Hybrid Improper Ferroelectric Pc Phase

• The Hybrid Improper Ferroelectric Pc phase is a polar state in layered perovskites induced by the coupling of nonpolar octahedral rotations and tilts.
• This mechanism exploits a trilinear free-energy invariant that triggers both strong in-plane and tunable out-of-plane polarization under strain and external fields.
• Its complex domain topology and persistent spin textures enable precise electric-field control of coupled ferroelectric and magnetic states for advanced spintronic applications.

Hybrid improper ferroelectricity is a mechanism in which spontaneous electric polarization emerges via coupling between two or more non-polar lattice modes, rather than arising as the primary order parameter. The “Pc phase” refers to a polar structural variant, frequently stabilized in layered Ruddlesden–Popper (RP) or related systems, in which this coupling induces ferroelectricity, frequently in combination with multiferroic and topological phenomena. In the Pc phase, the improper ferroelectric order is deeply tied to octahedral tilting/rotation patterns, trilinear invariants in the free-energy, domain wall topology, and (in low-dimensional derivatives) robust magnetoelectric and spin–orbit coupled effects. The following sections provide a comprehensive analysis of the structural origins, symmetry mechanisms, domain and switching behaviors, topological aspects, external control parameters, and functional implications of the hybrid improper ferroelectric Pc phase, with an emphasis on two-dimensional RP derivatives and their application-relevant characteristics.

1. Structural and Symmetry Principles of the Pc Phase

The layered RP perovskites with general formula Aₙ₊₁BₙO₃ₙ₊₁ (n = 2 in prototypical compounds) form the underlying lattice structure for hybrid improper ferroelectrics exhibiting the Pc phase. In these systems, the high-symmetry, nonpolar state is typically I4/mmm or a similar space group with inversion symmetry. Upon cooling or under suitable strain, two nonpolar lattice instabilities condense:

• An in-phase octahedral rotation (e.g., M₂⁺ mode),
• An out-of-phase octahedral tilt (e.g., M₅⁻ or X₃⁻/X₅⁻ depending on lattice context).

Crucially, neither distortion alone breaks inversion symmetry, but their trilinear coupling with a polar mode ($Q_P$) via the free-energy invariant

$F_\text{couple} = \alpha Q_P Q_{R} Q_{T}$

(where $Q_R$ and $Q_T$ are rotation and tilt amplitudes, respectively) triggers a polar distortion. In the Pc symmetry, this can produce both in-plane and (with anisotropic in-plane strain) out-of-plane polarization. The Pc phase may thus host both in-plane (IP-HIF) and out-of-plane hybrid improper ferroelectric (OP-HIF) order, the latter enabled and tunable by epitaxial strain or lattice distortion ratios ($x/y$) (Zhou et al., 24 Jul 2025).

In two-dimensional derivatives, the Pc phase is often realized by exfoliation or growth on symmetry-breaking substrates, leading to in-plane anisotropic lattice parameters and stabilization of the out-of-plane ferroelectric component. First-principles calculations show remanent values of $P_\text{ip} \approx 11.03$ μC/cm² and $P_\text{op} \approx 0.14$ μC/cm² for strained 2D-RP analogs, where the out-of-plane polarization overcomes substantial depolarization fields through the hybrid coupling route.

2. Phase Transitions, Polarization Switching, and Magnetoelectric Coupling

The Pc phase is energetically close to several competing polar or paraelectric phases (e.g., P2₁ symmetry with pure in-plane polarization). The transition from Pc to P2₁ is accessed via strain tuning or electric field control—effectively, by altering the relative amplitudes or phases of the coupled octahedral modes. This transition is characterized by:

• The loss (or rotation) of the out-of-plane polarization, leaving a robust in-plane component (IP-HIF),
• A 90° rotation of the weak ferromagnetic moment (wFM), enabled by the magnetization–polarization coupling mediated by octahedral tilts (Zhou et al., 24 Jul 2025, 1007.1003).

First-principles nudged elastic band (NEB) calculations delineate two classes of switching paths:

• Low-barrier ($\sim$40 meV/f.u.) IP-HIF path, in which the in-plane polarization reverses, anchored by an unchanged in-phase octahedral rotation (M₂⁺ mode), simultaneously rotating the weak ferromagnetism by 90°.
• Higher barrier ($\sim$120 meV/f.u.) OP-HIF path, where the out-of-plane polarization is reversed, requiring more substantial lattice reconfiguration.

The trilinear coupling allows for selective switching: changing only the octahedral tilt may reverse both polarization and magnetization, while altering the rotation often switches only polarization, as verified by first-principles and symmetry analysis (1007.1003, Zhou et al., 24 Jul 2025).

The result is a multiferroic coupling scheme where electric field control of lattice distortions effects both ferroelectric and magnetic state manipulation—central for magnetoelectric device concepts.

3. Domain Topology, Vortices, and Z₄×Z₂ Degeneracy

Hybrid improper ferroelectrics, and specifically the Pc phase in A₃B₂O₇ derivatives, exhibit a complex topological domain space resulting from the interplay of octahedral tilt and rotation order parameters. Domain states can be labeled as pairs $(\theta,\eta)$:

• $\theta\in\{1,2,3,4\}$ indexes tilt orientations,
• $\eta\in\{+,-\}$ indexes rotation phase, defining a Z₄×Z₂ space (Huang et al., 2016).

Within this topology:

• Each of four polarization directions is doubly degenerate (due to rotation sign),
• The domain wall network includes five archetypes (rotation, tilt, ferroelastic tilting, ferroelastic tilting+rotation, and antiphase boundaries),
• Z₃ vortices, where three domain walls meet, are universal and arise naturally from the group structure,
• Antiphase boundaries exhibit reversible zipper-like splitting into pairs of ferroelastic or ferroelectric domain walls during 90° or 180° polarization switching.

