In-Plane Polar Distortions in Materials

• In-plane polar distortions are symmetry-breaking atomic displacements within a crystallographic plane that influence ferroelectric and multiferroic behavior.
• They emerge from coupled nonpolar modes through proper, improper, or triggered mechanisms and can be quantitatively analyzed using metrics like membership coefficients.
• These distortions affect functionalities such as electronic transport, magnetoelectric coupling, and energy storage, paving the way for advanced device applications.

In-plane polar distortions refer to symmetry-breaking atomic displacements whose net polarization vector lies predominantly within a crystallographic plane, rather than along an out-of-plane axis. Such distortions can emerge in a wide range of crystalline and low-dimensional systems, often through entangled multi-mode instabilities or through local chemical, electronic, or interfacial driving forces. In-plane polar distortions fundamentally impact material functionalities, including ferroelectricity, magnetoelectric coupling, topological domain textures, electronic transport, and energy storage. Their origins and classification are highly system-dependent, often exhibiting hybrid character between proper, improper, and triggered ferroelectric mechanisms.

1. Definitions, Conceptual Framework, and Historical Evolution

Classical ferroelectrics (e.g., BaTiO₃) feature “proper” polar distortions: the off-center displacement of ions acts as the primary order parameter (OP), directly producing the emergence of an electric polarization vector $\mathbf{P}$, either in-plane or out-of-plane depending on structural details. However, in-plane polar distortions frequently arise in structurally or compositionally complex ferroelectrics through significant coupling with nonpolar modes. In “improper” ferroelectrics, $\mathbf{P}$ appears as a secondary OP, induced by symmetry-allowed couplings (e.g., trilinear couplings $p q_1 q_2$) among nonpolar structural distortions. In “triggered” cases, $\mathbf{P}$ exists only when a set of nonpolar distortions coexist, and is missing in any subset of those modes.

A key insight advanced in (Zhao et al., 15 Oct 2025) is that many modern systems—such as perovskite superlattices and polar orthorhombic hafnia—require a framework beyond purely single-mode (proper/improper) classifications. The nature of the in-plane polar distortion may be mixed: e.g., the polarization in SrTiO₃/CaTiO₃ superlattices contains large proper and improper components, while in LaGaO₃/YGaO₃ the improper mechanism dominates, as quantified by a “membership coefficient” $\eta$, with $\eta \simeq 0.21$ (mostly improper) vs. $\eta \simeq 0.56$ (mixed) depending on system and strain state.

2. Symmetry, Local Structure, and Microscopic Mechanisms

The symmetry properties and local structural origin of in-plane polar distortions depend sensitively on material class:

• Octahedral Tilt and Breathing Modes (Perovskites): In $A$-site ordered manganites such as SmBaMn$_2$O$_6$, in-plane ferroelectricity results from trilinear coupling between out-of-phase tilts (M$_5^-$), breathing distortions (M$_4^+$), and a small-amplitude polar mode (Γ$_5^-$). This is a “hybrid improper” mechanism (Nowadnick et al., 2019).
• Molecular and Hydrogen-bonded Ferroelectrics: In planar organic crystals such as 1-cyclobutene-1,2-dicarboxylic acid (CBDC), the in-plane component stems from coupled proton transfers (intermolecular) and a strong molecular buckling (twisting, π-bond switching) (Stroppa et al., 2011). The polarization can be decomposed into Wyckoff-site-resolved contributions, such as for CBDC (see $P_{\text{tot}} \approx (12.7, 0, -6.7)$ μC/cm², with $y$ as the in-plane direction).
• Layered and Low-Dimensional Materials: In 2D van der Waals systems (e.g., hBN, transition metal dichalcogenides), in-plane polar distortions can be engineered through stacking registry, twist, and strain. Symmetry analysis in incommensurate or strained bilayers reveals that breaking $C_3$ rotational symmetry and mirror planes enables substantial in-plane dipole moments of comparable magnitude to out-of-plane polarization (Yu et al., 2022).
• Interfaces and Heterostructures: At perovskite interfaces, interfacial polar discontinuity can be partly compensated by local in-plane (and out-of-plane) cation-anion displacements (“rumpling”), with clear impact on bond angles and electron bandwidth; see SrTiO$_3$/La$_{1-x}$Sr$_x$MnO$_3$ (Koohfar et al., 2017) or LaInO$_3$/BaSnO$_3$ (Aggoune et al., 2021).

