Piezoaxial coupling for strain-selected ferroaxial domain control

Technical Summary: Piezoaxial Coupling for Strain-Selected Ferroaxial Domain Control

Problem Statement
Ferroaxial order is a ferroic state characterized by an order parameter that transforms as an axial vector and is even under both spatial inversion ($P$) and time reversal ($T$). This symmetry distinguishes it from ferroelectric ($P$-odd) and ferromagnetic ($T$-odd) orders. Consequently, standard electric and magnetic fields cannot serve as universal linear conjugate fields to control the sign of a ferroaxial domain, as they do not share the requisite $P$-even, $T$-even axial-vector symmetry. While dynamical routes using circularly polarized terahertz pulses have been established to induce switching via non-equilibrium axial fields, a static, homogeneous conjugate field capable of biasing ferroaxial domains in equilibrium has remained elusive. Although experimental signatures of strain-ferroaxial coupling (e.g., in 1T-TiSe$_2$) have emerged, a general symmetry framework identifying the specific strain-derived fields for all point groups, coupled with first-principles verification of the resulting energy splitting, has not been established.

Methodology
The authors develop a symmetry-based hierarchy of static homogeneous strain polynomials that transform in the same irreducible representation as a selected ferroaxial order parameter ($A$).

1. Symmetry Formulation: The authors define a piezoaxial coupling term in the free energy, $F_A = -h(\boldsymbol{\varepsilon})A$, where $h(\boldsymbol{\varepsilon})$ is a strain-derived axial field. They systematically derive the lowest-order allowed strain polynomials for non-pyroaxial parent point groups, classifying them by crystal family (orthorhombic, tetragonal, trigonal, hexagonal, and cubic).
2. First-Principles Verification: To validate the theoretical framework, the authors perform density functional theory (DFT) calculations on the trigonal ferroaxial compound Na$_2$BaMg(PO$_4$)$_2$. In this material, the order parameter corresponds to the collective rotation of PO$_4$ tetrahedra about the trigonal axis.
3. Computational Protocols: Three complementary calculations are employed:
   • Clamped-coordinate domain splitting: Calculating the energy difference between pre-constructed ferroaxial states with opposite rotation signs under fixed strain.
   • Double-well potential scans: Mapping the energy profile along the rotation coordinate to observe the tilting of the potential.
   • Fixed-strain relaxation: Relaxing the atomic positions from the para-axial ($\phi=0$) structure under fixed strain to determine if the strain field alone selects a specific domain basin.

Key Contributions and Results

• Symmetry Hierarchy of Strain Fields: The paper establishes that the leading order of the piezoaxial field depends on the parent point group:
  • Orthorhombic: Linear in strain (shear field).
  • Tetragonal: Quadratic in strain.
  • Trigonal and Hexagonal: Cubic in strain.
  • Cubic: Divided into two classes. "Cubic-I" groups (23, $\bar{m}3$) allow linear full-strain fields for selected axes, while "Cubic-II" groups (432, $\bar{4}3m$, $m\bar{3}m$) forbid linear fields, requiring quadratic or cubic combinations depending on the selected axis ([001], [110], or [111]).
• Trigonal Case Specifics: For the trigonal system (e.g., Na$_2$BaMg(PO$_4$)$_2$), the leading basal-plane field is cubic: $h(\boldsymbol{\varepsilon}) \propto Y(3X^2 - Y^2)$, where $X$ and $Y$ are deviatoric strain components. In terms of signed strain amplitude $\varepsilon_u$ and principal-axis angle $\theta$, this yields $h \propto \varepsilon_u^3 \sin(6\theta)$.
• First-Principles Verification:
  • Angular and Amplitude Dependence: Calculated domain splittings ($\Delta E_{dom}$) strictly follow the predicted $\sin(6\theta)$ angular dependence and the cubic scaling $\Delta E_{dom} \propto \varepsilon_u^3$.
  • Potential Tilting: Energy scans confirm that the strain field introduces an odd-in-$\phi$ term that biases the double-well potential, lifting the degeneracy between the two domains.
  • Strain-Selected Relaxation: Starting from the para-axial structure, fixed-strain relaxations evolve into opposite ferroaxial minima solely based on the sign of the applied strain ($\varepsilon_u$) or the orientation of the strain axis ($\theta$).
• Experimental Feasibility: The authors estimate that for a correlated nucleus of $\sim 100$ unit cells, the strain-induced bias at a strain of $|\varepsilon_u| = 0.020$ can become thermodynamically significant near the transition temperature ($T_c \approx 540$ K), suggesting viability for strain-field cooling experiments.

Significance
The paper establishes static homogeneous strain as a symmetry-allowed conjugate field for ferroaxial order. It provides a general symmetry framework for identifying the specific strain polynomials required to control ferroaxial domains across various crystal systems. By demonstrating that strain can energetically select ferroaxial domains in equilibrium, the work proposes a "strain-field cooling" protocol analogous to magnetic field cooling. This offers a route to ferroaxial domain engineering and control without relying on non-equilibrium dynamical fields. The framework extends beyond simple structural transitions to include electronic and finite-wave-vector axial states (e.g., CDW states), provided the strain field is understood as coupling to an induced uniform axial composite. The authors note that while the framework is general, the most direct experimental realization is expected in proper structural ferroaxial transitions, such as the glaserite-type phosphates studied here.