Effective lifting of the topological protection of quantum spin Hall edge states by edge coupling

ARTICLE https://doi.org/10.1038/s41467-022-30996-z OPEN Effective lifting of the topological protection of quantum spin Hall edge states by edge coupling 1234567890():,; R. Stühler 1 ✉, A. Kowalewski1, F. Reis E. M. Hankiewicz2 & R. Claessen 1 ✉ 1, D. Jungblut2, F. Dominguez2,3, B. Scharf2, G. Li 2,4,5, J. Schäfer 1, The scientific interest in two-dimensional topological insulators (2D TIs) is currently shifting from a more fundamental perspective to the exploration and design of novel functionalities. Key concepts for the use of 2D TIs in spintronics are based on the topological protection and spin-momentum locking of their helical edge states. In this study we present experimental evidence that topological protection can be (partially) lifted by pairwise coupling of 2D TI edges in close proximity. Using direct wave function mapping via scanning tunneling microscopy/spectroscopy (STM/STS) we compare isolated and coupled topological edges in the 2D TI bismuthene. The latter situation is realized by natural lattice line defects and reveals distinct quasi-particle interference (QPI) patterns, identified as electronic Fabry-Pérot resonator modes. In contrast, free edges show no sign of any single-particle backscattering. These results pave the way for novel device concepts based on active control of topological protection through inter-edge hybridization for, e.g., electronic Fabry-Pérot interferometry. 1 Physikalisches Institut and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität Würzburg, D-97074 Würzburg, Germany. 2 Institut für Theoretische Physik und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität Würzburg, D-97074 Würzburg, Germany. 3 Institute for Mathematical Physics, TU Braunschweig, 38106 Braunschweig, Germany. 4 School of Physical Science and Technology, ShanghaiTech University, 201210 Shanghai, China. 5 ShanghaiTech Laboratory for Topological Physics, 200031 Shanghai, China. ✉email: ; NATURE COMMUNICATIONS | (2022)13:3480 | https://doi.org/10.1038/s41467-022-30996-z | www.nature.com/naturecommunications 1 ARTICLE I NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30996-z n 2D TIs the bulk-boundary correspondence enforces the existence of metallic gap states confined to the one-dimensional (1D) edges of the material. These edge states are spin-polarized and have their spin rigidly locked to the electron’s momentum1. As a consequence electrons moving along the edge cannot be backscattered by any non-magnetic defects, as schematically depicted in Fig. 1a. This topological edge state protection lies at the heart of the celebrated quantum spin Hall (QSH) effect2. It is obvious that the unique properties of the topological edge states lend themselves to a plethora of novel functionalities and application ideas, ranging from low-power consumption electronics to innovative spintronics devices to possible solid-state realizations of qubits3. Most theoretical device concepts are based on direct control of the topological edge states by external stimuli, such as electric or magnetic fields, or by bringing them into spatial proximity to ferromagnetic4 or superconducting materials. Ideas include in particular various types of field-effect transistors (FET). For example, the symmetry-breaking effect of an electric gate field can be used to trigger a phase transition of the 2D TI to a trivial insulator, as recently demonstrated for Na3Bi5. The on/off characteristics of such a FET is determined by the complete quenching of the current-carrying topological edge states6–8. In an alternative FET concept relying on much smaller gate fields the Fermi level is toggled between an in-gap position and the bulk band edges. In the former situation the edge states carry a dissipationless current, while in the latter they will couple to the dissipative bulk states, thereby effectively losing their topological protection. The resulting promotion of backscattering from impurities and phonons is estimated to allow on/off ratios of more than two orders of magnitude9. A more direct way of controlling and eventually lifting topological protection is achieved by tunneling between opposite edges of a 2D TI10–17. The resulting hybridization between right(left)-moving electrons on one edge with their left(right)moving partners of like spin on the opposite edge will open a small gap at the Dirac point and–even more importantly–create a channel for electron backscattering without the need to break time-reversal symmetry, as depicted in Fig. 1b. A possible realization of such a situation is a narrow constriction in a 2D TI as, e.g., engineered by nanopatterning or defined by suitable line defects in the atomic lattice18. Interedge tunneling will then lead to the formation of Fabry-Pérot-type resonances along the constriction and hence a modulated transmission through the device, controlled by the position of the Fermi level12. While numerous theoretical proposals along this line have been put forward, there exist surprisingly few experimental studies on edge coupling in a QSH insulator. Strunz et al.19 have recently studied the effect of Coulomb interaction between 2D TI edges in spatial proximity, whereas Jung et al.20 focus on the tunneling-induced gap opening in the 1D edge states of a topological crystalline insulator. Results and discussion In this paper we examine the effect of topological edge coupling by spatial mapping of the resulting wave function, thereby Fig. 1 Free vs. coupled helical edge states. a Edge segment of a generic 2D TI material. A pair of helical edge states is bound to the edge. Back-scattering from defects such as the edge kinks is impeded by topologically protected spin-momentum locking, i.e., left-moving spin-up states (blue) cannot scatter with right-moving spin-down states (red). The corresponding one-dimensional band structure is schematically depicted below, where the arrows represent the electron spin projection and the wavy line indicates (prohibited) single-particle backscattering. b Two opposing edge segments in close proximity. Spatial overlap of both pairs of helical edge states induces hybridization (or tunneling) between both edges, thereby allowing inter-edge scattering from one segment to the other across the boundary. Strikingly, this opens a channel for back scattering from the edge kinks, i.e., the topological protection becomes partially lifted due to inter-edge hybridization. The corresponding band structure and the now allowed single-particle backscattering is depicted below. c Tight-binding model of coupled edge states in the 2D TI bismuthene. The white area indicates a topologically trivial line defect along the xdirection embedded in the topological bulk (light brown). The green and pink boundaries of the finite-size bulk are connected via periodic boundary conditions, resulting in the depicted torus. The trivial line defect connects two opposite edges; the resulting wave function overlap b (...truncated)