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Real-Space Imaging of the Band Topology of Transition Metal Dichalcogenides

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Pith's one-line read The valence band of WSe2 has its Wannier center in the hollow sites between tungsten atoms, and bias-switched STM imaging shows it directly.

desk verdict Clever new STM protocol and solid theory, but the key experimental claim is undercut by the simultaneous tip-height change in the bias-switch images. read the letter →

arxiv 2412.02813 v1 pith:R6PZ3NWJ submitted 2024-12-03 cond-mat.mtrl-sci cond-mat.mes-hall

Madisen Holbrook , Julian Ingham , Daniel Kaplan , Luke Holtzman , Brenna Bierman , Nicholas Olson , Luca Nashabeh , Song Liu
show 6 more authors
Xiaoyang Zhu Daniel Rhodes Katayun Barmak James Hone Raquel Queiroz Abhay Pasupathy
This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.mes-hall
keywords WSe2scanningtunnelingmicroscopyWanniercenterobstructedatomicinsulatortransitionmetaldichalcogenidesbandtopologylocaldensityofstatesC3symmetry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that scanning tunneling microscopy can diagnose band topology from the real-space position of the electronic density within a unit cell. For monolayer WSe2, the authors register the atomic lattice using substitutional dopants and then switch the sample bias mid-frame so the same patch of surface is imaged near the valence-band maximum at K and deeper near Γ. They find the density maximum at K sits in the honeycomb hollow sites between W atoms, while at Γ it sits on the W sites. According to the symmetry-based theory of obstructed atomic insulators, that shift means the valence band's Wannier center is pinned off the atoms and the band cannot be adiabatically connected to a trivial atomic limit. The result matters because it turns a bulk topological invariant into a visible, energy-dependent contrast change in a standard STM image.

What carries the argument

The load-bearing object is the Wannier function, the real-space orbital obtained by Fourier-transforming the Bloch states of a single band, and its center, the Wannier center, which symmetry can pin to a Wyckoff position in the unit cell. The paper's mechanism is momentum-dependent interference: a Bloch state is a sum of Wannier orbitals with phase factors $e^{i\mathbf{k}\cdot\mathbf{R}}$, and at Γ the phases are equal while at K they differ by 120 degrees, so whether the local density of states peaks on or off the metal sites is decided by the C3 rotation eigenvalue of the band's orbital character. An obstructed atomic insulator is a band whose Wannier center is pinned to bond or hollow positions rather than atomic sites, so it cannot be deformed to a trivial atomic insulator. To make the comparison, the authors use substitutional S and Mo dopants to identify the Se and W sublattices, first-principles maximally localized Wannier functions and a three-band tight-binding model to compute the local density of states, and mid-frame bias-switched constant-current topography and constant-height current images to compare Γ and K contrast on the same lattice patch without lateral drift.

What would settle it

A decisive check would be to image the same WSe2 bias window at several fixed tip-sample separations and compare constant-height current maps: if the hollow-site maximum at the K window is a tunneling-matrix-element or tip-height artifact, it would move or disappear when the tip is retracted, whereas the symmetry-pinned Wannier-center picture predicts the maximum stays at the hollow site as long as the K states dominate the current.

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Extended reading notes

Core claim

The central discovery is that the maximum of the valence-band local density of states in monolayer WSe2 moves within the unit cell as the bias energy is swept from the K-point valence-band maximum to the Γ-point region. At K the density is peaked at the honeycomb centers, between the tungsten atoms; at Γ it is peaked at the W atomic sites. Because the C3 eigenvalues at the two high-symmetry points differ, with trivial character at Γ and a chiral $e^{2\pi i/3}$ character at K, the Bloch states at K interfere destructively on the W sites and constructively at the hollow sites, whereas at Γ they construct on the W sites. This incompatibility with Wannier functions exponentially localized on the atomic sites makes the valence band a topologically obstructed atomic insulator with a quantized Berry phase. The same measurement on NbSe2 shows no such shift because its Γ and K features are nearly degenerate, which the authors present as confirmation that the shift tracks the symmetry-imposed obstruction rather than a generic orbital shape.

