Hierarchical First-order and Second-order Topological States in Symmetry-Controllable Grain Boundary

doi:10.21203/rs.3.rs-3720367/v1

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Hierarchical First-order and Second-order Topological States in Symmetry-Controllable Grain Boundary

https://doi.org/10.21203/rs.3.rs-3720367/v1

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The discovery of topological states localized at the boundaries and dislocations is of paramount importance in the physics of topological phases. Structural imperfections can be a promising testbed to engineer the symmetries and the concomitant topological states, as they genuinely exist in solid-state matters. Here, we present the direct evidence of hierarchical first-order and second-order topological states in symmetry–enforced grain boundaries (GB) of 1T′–MoTe₂. Guided by exerted scanning tunneling microscope (STM)–tip press and pulse, we construct two distinct types of GBs, which are differentiated by the underlying symmorphic and nonsymmorphic symmetries. The GBs with the nonsymmorphic rotation symmetry harbor the first-order topological edge state protected by nonsymmorphic band degeneracy. On the other hand, the edge state of the symmorphic GBs attains the band gap. More interestingly, the gapped edge state realizes the second-order topological phase, shown by the additional corner states at the GB ends. We anticipate our experiment will pioneer the material platform for the hierarchical realization of the first-order and higher-order topology.

Physical sciences/Physics/Condensed-matter physics/Topological matter/Topological insulators

Physical sciences/Physics/Condensed-matter physics/Topological matter/Topological defects

Physical sciences/Nanoscience and technology/Nanoscale materials/Two-dimensional materials

The topology of the electronic band is an essential factor in understanding the materials’ electrical and quantum properties. Bulk–boundary correspondence, as a guiding principle of the topological phase, dictates the intimate relationship between the topological invariant of bulk and the metallic excitations at the boundary^1,2,3. A prototypical example of materials with non-trivial topology is found in topological insulators (TIs), which have inverted band gaps by large spin-orbit coupling and topological character protected by time-reversal symmetry^4,5,6,7. Moreover, a larger class of topological materials is available when we add crystalline symmetries, exemplified by topological crystalline insulators (TCIs)^8,9 and higher-order topological insulators (HOTIs) ^10,11,12. HOTIs are especially interesting because they broaden the concept of bulk-boundary correspondence not to be limited between N-dimensional bulk and (N-1)-dimensional boundary, but to be applicable between N-dimensional bulk and (N-n)-dimensional boundaries with n > 2. For example, second-order HOTI with 3D (2D) bulk should have topologically protected 1D hinge (0D corner) states.

Observing topological boundary states is a key to identifying the non-trivial bulk topology. Crystallographic defects such as lattice dislocations and grain boundaries (GBs) are of particular interest because they can trap the localized topological boundary states depending on the crystalline symmetry of the defects^13,14,15,16. In the case of HOTI, realizing variant crystallographic defects provides a promising route to engineer topological boundary states by creating different types of defects that either preserve or break the underlying crystalline symmetries^17,18. With a local probe that can characterize and manipulate these crystallographic defects, we can identify the non-trivial topology of the bulk as well as control the topological states by the symmetry of defects.

A scanning tunneling microscope (STM) is a unique atomic-scale microscopy tool that has been extensively used for observing topological states^1,2 as well as manipulating them with atom manipulation techniques¹⁹. The precise control of the bias pulse and the sample-tip distance allows STM to manipulate individual atoms and defects with atomic precision. However, the application of such nanoscale manipulation to HOTI and demonstration of hierarchical topological states with respect to the symmetries are challenging, because it requires finding the right bulk materials with non-trivial topology that at the same time can be manipulated by STM to create atomic defects with suitable crystalline symmetries.