The energetic landscape is such that domain switching is mediated not simply by Ising-like flips but by the reversible creation and annihilation of vortex–antivortex pairs and the unzipping/recombination of topologically protected boundaries. This leads to rich, mobile, and easily reconfigurable domain wall networks, with measurable consequences for conductivity, switching kinetics, and emergent wall-bound phenomena (Huang et al., 2016, Smith et al., 2019).

4. Spin Texture, Altermagnetism, and Spintronics in 2D Pc Phases

Central to the functional impact of the Pc hybrid improper ferroelectric phase is the emergence of persistent spin textures (PSTs) and tunable altermagnetism, especially in two-dimensional Ruddlesden–Popper derivatives:

• In the Pc phase, effective $k\cdot p$ Hamiltonians are dominated by quadratic momentum terms ($\beta$ terms) in the spin–orbit coupling,

$$H = \sigma_x \alpha_1 k_x k_y + \sigma_y (\beta_1 k_x^2 + \beta_2 k_y^2) + \sigma_z \gamma_1 k_x k_y$$

with PST aligned along the polarization direction and consistent with the orientation of weak ferromagnetism.

• Upon switching to P2₁ symmetry (pure in-plane polarization), the principal spin-texture axis rotates by 90°, with the dominant term shifting to $\sigma_x (\alpha_1 k_x^2 + \alpha_2 k_y^2)$, reflecting the realignment of wFM and the PST axis (Zhou et al., 24 Jul 2025).
• The Pc phase hosts altermagnetism; the P2₁ phase displays full-Brillouin-zone band splitting.

This tunable and persistent spin texture, protected by phase symmetry and crystal distortions, strongly suppresses spin relaxation and supports long spin lifetimes, making these materials natural candidates for electrical manipulation and robust spin–orbit devices. Band splitting controlled by electric field or strain enables additional functionalities relevant for nonvolatile logic, spin-current generation, and spin-orbit transistors.

5. Effects of Strain, Pressure, and Substitutions

The Pc phase, and HIF order more generally, is highly susceptible to external tuning:

• Strain Engineering: Both biaxial and uniaxial strain directly influence the amplitude and order parameters of the rotational and tilt modes, shifting phase boundaries and dramatically lowering the intrinsic switching barrier by proximity to nonpolar or antipolar phases. The strain–tolerance factor phase diagram delineates regimes where switching barriers are suppressed and polar/nonpolar phases compete (Li et al., 2020, Zhou et al., 2013, Li et al., 2018).
• Pressure Effects: Hydrostatic pressure can stabilize or destabilize the Pc phase relative to higher-symmetry phases (e.g., Acaa), with unique regimes where applied pressure enhances polar amplitudes (due to differential response of the coupled nonpolar modes), in contrast to monotonic suppression seen in proper ferroelectrics (Clarke et al., 2021, Liu et al., 2019).
• Site-Selective Substitution: Iso-valent (e.g., Mg, Sr) and heterovalent substitutions provide chemical routes to alter the relative stability of polar phases, amplitude of polarization, and coercive field, offering another axis for functional tuning and optimization (Li et al., 2018).

In ultrathin limits, stabilization of the primary order parameter (e.g., via engineered bridging layer) can maintain improper ferroelectricity with undiminished Curie temperature, overcoming the critical thickness limit in proper ferroelectrics (Li et al., 8 Mar 2025).

6. Domain Wall Conduction, Topological Defects, and Device Implications

Hybrid improper ferroelectric Pc phases support conducting and charged domain walls of remarkable stability, abundance, and variability. The stabilization mechanisms involve both intrinsic topological constraints (e.g., vortex lines in ErMnO₃, Z₃ vortices in (Ca,Sr)₃Ti₂O₇) and extrinsic defect chemistry (e.g., stacking faults with Sr segregation in Ca₃₋ₓSrₓTi₂O₇) (Nakajima et al., 2021, Zahn et al., 13 Oct 2024):

• Meandering charged domain walls can show conduction contrast of up to 2 orders of magnitude depending on wall polarity,
• Out-of-phase boundaries (stacking faults) compensate bound charge and stabilize sharp charged walls,
• Phase-field and Landau-Ginzburg modeling confirms that restoring forces from the improper ferroelectric order enable reversible wall motion over distances >250 nm—much greater than in proper ferroelectrics—under electric-field cycling, with the robustness set by topological pinning centers (Zahn et al., 13 Oct 2024).

These features enable predictable domain wall location and movement, essential for nonvolatile memory, logic, and capacitor applications. The ability to modulate domain wall conductivity, orientation, and motion by electric field or strain expands the operational space for all-oxide electronics.

7. Functional Prospects and Future Directions

The unique interplay between lattice, polarization, magnetism, and spin in the hybrid improper ferroelectric Pc phase permits the realization of:

• Electrically and mechanically switchable multiferroics with nonvolatile memory and logic capabilities,
• Spintronic devices with PST-enabled high spin coherence and electrically tunable band splitting,
• Nanoscale devices leveraging switchable domain wall conduction pathways,
• Ultrathin ferroelectric or multiferroic devices with room-temperature stability and absent critical thickness limits,
• Systematic material design leveraging group-theoretical and symmetry-guided strategies to create hybrid improper (multi)ferroelectrics beyond perovskite lattices (Xu et al., 2016, Boström et al., 2017, Griesi et al., 2021).

Continued research is poised to address open questions regarding dynamic switching pathways, compositional tuning for room-temperature multiferroicity, control of vortex topologies, and integration into functional heterostructures for low-power, multifunctional electronic, spintronic, and memory applications.