Polarization Engineering of the Orbital Hall Conductivity in Two-dimensional Ferroelectric Higher-Order Topological Insulator Tl$_2$S and SnS

Abstract: Ferroelectric higher-order topological insulators (HOTIs) exhibit tunable physical properties arising from the interplay between ferroelectric polarization and band topology. This work investigates the topological origin of two classes of two-dimensional (2D) ferroelectric HOTIs with out-of-plane or in-plane polarization, revealing their distinct orbital transport behaviors and the mechanism for engineering orbital Hall conductivity (OHC) via polarization control. Our results demonstrate the unique role of polarization in modulating both the higher-order band topology and orbital transport. A strong coupling between in-plane polarization and higher-order topology is identified, establishing in-plane polarization as an intrinsic means to reversibly switch the OHC plateau within the band gap. Using Tl$_2$S and SnS as representative models of the two HOTI types, we demonstrate persistent and electrically switchable orbital transport, respectively. Our study advances the understanding of the coupling among ferroelectricity, higher-order topology, and orbital transport, offering new avenues for controllable orbitronics.

• The paper demonstrates that polarization engineering enables control of orbital Hall conductivity in 2D ferroelectric HOTIs, with distinct roles for out-of-plane and in-plane polarization.
• Using DFT, tight-binding Wannier models, and Berry curvature analysis, the study quantifies topological transitions and the emergence of corner modes in Tl₂S and SnS.
• The findings suggest practical applications in orbitronics by allowing reversible, non-volatile switching of topological states through ferroelectric polarization.

Polarization Engineering of the Orbital Hall Conductivity in Two-Dimensional Ferroelectric Higher-Order Topological Insulators

Introduction

This work provides a comprehensive theoretical investigation of two-dimensional (2D) ferroelectric higher-order topological insulators (HOTIs)—specifically Tl$_2$S and SnS—with a focus on the interplay between ferroelectric polarization, higher-order topology, and orbital transport phenomena. The role of polarization, both out-of-plane and in-plane, is elucidated regarding the modulation of the orbital Hall conductivity (OHC), with deep analysis of the underlying band topology and associated boundary states. The paper leverages density functional theory (DFT), tight-binding (TB) Wannier models, and topological invariants derived from crystal symmetries to systematically address the tunability and detection of HOTI phases via orbital transport measurements.

Out-of-Plane Ferroelectric HOTI: Tl$_2$S

Structural and Topological Characteristics

Tl$_2$S crystallizes in a layered anti-TMD motif, with the high-symmetry parent structure (space group $P\bar{3}m1$) dynamically unstable, yielding two stable out-of-plane ferroelectric phases ($P3$ and $P31m$). Both retain $C_3$ rotational symmetry; $P31m$ is further enriched by mirror symmetries. DFT calculations confirm structural dynamical stability and the existence of significant out-of-plane polarization. However, the $C_3$ symmetry precludes in-plane polarization.

Figure 1: Crystal structures of Tl$_2$S in high-symmetry and ferroelectric phases, highlighting space group symmetry reduction and mirror operations.

From a topological perspective, both ferroelectric polymorphs are shown to belong to nontrivial higher-order topological classes, as discerned from the calculated irreducible representations of their occupied bands. For the $_2$0 phase, the topological indices $_2$1 signal quantized fractional corner charge ($_2$2), protected by the residual $_2$3 symmetry. The associated corner-localized in-gap modes are evidenced via finite nanoflake calculations.

Figure 2: (a) Nanoflake geometry for corner state analysis, (b) edge states, (c) quantized bulk-boundary fractionalization in energy spectrum, (d) band structure with OHC, (e) OHC under polarization switching.

Orbital Hall Conductivity and Polarization Switching

Analysis of the OHC, as calculated from the orbital Berry curvature, reveals a finite plateau within the band gap—a direct fingerprint of the HOTI phase and the existence of orbital Chern phases in this class of materials. The magnitude of the OHC plateau in Tl$_2$4S, however, is modest, ascribed to both the orbital character near the Fermi level (primarily S-$_2$5 and Tl-$_2$6) and the indirect nature of the band gap.

During ferroelectric polarization switching (traced via the nudged elastic band method), an intermediate centrosymmetric phase ($_2$7) is traversed. Topologically, the system remains a HOTI, with $_2$8 symmetry acting as the protector of higher-order signatures. The OHC plateau stays virtually unchanged throughout, reflecting the insensitivity of the higher-order topology (and associated orbital transport) to out-of-plane polarization reversal.

Figure 3: Orbital Berry curvature in intermediate and ferroelectric Tl$_2$9S, confirming the topological continuity and robustness of the OHC.

In-Plane Ferroelectric HOTI: SnS

Symmetry Breaking and Topological Switching

Contrastingly, SnS features a well-defined in-plane ferroelectric phase transition between a high-symmetry $_2$0 (nonpolar) and a lower-symmetry polar $_2$1 phase. The onset of in-plane polarization breaks both inversion and $_2$2 rotational symmetries, while preserving $_2$3. This symmetry breaking directly enables a higher-order topological phase transition.

In the absence of polarization (intermediate phase), HOTI corner modes are absent, and the OHC plateau within the band gap vanishes. Upon entering the in-plane ferroelectric phase, symmetry-enforced corner states appear, with a concomitant emergence of a nonzero OHC plateau—a direct manifestation of HOTI order mediated by in-plane polarization.

Figure 4: Structural and electronic features of SnS phases; (a,b,c) intermediate phase (no OHC plateau); (d,e,f) polar phase with higher-order bands and finite OHC.

A sharp polarization-driven transition of the OHC plateau, from zero to finite values, is calculated, confirming the direct and reversible polarization engineering of topological transport in SnS.

Berry Curvature and Topological Mechanism

Calculation of the orbital Berry curvature elucidates this transition's microscopic mechanism. In the nonpolar phase, the Berry curvature integrates to zero due to full compensation across the BZ. Polarization breaks this compensation, yielding a finite total OHC in the polar structure.

Figure 5: Orbital Berry curvature of SnS, contrasting full compensation in the intermediate phase (zero OHC) with uncompensated contributions yielding finite OHC in the polar phase.

Implications and Future Directions

This work identifies ferroelectric polarization—especially in-plane—as a decisive symmetry parameter for modulating higher-order topology and its observable correlates in orbital transport metrics. The demonstration that the OHC plateau can be robustly switched on and off in SnS by polarization reversal establishes a clear mechanism for all-electric, non-volatile control of orbitronics in quantum materials.

From a theoretical perspective, the symmetry-topology coupling described here generalizes the role of ferroelectric order as both companion and controller of topological quantum numbers in low-dimensional systems. The practical implications are significant: device architectures leveraging HOTI-based orbitronics could exploit polarization states for robust, switchable quantum functionalities, advancing non-charge-based information processing paradigms.

Anticipated future research avenues include extension to other material systems (e.g., magnetic HOTIs), explicit investigation of dynamical switching processes, potential realization of orbital devices, and experimental explorations via orbital-resolved spectroscopy and transport.

Conclusion

A systematic distinction between out-of-plane and in-plane ferroelectric HOTIs is established: in Tl$_2$4S, higher-order topological and orbital transport properties are invariant under polarization switching, reflecting topological robustness. In SnS, in-plane polarization acts as a symmetry-breaking field, driving genuine higher-order topological phase transitions and reversible OHC plateau switching. These findings reinforce the central role of polarization engineering in next-generation orbitronics, providing robust theoretical grounds for practical manipulation of topological quantum matter.

(2604.18093)