Probing the Electronic Structure at the Boundary of Topological Insulators in the $\mathrm{Bi}_2\mathrm{Se}_3$ Family by Combined STM and AFM

Adrian Weindl; Andrea Donarini; Christoph S. Setescak; Franz J. Giessibl; Irene Aguilera; Matthias Kronseder

REVIEW 3 major objections 5 minor 78 references

Probing the Electronic Structure at the Boundary of Topological Insulators in the $\mathrm{Bi}_2\mathrm{Se}_3$ Family by Combined STM and AFM

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A GW-based tight-binding scheme quantitatively reproduces measured STM tunneling spectra of three topological insulators, showing that bare DFT cannot describe this material family.

desk verdict Solid methods paper with a real GW-vs-DFT result buried under an overclaimed 'excellent agreement' — worth refereeing, needs toning down. read the letter →

arxiv 2411.13529 v1 pith:KKLOOD2K submitted 2024-11-20 cond-mat.mes-hall

Christoph S. Setescak , Irene Aguilera , Adrian Weindl , Matthias Kronseder , Andrea Donarini , Franz J. Giessibl This is my paper · ORCID

classification cond-mat.mes-hall PACS 71.15.Mb71.15.Qe73.20.-r68.37.Ef

keywords topologicalinsulatorscanningtunnelingspectroscopydifferentialconductanceGWapproximationWannierfunctionsChen'sderivativeruleBi2Se3Bi2Te3

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 establishes that scanning tunneling spectroscopy on the topological insulators Bi2Se3, Bi2Te2Se, and Bi2Te3 can be predicted quantitatively from first-principles many-body calculations. The authors build half-space tight-binding Hamiltonians from GW-corrected, Wannier-interpolated band structures and use Chen's derivative rule to compute the tunneling current and differential conductance for both metal and CO-terminated tips. The calculated spectra closely track 4.4 K STM/AFM experiments across the Dirac point, bulk band edges, and Van Hove singularities. Direct comparison shows that plain density functional theory shifts the spectral peaks and underestimates the trivial band gaps, whereas the GW Hamiltonians reproduce the measured positions and amplitudes. If correct, the scheme turns STS into a quantitative probe of many-body quasiparticle band structures.

What carries the argument

The key machinery is a four-step pipeline: maximally localized Wannier functions and Wannier-interpolated tight-binding Hamiltonians built from first-principles DFT+GW calculations; half-space Hamiltonians obtained by Dirichlet truncation of the bulk hopping matrices between quintuple layers; propagation of the resulting Bloch states through a laterally averaged vacuum potential with a work-function step; and Chen's derivative rule, which turns the tunneling matrix element into differential operators acting on the sample wavefunction at the tip apex. For CO-terminated tips the orbital mix includes s and p orbitals, with the CO tilt included through a mechanistic probe-particle model. The pipeline converts a quasiparticle band structure into the experimental dI/dV signal with no free parameters other than a global Fermi-level shift and an overall normalization.

What would settle it

Measure dI/dV across a sample region where the local work function or band bending is known to vary, for example near a step edge or on a sample intentionally doped to move the Fermi level far from the Dirac point. The fixed vacuum-barrier model uses a single work function and a single global Fermi-level shift; if the conduction-band onset shifts relative to the bulk-gap features by more than the fit tolerance as the local band bending changes, the laterally averaged step-potential approximation is falsified. Alternatively, a self-consistent surface calculation that moves the Dirac point or conduction band edge relative to the bulk valence band by more than the fit tolerance would break the scheme's central agreement.

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Extended reading notes

Core claim

The central claim is that the measured dI/dV spectra of the Bi2Se3 family are quantitatively reproduced by a tight-binding Hamiltonian whose parameters are taken from GW many-body calculations, evaluated with Chen's derivative rule for the tip-sample tunneling matrix element. The paper shows that a DFT-based tight-binding model systematically underestimates the trivial band gaps at the Brillouin-zone boundary and places the spectral peaks at the wrong energies, while the GW-based model places them at the measured positions and captures the linear conductance increase inside the bulk gap, the hybridization of the topological surface state with bulk bands, and the suppression of tunneling matrix elements near the conduction-band edge. A specific finding is that the linear increase of dI/dV below the Dirac point in Bi2Se3 comes from tunneling into bulk valence bands hybridized with the surface state rather than from an isolated lower Dirac cone, modifying the interpretation of previous step-edge and gap-opening studies.

Load-bearing premise

The load-bearing assumption is that a hard truncation of the bulk GW tight-binding Hamiltonian plus a laterally averaged vacuum barrier with a literature value of the work function is an accurate model of the actual surface, so that the only free adjustments—an overall bias shift and a normalization—are enough to match every measured spectrum.

Editorial extensions

If this is right

STS spectra computed from GW Hamiltonians can be used to read off the energy position of the Dirac point and the bulk band edges in all three compounds, even where the Dirac point is buried in the valence bands.
The lower linear segment in the Bi2Se3 spectrum is attributed to bulk valence bands hybridized with the surface state, not an isolated lower Dirac cone, which changes how to interpret future gap-opening experiments.
Calculated dI/dV maps distinguish the orbital character of Bloch bands at the atomic scale, making the band inversion at the Gamma point visible without ARPES.
The method works over wide bias ranges (up to plus or minus 2 V on Bi2Te3), where Van Hove singularities from flat bands at the Brillouin-zone boundary produce characteristic peaks that can be used to quantify trivial band gaps.
Because the tight-binding model is computationally cheap, the Brillouin-zone integration can be done on dense 80-by-80 or 90-by-90 grids, eliminating the numerical artifacts that limited earlier first-principles comparisons.

