Atomically precise engineering of spin–orbit polarons in a kagome magnetic Weyl semimetal

Abstract

Atomically precise defect engineering is essential to manipulate the properties of emerging topological quantum materials for practical quantum applications. However, this remains challenging due to the obstacles in modifying the typically complex crystal lattice with atomic precision. Here, we report the atomically precise engineering of the vacancy-localized spin–orbit polarons in a kagome magnetic Weyl semimetal Co₃Sn₂S₂, using scanning tunneling microscope. We achieve the step-by-step repair of the selected vacancies, leading to the formation of artificial sulfur vacancies with elaborate geometry. We find that that the bound states localized around these vacancies undergo a symmetry dependent energy shift towards Fermi level with increasing vacancy size. As the vacancy size increases, the localized magnetic moments of spin–orbit polarons become tunable and eventually become itinerantly negative due to spin–orbit coupling in the kagome flat band. These findings provide a platform for engineering atomic quantum states in topological quantum materials at the atomic scale.

Introduction

Creation and manipulation of many–body quantum states are crucial for develo** advanced technologies in quantum computation, communications, security, and sensing^1,2,3,4. Individual atomic-scale defects in a solid material provides one of the ideal candidates for generating localized quantum states due to the introduction of symmetry breaking, degeneracy lifting and scattering sources in the vicinity of the defects^{5,6,7,8,9,10,11}. The atomically precise engineering of bound states has been realized in the vacancies of a few material platforms such as insulating film^12,13,14, diamond¹⁵, graphene¹⁶, h–BN¹⁷ and two–dimensional transition-metal dichalcogenides¹⁸, which is appealing for practical quantum applications. However, so far, the engineering of defect bound states is limited to a few material platforms due to the challenges in modifying the atomic defects of complex lattice.

Topological quantum materials have recently attracted considerable attention due to their fascinating symmetry-protected band structures and cooperative effects involving the interplay of multiple degrees of freedom (charge, spin, orbital, lattice)^19,20. The interactions of multiple degrees of freedom in quantum materials are dynamically intertwined with each other, which results in exotic quantum states^21,22. In recent years, the transition-metal kagome lattice materials which host Dirac points and nearly flat bands that naturally promote topological and correlation effects^23,24 are discovered, providing exciting opportunities for exploring frustrated, correlated, and topological quantum states of matter^{25,26,27,28,29,30,31,32,33}. Remarkably, quantum states including the magnetic polarons have been discovered in magnetic transition-metal kagome shandites, which provides a promising way to engineer bound states for dilute magnetic topological materials and kagome–lattice–based devices^34,

Fig. 1: Atomically precise engineering of vacancies at S–terminated surface of Co₃Sn₂S₂.

a Schematics of tip-assisted atomically precise vacancy engineering at a S surface over the kagome layer, showing that the vacancies with various shapes are transformed into the ones with well-organized geometries. b,c STM images showing the topography before (b) and after (c) vacancy repairing, demonstrating the filling of S atom. The red arrow indicates the position of tip pulse for the vacancy repair. d Schematic showing the filling of S atom from bottom S layer. e, f STM images showing the topography before (e) and after (f) vacancy creation, demonstrating the removal of S atom at surface. The black cross indicates the position of tip pulse for the vacancy creation. g Schematic showing the removal of S atom from top S layer to fill the vacancy at the bottom S layer. h Series of STM images showing that a long vacancy chain is gradually shortened by the vacancy repairing method. i Series of STM images showing that a cross-shaped vacancy consisting of four S absences leads to the formation of a triangular vacancy. The red arrows in (b), (h) and (i) mark the position of tip pulse during the vacancy repairing process. The black cross in (e) denotes the position of tip pulse during the vacancy creation process. STM scanning parameters for (b, c), (e, f) and (h, i): Sample bias V_s = −400 mV; Current setpoint I_t = 500 pA.

Results

Atomically precise construction of sulfur vacancies

The crystal structure of Co₃Sn₃S₂ consist of the rhombohedral lattice where the kagome Co₃Sn layer are sandwiched between two triangular S layers, which are further encapsulated by two separated triangular Sn layers³². Cleavage in vacuum typically results in Sn and S terminated surfaces with kagome Co₃Sn surfaces rarely obtained²⁹.

We start with the S terminated surface, which has been identified by STM and atomic force microscopy (AFM) in previous works^29,

In addition to the single-chain vacancy, the tunability of the bound states extends to the vacancies with more elaborate shapes, including double column vacancy chains (Fig. 3c), equilateral triangle (Fig. 3e and Supplementary Fig. 11) and equilateral-hexagon vacancies (Fig. 3g). All vacancies exhibit series of several peaks with equally spacing energy and the sharpest peak with highest energy position is similarly assigned as the primary bound states P(N). As summarized in Fig. 3i, the evolution of P(N) for each symmetric shape follows an exponential function, with all P(N) shifting exponentially towards a critical energy value near the Fermi level as the size increases. The critical energy level of P(N) depends on the vacancy shape (highlighted by different color in Fig. 3), with higher symmetry shapes possessing higher critical energy levels (Fig. 3i). For instance, the critical energy of P(N) of single chain vacancies is about −240 meV while one of the hexagonal vacancies is almost at Fermi levels.

