Evidence for monopole-like topological magnetoelectric effect in image potential states

Baojie Feng; Chenmin Liu; Hong-Jun Gao; Huaiyu Zhang; Jian-Qiang Zhong; Lan Chen; Peng Cheng; Yang Zhan; Yeliang Wang; Yi-Qi Zhang

arxiv: 2509.24648 · v2 · submitted 2025-09-29 · ❄️ cond-mat.mtrl-sci

Evidence for monopole-like topological magnetoelectric effect in image potential states

Yang Zhan , Huaiyu Zhang , Yu Wang , Chenmin Liu , Jian-Qiang Zhong , Yeliang Wang , Hong-Jun Gao , Yi-Qi Zhang

show 3 more authors

Baojie Feng Peng Cheng Lan Chen

This is my paper

Pith reviewed 2026-05-18 12:59 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords image potential statestopological magnetoelectric effectmagnetic monopoletopological insulatorscanning tunneling microscopyBi(111)field emissionsurface states

0 comments

The pith

An anomalous splitting of image potential states on a topological insulator surface is evidence for a monopole-like topological magnetoelectric effect.

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

The paper aims to demonstrate that applying an active electric field in the field-emission regime at the surface of bismuth, a higher-order topological insulator, produces a splitting in the image potential states that matches the signature of an effective magnetic monopole. This builds on the theoretical construction of real-space monopoles at a topological insulator-vacuum interface. If the attribution holds, field-emission spectroscopy would serve as a practical way to generate and detect these monopole-like objects through their influence on electron states near the surface. A reader would care because it connects everyday surface measurements to electromagnetic responses that theories beyond the standard model have long predicted but rarely made accessible in the lab.

Core claim

The authors observe an anomalous splitting of image potential states using scanning tunneling microscopy in the field-emission regime on Bi(111). By tuning dielectric properties of the substrate and film thickness, they link the splitting to the radially active electric field interacting with topological surface states. Phenomenological analysis then attributes the peak splitting to the equivalent magnetic field of a monopole-like topological magnetoelectric response.

What carries the argument

The monopole-like topological magnetoelectric response, in which an active electric field at the topological insulator-vacuum interface generates an effective magnetic field that splits the image potential states.

If this is right

Field-emission image potential state spectroscopy becomes a platform for generating active electric fields and detecting the resulting image magnetic monopoles at topological surfaces.
The magnitude of the splitting should vary predictably with electric-field strength, substrate dielectric constant, and film thickness through their effects on the topological response.
Suppression of topological surface states, for instance by doping or temperature, should eliminate or reduce the anomalous splitting component.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same field-emission approach could be applied to other higher-order topological materials to test whether monopole-like signatures appear more generally.
If the effective monopoles prove controllable, the geometry might allow engineering of localized electromagnetic responses in thin-film devices.
Comparison with related surface-sensitive techniques such as photoemission could clarify whether the splitting is unique to the field-emission geometry or appears more broadly.

Load-bearing premise

That the observed splitting of image potential states cannot be fully explained by ordinary electrostatic or dielectric effects and instead requires the topological magnetoelectric contribution from the surface states and active field.

What would settle it

A detailed model or control experiment that reproduces the full splitting using only conventional electrostatic and dielectric contributions, without any topological surface-state input.

read the original abstract

Magnetic monopoles, hypothetical particles behaving as isolated magnetic charges, have long been predicted by theories beyond the standard model but remain elusive in experimental detection. Subsequently, Xiaoliang Qi et al. proposed that magnetic monopoles can be constructed in real space by introducing an active electric field at the interface between a topological insulator and vacuum [Science 323, 1184 (2009)]. Here we use scanning tunneling microscopy in the field-emission regime to realize an active electric-field geometry at the surface of the higher-order topological insulator Bi(111), and observe an anomalous splitting of the image potential states (IPSs). By tuning the dielectric properties of the substrate and film thickness, we considered that the peak-splitting of IPSs is related to the radially active electric field and topological surface states. Combined with phenomenological analysis, this peaksplitting can be attributed to the equivalent magnetic field of monopole-like topological magnetoelectric response. This work establishes field-emission IPS spectroscopy as a sensitive platform for generating active electric fields and probing the resulting image magnetic monopoles at topological surfaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper reports a clear experimental splitting of image potential states on Bi(111) via field-emission STM, with tuning data that correlates to dielectric and thickness changes, but the monopole-like topological magnetoelectric attribution rests on phenomenological fitting without quantitative exclusion of conventional electrostatic models.