3. Topological, Domain, and Texture Phenomena

In-plane polar distortions often give rise to complex domain structures and topological objects:

• Moiré Meron and Antimeron Networks: In twisted or strained bilayers (e.g., hBN), the spatial modulation of local stacking produces a continuous, topologically nontrivial in-plane polarization field. Domains can host Bloch-type merons (vortex-like, winding number $+1$ for twist), Néel-type merons (radial/hedgehog, winding $-1$ for biaxial heterostrain), and anti-merons (winding $-1$, divergence- and curl-free for area-conserving uniaxial heterostrain) (Bennett et al., 2022, Yu et al., 2022). Topological charge is computed as

$$Q = \frac{1}{4\pi} \int \hat{\mathbf{P}} \cdot (\partial_x \hat{\mathbf{P}} \times \partial_y \hat{\mathbf{P}}) \, ds$$

• Magnetic Skyrmions with Asymmetric In-Plane Structure: In polar magnets with easy-plane anisotropy, in-plane distortions modify the skyrmion internal structure, leading to energy landscapes with alternating positive and negative asymptotics relative to the surrounding background; the resulting anisotropic inter-skyrmion interactions can be attractive or repulsive (Leonov et al., 2017).
• Domain Wall Vortices and Multiferroic Domain Engineering: Group theoretical and DFT studies in SmBaMn$_2$O$_6$ reveal a network of as many as $16$ structural domains, with complex domain walls and vortex-like defects hosting distinct structural and electronic states that can stabilize competing magnetic or metallic phases (Nowadnick et al., 2019).

4. Coupling to Magnetism, Transport, and Multifunctional Properties

In-plane polar distortions profoundly influence diverse functional responses:

• Electronic Band Structure and Carrier Gases: At perovskite oxide interfaces, in-plane polar distortions (e.g., from octahedral tilts) generate depolarization fields that renormalize the critical thickness for 2DEG formation (e.g., LaInO$_3$/BaSnO$_3$), and control band offsets, mobility, and spatial confinement (Aggoune et al., 2021).
• Rashba Spin Splitting in Polar Metals: In layered metals such as Ca$_3$Ru$_2$O$_7$, pressure-induced enhancement of in-plane polar distortions (via octahedral rotations) does not necessarily lead to an increase in the Rashba spin splitting; DFT and diffraction show a nontrivial, anticorrelated dependence, highlighting the complex interplay between structural and spin-orbit-driven phenomena (Ladbrook et al., 30 Jan 2025).
• Magnetoelectric Coupling and Anisotropies: Local in-plane polar distortions induced by local lattice relaxation (e.g., at Mn$^{2+}$ in Ga(Mn)As) break tetrahedral symmetry and, through spin-orbit-mediated p–d hybridization, stabilize magnetic easy axes along specific in-plane directions ($\langle 110\rangle$), explaining anisotropic ferromagnetic behavior (Subramanian et al., 2012).
• Energy Storage: Thin film relaxor ferroelectrics with engineered in-plane polar domains (e.g., La/Si co-doped BaTiO$_3$) exhibit increased maximum polarization, reduced remanence, enhanced breakdown fields, and high energy density (e.g., 203.7 J/cm³ at $\approx$6.15 MV/cm) (Lei et al., 13 Oct 2025). Phase-field simulations validate that in-plane domain formation is key to superior recoverable energy storage (cf. $$W_\mathrm{r} = \int_{P_\mathrm{r}}^{P_\mathrm{m}} E\,dP$$).