Reading between the lines

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

Editorial inference: if the scheme holds, the same pipeline could be applied to other layered or weakly bonded materials where GW corrections are large, turning STS into a routine test of many-body band structures.
Editorial inference: the hard-Dirichlet boundary condition implies that surface-specific reconstructions or band bending at the top quintuple layer would show up as systematic deviations; comparing spectra on different surface terminations or under an externally tuned two-dimensional electron gas would test this.
Editorial inference: the energy-dependent tip-orbital mix was held constant, so making it energy-dependent could improve the quantitative agreement of the dI/dV maps and would probe how much of the residual deviation comes from the tip model.
Editorial inference: adding defects to the tight-binding model, as suggested in the paper, could predict impurity resonances at the atomic scale and directly connect the observed dI/dV maps to specific native point defects.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a solid methods paper whose headline overclaims. The scheme — GW-derived Wannier tight-binding Hamiltonians plus Chen's derivative rule for metal and CO tips — is a genuine step forward for interpreting dI/dV on the Bi2Se3 family, and the large-bias data on Bi2Te3 is genuinely new. The reinterpretation of the gap-region linear dI/dV as partly due to bulk valence bands rather than a clean lower Dirac cone is also a nice correction to prior work. What the paper does well: the pipeline from first-principles GW to a half-space TB Hamiltonian to real-space tunneling matrix elements is described clearly, and the orbital-resolved maps plus CO-tip tilting via the probe-particle model show real craftsmanship. The DFT-vs-GW comparison is the strongest part: the peak-separation inequality (Eq. 21) is far less sensitive to the fitted Fermi-level shifts and normalization than the absolute spectra, and it does support the claim that GW, not bare DFT, is needed for this family. Now the soft spots. The phrase 'excellent agreement' is doing more work than the evidence supports. Each experimental spectrum is shifted by an adjustable ΔEF and normalized; the surface model is a hard Dirichlet truncation with a laterally averaged potential step and an ad hoc boundary position; and the text itself concedes discrepancies at ~0.5 eV in Fig. 3, above 0.2 eV in Fig. 5, and in the third peak of Fig. 8. These are not fatal — the authors are honest about the surface approximation (observation xii) and about defect issues on Bi2Te2Se — but they mean the 'accurate reproduction' claim should be downgraded to 'good qualitative and semi-quantitative agreement, with the main features and the GW-vs-DFT ordering robust.' The lack of public code or data is a minor irritation; the Hamiltonians come from published Ref. [20] and the method is described well enough to reproduce in principle. My overall take: the contribution is real and publishable, but the conclusion needs recalibration. The physics result — many-body corrections are required to explain these STS spectra — is probably right. The surface model is the main uncertainty, and a referee should push for a sensitivity check on the truncation and boundary position. Recommendation: send it to peer review. It deserves serious refereeing, and with revisions toning down the absolute claim and quantifying the discrepancies, it should be accepted. I would cite the methods part in work on STS modeling. For a reading group, it's worth a look if you work on topological insulators or scanning probe theory, but it's not a must-read.

Referee Report

3 major / 5 minor

Summary. The paper presents a numerical scheme for computing STM tunneling current and differential conductance dI/dV for the topological insulators Bi2Se3, Bi2Te2Se, and Bi2Te3, using Wannier-interpolated tight-binding Hamiltonians derived from DFT+GW calculations. The method combines half-space band structures obtained by Dirichlet truncation of the bulk TB Hamiltonian, a simple step-like vacuum barrier, and Chen's derivative rule for s-wave metal tips and s/p-wave CO-terminated tips. The authors compare calculated and experimentally measured dI/dV spectra over bias ranges from near the band gap to roughly 2 eV, identify signatures of topological boundary modes, bulk-band hybridizations, Van Hove singularities, and orbital character, and argue that GW-corrected Hamiltonians reproduce the data substantially better than bare DFT Hamiltonians.

Significance. If fully established, this would be a valuable quantitative bridge between GW quasiparticle band structures and atomic-scale STM/STS measurements, with the practical advantage that the tight-binding scheme permits dense Brillouin-zone sampling that is computationally infeasible in direct first-principles surface calculations. The paper has several genuine strengths: the GW-TB Hamiltonians are independent first-principles inputs from prior work, the k-grid integration is far denser than in previous first-principles comparisons, the CO-tip treatment includes orbital mixing and tip tilt through the probe-particle model, and the GW-versus-DFT discrimination in Section V is based on a relative peak-separation criterion that is less sensitive to the fitted shifts. The inclusion of runnable-script cues in figure captions also signals a welcome degree of reproducibility. The main weakness is that the headline claims of 'excellent agreement' and 'accurate reproduction' are stronger than the evidence, given acknowledged discrepancies and several per-spectrum adjustable parameters.

major comments (3)