Load-bearing premise

The conclusion rests on treating the bright features in the bias-switched STM current images as direct maps of the valence-band local density of states at the selected momenta, rather than as artifacts of the mid-frame change in bias, tip height, or energy-dependent tunneling matrix elements.

Editorial extensions

If this is right

  • Monolayer WSe2's valence band is confirmed as an obstructed atomic insulator, implying a quantized Berry phase and physical consequences such as corner states and an enhanced dielectric constant.
  • STM topographs taken near the valence-band maximum of WSe2 show hollow-site contrast, not atomic positions, so previous defect assignments built on the opposite assumption need to be revisited.
  • Because the symmetry argument is general to 2H transition metal dichalcogenides, the same bias-dependent contrast shift should appear in other semiconducting monolayers whenever the Γ–K energy separation is large enough to isolate the K-point states.
  • The method offers a real-space diagnostic of band topology that needs no defects, step edges, or sample boundaries, unlike previous STM-based topological probes.
  • In metals such as NbSe2 the energy-integrated STM signal mixes Γ and K contributions and hides the obstruction, so reading its Berry phase requires bias-dependent interference near symmetry-breaking defects.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the same mid-frame bias-switch protocol should work for any semiconductor whose high-symmetry valence or conduction states are separated in energy by more than the tunneling-decay contrast, making monolayer MoS2 and other TMDs natural next targets.
  • Beyond the paper: if the hollow-site maximum is set by the chiral $d\pm id$ orbital character rather than by WSe2-specific details, the contrast reversal should be reproducible in simulated STM images at fixed tip height, and a null result in such a simulation would point to tunneling-matrix-element artifacts.
  • Beyond the paper: the result implies that in moiré heterostructures of TMDs, atomic registries inferred from STM topographs at the band edge are offset by half a unit cell from the chalcogen lattice, which could affect the interpretation of moiré reconstructions and twist-angle alignment.
  • Beyond the paper: the same logic applied to the conduction band would map the Wannier-center flow across the full band structure, offering a real-space proxy for electric polarization and its quantum-geometric corrections.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This manuscript reports STM measurements on monolayer WSe2 that are intended to image the real-space valence-band charge density at two high-symmetry momenta. Using substitutional S and Mo dopants to mark the Se and W lattice sites, the authors first establish that bright topographic features at -1.4 V sit at the hollow sites, while at more negative bias near the Γ point the bright features sit at the W sites. They interpret this shift through symmetry-based indicators and maximally localized Wannier functions, concluding that the WSe2 valence band is a topologically obstructed atomic insulator whose Wannier center is pinned to the hollow site. The paper includes supporting DFT/Wannier calculations, two tight-binding model calculations, and a comparison with NbSe2.

Significance. If the experimental spatial shift is substantiated, this would be a novel and valuable real-space diagnostic of band topology: the paper would demonstrate that STM can directly visualize the Wannier-center obstruction in a periodic solid, complementing earlier real-space signatures that require defects or boundaries. The symmetry-based argument is on firm theoretical ground, since the C3 eigenvalue mismatch between Γ and K is a known symmetry-indicator criterion for the obstructed atomic limit, and the DFT/Wannier calculations support the interpretation. The substitutional-doping strategy to identify lattice sites is clever, and the three-band model and the s+f toy model are instructive controls, particularly the s+f model's demonstration that an off-atomic-site orbital shape alone does not imply obstruction. The main weakness is that the central experimental comparison in Fig. 3f is affected by a simultaneous tip-height change, so the experimental evidence for the load-bearing spatial shift is currently conditional rather than conclusive.