read the letter

The main thing to know is that this work measures an anomalous splitting of image potential states on Bi(111) using scanning tunneling microscopy in the field-emission regime and ties it, through tuning experiments and a fit, to an effective magnetic field from a monopole-like topological magnetoelectric response at the surface-vacuum interface. The observation itself is new relative to the 2009 Qi et al. theory it cites, and the tuning of substrate dielectric properties plus film thickness shows a correlation that supports some role for the active electric field and topological surface states. That experimental control is useful and gives the data more weight than a single-condition measurement would have. The setup extends the earlier proposal with concrete field-emission STM spectra on a higher-order topological insulator, which is a reasonable next step. The softer spot is the interpretation step. The equivalent magnetic field is extracted by fitting the observed energy shifts, so the claimed monopole-like response is defined from the same data it is meant to explain. The manuscript does not include an explicit calculation showing that a standard image-potential Schrödinger solution with position-dependent permittivity and finite-thickness boundaries fails to reproduce the splitting size or field dependence. Without that comparison, conventional dielectric screening or field-induced modifications remain possible alternatives. The tuning experiments help narrow the options but do not quantitatively close them. This paper is aimed at researchers working on topological surface states, magnetoelectric effects, and STM-based probes of image potentials. Someone already following the 2009 proposal or looking for new spectroscopic signatures in topological materials would find the raw spectra and tuning trends worth examining. The experimental result is grounded enough and the connection to prior theory is direct enough that it deserves a serious referee, even though the central attribution will need more modeling work to hold up. I would recommend sending it for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript reports STM measurements in the field-emission regime on Bi(111), observing anomalous splitting of image potential states (IPS). Tuning experiments varying substrate dielectric properties and film thickness are used to correlate the splitting with a radially active electric field and topological surface states. Phenomenological analysis then attributes the splitting to an equivalent magnetic field arising from a monopole-like topological magnetoelectric response, as predicted for topological insulator-vacuum interfaces.

Significance. If the attribution to a monopole-like effect is substantiated with quantitative support, the work would realize a long-predicted construction of magnetic monopoles in real space via active electric fields at topological surfaces and establish field-emission IPS spectroscopy as a new probe. The tuning experiments provide correlative evidence linking splitting to dielectric response and thickness, strengthening the case for an active-field role. The absence of a quantitative model or explicit exclusion of conventional mechanisms currently limits the strength of the central claim.

major comments (2)

[Abstract] Abstract: The claim that peak-splitting 'can be attributed to the equivalent magnetic field of monopole-like topological magnetoelectric response' rests on phenomenological analysis; because the equivalent magnetic field is obtained by fitting the observed energy shifts, the interpretation is circular—the quantity invoked to explain the data is defined from those same data.
[Tuning experiments] Tuning experiments: While dielectric and thickness tuning correlates splitting with the active electric field, no explicit calculation is shown demonstrating that a conventional electrostatic model (Schrödinger equation for the image potential with position-dependent permittivity and finite-thickness boundary conditions) fails to reproduce the splitting magnitude and field dependence.

minor comments (2)

[Abstract] Abstract: 'peaksplitting' should be written as two words for clarity.
[Abstract] Abstract: The sentence 'we considered that the peak-splitting of IPSs is related to...' is vague; rephrase to state the specific inference drawn from the tuning data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the presentation of our results. We address each major comment point by point below, providing the strongest honest defense based on the existing data and analysis while acknowledging where revisions are warranted.

read point-by-point responses

Referee: [Abstract] Abstract: The claim that peak-splitting 'can be attributed to the equivalent magnetic field of monopole-like topological magnetoelectric response' rests on phenomenological analysis; because the equivalent magnetic field is obtained by fitting the observed energy shifts, the interpretation is circular—the quantity invoked to explain the data is defined from those same data.

Authors: The phenomenological analysis extracts an effective magnetic field strength from the measured energy splitting by applying the established relation between magnetic field and Landau-level-like shifts in image potential states. This extraction follows directly from the theoretical framework of Qi et al. for monopole-like magnetoelectric effects at topological insulator-vacuum interfaces under an active electric field. The approach is not circular because the theory independently predicts both the existence of an equivalent magnetic field and its functional dependence on the radial electric field and topological surface states; the fit merely quantifies the magnitude for comparison with that prediction. The dielectric and thickness tuning experiments provide independent correlative support by demonstrating that the splitting appears only when both the active-field geometry and topological surface states are present, behaviors not generically expected from conventional electrostatics. We will revise the abstract to explicitly distinguish the theoretical prediction from the phenomenological quantification. revision: yes
Referee: [Tuning experiments] Tuning experiments: While dielectric and thickness tuning correlates splitting with the active electric field, no explicit calculation is shown demonstrating that a conventional electrostatic model (Schrödinger equation for the image potential with position-dependent permittivity and finite-thickness boundary conditions) fails to reproduce the splitting magnitude and field dependence.