[Section III B, Eq. (8), observation (xii)] The half-space description is a hard Dirichlet truncation of the bulk GW-TB Hamiltonian combined with an abrupt, laterally averaged vacuum barrier with literature work functions and an ad hoc boundary position z_b = 1.3 Å beyond the outermost atomic cores. This neglects surface relaxation, band bending, and boundary-localized orbital shifts, as the authors concede in observation (xii). Because Eq. (11) makes the vacuum decay constant kappa(n,k_parallel,Delta) directly dependent on Phi and z_b, any error in the surface potential can be partially absorbed by the per-spectrum Fermi-level shift and the overall normalization. Since the absolute 'excellent agreement' claim rests on the shape of dI/dV over eV-scale ranges, I request a sensitivity check: for example, vary z_b and Phi within reasonable bounds and show that the calculated peak positions and relative intensities move by less than the claimed agreement. Without such a check, the quantitative claim is not fully established; the robust statement is the relative-gap comparison of Section V.
[Section IV A, Section IV B, Section V; Figs. 3, 5, 8] The abstract and observation (iv) claim 'excellent agreement' and 'accurately reproduced', but the manuscript itself reports discrepancies at load-bearing points. In Fig. 3(a) the calculated dI/dV is significantly lower than experiment for E = 0.5 ± 0.1 eV. In Fig. 5(a) there are deviations above E ≈ 0.2 eV that are not reproduced, with a surface 2DEG suggested as a possible cause. In Fig. 8(a) the third GW peak lies at 2.25 eV versus 2.5 eV in experiment, a 0.25 eV shift. These are acknowledged in the text and are not incidental. No error bars or uncertainty estimates are given for the spectra or for the extracted peak positions. The headline claim should be downgraded to a qualitative or relative statement, or the agreement should be quantified with a stated error metric per energy range.
[Section IV A, Fig. 3(b)] The quantitative comparison is not parameter-free. Each experimental spectrum is shifted by a fitted per-spectrum Delta_EF, and the theoretical and experimental dI/dV are normalized through the integrated current traces as described in Section IV A; the Brillouin-zone broadening or smoothing and the boundary position z_b are also adjustable. These degrees of freedom can compensate for systematic errors in the barrier model and in the absolute energy alignment. The GW-versus-DFT inference in Eq. (21) is less sensitive to these choices because it uses relative peak separations, but the paper should state this distinction explicitly and report the sensitivity of the spectra to each fitted parameter, for example by tabulating Delta_EF values across spectra and locations on the same sample.

minor comments (5)

[Section V, Fig. 8] The text describing Fig. 8(a) says the DFT calculation is the green line, while the figure caption says the DFT-Hamiltonian spectrum is red; these color assignments need to be reconciled.
[Figure captions, Figs. 4, 5, 6] Several figure captions contain 'Run the script ...' commands (for example, 'Run the script Bi2Te3_WF_Eval_25Feb2023.m to get this figure'); such reproducibility instructions belong in a code-availability section or supplement rather than in the printed figure captions.
[Section IV B] In the paragraph discussing deviations above E = 0.2 eV, the text refers to 'Figure IV B (a)', which should be 'Figure 5(a)'.
[Section III C] The text uses 'Chens derivative rule' without the possessive apostrophe; it should read 'Chen's derivative rule'.
[Section VI A] The sentence 'as discussed in Section IV C and Section 8' contains a broken cross-reference: there is no Section 8, and the intended reference is likely Section IV C, where the corresponding peaks are discussed.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central GW-vs-DFT comparison is independent of the fitted per-spectrum shifts, and the cited prior GW Hamiltonians are parameter-free first-principles inputs, not fits to the STM data.

full rationale

The derivation chain is self-contained at the level that matters. The dI/dV calculation starts from Wannier-interpolated GW and DFT tight-binding Hamiltonians (Eq. 3) obtained in prior work [20] with overlapping authorship (I. Aguilera). This is a self-citation, but [20] is a published, parameter-free first-principles calculation; it is not fitted to the STM spectra reported here, and the current paper independently tests it against new experimental dI/dV data. The only per-experiment free parameters are the Fermi-level shift ΔEF (Sec. II, observation (iii); Sec. IV A) and an overall amplitude normalization via the integrated tunneling current (Sec. IV A). These affect global energy alignment and scale, but the decisive quantitative test — Eq. (21), ΔE_K^GW < ΔE_exp < ΔE_M^GW while both DFT gaps are smaller — uses the difference between peak positions in a single experimental spectrum, which is invariant to ΔEF and to normalization. The surface model (hard Dirichlet truncation, Eq. 5, and step barrier, Eq. 8) uses literature work functions and a fixed boundary position, not parameters optimized to the dI/dV data. The manuscript explicitly acknowledges the non-self-consistent surface treatment and remaining discrepancies (e.g., E≈0.5 eV in Fig. 3, E>0.2 eV in Fig. 5, third peak at 2.5 eV vs 2.25 eV in Fig. 8), which weakens the absolute quantitative claim but does not make the GW-vs-DFT ranking circular. No equation in the paper reduces by construction to a fitted quantity, and no load-bearing claim rests solely on a self-citation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the GW-TB Hamiltonians from prior work (with shared authorship), a non-self-consistent vacuum model, and Chen's derivative rule. The only numerics fitted to the new experimental data are the per-spectrum Fermi-level shifts and an overall amplitude normalization, which do not by themselves determine spectral features.

free parameters (4)

Fermi-level shift ΔEF per experimental spectrum = 0.24 eV (Bi2Se3, Fig 1a), 0.22 eV (Bi2Se3 metal tip, Fig 3), -0.010 eV (Bi2Te2Se, Fig 1b), +0.195 eV (Bi2Te3, Fig…
A rigid energy shift applied to align the theoretical energy axis (relative to the calculated Fermi level) with the experimental bias voltage, accounting for sample doping. The shift is chosen per dataset to maximize agreement; it is not predicted.
Vertical normalization constant for I and dI/dV = Not stated, determined per spectrum by matching integrated tunneling current
In Section IV A, the theoretical current is integrated from the experimental Fermi energy and used to normalize the dI/dV spectra, fixing the overall amplitude.
BZ integration broadening/smoothing parameter = Not stated, described as 'physically realistic smoothing parameters'
Mentioned in observation (xii) as needed to eliminate numerical artefacts; the specific value is not reported, making it an effective free parameter for spectral smoothing.
Sample boundary position z_b = 1.3 Å outside the outermost atomic cores
Hand-chosen geometry parameter in Eq. (8) and Section III B that sets the origin of the vacuum potential step and affects the decay of Bloch functions.