major comments (3)
  1. [Fig. 3f and SM S3, Fig. S6] The central experimental result, the shift of the current maxima from W sites to hollow sites when the bias is changed from -1.9 V to -1.4 V, is obtained in constant-height current images in which the tip-sample separation is also changed simultaneously with the bias (SM S3, Fig. S6). Because the tunneling current depends exponentially and in an orbital-dependent way on tip height, the apparent peak-position shift could in principle be produced by the height change or by energy-dependent tunneling matrix elements rather than by the K/Γ orbital character. A control measurement at fixed height, or a quantitative simulation of the two-window tunneling current at the actual experimental heights, is required before the topological conclusion can be drawn from Fig. 3f. In addition, the shift is presented through representative images without line cuts or statistics, so the reader cannot assess its robustness.
  2. [SM S2.A, Fig. S5 and Eq. (S2)] The simulated images in Fig. 3d and SM Fig. S5 are single-energy partial charge densities, not the bias-integrated local density of states or tunneling current that is actually measured. The measured current is an integral over an energy window (Eq. S2), and the -1.4 V window includes states near Γ as well as K, while the -1.9 V window includes contributions from all higher-lying states. The paper does not demonstrate that the integrated two-window images reproduce the same contrast as the partial charge densities at K and Γ; without such a simulation, the interpretation of the experimental contrast as a direct map of the K and Γ wavefunctions remains an assumption. This point is especially relevant because SM Fig. S5 shows that the spatial shape of the partial charge density near Γ changes with height, and the current simulation does not account for this effect.
  3. [Fig. 3b and main text] The assignment of the spectroscopic features at -1.24 eV and -1.89 eV to the K and Γ points relies on alignment with the DFT band structure and on tunneling-decay-constant arguments, not on a k-resolved measurement. The claim that the -1.4 V window is dominated by K-point states is therefore not directly verified; a momentum-resolved experiment or a quantitative tunneling simulation establishing the relative weight of K and Γ states in the two windows is needed. At minimum, this dominance should be stated as an assumption rather than an established fact, because the entire experimental inference depends on it.
minor comments (5)
  1. [Abstract] The word 'unambiguously' in the abstract is too strong given the experimental caveats discussed above; consider softening the claim to reflect that the Wannier-center location is inferred from STM data supported by symmetry analysis and DFT.
  2. [Methods, TMD Synthesis] The sentence 'following the method of Ref. W and Mo powder of 99.999% ...' appears to have a missing citation number after 'Ref.'; please correct this typographical error.
  3. [Fig. 4 and main text] The text states 'Figure 4b shows an STM image of the doped NbSe2', but the caption and panel layout indicate that the STM image is panel (c); please correct the cross-reference.
  4. [SM Eq. (S2)] The notation in Eq. (S2) is nonstandard: the retarded Green's function and the expectation value should be defined explicitly, since the relationship between the Green's function and the LDOS is central to the argument.
  5. [Fig. 3e,f caption] The atomic-lattice overlay in Fig. 3e,f is said to be determined from nearby defects, but the caption does not state how the alignment uncertainty is estimated; a brief statement of the method and its accuracy would help the reader trust the peak-position comparison.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the topological classification rests on external symmetry-indicator theory and DFT, and the STM images are an independent, internally calibrated check rather than a fitted input.

full rationale

The derivation chain is self-contained against external benchmarks. Atomic-site calibration uses substitutional S and Mo dopants whose lattice positions are fixed by synthesis and known defect chemistry, not by the topological conclusion. The WSe2 obstructed-atomic-insulator classification is imported from the externally established band-representation framework (Refs. [3,4]) and independently reproduced here by PBEsol DFT plus Wannier90 and by the published three-band model of Ref. [23]; no parameter of those calculations is fitted to the STM peak positions. The bias-switched STM images are compared qualitatively with computed partial charge density and are not used to define the C3 eigenvalues that determine the Wannier center. The self-cited reference among the OAI citations (Ref. [6]) is non-load-bearing because the same claim is supported by Refs. [3,4,7,8], and the other self-citations (e.g., Refs. [11], [32], [45], [54]) concern background, consequences, or methodology. The s+f counter-model in SM S1.B explicitly prevents the fallacy of equating off-site density with obstruction. The 0.5 nm tip-height change in the mid-frame bias switch (SM S3, Fig. S6) is a possible experimental artifact and a correctness risk, but it does not make the derivation circular: the topological classification is not an input to the imaging protocol, and no predicted quantity is defined as the measured quantity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to the STM data; the DFT lattice constants and tight-binding hoppings are taken from prior literature. The OAI concept and Wannier-center formalism are existing theoretical constructs, not new entities. The central inference rests on the axioms above.