In this work, we apply the tensile strain on 1T′-MoTe₂ by poking the surface with an STM tip in a controlled fashion to create and stabilize symmetry differentiable GBs. The boundary matching of the atomic structure admits the two possible types of GBs, each of which is characterized by the distinct crystalline symmetries along the GBs: symmorphic C2– (SymC2) and non-symmorphic C2– (NonSymC2) rotational symmetries. The strain induces domain switching between three orientation variants of the 1T′ phase and creates symmetry-dictated GBs^1, 20. To create GBs, we design a novel STM manipulation method that induces ferroelastic switching between three orientational variants of the 1T′ phase of MoTe₂ in a controlled and reproducible way. The GBs created by STM manipulation exhibit different topological states depending on a type of crystalline symmetry, i.e. the first-order topological edge states and the second-order topological corner states for non-NonSymC2 and SymC2 rotational symmetries, respectively. Our results demonstrate that STM manipulation is a promising route to achieve the controllable manipulation of the HOTI states by ferroelastic switching.

Creation of symmetry-enforced grain boundaries

Due to the quantum spin Hall phase of 1T’–MoTe₂, the GBs of 1T’–MoTe₂ harbor the pair of helical edge states originating from each side of the quantum spin Hall insulating bulk. In general, these pairs can be gapped out, but NonSymC2 GBs stabilize the double helical edge mode, whose band degeneracy is protected by the nonsymmorphic symmetry (Fig. 1a top)¹⁶. On the other hand, the SymC2 GBs lose their topological protection and acquire a finite band gap. Nevertheless, SymC2 symmetry gives rise to the higher-order topological classifications of the gapped edge. The resulting electronic states of the SymC2 GB are analogous to the well-known Su-Schrieffer-Heeger chain²¹, which manifests as the second-order topological corner states (Fig. 1a bottom)²².

To create such GBs and to characterize their atomic and electronic structures, we employ STM and scanning tunneling spectroscopy (STS). A tensile strain can be applied by pressing the STM tip to the surface, and furthermore, electrical and/or thermal energy can be transmitted by applying a tip voltage pulse at a desired position (Fig. 1c). Inset of Fig. 1b shows a representative topographic image of the SymC2 GB between NonSymC2 GBs created by such STM manipulation. The differential conductance (dI/dV) spectra in Fig. 1b show a strong contrast between the ones taken at the center of GB (cyan), ends of GB (magenta and blue), and bulk (black), which we will later show to originate from the second-order topological states of SymC2 GB. Figure 1d shows a representative large-scale derivative STM image of the strain-engineered GBs in 1T’–MoTe₂. In the pristine surface of 1T’–MoTe₂, the Te atoms form quasi-one-dimensional chains with a preferred direction due to the dimerization of Mo atoms from the high symmetry phase (1T) (denoted by region G1 in Fig. 1c). By pressing the STM tip to the surface with applying a voltage pulse of − 2.5 V, the pressed G1 region transforms to the diamond shape of different types of grains (G2 and G3), corresponding to the two of the other six–fold rotation variants of the 1T’ phase with different Te–chain direction (Fig. 1d).

To characterize the optimal condition for obtaining the grain-switched diamond-shaped structure shown in Fig. 1e, we investigated several experimental parameters: (1) applied voltage pulse (V_p); (2) depth of surface indentation (D); (3) time to lower the tip to the surface (T1, varied from 0.5 s to 10 s); (4) duration at the lowest position for tip pulse (T2, varied from 0.2 s to 0.5 s); and (5) time to lift the tip (T3, varied from 0.02 s to 10 s.). The various results from changing the parameters are presented in the Supplementary Information. Subsequently, we determined the following optimal condition: V_p = − 2.5 ~ − 3 V, D = − 0.1 ~ − 0.2 nm, T1 = 0.5 ~ 3 s, T2 = 200 ms and T3 = 20 ms (Supplementary Fig. S1). We note that applying only voltage pulse V_p did not generate new grains, and applying tensile strain by pressing down the surface with certain D and T1 is essential to form the diamond-shaped structure. Further details of their dependence are presented in Supplementary Note 1 and Fig. S2–S5.

First-order and second-order topological states

The relative angles of Te-chain directions characterize the GBs between each region. Adjacent G2 and G3 grains bear a GB with the angle of 120° (G2–G3), and G2 and G3 regions form mostly GB with the angles of 60° with respect to the original G1 phase (G1–G2 and G1–G3) (See supplementary Fig. S6 for full geometric characterizations of the GBs). Further inspection of the STM image on the 120° boundary reveals the coexistence of both the NonSymC2 and SymC2 GBs (Fig. 2a). Furthermore, the distinctive behaviors in the dI/dV spectra are identified for different GBs. The dI/dV spectrum at the NonSymC2 GB, marked by the dotted cyan line in Fig. 2a (also shown as the cyan line in Fig. 2b), reveals a differential conductance peak near + 28 mV, whereas no such features are shown inside the grain far away from GBs (black line in Fig. 2b) as well as the SymC2 GB (yellow line in Fig. 2b).