assumptions (5)

domain assumption Chen's derivative rule (Eq. 18) accurately approximates the Bardeen tunneling matrix elements for s- and p-type tip orbitals.
Used in Section III C to compute dI/dV; relies on tip wavefunction expansion into spherical harmonics and neglects the explicit surface integral of Eq. (15).
domain assumption The GW-corrected Wannier-interpolated tight-binding Hamiltonians from Ref. [20] accurately describe the quasiparticle band structure of the Bi2Se3 family over the full Brillouin zone and relevant energy range.
These Hamiltonians are the foundation for all spectra, and the claim that GW is necessary rests on their accuracy. They are taken from a prior paper with overlapping authorship (Aguilera et al.).
domain assumption A laterally averaged potential step with literature work functions (Eq. 8) and Dirichlet truncation of the bulk Hamiltonian describes the surface and vacuum barrier without self-consistent surface effects.
Stated in Section III B and observation (xii); the validity of neglecting surface band bending, relaxation, and boundary-localized potential shifts is assumed.
domain assumption The CO-functionalized tip can be modeled by a fixed linear combination of s and p orbitals with decay constant kν = 1 Å and coefficients from Table II.
Based on Refs. [54,55]; no energy or distance dependence of the coefficients is included.
domain assumption The tip has a constant density of states over the measured bias range, so dI/dV is proportional to the sum of |M|^2.
Implicit in Eq. (14); the tip DOS is not included as an energy-dependent factor.

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Cite this review

Pith. "Pith review of Probing the Electronic Structure at the Boundary of Topological Insulators in the $\mathrm{Bi}_2\mathrm{Se}_3$ Family by Combined STM and AFM." pith.science (2026). https://pith.science/paper/KKLOOD2K

@misc{pith2026241113529,
  author       = {Pith},
  title        = {Pith review of: Probing the Electronic Structure at the Boundary of Topological Insulators in the $\mathrmBi_2\mathrmSe_3$ Family by Combined STM and AFM},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KKLOOD2K}},
  note         = {Machine review of arXiv:2411.13529}
}

abstract

We develop a numerical scheme for the calculation of tunneling current $I$ and differential conductance $\mathsf{d}I/\mathsf{d}V$ of metal and CO-terminated STM tips on the topological insulators $\mathrm{Bi}_2\mathrm{Se}_3$, $\mathrm{Bi}_2\mathrm{Te}_2\mathrm{Se}$ and $\mathrm{Bi}_2\mathrm{Te}_3$ and find excellent agreement with experiment. The calculation is an application of Chen's derivative rule, whereby the Bloch functions are obtained from Wannier interpolated tight-binding Hamiltonians and maximally localized Wannier functions from first-principle DFT+$GW$ calculations. We observe signatures of the topological boundary modes, their hybridization with bulk bands, Van Hove singularities of the bulk bands and characterize the orbital character of these electronic modes using the high spatial resolution of STM and AFM. Bare DFT calculations are insufficient to explain the experimental data, which are instead accurately reproduced by many-body corrected $GW$ calculations.

Figures

Figures reproduced from arXiv: 2411.13529 by the authors.

**Figure 2.** FIG. 2. Positive (red) and negative (blue) isosurfaces of the Wannier functions of Bi [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** (c) shows that this minimum in the dI/dV spectrum corresponds to the energy of the Dirac point ED. Other important markers in the experimental dI/dV spectrum are at E = −0.7 eV, E = −0.25 eV and E = 0.75 eV, where the experimental dI/dV signal significantly increases towards lower and higher energy, respectively. These features are reproduced by the theoretical dI/dV spectrum in [PITH_FULL_IMAGE:figures/full_fig… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4. (a) Half-space band structure of Bi [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: (a) shows the experimental and calculated dI/dV spectrum of a CO terminated tip positioned on a Te atom on the surface of Bi2Te2Se. An atomic site with maximal distance to substitutional defects was chosen for this. The spectrum was recorded on a bias range from −0.4 …

**Figure 6.** Figure 6: (a) shows the dI/dV spectrum of CO terminated tip positioned above an hollow site on the Bi2Te3 surface over a bias voltage range of −2 V to 2 V. The experimental spectrum exhibits peaks in the differential conductance at −1.2 V, −0.9 V, 1.1 V and 1.75 V. The orbital …

**Figure 7.** Figure 7: FIG. 7. (a) [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: FIG. 8. (a) Comparison of experimentally obtained [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

**Figure 9.** Figure 9: FIG. 9. (a) [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: FIG. 10. Orbital weighted half-space band structure of [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

**Figure 11.** Figure 11: FIG. 11. (a) - (e) Experimentally obtained [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

78 extracted references · 64 canonical work pages

[20]

Aguilera, C

I. Aguilera, C. Friedrich, and S. Bl¨ ugel, Many-body cor- rected tight-binding Hamiltonians for an accurate quasi- particle description of topological insulators of the Bi2Se3 family, Phys. Rev. B 100, 155147 (2019)

work page 2019
[1]

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized hall conductance in a two- dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982)

1982
[2]

C. L. Kane and E. J. Mele, Quantum spin hall effect in graphene, Phys. Rev. Lett. 95, 226801 (2005)

2005
[3]

C. L. Kane and E. J. Mele, Z2 Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95, 146802 (2005)

2005
[4]

L. Fu, C. L. Kane, and E. J. Mele, Topological insulators in three dimensions, Phys. Rev. Lett. 98, 106803 (2007)

2007
[5]

B. Lv, T. Qian, and H. Ding, Angle-resolved photoemis- sion spectroscopy and its application to topological ma- terials, Nature Reviews Physics 1, 609 (2019)

2019
[6]