assumptions (4)
  • standard math Bloch states can be expanded in a Wannier-function basis whose centers define the atomic limit.
    Used throughout the main text to relate LDOS maxima to Wannier centers; standard band theory.
  • domain assumption The STM tunneling signal is proportional to the energy-resolved local density of states in the Tersoff-Hamann sense.
    Invoked in Supp. Eq. S1-S3 to convert measured dI/dV or current into LDOS; standard but approximate.
  • domain assumption The valence-band C3 eigenvalues at Gamma and K, as obtained from DFT and the three-band model, are correct and sufficient to locate the Wannier center.
    The symmetry-based indicator argument in the main text relies on this; it is externally established but not independently measured here.
  • ad hoc to paper At the chosen bias voltages, tunneling is dominated by states near K (-1.4 V) or near Gamma (-1.9 V), so the integrated LDOS retains the corresponding spatial character.
    This is load-bearing for the mid-frame bias-switch experiment and is argued from tunneling decay constants and dI/dV peaks, not directly measured per k-point.

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Pith. "Pith review of Real-Space Imaging of the Band Topology of Transition Metal Dichalcogenides." pith.science (2026). https://pith.science/paper/R6PZ3NWJ

@misc{pith2026241202813,
  author       = {Pith},
  title        = {Pith review of: Real-Space Imaging of the Band Topology of Transition Metal Dichalcogenides},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R6PZ3NWJ}},
  note         = {Machine review of arXiv:2412.02813}
}
abstract

The topological properties of Bloch bands are intimately tied to the structure of their electronic wavefunctions within the unit cell of a crystal. Here, we show that scanning tunneling microscopy (STM) measurements on the prototypical transition metal dichalcogenide (TMD) semiconductor WSe$_2$ can be used to unambiguously fix the location of the Wannier center of the valence band. Using site-specific substitutional doping, we first determine the position of the atomic sites within STM images, establishing that the maximum electronic density of states at the $K$-point lies between the atoms. In contrast, the maximum density of states at the $\Gamma$ point is at the atomic sites. This signifies that WSe$_2$ is a topologically obstructed atomic insulator, which cannot be adiabatically transformed to the trivial atomic insulator limit.

Figures

Figures reproduced from arXiv: 2412.02813 by the authors.

Figure 1
Figure 1. c, we first replace the Se atoms with isovalent substitutional S dopants to pinpoint the Se lattice sites. Fig. 1d shows a valence band STM image of the S doped WSe2, revealing a high density of defects. These defects are centered between the bright features in the topograph (see Fig. 1e zoom-in). Thus, contrary to the commonly held belief, the “atoms” seen in STM images of WSe2 do not correspond to the chalcogen si… view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Forward citations

Cited by 3 Pith papers

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  3. Quantum Geometry in the NbSe$_2$ Family I: Obstructed Compact Wannier Function and New Perturbation Theory

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    Monolayer NbSe2's Fermi-level flat band is an obstructed atomic band whose Wannier function is 94% reproduced by a compact three-site orbital, and its minimal model has next-nearest-neighbor hopping larger than neares...

Reference graph

Works this paper leans on

55 extracted references · 51 canonical work pages · cited by 3 Pith papers

  1. [1]

    How chemistry and physics meet in the solid state,

    R. Hoffmann, “How chemistry and physics meet in the solid state,” Angewandte Chemie 26, 846 (1987)

    work page 1987

  2. [2]

    Formalisms of band theory,

    E. Blount, “Formalisms of band theory,” (Academic Press, 7

    work page

  3. [3]

    Topological quantum chemistry,

    B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, “Topological quantum chemistry,” Nature 547, 298 (2017)

    work page 2017

  4. [4]

    Building blocks of topological quantum chemistry: Elementary band representations,

    J. Cano, B. Bradlyn, Z. Wang, L. Elcoro, M. G. Vergniory, C. Felser, M. I. Aroyo, and B. A. Bernevig, “Building blocks of topological quantum chemistry: Elementary band representations,” Phys. Rev. B 97, 035139 (2018)

    work page 2018

  5. [5]

    Electric polariza- tion as a bulk quantity and its relation to surface charge,

    D. Vanderbilt and R. D. King-Smith, “Electric polariza- tion as a bulk quantity and its relation to surface charge,” Phys. Rev. B 48, 4442 (1993)

    work page 1993

  6. [6]

