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

Technical Summary: Structural Chirality and Natural Optical Activity across the $\alpha$-to-$\beta$ Phase Transition in SiO$_2$ and AlPO$_4$

Problem Statement
Natural optical activity (NOA), the rotation of the plane of polarized light, is traditionally associated with structural chirality. However, a persistent conceptual oversimplification exists in the literature: the assumption that the sign of optical rotation is directly determined by the handedness of the space group symbol (e.g., $P6_422$ vs. $P6_222$). This correlation fails in specific cases, such as the $\alpha$-to-$\beta$ phase transitions in quartz (SiO$_2$) and berlinite (AlPO$_4$). In these transitions, the space group symmetry changes from a high-symmetry hexagonal phase ($P6_422$ or $P6_222$) to a low-symmetry trigonal phase ($P3_121$ or $P3_221$), reversing the nominal handedness of the principal screw axis. Paradoxically, the sign of the optical activity remains unchanged across this transition. The precise microscopic origin of this behavior and the factors governing the sign of NOA in these systems require clarification beyond simple space group assignment.

Methodology
The authors conducted a first-principles investigation using density functional theory (DFT) and density functional perturbation theory (DFPT) implemented in the Abinit code.

• Structural Modeling: The study utilized the PBEsol exchange-correlation functional with optimized norm-conserving pseudopotentials (PseudoDojo v.4) and a 40 Hartree energy cutoff. Structural relaxations were performed with a $6\times6\times6$ k-mesh.
• Phonon Analysis: DFPT was employed to identify unstable phonon modes at the Brillouin zone center ($\Gamma$ point) in the high-symmetry phases. The transition pathway was modeled by condensing the eigenvector of the unstable mode.
• Optical Activity Calculation: The study focused on the low-frequency limit of the rotatory power, $\bar{\rho}(\omega) = \rho(\omega)/(\hbar\omega)^2$, calculated using the linear-response approach grounded in DFPT developed by Zabalo and Stengel. Calculations assumed light propagation along the optic axis (z-direction).
• Comparative Analysis: To isolate the contributions of specific atomic species, a computational experiment was performed where Si atoms in SiO$_2$ were substituted into the Al and P sites of the AlPO$_4$ structure to observe changes in optical response.

Key Results

1. Mechanism of Phase Transition: The $\alpha$-to-$\beta$ transition in both SiO$_2$ and AlPO$_4$ is driven by the condensation of a zone-center $\Gamma_3$ phonon mode. This mode is silent (neither Raman nor infrared active) and induces a displacive phase transition. The distortion breaks the $6_4$ screw-axis symmetry of the high-symmetry phase, resulting in a $3_1$ screw axis in the low-symmetry phase.
2. Discrepancy Between Space Group and Optical Sign:
   • For SiO$_2$, the calculations predict a positive rotatory power for both the high-symmetry ($P6_422$) and low-symmetry ($P3_121$) phases.
   • For AlPO$_4$, the calculations predict a negative rotatory power for both phases.
   • Crucially, despite the reversal of the screw axis handedness during the transition (e.g., from $P6_422$ to $P3_121$), the sign of the optical activity remains constant for each material.
3. Origin of Optical Activity Sign: The study demonstrates that the sign of NOA is not dictated by the nominal handedness of the space group screw axis. Instead, it is determined by the atomic-scale handedness of the most polarizable atoms in the structure.
   • In SiO$_2$, the oxygen atoms form a helical chain that dictates the positive rotation.
   • In AlPO$_4$, the ordering of Al and P atoms doubles the unit cell along the c-axis, altering the relative positions of the oxygen atoms. This results in a left-handed helical arrangement of the oxygen sites (similar to the $P6_222$ SiO$_2$ structure), which dictates the negative rotation.
   • Substitution experiments confirmed that Si, Al, and P atoms contribute minimally to the optical activity; the signal is dominated by the oxygen sublattice.
4. Evolution of Rotatory Power: As the $\Gamma_3$ distortion amplitude increases from the high-symmetry to the low-symmetry phase, the magnitude of the rotatory power decreases linearly. The authors note that while further distortion could theoretically revert the sign, such configurations are physically inaccessible due to a significant increase in Landau energy.

Significance and Claims
The paper claims to resolve the apparent contradiction between structural symmetry changes and the preservation of optical activity signs in SiO$_2$ and AlPO$_4$. The primary contribution is the clarification that the sign of natural optical activity is governed by the chirality of the most polarizable atomic species (oxygen in these oxides) rather than the global space group symbol.

The authors emphasize that while structural chirality is a sufficient condition for NOA, the specific sign of rotation cannot be inferred solely from the handedness of the space group screw axis. This work serves as a caution against simplistic interpretations of optical activity based on symmetry alone, highlighting the necessity of analyzing the microscopic arrangement of polarizable atoms to understand the optical properties of chiral crystals. The findings align with and provide a first-principles microscopic foundation for previous experimental observations by Glazer and Stadnicka regarding the preservation of optical activity despite screw-axis reversal.