Knispel, W

T. Knispel, W. Jolie, N. Borgwardt, J. Lux, Z. Wang, Y. Ando, A. Rosch, T. Michely, and M. Gr¨ uninger, Charge puddles in the bulk and on the surface of the topological insulator BiSbTeSe2 studied by scanning tun- neling microscopy and optical spectroscopy, Phys. Rev. B 96, 195135 (2017)

2017
[7]

Mayer, H

T. Mayer, H. Werner, F. Schmid, R. Diaz-Pardo, J. Fujii, I. Vobornik, C. H. Back, M. Kro- nseder, and D. Bougeard, Transport properties of band engineered p−n heterostructures of epitaxial Bi2Se3/(Bi1−xSbx)2(Te1−ySey)3 topological insulators, Phys. Rev. Mater. 5, 014202 (2021)

work page 2021

Show all 78 references

[8]

Pietanesi, M

L. Pietanesi, M. Marganska, T. Mayer, M. Barth, L. Chen, J. Zou, A. Weindl, A. Liebig, R. D ´ ıaz- Pardo, D. Suri, F. Schmid, F. J. Gießibl, K. Richter, Y. Tserkovnyak, M. Kronseder, and C. H. Back, Tracing dirac points of topological surface states by ferromagnetic resonance,...

2024
[9]

Urazhdin, D

S. Urazhdin, D. Bilc, S. D. Mahanti, S. H. Tessmer, T. Kyratsi, and M. G. Kanatzidis, Surface effects in lay- ered semiconductors Bi 2Se3 and Bi 2Te3, Phys. Rev. B 69, 085313 (2004)

2004
[10]

Y. S. Hor, A. Richardella, P. Roushan, Y. Xia, J. G. Checkelsky, A. Yazdani, M. Z. Hasan, N. P. Ong, and R. J. Cava, p-type Bi 2Se3 for topological insulator and low-temperature thermoelectric applications, Phys. Rev. B 79, 195208 (2009)

2009
[11]

Romanowich, M.-S

M. Romanowich, M.-S. Lee, D.-Y. Chung, S. D. Mahanti, M. G. Kanatzidis, and S. H. Tessmer, Interplay of topo- logical surface and bulk electronic states in Bi2Se3, Phys. Rev. B 87, 085310 (2013)

2013
[12]

Jurczyszyn, M

M. Jurczyszyn, M. Sikora, M. Chrobak, and L. Jur- czyszyn, Studies of surface states in Bi 2Se3 induced by the BiSe substitution in the crystal subsurface structure, Applied Surface Science 528, 146978 (2020)

2020
[13]

Teague, H

M. Teague, H. Chu, F.-X. Xiu, L. He, K.-L. Wang, and N.-C. Yeh, Observation of Fermi-energy dependent uni- tary impurity resonances in a strong topological insulator Bi2Se3 with scanning tunneling spectroscopy, Solid State Communications 152, 747 (2012)

2012
[14]

Netsou, D

A.-M. Netsou, D. A. Muzychenko, H. Dausy, T. Chen, F. Song, K. Schouteden, M. J. Van Bael, and C. Van Hae- sendonck, Identifying Native Point Defects in the Topo- logical Insulator Bi 2Te3, ACS Nano 14, 13172 (2020)

2020
[15]

Liebig, C

A. Liebig, C. Setescak, A. Weindl, and F. J. Giessibl, Structural Characterization of Defects in the Topological Insulator Bi 2Se3 at the Picometer Scale, The Journal of Physical Chemistry C 126, 21716 (2022)

2022
[16]

Alpichshev, J

Z. Alpichshev, J. G. Analytis, J.-H. Chu, I. R. Fisher, Y. L. Chen, Z. X. Shen, A. Fang, and A. Kapitulnik, STM Imaging of Electronic Waves on the Surface of Bi 2Te3: Topologically Protected Surface States and Hexagonal Warping Effects, Phys. Rev. Lett. 104, 016401 (2010)

2010
[17]

A. Y. Dmitriev, N. I. Fedotov, V. F. Nasretdinova, and S. V. Zaitsev-Zotov, Effect of surface defects and few-atomic steps on the local density of states of the atomically-clean surface of Bi 2Se3 topological insulator, JETP Letters 100, 398 (2014)

2014
[18]

N. I. Fedotov and S. V. Zaitsev-Zotov, Experimental search for one-dimensional edge states at surface steps of the topological insulator Bi 2Se3: Distinguishing between effects and artifacts, Phys. Rev. B 95, 155403 (2017). 19

2017
[19]

Queiroz, R

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

2024 arXiv
[21]

Marzari, A

N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized Wannier functions: Theory and applications, Rev. Mod. Phys. 84, 1419 (2012)

2012
[22]

Bardeen, Tunnelling from a Many-Particle Point of View, Phys

J. Bardeen, Tunnelling from a Many-Particle Point of View, Phys. Rev. Lett. 6, 57 (1961)

1961
[23]

C. J. Chen, Tunneling matrix elements in three- dimensional space: The derivative rule and the sum rule, Phys. Rev. B 42, 8841 (1990)

1990
[24]

Zhang, C

H. Zhang, C. Liu, X. Qi, and et al., Topological insulators in Bi 2Se3, Bi 2Te3 and Sb 2Te3 with a single Dirac cone on the surface., nature physics 5, 438 (2009)

2009
[25]

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Cengage Learning, 1976)

1976
[26]

Leung and E

B. Leung and E. Prodan, Effect of strong disorder in a three-dimensional topological insulator: Phase diagram and maps of the Z2 invariant, Phys. Rev. B 85, 205136 (2012)

2012
[27]

C. M. Acosta, M. P. Lima, A. J. R. da Silva, A. Fazzio, and C. H. Lewenkopf, Tight-binding model for the band dispersion in rhombohedral topological insulators over the whole Brillouin zone, Phys. Rev. B98, 035106 (2018)

2018
[28]

I. A. Nechaev, R. C. Hatch, M. Bianchi, D. Guan, C. Friedrich, I. Aguilera, J. L. Mi, B. B. Iversen, S. Bl¨ ugel, P. Hofmann, and E. V. Chulkov, Evidence for a direct band gap in the topological insulator Bi 2Se3 from theory and experiment, Phys. Rev. B 87, 121111 (2013)

2013
[29]

Orlita, B

M. Orlita, B. A. Piot, G. Martinez, N. K. S. Ku- mar, C. Faugeras, M. Potemski, C. Michel, E. M. Hankiewicz, T. Brauner, i. c. v. Draˇ sar, S. Schreyeck, S. Grauer, K. Brunner, C. Gould, C. Br¨ une, and L. W. Molenkamp, Magneto-Optics of Massive Dirac Fermions in Bulk Bi 2Se3,...