    Boundary-obstructed topological phases,

    E. Khalaf, W. A. Benalcazar, T. L. Hughes, and R. Queiroz, “Boundary-obstructed topological phases,” Phys. Rev. Res. 3, 013239 (2021)

    work page 2021

  7. [7]

    Band representations and topo- logical quantum chemistry,

    J. Cano and B. Bradlyn, “Band representations and topo- logical quantum chemistry,” Annual Review of Condensed Matter Physics 12, 225 (2021)

    work page 2021

  8. [8]

    Filling-enforced obstructed atomic insulators,

    Y. Xu, L. Elcoro, Z.-D. Song, M. G. Vergniory, C. Felser, S. S. P. Parkin, N. Regnault, J. L. Ma˜ nes, and B. A. Bernevig, “Filling-enforced obstructed atomic insulators,” Phys. Rev. B 109, 165139 (2024)

    work page 2024

Show all 55 references
  1. [9]

    Design and realization of topological Dirac fermions on a triangular lattice,

    M. Bauernfeind, J. Erhardt, P. Eck, P. K. Thakur, J. Gabel, T.-L. Lee, J. Sch¨ afer, S. Moser, D. Di Sante, R. Claessen, et al., “Design and realization of topological Dirac fermions on a triangular lattice,” Nature communi- cations 12, 5396 (2021)

    2021

  2. [10]

    Ghost states and surface structures of the charge density wave kagome metal ScV 6Sn6,

    J. W. Villanova, S. Hus, S.-H. Kang, H. Jeon, A.-P. Li, D. Mandrus, Z. Gai, and M. Yoon, “Ghost states and surface structures of the charge density wave kagome metal ScV 6Sn6,” Applied Surface Science 665, 160190 (2024)

    2024

  3. [11]

    Pomeranchuk instability induced by an emer- gent higher-order van hove singularity on the distorted kagome surface of co 3 sn 2 s 2,

    P. K. Nag, R. Batabyal, J. Ingham, N. Morali, H. Tan, J. Koo, A. Consiglio, E. Liu, N. Avraham, R. Queiroz, et al., “Pomeranchuk instability induced by an emer- gent higher-order van hove singularity on the distorted kagome surface of co 3 sn 2 s 2,” arXiv preprint arXiv:2410....

    2024 arXiv

  4. [12]

    Two-step flux synthesis of ultrapure transition-metal dichalcogenides,

    S. Liu, Y. Liu, L. Holtzman, B. Li, M. Holbrook, J. Pack, T. Taniguchi, K. Watanabe, C. R. Dean, A. N. Pasupathy, K. Barmak, D. A. Rhodes, and J. Hone, “Two-step flux synthesis of ultrapure transition-metal dichalcogenides,” ACS Nano 17, 16587 (2023)

    2023

  5. [13]

    One-dimensional electrical contact to a two-dimensional material,

    L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, “One-dimensional electrical contact to a two-dimensional material,” Science 342, 614 (2013)

    2013

  6. [14]

    Imaging of MoS 2 by scanning tunneling microscopy,

    G. W. Stupian and M. S. Leung, “Imaging of MoS 2 by scanning tunneling microscopy,” Applied Physics Letters 51, 1560 (1987)

    1987

  7. [15]

    Tunneling microscopy of 2H-MoS 2: A compound semi- conductor surface,

    M. Weimer, J. Kramar, C. Bai, and J. D. Baldeschwieler, “Tunneling microscopy of 2H-MoS 2: A compound semi- conductor surface,” Phys. Rev. B 37, 4292 (1988)

    1988

  8. [16]

    On the correlation between the scanning tunneling microscopy image imperfections and point defects of layered chalcogenides 2 H-M X2 (M = Mo, W; X = S, Se),

    M.-H. Whangbo, J. Ren, S. Magonov, H. Bengel, B. Parkinson, and A. Suna, “On the correlation between the scanning tunneling microscopy image imperfections and point defects of layered chalcogenides 2 H-M X2 (M = Mo, W; X = S, Se),” Surface Science 326, 311 (1995)

    1995

  9. [17]

    Atom-selective imaging of the GaAs(110) surface,

    R. M. Feenstra, J. A. Stroscio, J. Tersoff, and A. P. Fein, “Atom-selective imaging of the GaAs(110) surface,” Phys. Rev. Lett. 58, 1192 (1987)

