the optical response along the transition pathway, we clarify the microscopic origin of this behavior. We demonstrate that the sense of optical rotation is determined not by the nominal helicity of the screw axis given in the space group symbol, but by the atomic-scale helicity of the most polarizable atoms of the structure.

Figures (4)

• Figure 1: Top views of the in-plane structural distortion in SiO$_2$ from the (a) $P6_4$22 to the (b) $P3_1$21 space groups with the movements of the Si atoms represented with black arrows in the atoms. Curved arrows are a guide to the eye indicating clockwise and counter-clockwise helix displayed by the Si atoms. (c) Side view illustrating the twisting of SiO$_4$ tetrahedra as a function of the $\Gamma_3$ mode amplitude. The amplitude $A = 1.0$ corresponds to the optimal distortion, yielding the maximum energy gain.
• Figure 2: Top (first left column figures) and side views of the progressive structural distortion in AlPO$_4$ from the P$6_4$22 to the P$3_1$21 space groups. The distortion amplitude increases from left to right and from top to bottom, in increments of 0.2, ranging from 0 (undistorted high symmetry phase) to 1 (relaxed low symmetry phase). Black arrows in the first panels indicate the directions of atomic displacements associated with the distortion. Red, blue and pink atoms correspond to O, Al and P respectively.
• Figure 3: (a) Evolution of the rotatory power (in units of $\frac{\rm{deg}}{\rm{mm} \cdot (\rm{eV})^2}$) as a function of the condensation of the $\Gamma_3$ mode distortion connecting the $P6_422$ and $P3_121$ phases of AlPO$_4$ and SiO$_2$. Red (left) and blue (right) axis correspond respectively to the rotatory power associated to the AlPO$_4$ and SiO$_2$ crystals. (b) Double well associated with the condensation of the unstable $\Gamma_3$ phonon mode eigendisplacement as calculated in SiO$_2$. The crystal cell parameters are linearly interpolated between the two minima to model intermediate configurations.
• Figure 4: Comparison of the crystal structures of $P6_422$ AlPO$_4$, P$6_2$22 SiO$_2$ and $P6_422$ SiO$_2$ ordered from top to bottom or left to right respectively in each case. The SiO$_2$ crystal structure has been duplicated along the $c$-axis for comparison purposes. (a-d) different views of the crystal structures as indicated by the corresponding cartesian axis. Dark blue, light blue, pink and red balls correspond to Si, Al, P and O atoms respectively.