2015
[30]

S´ anchez-Barriga, I

J. S´ anchez-Barriga, I. Aguilera, L. V. Yashina, D. Y. Tsukanova, F. Freyse, A. N. Chaika, C. Callaert, A. M. Abakumov, J. Hadermann, A. Varykhalov, E. D. L. Rienks, G. Bihlmayer, S. Bl¨ ugel, and O. Rader, Anomalous behavior of the electronic structure of (Bi1−xInx)2Se3 acro...

2018
[31]

Kobayashi, Electron transmission through atomic steps of Bi 2Se3 and Bi 2Te3 surfaces, Phys

K. Kobayashi, Electron transmission through atomic steps of Bi 2Se3 and Bi 2Te3 surfaces, Phys. Rev. B 84, 205424 (2011)

2011
[32]

M. K. Fuccillo, S. Jia, M. E. Charles, and R. J. Cava, Thermoelectric Properties of Bi 2Te2Se Compensated by Native Defects and Sn Doping, Journal of Electronic Ma- terials 42, 1246 (2013)

2013
[33]

Neupane, S.-Y

M. Neupane, S.-Y. Xu, L. A. Wray, A. Petersen, R. Shankar, N. Alidoust, C. Liu, A. Fedorov, H. Ji, J. M. Allred, Y. S. Hor, T.-R. Chang, H.-T. Jeng, H. Lin, A. Bansil, R. J. Cava, and M. Z. Hasan, Topological sur- face states and Dirac point tuning in ternary topological insul...

2012
[34]

Bruus and K

H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics: An Introduction , Oxford Graduate Texts (Oxford University Press, Oxford, 2004)

2004
[35]

Zhang, K

Y. Zhang, K. He, C.-Z. Chang, C.-L. Song, L.-L. Wang, X. Chen, J.-F. Jia, Z. Fang, X. Dai, W.-Y. Shan, S.- Q. Shen, Q. Niu, X.-L. Qi, S.-C. Zhang, X.-C. Ma, and Q.-K. Xue, Crossover of the three-dimensional topologi- cal insulator Bi2Se3 to the two-dimensional limit, Nature Ph...

2010
[37]

Friedrich, S

C. Friedrich, S. Bl¨ ugel, and A. Schindlmayr, Efficient im- plementation of the GW approximation within the all- electron FLAPW method, Physical Review B 81, 125102 (2010)

2010
[38]

Aguilera, C

I. Aguilera, C. Friedrich, and S. Bl¨ ugel, Spin-orbit cou- pling in quasiparticle studies of topological insulators, Phys. Rev. B 88, 165136 (2013)

2013
[39]

Aguilera, C

I. Aguilera, C. Friedrich, G. Bihlmayer, and S. Bl¨ ugel, GW study of topological insulators Bi 2Se3, Bi 2Te3, and Sb2Te3: Beyond the perturbative one-shot approach, Phys. Rev. B 88, 045206 (2013)

2013
[40]

Michiardi, I

M. Michiardi, I. Aguilera, M. Bianchi, V. E. de Carvalho, L. O. Ladeira, N. G. Teixeira, E. A. Soares, C. Friedrich, S. Bl¨ ugel, and P. Hofmann, Bulk band structure of bi2te3, Phys. Rev. B 90, 075105 (2014)

2014
[41]

Kuo, S.-C

C.-T. Kuo, S.-C. Lin, J.-P. Rueff, Z. Chen, I. Aguilera, G. Bihlmayer, L. Plucinski, I. L. Graff, G. Conti, I. A. Vartanyants, C. M. Schneider, and C. S. Fadley, Orbital contributions in the element-resolved valence electronic structure of Bi 2Se3, Phys. Rev. B 104, 245105 (2021)

2021
[42]

Plank, J

H. Plank, J. Pernul, S. Gebert, S. N. Danilov, J. K¨ onig- Otto, S. Winnerl, M. Lanius, J. Kampmeier, G. Mus- sler, I. Aguilera, D. Gr¨ utzmacher, and S. D. Ganichev, Infrared/terahertz spectra of the photogalvanic effect in (Bi,Sb)Te based three-dimensional topological insula...

2018
[43]

Mohelsky, J

I. Mohelsky, J. Wyzula, F. Le Mardel´ e, F. Abadizaman, O. Caha, A. Dubroka, X. D. Sun, C. W. Cho, B. A. Piot, M. F. Tanzim, I. Aguilera, G. Bauer, G. Springholz, and M. Orlita, Electronic band structure of Sb 2Te3, Phys. Rev. B 109, 165205 (2024)

2024
[44]

I. A. Nechaev, I. Aguilera, V. De Renzi, A. di Bona, A. Lodi Rizzini, A. M. Mio, G. Nicotra, A. Politano, S. Scalese, Z. S. Aliev, M. B. Babanly, C. Friedrich, S. Bl¨ ugel, and E. V. Chulkov, Quasiparticle spectrum and plasmonic excitations in the topological insulator Sb2Te3,...