The geometry dependent bound states suggest the strong couplings between adjacent single vacancies. Furthermore, the spatial distributions of the bound states across the single chain vacancies (Supplementary Fig. 12) exhibit quasi one-dimensional band behaviors^40,41,42, indicating the existences of vacancy–vacancy couplings. The dI/dV maps at large-size triangular vacancy show quantum confinement effect, featuring a quantum antidot (Supplementary Fig. 13). To gain insight into the shape dependent energy shift behaviors of P(N), we develop a simple model (see method in Supplementary Information and Supplementary Fig. 14) to simulate the bound states around vacancies. We simulate vacancies with a simple tight-binding model with a nearest neighbor hop** t by considering the hybridization between vacancies (Fig. 3j). We construct four types of vacancy patterns with different number of S vacancies, which consist of single chain, double column chain, triangle and hexagon. In each vacancy configuration, the highest energy level is extracted as the vacancy state. We find that the vacancy state undergoes a similar exponential shift towards higher energy (Fig. 3k), which is consistent with experimental observations in Fig. 3i.

The size dependent magnetic moment of vacancies

To gain deep understanding of nature of localized states at vacancies, we further study the shape dependent magnetic moment of the bound states. We focus on the magnetic moment of triangular vacancy due to their high yields and relatively–large lattice distortions (Supplementary Fig. 15). The bound states of triangular vacancy present the anomalous Zeeman effect with external magnetic field (Fig. 4a, b). Fitting the energy position (details see Supplementary Fig. 16) as a function of the magnetic field, we obtained the effective magnetic moment value |µ(N)|. For example, |µ(N = 3)| = 0.09 meV/T = 1.55 µ_B (Fig. 4a) and |µ(N = 10)| = 0.17 meV/T  = 2.93 µ_B (Fig. 4b). These results indicate that the magnetic moment of bound states localized at vacancy is directly related to the vacancy size.

Fig. 4: Tunability of the magnetic moment of bound states through controlling the size of triangular S vacancies.

a, b Intensity plot of field dependent dI/dV spectra (top) and corresponding energy shift of primary peak position with magnetic field (bottom) for triangular vacancy of N = 3 (a) and N = 10 (b). A linear function is applied to fit the energy shift of peak position, with the slope marked. c STM image (top) and corresponding line profile (bottom), showing a narrow Co₃Sn terrace confined by the step edges of adjacent S terraces. d Intensity plot of field dependent dI/dV spectra (top) and corresponding energy shift of peak position with magnetic field (bottom) of the bound states at narrow Co₃Sn terrace in (c), revealing a magnetic moment of approximately 0.19 meV/T = 3.28 µ_B for the bound state near Fermi level. e Evolution of the magnetic moment with the atomic number of vacancies, showing that the magnetic moments extend from localized magnetic moment around vacancies (spin–orbit polarons, SOPs) to flat band negative magnetism from Co₃Sn kagome layer. The colored curve is a visual guide to the trend in the evolution of the moment with increasing atomic number N. The error bars in (e) are determined by linear fitting as shown in bottom panel in (a), (b) and (d). The red circling arrows in (e) indicate the circulating current in kagome lattice. STM parameters: (a): V_s = −250 mV, I_t = 500 pA; (b) V_s = −50 mV, I_t = 500 pA; (d) V_s = 20 mV, I_t = 500 pA; V_mod = 0.2 mV; (c) V_s = −400 mV, I_t = 50 pA.

The Co₃Sn terrace, confined by the step edges of adjacent S terraces (Fig. 4c), is considered as a naturally occurring vacancy with an enormous size (N = ∞). The spatially-averaged dI/dV spectrum obtained at the Co₃Sn terrace shows sharp peaks in the vicinity of Fermi level, which is consistent with previous STM results on the Co₃Sn surface^29,36. The magnetic field dependent dI/dV curves (Fig. 4d) show similar anomalous Zeeman effect with an effective magnetic moment of µ(N = ∞ ) = −0.19 meV/T = −3.28 µ_B (the negative value is aiming to highlight the moment is negative in anomalous Zeeman effect and differentiate it from the positive moment in Zeeman effect). The negative orbital magnetic moment results from the spin–orbit coupling in the kagome flat band considering the non-trivial Berry phase of the flat band⁴³. Evolution of the magnetic moment with the atomic number of vacancies shows that the magnetic moments extend from localized magnetic moment around vacancies to the flat band negative magnetism from Co₃Sn kagome layer (Fig. 4e).