Facet-dependent chiral edge and corner states in uniformly magnetized three-dimensional topological insulators: experimental realization, detection, and quantitative characterization of the exchange and hybridization parameters

Zhang, Kane, and Mele [Phys. Rev. Lett. 110, 046404 (2013)] predicted that a spatially uniform out-of-plane magnetization on the surface of a three-dimensional topological insulator (TI) opens an exchange gap in the Dirac surface states and binds chiral edge modes to every edge across which the z-projection of the surface normal reverses — a chiral 'equator' on a magnetized TI sphere, side-face channels on a slab, and staggered chiral corner modes on zigzag faceted surfaces — without requiring any magnetic domain wall. They further identified the competition between the exchange-induced mass $\Delta_z$ and the finite-size inter-surface hybridization $m_t$ as the control parameter of the topological transition, and noted that the magnitude of the anisotropic exchange term $\sigma_x\tau_z$ that appears on side (non-cleavage) faces is material-dependent and must be measured. The thin-film limit (two parallel cleavage faces, $|\Delta_z|>|m_t|$) has since been confirmed experimentally as the quantum anomalous Hall (QAH) effect, but that case was explicitly flagged by the authors as a very special case of their proposal. The general facet-dependent prediction — realization and direct detection of the chiral corner/edge states under uniform magnetization in an electronic TI, together with quantitative experimental characterization of the side-face exchange term, the face-dependent exchange parameters, and the thickness/hybridization crossover — remains open.

Background

A topological insulator (TI) is a bulk insulator whose boundary hosts an odd number of gapless Dirac surface states protected by time-reversal symmetry; the canonical three-dimensional examples are the bismuth chalcogenides Bi$_2$Se$_3$ and Bi$_2$Te$_3$. When time-reversal symmetry is broken — e.g. by magnetic atoms deposited on, or alloyed into, the surface — the Dirac cone acquires a mass gap $\Delta_z$, and the surface becomes a half-quantum anomalous Hall conductor (each such surface carries Hall conductance $e^2/2h$) whose transitions across mass sign changes carry chiral (one-way) edge modes. The quantum anomalous Hall (QAH) effect is the resulting quantized Hall conductance $e^2/h$ without an external magnetic field. First-principles work by Yu et al. [Science 329, 61 (2010)] predicted that uniformly magnetized TI thin films become QAH insulators, and Chang et al. [Science 340, 167 (2013)] observed the effect in Cr-doped (Bi,Sb)$_2$Te$_3$; the thickness-driven transition between the ordinary and QAH regimes, controlled by the competition between the exchange mass $\Delta_z$ and the inter-surface hybridization $m_t$, was quantitatively mapped by Ji et al. [ACS Nano 16, 1134 (2022)]. Zhang, Kane, and Mele [Phys. Rev. Lett. 110, 046404 (2013)] went beyond the thin-film geometry: for a magnetized TI of arbitrary shape with uniform out-of-plane magnetization $\Delta_z\sigma_z$ (where $\sigma$ and $\tau$ are Pauli matrices acting on electron spin and surface orbital/normal-sector degrees of freedom), they showed that chiral modes appear precisely on the edges where the z-projection of the surface normal reverses sign — the equator of a sphere, the side faces of a slab, and the corners of a zigzag faceted surface — so that no magnetic domain wall is needed to nucleate chiral transport. On side (non-cleavage) faces the exchange field also generates an anisotropic term proportional to $\sigma_x\tau_z$, whose magnitude (set by the difference of electron spin g-factors of, e.g., Bi and Se) the paper estimated to be small but left for experiments to determine; their discussion explicitly proposes detecting the predicted chiral corner modes in topological-insulator/ferromagnet intergrowths by scanning tunneling spectroscopy or multi-terminal transport. Inter-surface hybridization $m_t$ on side faces is negligible once the sample dimension exceeds the surface-state penetration length but is decisive for thin samples. The subsequent theory literature reformulated the facet- and corner-localized chiral states within higher-order topology (topological phases of a $d$-dimensional crystal whose gapless boundary modes, instead of covering the whole surface, localize on the boundary of the surface itself, i.e. on hinges and corners): Schindler et al. [Sci. Adv. 4, eaat0346 (2018)] described chiral hinge states of axion insulators (an 'axion insulator' is a TI whose opposing surfaces carry opposite exchange masses, realizing the magnetoelectric topological response without a net Hall effect), and Yue et al. [Nat. Phys. 15, 577 (2019)] predicted symmetry-enforced chiral hinge states in the magnetic axion insulator Bi$_{2-x}$Sm$_x$Se$_3$ — a mechanism that is related but distinct, since it rests on crystalline-symmetry-enforced surface Dirac masses rather than the uniform-magnetization facet criterion of Zhang, Kane, and Mele. Photonic lattices have realized analogs of non-coplanar chiral hinge transport (Lai et al., Nat. Commun. 16, 3826 (2025)), but these are not electronic TIs. Finally, Virk and Yazyev [Sci. Rep. 6, 20220 (2016)] established theoretically that high-index (non-cleavage) surfaces of bismuth-chalcogenide nanostructures — surface orientations other than the natural quintuple-layer cleavage plane — host their own Dirac surface states, a prerequisite for the side-face physics, but no experimental characterization of exchange parameters on such faces exists.

Problem Statement

Can the facet-dependent chiral edge and corner states predicted by Zhang, Kane, and Mele for a three-dimensional topological insulator with a spatially uniform out-of-plane surface magnetization $\Delta_z\sigma_z$ be realized and directly detected in an electronic material, and can the governing material parameters be quantitatively determined? Concretely, the problem has four linked components under one objective — experimental verification of the uniform-magnetization chiral-edge criterion: (1) realize the required system, i.e. a faceted topological-insulator crystal or TI/ferromagnet intergrowth in which a spatially uniform out-of-plane surface magnetization opens Dirac mass gaps on all relevant facets, including non-cleavage (side and high-index) faces; (2) detect the predicted chiral modes localized on the edges and corners at which the z-projection of the surface normal reverses — e.g. staggered chiral corner modes on zigzag sides and side-face channels on slabs — disentangling them from side-face and bulk contributions; (3) measure the exchange-induced surface terms material by material, including the anisotropic $\sigma_x\tau_z$ term on side faces whose magnitude the source paper explicitly left to experiment, and establish its relative importance against the dominant Zeeman-like $\Delta_z\sigma_z$ term; and (4) characterize the inter-surface hybridization $m_t$ and the crossover thickness (relative to the surface-state penetration length) on non-cleavage faces, fixing the $|\Delta_z|>|m_t|$ threshold for the topological transition in geometries beyond the already-resolved parallel-cleavage thin-film case. The common scientific objective is the experimental validation (or refutation) of the face-dependent exchange framework and of chiral edge formation without magnetic domain walls in a real faceted topological insulator. The question is whether, for a specific material system, the chiral modes appear where the criterion predicts them and the exchange and hybridization parameters governing the transition are quantitatively known, or whether a controlled negative result shows that the predicted facet-dependent modes and parameter values cannot be realized there.

References

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