5. Methodological Developments and Theoretical Models

Describing and classifying in-plane polar distortions in complex systems has prompted new theoretical tools:

• Graph Theory of Polar Distortion Pathways: (Zhao et al., 15 Oct 2025) introduces hierarchy graphs constructed from the power set of nonpolar order parameters $\mathbb{P}(Q)$. Directed edges encode condensation of a mode; comparison of “polar” and “nonpolar” graphs exposes “improper paths” for complex ferroelectrics. The “proper membership coefficient” $\eta = \min(|P|,|P_0|)/\max(|P|,|P_0|)$ quantitatively distinguishes the degree of proper/improper character of the polarization.
• First-Principles Mode Decomposition and Symmetry Analysis: Quantitative symmetry-mode decomposition (e.g., via Amplimodes or ISODISTORT) combined with DFT energetics allows partitioning of polarization contributions among structural modes or Wyckoff orbits; this enables both physical interpretation and prediction of emergent in-plane polarization (Stroppa et al., 2011, Nowadnick et al., 2019, Fabini et al., 15 Jan 2024).
• Phase-Field Simulation for Domain Engineering: Continuum modeling and phase-field simulations describe how local domain configuration, co-doping, and strain engineering promote desired in-plane polar architectures for enhanced functional response, especially in thin-film dielectrics (Lei et al., 13 Oct 2025).
• Berry Connection Formalism for 2D Materials: In van der Waals bilayers, ab initio Berry-phase polarization calculations combined with symmetry-enforced constraints enable accurate mapping of in-plane polarization textures as a function of local stacking registry (Bennett et al., 2022, Yu et al., 2022):

$$\mathbf{P}_\parallel = \frac{1}{(2\pi)^2} \int d\mathbf{k} \left(i\langle u^v_\mathbf{k} | \nabla_\mathbf{k} u^v_\mathbf{k} \rangle + i\langle u^{v'}_\mathbf{k} | \nabla_\mathbf{k} u^{v'}_\mathbf{k} \rangle \right)$$

6. Applications, Device Implications, and Future Directions

• Energy Storage: In-plane polar design strategies allow high-density, reversible dielectric capacitors with high efficiency and reduced hysteresis, enabled by engineered in-plane domain structures (Lei et al., 13 Oct 2025).
• Information Storage and Topological Electronics: The realization of switchable, topological in-plane polarization textures (merons, antimerons, domain vortices) suggests platforms for high-density nonvolatile memory, topological logic, and robust domain-wall-based devices (Bennett et al., 2022, Nowadnick et al., 2019).
• Adaptive Optics and Piezoelectrics: In-plane polarized piezo actuators with topological buckling enhance tuning speed in adaptive lenses, demonstrating the direct impact of in-plane distortions on device performance (Lemke et al., 2016).
• Spintronic and Rashba Devices: Complex polar metals and chiral ferroelectrics with intrinsic in-plane polar patterns (e.g., CsSnBr$_3$) represent promising candidates for Rashba-type spin splitting and spintronic applications (Fabini et al., 15 Jan 2024, Ladbrook et al., 30 Jan 2025).
• Material Search and Engineering: Understanding the multi-natured origin of in-plane polar distortions through graph-based and first-principles methods enables systematic discovery and optimization of ferroelectrics and multifunctional materials with predesigned properties (Zhao et al., 15 Oct 2025).

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In-plane polar distortions, encompassing proper, improper, and triggered character, underlie a rich spectrum of phenomena in classical and quantum materials. Their thorough classification, quantitative modeling, and targeted manipulation are central to advancing new device paradigms in electronics, spintronics, and energy technologies. Recent advances in theoretical tools (graph theory, ab initio mode analysis) and experimental synthesis (precision epitaxy, local probe microscopy) continue to expand the scope of controllable in-plane ferroelectric and multiferroic functionalities.