Material’s Edges Host Stable Electrical Currents Despite Boundary Imperfections

Researchers investigate the behaviour of edge states within the two-dimensional kagome lattice, exploring the interplay of lattice termination, spin-orbit coupling, and magnetic order. Sajid Sekh from the Institute of Nuclear Physics, Polish Academy of Sciences, Annica M. Black-Schaffer from the Department of Physics and Astronomy, Uppsala University, and Andrzej Ptok, also of the Institute of Nuclear Physics, Polish Academy of Sciences, demonstrate how the existence of these edge states is acutely dependent on boundary geometry and can be dramatically influenced by spin-orbit interactions. Their work reveals that Kane-Mele spin-orbit coupling establishes robust, protected helical edge states, creating an insulating phase unaffected by termination details, while combined Zeeman and Rashba coupling induces Chern insulating phases with corresponding chiral edge modes. These findings are significant as they offer crucial insights into manipulating edge states within kagome lattices, potentially paving the way for the design of materials with novel and tunable electronic properties.

Scientists are unlocking new potential in two-dimensional materials through detailed analysis of edge states within the kagome lattice, a structure formed by corner-sharing triangles arranged in a hexagonal pattern. This research demonstrates how the properties of these edge states, which reside at the boundaries of the material, can be precisely tuned by manipulating lattice termination, spin-orbit coupling, and magnetic order. The work reveals a surprising sensitivity of edge state existence to the precise geometry of the lattice edge, with certain terminations effectively eliminating these crucial states. Introducing spin-orbit coupling, an interaction between an electron’s spin and its motion, stabilizes topologically protected helical edge states, creating a robust insulating phase largely unaffected by boundary details. In contrast, combining a Zeeman field, representing an applied magnetic field, with Rashba spin-orbit coupling drives the system into Chern insulating phases, where the number of chiral edge modes directly corresponds to the Chern number, a topological invariant. Researchers employed a tight-binding approach to model the kagome lattice, systematically investigating the interplay between these key parameters. This computational method, which constructs electronic band structures from atomic orbitals, was chosen for its efficiency in modelling complex lattice geometries and incorporating relativistic effects crucial to understanding topological phases. The study began by establishing a pristine kagome lattice to assess the inherent sensitivity of edge states to boundary configurations, systematically varying the termination to determine its impact on edge mode existence. Kane-Mele spin-orbit coupling, a mechanism describing the interplay between electron spin and momentum, was then introduced to open a bulk gap and subsequently stabilise protected helical edge states. This implementation of Kane-Mele coupling creates a robust insulating phase largely independent of termination details, simplifying the analysis of edge state behaviour. The combined influence of a Zeeman field and Rashba spin-orbit coupling, another spin-momentum locking effect, was explored to induce Chern insulating phases. The system’s topological properties were characterised by calculating Chern numbers, which directly correlate with the number of chiral edge modes present. Further methodological innovation involved the modelling of non-coplanar magnetic textures, generating multiple Chern phases through finite scalar spin chirality. The Wilson loop formalism was employed to compute the Z2 topological invariant, a key indicator of a quantum spin Hall (QSH) phase, by tracking the winding of phase factors derived from transfer matrix calculations. This approach, equivalent to locating maximally localised Wannier centres, provided a robust method for classifying the topology of time-reversal symmetric bands. The inclusion of out-of-plane Zeeman fields and Rashba spin-orbit coupling, expressed through a minimal Hamiltonian, enabled the investigation of quantum anomalous Hall (QAH) phases and the associated quantized anomalous Hall conductivity. This Hamiltonian incorporated terms representing kinetic energy, Rashba coupling, and Zeeman splitting, allowing precise control over spin polarisation and band structure manipulation. Lattice termination profoundly influences edge state properties, with specific configurations completely suppressing edge modes. The research demonstrates that certain boundary geometries eliminate these states, highlighting a strong sensitivity to termination details. Introducing Kane-Mele spin-orbit coupling opens a bulk gap and simultaneously stabilizes protected helical edge states, resulting in a robust Z2 insulating phase that remains unaffected by variations in termination. The observed Chern numbers directly correspond to the number of chiral edge modes present in the system, confirming a clear relationship between topological invariants and edge state behaviour. Further investigation reveals that non-coplanar magnetic textures generate multiple Chern phases through finite scalar spin chirality. The magnitude of the topological gaps is strongly tuned by the Kane-Mele coupling, demonstrating precise control over the electronic band structure. This intricate control over edge states opens avenues for creating materials with novel electronic characteristics and exploring exotic topological phases. The relentless pursuit of robust edge states has yielded an intriguing result concerning the kagome lattice, addressing the long-standing challenge of engineering predictable and resilient edge states. What distinguishes this research is the emphasis on ‘termination’ as a design parameter, capable of either suppressing or enabling edge states. This level of tunability is significant, potentially paving the way for novel electronic devices where information is carried by these robust edge states. However, the theoretical nature of this work is a key limitation; translating these findings into real materials will require overcoming significant synthetic challenges. The precise control over lattice termination and magnetic textures demanded by these simulations may prove difficult to achieve in practice, and the influence of material defects and interactions not accounted for in the tight-binding model remain open questions. Looking ahead, this work should inspire further exploration of similar lattice structures and the development of new materials with tailored edge states, requiring a combination of computational modelling and experimental synthesis. Beyond device applications, understanding these topological phases could also offer insights into fundamental physics, potentially revealing new states of matter and expanding our understanding of quantum materials.