The atomic configurations of both types of 120° GBs are characterized by a C2 rotation along the y-axis, offering greater energetic stability compared to the y-axis mirror reflection in 120° GBs¹⁶. These configurations can be further categorized into NonSymC2 and SymC2 based on the presence or lack of half translation, illustrated in Fig. 2c and e. Their electronic properties are depicted in Fig. 2d and f. Here, the blue color represents the state exclusive to the GB atom. Notably, the NonSymC2 exhibits a Weyl state within the GB, attributed to its nonsymmorphic symmetry. This state hinders gap formation resulting from the interaction between the left and right TI edge states of MoTe₂. Conversely, SymC2 lacks this symmetry protection, leading to a gap opening.

The spatial dI/dV maps of the 120° GB clearly show the one-dimensional boundary state at the NonSymC2 GBs (Fig. 3b). The boundary mode is well–localized at the NonSymC2 GB as seen in the dI/dV map acquired at 34 mV in the upper right panel in Fig. 3b (further dI/dV maps are presented in Supplementary Fig. S7). The observed boundary mode in the NonSymC2 GB agrees with the theoretically predicted bowtie shape in-gap metallic states (Fig. 2d). The boundary states originated from the pair of the helical edge states of each quantum spin Hall grain. In general, these boundary states are allowed to be gapped out by the interactions between the helical edges, since each side of the GB has the same Z₂ index. However, the additional NonSymC2 symmetry gives rise to the Young–Kane type nonsymmorphic band degeneracy²³, which protects the band crossings of the double-helical edge states.

In contrast, the SymC2 GB shows no sign of the in-gap edge states, and it suggests the avoided level crossings between edge modes in the absence of symmetry protection. However, interestingly, the measured dI/dV spectrum at the end of SymC2 GB (violet line in Fig. 3b) reveals the additional peak near − 4 mV (further clearly shown in the dI/dV spectra in Fig. 1b). The peaks are spatially localized at both ends of the GBs, as shown in the spatial dI/dV map in the left panel in Fig. 3c, and it signifies the presence of the zero-dimensional corner states.

Our theoretical calculation also verifies the gapped spectra of the edge mode (Fig. 2f). Further symmetry analysis reveals the higher-order topological classifications of the edge mode. The product of the SymC2 symmetry and the time-reversal symmetry forms the effective time-reversal symmetry with T² = 1. The additional approximate inversion symmetry of Te atoms along the grain boundary gives Z₂ classification of the gapped one–dimensional edge (AI class). The corresponding symmetry classification is formally equivalent to that of the SSH model, where the dimerizations of the hoppings give rise to the unpaired boundary states at each end²⁴. In SymC2 GB, the dimerization of the Mo atoms (shown by the purple stripes in Fig. 2e) gives rise to the effective dimerization of the hoppings. Using the effective model, we also calculated LDOS near both SymC2 and NonSymC2 GBs (Supplementary Figs. S10 and S11).

Next, we examine the possible hybridization between the first–order topological boundary states and the second–order topological corner states. Despite their positional proximity, the local dI/dV map on the side of the SymC2 GBs shows the diminishing amplitude as it approaches the end of the chain. This behavior resembles a typical quantum confinement phenomenon^25,26,27 with weak hybridization with the defect states. As a result, we conclude the vanishing tunneling of the corner state to the NonSymC2 GBs. We can read out information about the hybridization by analyzing the confined state by analyzing NonSymC2 GBs with a finite length (Fig. 4a). The dI/dV maps reveal the quantum well–like resonant bound states with the first (n = 1) and the second harmonics (n = 2) for the corresponding applied bias voltages 20 mV and 58 mV, respectively (Fig. 4b). The change of the local density of states (LDOS) maxima depending on the measured position within the NonSymC2 GB shown in a series of dI/dV spectra in Fig. 4c exhibits a typical quantum confinement phenomenon^3–5. These confinements are again clearly seen from dI/dV maps in Fig. 4d. The wave function of the confined state with n = 1 vanishes strongly at both ends of the NonSymC2 GB, which indicates the physical separation from that of the SymC2.