2015
[45]

Battiato, I

M. Battiato, I. Aguilera, and J. S´ anchez-Barriga, Gener- alized GW +Boltzmann Approach for the Description of Ultrafast Electron Dynamics in Topological Insulators, Materials 10, 810 (2017)

2017
[46]

Pizzi, V

G. Pizzi, V. Vitale, R. Arita, S. Bl¨ ugel, F. Freimuth, G. G´ eranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Iba˜ nez-Azpiroz, H. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, S. Ponc´ e, T. Pon- wei...

2020
[47]

Takane, S

D. Takane, S. Souma, T. Sato, T. Takahashi, K. Segawa, and Y. Ando, Work function of bulk-insulating topologi- cal insulator Bi 2–xSbxTe3–ySey, Applied Physics Letters 109, 091601 (2016)

2016
[48]

Ryu, Work function of Bismuth Telluride: First- principles approach, Journal of the Korean Physical So- 20 ciety 72, 122 (2018)

B. Ryu, Work function of Bismuth Telluride: First- principles approach, Journal of the Korean Physical So- 20 ciety 72, 122 (2018)

2018
[49]

Gundlach, Zur Berechnung des Tunnelstroms durch eine trapezf¨ ormige Potentialstufe, Solid-State Electronics 9, 949 (1966)

K. Gundlach, Zur Berechnung des Tunnelstroms durch eine trapezf¨ ormige Potentialstufe, Solid-State Electronics 9, 949 (1966)

1966
[50]

Binnig, K

G. Binnig, K. H. Frank, H. Fuchs, N. Garcia, B. Reihl, H. Rohrer, F. Salvan, and A. R. Williams, Tunneling Spectroscopy and Inverse Photoemission: Image and Field States, Phys. Rev. Lett. 55, 991 (1985)

1985
[51]

Huang, W.-Y

H.-S. Huang, W.-Y. Chan, W.-B. Su, G. Hoffmann, and C.-S. Chang, Measurement of work function difference between Pb/Si(111) and Pb/Ge/Si(111) by high-order Gundlach oscillation, Journal of Applied Physics 114, 214308 (2013)

2013
[52]

See Supplemental Material
[53]

Bl¨ ugel, Theorie der Rastertunnelmikroskopie, in Physik der Nanostrukturen , Band 01 (Verlag des Forschungszentrums J¨ ulich, J¨ ulich, 1998) Chap

S. Bl¨ ugel, Theorie der Rastertunnelmikroskopie, in Physik der Nanostrukturen , Band 01 (Verlag des Forschungszentrums J¨ ulich, J¨ ulich, 1998) Chap. A3, pp. A3.1–A3.56

1998
[54]

M´ andi and K

G. M´ andi and K. Palot´ as, Chen’s derivative rule revis- ited: Role of tip-orbital interference in STM, Phys. Rev. B 91, 165406 (2015)

2015
[55]

Gross, N

L. Gross, N. Moll, F. Mohn, A. Curioni, G. Meyer, F. Hanke, and M. Persson, High-Resolution Molecular Orbital Imaging Using a p-Wave STM Tip, Phys. Rev. Lett. 107, 086101 (2011)

2011
[56]

Oinonen, A

N. Oinonen, A. V. Yakutovich, A. Gallardo, M. Ondr´ aˇ cek, P. Hapala, and O. Krejˇ c ´ ı, Advancing scanning probe microscopy simulations: A decade of development in probe-particle models, Computer Physics Communications 305, 109341 (2024)

2024
[57]

Krejˇ c ´ ı, P

O. Krejˇ c ´ ı, P. Hapala, M. Ondr´ aˇ cek, and P. Jel ´ ınek, Prin- ciples and simulations of high-resolution STM imaging with a flexible tip apex, Phys. Rev. B 95, 045407 (2017)

2017
[58]

Hapala, R

P. Hapala, R. Temirov, F. S. Tautz, and P. Jel ´ ınek, Ori- gin of High-Resolution IETS-STM Images of Organic Molecules with Functionalized Tips, Phys. Rev. Lett. 113, 226101 (2014)

2014
[59]

Hapala, G

P. Hapala, G. Kichin, C. Wagner, F. S. Tautz, R. Temirov, and P. Jel ´ ınek, Mechanism of high-resolution STM/AFM imaging with functionalized tips, Phys. Rev. B 90, 085421 (2014)

2014
[60]

F. J. Giessibl, The qPlus sensor, a powerful core for the atomic force microscope, Review of Scientific Instruments 90, 011101 (2019)

2019
[61]

Liebig, A Matlab program for scanning-probe spec- troscopy data analysis (Bachelor Thesis, Universit¨ at Re- gensburg, 2014)

A. Liebig, A Matlab program for scanning-probe spec- troscopy data analysis (Bachelor Thesis, Universit¨ at Re- gensburg, 2014)

2014
[62]

Hofmann, Hochaufl¨ osende Rasterkraftmikroskopie auf Graphen und Kohlenmonoxid (PhD Thesis, Universit¨ at Regensburg, 2014)

T. Hofmann, Hochaufl¨ osende Rasterkraftmikroskopie auf Graphen und Kohlenmonoxid (PhD Thesis, Universit¨ at Regensburg, 2014)

2014
[63]

Liebig, A

A. Liebig, A. Peronio, D. Meuer, A. J. Weymouth, and F. J. Giessibl, High-precision atomic force mi- croscopy with atomically-characterized tips, New Journal of Physics 22, 063040 (2020)

2020
[64]

Bartels, G

L. Bartels, G. Meyer, and K.-H. Rieder, Controlled ver- tical manipulation of single CO molecules with the scan- ning tunneling microscope: A route to chemical contrast, Applied Physics Letters 71, 213 (1997)

1997
[65]

Z.-H. Zhu, G. Levy, B. Ludbrook, C. N. Veenstra, J. A. Rosen, R. Comin, D. Wong, P. Dosanjh, A. Ubaldini, P. Syers, N. P. Butch, J. Paglione, I. S. Elfimov, and A. Damascelli, Rashba spin-splitting control at the sur- face of the topological insulator Bi 2Se3, Phys. Rev. Lett....