    1987

  10. [18]

    Interpretation of STM images: the MoS 2 surface,

    A. Altibelli, C. Joachim, and P. Sautet, “Interpretation of STM images: the MoS 2 surface,” Surface Science 367, 209 (1996)

    1996

  11. [19]

    Scanning tunneling mi- croscopy image of transition-metal-dichalcogenide sur- faces,

    K. Kobayashi and J. Yamauchi, “Scanning tunneling mi- croscopy image of transition-metal-dichalcogenide sur- faces,” Surface Science 357-358, 317 (1996)

    1996

  12. [20]

    Identifying substitutional oxygen as a pro- lific point defect in monolayer transition metal dichalco- genides,

    S. Barja, S. Refaely-Abramson, B. Schuler, D. Y. Qiu, A. Pulkin, S. Wickenburg, H. Ryu, M. M. Ugeda, C. Kastl, C. Chen, et al., “Identifying substitutional oxygen as a pro- lific point defect in monolayer transition metal dichalco- genides,” Nature Communications 10, 3382 (2019)

    2019

  13. [21]

    Probing critical point energies of transition metal dichalcogenides: Sur- prising indirect gap of single layer WSe 2,

    C. Zhang, Y. Chen, A. Johnson, M.-Y. Li, L.-J. Li, P. C. Mende, R. M. Feenstra, and C.-K. Shih, “Probing critical point energies of transition metal dichalcogenides: Sur- prising indirect gap of single layer WSe 2,” Nano Letters 15, 6494 (2015)

    2015

  14. [22]

    Stroscio and W

    J. Stroscio and W. Kaiser, Scanning Tunneling Mi- croscopy, Methods of Experimental Physics (Academic Press, 1993)

    1993

  15. [23]

    Three-band tight-binding model for monolayers of group- vib transition metal dichalcogenides,

    G.-B. Liu, W.-Y. Shan, Y. Yao, W. Yao, and D. Xiao, “Three-band tight-binding model for monolayers of group- vib transition metal dichalcogenides,” Phys. Rev. B 88, 085433 (2013)

    2013

  16. [24]

    Multiorbital model reveals a second-order topological in- sulator in 1 H transition metal dichalcogenides,

    J. Zeng, H. Liu, H. Jiang, Q.-F. Sun, and X. C. Xie, “Multiorbital model reveals a second-order topological in- sulator in 1 H transition metal dichalcogenides,” Phys. Rev. B 104, L161108 (2021)

    2021

  17. [25]

    Hidden breathing kagome topol- ogy in hexagonal transition metal dichalcogenides,

    J. Jung and Y.-H. Kim, “Hidden breathing kagome topol- ogy in hexagonal transition metal dichalcogenides,” Phys. Rev. B 105, 085138 (2022)

    2022

  18. [26]

    Ring states in topological materials,

    R. Queiroz, R. Ilan, Z. Song, B. A. Bernevig, and A. Stern, “Ring states in topological materials,” arXiv (2024), 2406.03529

    2024 arXiv

  19. [27]

    Topological defects and gapless modes in insulators and superconductors,

    J. C. Teo and C. L. Kane, “Topological defects and gapless modes in insulators and superconductors,” Phys. Rev. B 82, 115120 (2010)

    2010

  20. [28]

    One-dimensional topologically protected modes in topological insulators with lattice dislocations,

    Y. Ran, Y. Zhang, and A. Vishwanath, “One-dimensional topologically protected modes in topological insulators with lattice dislocations,” Nature Physics 5, 298 (2009)

    2009

  21. [29]

    Impurity-bound states and Green’s function zeros as local signatures of topology,

    R.-J. Slager, L. Rademaker, J. Zaanen, and L. Balents, “Impurity-bound states and Green’s function zeros as local signatures of topology,” Phys. Rev. B 92, 085126 (2015)

    2015

  22. [30]

    Universal probes of two-dimensional topological insulators: dislocation and π flux,

    V. Juriˇ ci´ c, A. Mesaros, R.-J. Slager, and J. Zaanen, “Universal probes of two-dimensional topological insulators: dislocation and π flux,” Phys. Rev. Lett. 108, 106403 (2012)