Electric-field induced grain boundary formation

We further investigated another method of GB creation, wherein we applied an electric field along the in-plane direction of the top layer of 1T’–MoTe₂. To achieve an in-plane electric field, first, we created a hole with a diameter of approximately 10 nm and a depth of over 2 nm by pressing and applying a voltage pulse with an STM tip (Fig. 5b)²⁸. Next, we placed the tip at the center of the hole as indicated by the red cross in Fig. 5b but less deep. Lowering the tip by a smaller depth allows full contact between the circumference of the tip and the edge of the formerly created hole. Thus, when a voltage pulse is applied, the electric field is applied along the in-plane direction and induces electrically strain-driven phase switching. As a result, the characteristic diamond structure is formed without any mechanical stress (Fig. 5c).

To support our hypothesis of GB formation by the in-plane electric field, we demonstrated a more delicately controlled GB formation process by carefully positioning the STM tip at a desired side of the edge of an existing hole (Fig. 5e). For example, in Fig. 5e, we placed the tip very close to the upper side of the edge as indicated by the red cross. Then, we applied a voltage pulse to create GBs. As a result, we obtained only the upper half of the previously observed diamond structure, because grain switching occurred only at the region next to the contact between the tip and the hole edge (Fig. 5f). The result substantiates that the electric field-induced phase switching requires a pre-existing hole structure with exposed step edges which allows the pulse from the tip to transmit the strain in the in-plane direction.

In conclusion, we demonstrate hierarchical first–order and second–order topological states created in the NonSymC2 and SymC2 GBs created in a 1T’-MoTe₂ by mechanically and electrically induced tensile strain using the STM tip. While the 1D topological edge mode appears on the NonSymC2 GB, the SymC2 GB attains the topological band gap and realizes the higher-order topological states at the GB ends. HOTIs have been theoretically well studied, however, their experimental observations have been rarely reported^29,30. Thus, our experimental demonstration not only provides direct evidence for the physical realization of higher–order topological states but also a novel method for creating and controlling symmetry–dictated GBs.

Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045 (2010).
Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057 (2011).
Isaev, L., Moon, Y. H. & Ortiz, G. Bulk-boundary correspondence in three-dimensional topological insulators. Phys. Rev. B 84, 075444 (2011).
Kane, C. L. & Mele, E. J. Quantum Spin Hall Effect in Graphene. Phys. Rev. Lett. 95, 226801 (2005).
Kane, C. L. & Mele, E. J. Z₂ Topological Order and the Quantum Spin Hall Effect. Phys. Rev. Lett. 95, 146802 (2005).
Bernevig, B. A. & Zhang, S.-C. Quantum Spin Hall Effect. Phys. Rev. Lett. 96, 106802 (2006).
Fu, L., Kane, C. L. & Mele, E. J. Topological Insulators in Three Dimensions. Phys. Rev. Lett. 98, 106803 (2007).
Fu, L. Topological Crystalline Insulators. Phys. Rev. Lett. 106, 106802 (2011).
Ando, Y. & Fu, L. Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials. Annu. Rev. Condens. Matter Phys. 6:1, 361-381 (2015).
Xie, B., Wang, H.-X., Zhang, X., Zhan, P., Jiang, J.-H., Lu, M. & Chen, Y. Higher-order band topology. Nat. Rev. Phys. 3, 520–532 (2021).
Schindler, F., Cook, A. M., Vergniory, M. G., Wang, Z., Parkin, S. S. P., Bernevig, B. A. & Neupert, T. Higher-order topological insulators. Sci. Adv. 4, eaat0346 (2018).
Trifunovic, L. & Brouwer, P. W. Higher-Order Bulk-Boundary Correspondence for Topological Crystalline Phases. Phys. Rev. X 9, 011012 (2019).
Ran, Y., Zhang, Y. & Vishwanath, A. One-dimensional topologically protected modes in topological insulators with lattice dislocations. Nature Phys 5, 298–303 (2009).
Liu, C.-X., Zhang, R.-X. & VanLeeuwen, B. K. Topological nonsymmorphic crystalline insulators Phys. Rev. B 90, 085304 (2014).
Nayak, A. K., Reiner, J., Queiroz, R., Fu, H., Shekhar, C., Yan, B., Felser, C., Avraham, N. & Beidenkopf, H. Resolving the topological classification of bismuth with topological defects. Sci. Adv. 5, eaax6996 (2019).
Kim, H. W., Kang, S.-H., Kim, H.-J., Chae, K., Cho, S., Ko, W., Jeon, S., Kang, S. H., Yang, H., Kim, S. W., Park, S., Hwang, S., Kwon, Y.-K. & Son, Y.-W. Symmetry Dictated Grain Boundary State in a Two-Dimensional Topological Insulator. Nano Lett. 20, 8, 5837–5843 (2020).
Juričić, V., Mesaros, A., Slager, R.-J. & Zaanen, J. Universal Probes of Two-Dimensional Topological Insulators: Dislocation and π Flux. Phys. Rev. Lett. 108, 106403 (2012).
Slager, R.-J., Mesaros, A., Juričić, V. & Zaanen, J. Interplay between electronic topology and crystal symmetry: Dislocation-line modes in topological band insulators. Phys. Rev. B 90, 241403(R) (2014).
Ko, W., Ma, C., Nguyen, G. D., Kolmer, M. & Li, A.-P. Atomic-Scale Manipulation and In Situ Characterization with Scanning Tunneling Microscopy. Adv. Funct. Mater., 29, 190377 (2019).
Li, W. & Li, J. Ferroelasticity and domain physics in two-dimensional transition metal dichalcogenide monolayers. Nat Commun 7, 10843 (2016).
Su, W. P. Schrieffer, J. R. & Heeger, A. J. Solitons in Polyacetylene. Phys. Rev. Lett. 42, 1698 (1979).
Li, L. Xu, Z. Xu & Chen. S. Topological phases of generalized Su-Schrieffer-Heeger models. Phys. Rev. B 89, 085111 (2014).
Young, S. M. & Kane, C. L. Dirac semimetals in two dimensions, Phys. Rev. Lett. 115, 126803 (2015).
Chiu, C. K., Teo, J. C., Schnyder, A. P., & Ryu, S. Classification of topological quantum matter with symmetries. Reviews of Modern Physics, 88(3), 035005 (2016).
Nilius, N., Wallis, T. M. & Ho, W. Development of one-dimensional band structure in artificial gold chains, Science 297, 1853 (2002),
Do, E. H. & Yeom, H. W. Electron quantization in broken atomic wires, Phys. Rev. Lett. 115, 266803 (2015).
Howard, S., Jiao, L., Wang, Z., Morali, N., Batabyal, R., Kumar-Nag, P., Avraham, N., Beidenkopf, H., Vir, P., Liu, E., Shekhar, C., Felser, C., Hughes, T. & Madhavan, V. Evidence for one-dimensional chiral edge states in a magnetic Weyl semimetal Co₃Sn₂S₂. Nat Commun 12, 4269 (2021).
Gu, M. S. Ku, J. Jang, W, Lee, C. Y. Kim, S. H & Kim, H. W. Electric field-assisted patterning of few-layer MoTe2 by scanning probe lithography, J. Kor. Phys. Soc. 82, 274 (2023).
Schindler, F., Wang, Z., Vergniory, M.G. et al. Higher-order topology in bismuth. Nature Phys 14, 918–924 (2018).
Noguchi, R., Kobayashi, M., Jiang, Z. et al. Evidence for a higher-order topological insulator in a three-dimensional material built from van der Waals stacking of bismuth-halide chains. Nat. Mater. 20, 473–479 (2021).

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Hierarchical First-order and Second-order Topological States in Symmetry-Controllable Grain Boundary

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Introduction

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Creation of symmetry-enforced grain boundaries

First-order and second-order topological states

Electric-field induced grain boundary formation

References

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Supplementary Files

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