2011
[66]

Grasser, Native point defects and standing waves - scanning probe microscopy of the topological insulator Bi2Te3 (Bachelor Thesis, Universit¨ at Regensburg, 2021)

E. Grasser, Native point defects and standing waves - scanning probe microscopy of the topological insulator Bi2Te3 (Bachelor Thesis, Universit¨ at Regensburg, 2021)

2021
[67]

Bianchi, D

M. Bianchi, D. Guan, S. Bao, J. Mi, B. B. Iversen, P. D. King, and P. Hofmann, Coexistence of the topological state and a two-dimensional electron gas on the surface of bi2se3, Nature Communications 1, 128 (2010)

2010
[68]

N. D. Lang, Spectroscopy of single atoms in the scanning tunneling microscope, Phys. Rev. B 34, 5947 (1986)

1986
[69]

J. A. Stroscio, R. M. Feenstra, and A. P. Fein, Elec- tronic Structure of the Si(111) 2× 1 Surface by Scanning- Tunneling Microscopy, Phys. Rev. Lett. 57, 2579 (1986)

1986
[70]

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
[71]

A. J. Weymouth, T. Wutscher, J. Welker, T. Hofmann, and F. J. Giessibl, Phantom Force Induced by Tunneling Current: A Characterization on Si(111), Phys. Rev. Lett. 106, 226801 (2011)

2011
[72]

Liu and D

J. Liu and D. Vanderbilt, Topological phase transitions in (Bi1−xInx)2Se3 and (Bi1−xSbx)2Se3, Phys. Rev. B 88, 224202 (2013)

2013
[73]

Schouteden, Z

K. Schouteden, Z. Li, T. Chen, F. Song, B. Partoens, C. Van Haesendonck, and K. Park, Moir´ e superlattices at the topological insulator Bi 2Te3, Scientific Reports 6, 20278 (2016)

2016
[74]

Cheng, C

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, J.- F. Jia, J. Wang, Y. Wang, B.-F. Zhu, X. Chen, X. Ma, K. He, L. Wang, X. Dai, Z. Fang, X. Xie, X.-L. Qi, C.-X. Liu, S.-C. Zhang, and Q.-K. Xue, Landau Quantization of Topological Surface States in Bi 2Se3, Phys. Rev. Lett. 105...

2010
[75]

Jiang, Y

Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Q.-K. Xue, Landau Quan- tization and the Thickness Limit of Topological Insula- tor Thin Films of Sb 2Te3, Phys. Rev. Lett. 108, 016401 (2012)

2012
[76]

Hanaguri, K

T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa, Momentum-resolved Landau-level spec- troscopy of Dirac surface state in Bi 2Se3, Phys. Rev. B 82, 081305 (2010)

2010
[77]

Y. Liu, Y. Y. Li, S. Rajput, D. Gilks, L. Lari, P. L. Galindo, M. Weinert, V. K. Lazarov, and L. Li, Tuning Dirac states by strain in the topological insulator Bi2Se3, Nature Physics 10, 294 (2014)

2014
[78]

Ludewig and G

M. Ludewig and G. Chuan Thiang, Cobordism invari- ance of topological edge-following states, ATMP 26, 673 (2022)

2022
[79]

E. E. Ewert and R. Meyer, Coarse geometry and topo- logical phases, Communications in Mathematical Physics 366, 1069 (2019)

2019

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [20]

Aguilera, C

I. Aguilera, C. Friedrich, and S. Bl¨ ugel, Many-body cor- rected tight-binding Hamiltonians for an accurate quasi- particle description of topological insulators of the Bi2Se3 family, Phys. Rev. B 100, 155147 (2019)

work page 2019

[2] [1]

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized hall conductance in a two- dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982)

1982

[3] [2]

C. L. Kane and E. J. Mele, Quantum spin hall effect in graphene, Phys. Rev. Lett. 95, 226801 (2005)

2005

[4] [3]

C. L. Kane and E. J. Mele, Z2 Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95, 146802 (2005)

2005

[5] [4]

L. Fu, C. L. Kane, and E. J. Mele, Topological insulators in three dimensions, Phys. Rev. Lett. 98, 106803 (2007)

2007

[6] [5]

B. Lv, T. Qian, and H. Ding, Angle-resolved photoemis- sion spectroscopy and its application to topological ma- terials, Nature Reviews Physics 1, 609 (2019)

2019

[7] [6]

Knispel, W

T. Knispel, W. Jolie, N. Borgwardt, J. Lux, Z. Wang, Y. Ando, A. Rosch, T. Michely, and M. Gr¨ uninger, Charge puddles in the bulk and on the surface of the topological insulator BiSbTeSe2 studied by scanning tun- neling microscopy and optical spectroscopy, Phys. Rev. B 96, 195135 (2017)

2017

[8] [7]

Mayer, H

T. Mayer, H. Werner, F. Schmid, R. Diaz-Pardo, J. Fujii, I. Vobornik, C. H. Back, M. Kro- nseder, and D. Bougeard, Transport properties of band engineered p−n heterostructures of epitaxial Bi2Se3/(Bi1−xSbx)2(Te1−ySey)3 topological insulators, Phys. Rev. Mater. 5, 014202 (2021)

work page 2021