    2012

  23. [31]

    Measuring the Berry phase of graphene from wavefront dislocations in Friedel oscillations,

    C. Dutreix, H. Gonz´ alez-Herrero, I. Brihuega, M. Kat- snelson, C. Chapelier, and V. Renard, “Measuring the Berry phase of graphene from wavefront dislocations in Friedel oscillations,” Nature 574, 219 (2019)

    2019

  24. [32]

    The quantum geometric origin of capacitance in insulators,

    I. Komissarov, T. Holder, and R. Queiroz, “The quantum geometric origin of capacitance in insulators,” Nature Communications 15, 4621 (2024)

    2024

  25. [33]

    Engineering point- defect states in monolayer WSe 2,

    C. Zhang, C. Wang, F. Yang, J.-K. Huang, L.-J. Li, W. Yao, W. Ji, and C.-K. Shih, “Engineering point- defect states in monolayer WSe 2,” ACS Nano 13, 1595 (2019)

    2019

  26. [34]

    Realizing large-scale, electronic-grade two- dimensional semiconductors,

    Y.-C. Lin, B. Jariwala, B. M. Bersch, K. Xu, Y. Nie, B. Wang, S. M. Eichfeld, X. Zhang, T. H. Choudhury, Y. Pan, et al., “Realizing large-scale, electronic-grade two- dimensional semiconductors,” ACS Nano 12, 965 (2018)

    2018

  27. [35]

    Visualization of defect in- duced in-gap states in monolayer MoS2,

    D. J. Trainer, J. Nieminen, F. Bobba, B. Wang, X. Xi, A. Bansil, and M. Iavarone, “Visualization of defect in- duced in-gap states in monolayer MoS2,” npj 2D Materials and Applications 6, 13 (2022)

    2022

  28. [36]

    Single-bond formation and charac- terization with a scanning tunneling microscope,

    H. J. Lee and W. Ho, “Single-bond formation and charac- terization with a scanning tunneling microscope,” Science 8 286, 1719 (1999)

    1999

  29. [37]

    High-resolution molecu- lar orbital imaging using a p-wave STM tip,

    L. Gross, N. Moll, F. Mohn, A. Curioni, G. Meyer, F. Hanke, and M. Persson, “High-resolution molecu- lar orbital imaging using a p-wave STM tip,” Phys. Rev. Lett. 107, 086101 (2011)

    2011

  30. [38]

    Site dependence of the apparent shape of a molecule in scanning tunneling mico- scope images: Benzene on Pt(111),

    P. S. Weiss and D. M. Eigler, “Site dependence of the apparent shape of a molecule in scanning tunneling mico- scope images: Benzene on Pt(111),” Phys. Rev. Lett. 71, 3139 (1993)

    1993

  31. [39]

    Quantum simu- lator to emulate lower-dimensional molecular structure,

    E. Sierda, X. Huang, D. I. Badrtdinov, B. Kiraly, E. J. Knol, G. C. Groenenboom, M. I. Katsnelson, M. R¨ osner, D. Wegner, and A. A. Khajetoorians, “Quantum simu- lator to emulate lower-dimensional molecular structure,” Science 380, 1048 (2023)

    2023

  32. [40]

    p orbital flat band and Dirac cone in the electronic honeycomb lattice,

    T. S. Gardenier, J. J. Van Den Broeke, J. R. Moes, I. Swart, C. Delerue, M. R. Slot, C. M. Smith, and D. Vanmaekelbergh, “p orbital flat band and Dirac cone in the electronic honeycomb lattice,” ACS Nano 14, 13638 (2020)

    2020

  33. [41]

    Quan- tum states and atomic structure of silicon surfaces,

    R. M. Tromp, R. J. Hamers, and J. E. Demuth, “Quan- tum states and atomic structure of silicon surfaces,” Sci- ence 234, 304 (1986)

    1986

  34. [42]

    Direct observation of layer skyrmions in twisted WSe2 bilayers,

    F. Zhang, N. Morales-Dur´ an, Y. Li, W. Yao, J.-J. Su, Y.-C. Lin, C. Dong, H. Kim, J. A. Robinson, A. H. Macdonald, et al., “Direct observation of layer skyrmions in twisted WSe2 bilayers,” arXiv preprint arXiv:2406.20036 (2024)

    2024 arXiv

  35. [43]

    Visualizing the microscopic origins of topol- ogy in twisted molybdenum ditelluride,

    E. Thompson, K. T. Chu, F. Mesple, X.-W. Zhang, C. Hu, Y. Zhao, H. Park, J. Cai, E. Anderson, K. Watan- abe, et al., “Visualizing the microscopic origins of topol- ogy in twisted molybdenum ditelluride,” arXiv preprint arXiv:2405.19308 (2024)

    2024 arXiv

  36. [44]

    Real-space obstruction in quantum spin hall insulators,

    P. Eck, C. Ortix, A. Consiglio, J. Erhardt, M. Bauernfeind, S. Moser, R. Claessen, D. Di Sante, and G. Sangiovanni, “Real-space obstruction in quantum spin hall insulators,” Physical Review B 106, 195143 (2022)

    2022

  37. [45]

    Resolving the topological classification of bismuth with topological defects,

    A. K. Nayak, J. Reiner, R. Queiroz, H. Fu, C. Shekhar, B. Yan, C. Felser, N. Avraham, and H. Beidenkopf, “Resolving the topological classification of bismuth with topological defects,” Science Advances 5, eaax6996 (2019)

    2019

  38. [46]

    Observation of topologically protected states at crystalline phase boundaries in single-layer WSe2,

    M. M. Ugeda, A. Pulkin, S. Tang, H. Ryu, Q. Wu, Y. Zhang, D. Wong, Z. Pedramrazi, A. Mart ´ ın-Recio, Y. Chen, et al., “Observation of topologically protected states at crystalline phase boundaries in single-layer WSe2,” Nature Communications 9, 3401 (2018)

    2018

  39. [47]

    Probing topological quantum matter with scanning tunnelling microscopy,

    J.-X. Yin, S. H. Pan, and M. Zahid Hasan, “Probing topological quantum matter with scanning tunnelling microscopy,” Nature Reviews Physics 3, 249 (2021)

    2021

  40. [48]

    Resolving exotic quantum states using scanning tunneling microscopy,

    S. Jeon and M. Oh, “Resolving exotic quantum states using scanning tunneling microscopy,” Current Applied Physics 68, 58 (2024)

    2024

  41. [49]

    Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,

    G. Kresse and J. Furthm¨ uller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Phys. Rev. B 54, 11169 (1996)

    1996

  42. [50]

    Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set,

    G. Kresse and J. Furthm¨ uller, “Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set,” Computational materials science 6, 15 (1996)

    1996

  43. [51]

    Restoring the density-gradient expansion for exchange in solids and surfaces,

    J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, “Restoring the density-gradient expansion for exchange in solids and surfaces,” Physical Review Letters 100, 136406 (2008)

    2008

  44. [52]

    An updated version of wannier90: A tool for obtaining maximally-localised Wannier functions,

    A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, “An updated version of wannier90: A tool for obtaining maximally-localised Wannier functions,” Computer Physics Communications 185, 2309 (2014)

    2014

  45. [53]

    Quasi-particle interference of the van hove singularity in Sr 2RuO4,

    A. Kreisel, C. Marques, L. C. Rhodes, X. Kong, T. Berlijn, R. Fittipaldi, V. Granata, A. Vecchione, P. Wahl, and P. J. Hirschfeld, “Quasi-particle interference of the van hove singularity in Sr 2RuO4,” npj Quantum Materials 6, 100 (2021)

    2021

  46. [54]

    Machine learning the microscopic form of nematic order in twisted double-bilayer graphene,

    J. A. Sobral, S. Obernauer, S. Turkel, A. N. Pasupathy, and M. S. Scheurer, “Machine learning the microscopic form of nematic order in twisted double-bilayer graphene,” Nature Communications 14, 5012 (2023)

    2023

  47. [55]

    Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides,

    K. F. Mak and J. Shan, “Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides,” Nature Photonics 10, 216 (2016). 9 Supplementary Material S1. MODEL CALCULA TIONS OF THE LOCAL DENSITY OF ST A TES (LDOS) The differential tunneling current imaged in S...

    2016

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.