Authors: We agree that an explicit comparison to conventional electrostatic models would strengthen the exclusion of alternative explanations. The tuning experiments were designed to vary the substrate dielectric constant and film thickness while keeping the STM tip geometry fixed, thereby modulating the radial component of the active electric field and the contribution of topological surface states. These changes produce systematic variations in splitting that track the expected conditions for the monopole-like response. Nevertheless, to directly address the concern, we will add calculations solving the one-dimensional Schrödinger equation for the image potential with position-dependent permittivity and finite-thickness boundary conditions, demonstrating that such models yield neither the observed splitting magnitude nor its dependence on applied field without invoking the additional topological magnetoelectric term. revision: yes

Circularity Check

1 steps flagged

Phenomenological fit of equivalent magnetic field to IPS splitting makes monopole-like attribution circular by construction

specific steps

fitted input called prediction [Abstract (phenomenological analysis)]
"Combined with phenomenological analysis, this peak-splitting can be attributed to the equivalent magnetic field of monopole-like topological magnetoelectric response."

The equivalent magnetic field is obtained by fitting the observed IPS energy shifts; the monopole-like interpretation is therefore defined from the same data it is invoked to explain, satisfying the fitted-input-called-prediction pattern.

full rationale

The paper observes IPS peak splitting under field-emission conditions on Bi(111), tunes dielectric properties and thickness to link it to radially active electric field plus topological surface states, then invokes phenomenological analysis to attribute the splitting to an 'equivalent magnetic field of monopole-like topological magnetoelectric response.' This reduces to fitting the effective B-field parameter directly to the measured energy shifts, so the claimed topological magnetoelectric effect is a re-description of the input data rather than an independent derivation. No explicit Schrödinger-equation solution with conventional position-dependent permittivity and finite-thickness boundaries is shown to fail quantitatively, leaving the exclusion of standard electrostatic mechanisms unverified and the fit load-bearing.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on interpreting the splitting through a topological magnetoelectric lens after phenomenological matching; this introduces an effective monopole without first-principles derivation or independent falsifiable signature beyond the fit itself.

free parameters (1)

effective magnetic field strength
Chosen to reproduce the magnitude of the observed IPS splitting under the monopole-like model

axioms (2)

domain assumption Bi(111) possesses higher-order topological surface states that couple to an external electric field to produce a magnetoelectric response
Invoked when the authors attribute the splitting to the combination of active electric field and topological surface states
domain assumption The field-emission regime of STM produces a radially directed active electric field at the surface
Assumed from tip geometry and operating mode

invented entities (1)

monopole-like topological magnetoelectric response no independent evidence
purpose: To generate the equivalent magnetic field that accounts for the IPS peak splitting
Postulated on the basis of phenomenological analysis to explain the data

pith-pipeline@v0.9.0 · 5749 in / 1644 out tokens · 48293 ms · 2026-05-18T12:59:29.804174+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

magnetic monopoles can be constructed in real space by introducing an active electric field at the interface between a topological insulator and vacuum... ∇×B ∝ n×E ... pair of inverse magnetic monopoles ... only occur in 3D TIs with thicknesses beyond the 2D limit
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the IPS peaks were re-fitted ... splitting energy ΔE(n) ... fitting ... using Eq. 10 ... ΔE(n) ∝ √n(n−1)/[n³(n−1/2)]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

[1]

Dirac, P. A. M. Quantised singularities in the electromagnetic field. Proc. R. Soc. London A 133, 60–72 (1931)

work page 1931
[2]

& Witten, E

Wen, X.G. & Witten, E. Electric and magnetic charges in superstring models. Nucl. Phys. B, 261, 651–677 (1985)

work page 1985
[3]

Mavromatos, N. E. & Mitsou, V . A. Magnetic monopoles revisited: models and searches at colliders and in the cosmos. Int. J. Mod. Phys. A 35, 2030012 (2020)

work page 2020
[4]

Bramwell, S. T. et al. Measurement of the charge and current of magnetic monopoles in spin ice. Nature 461, 956–959 (2009)

work page 2009
[5]

W., Ruokokoski, E., Kandel, S., Mottonen, M

Ray, M. W., Ruokokoski, E., Kandel, S., Mottonen, M. & Hall, D. S. Observation of Dirac monopoles in a synthetic magnetic field. Nature 505, 657 (2014)

work page 2014
[6]

W., Ruokokoski, E., Kandel, S., Mottonen, M

Ray, M. W., Ruokokoski, E., Kandel, S., Mottonen, M. & Hall, D. S. Observation of isolated monopoles in a quantum field. Science 348, 544 (2015)

work page 2015
[7]

Milton, K. A. Theoretical and experimental status of magnetic monopoles. Reports on Progress in Physics 69, 1637–1711 (2006)

work page 2006
[8]

Qi, X.-L., Hughes, T. L. & Zhang, S. -C. Topological field theory of time -reversal invariant 13 insulators. Phys. Rev. B 78, 195424 (2008)

work page 2008
[9]

& Zhang, S

Qi, X.-L., Li, R., Zang, J. & Zhang, S. -C. Inducing a magnetic monopole with topological surface states, Science 323, 1184 (2009)

work page 2009
[10]

Chang, C.-Z. et al. Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator. Science 340, 167-170(2013)

work page 2013
[11]

Deng, Y . et al. Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4. Science 367, 895-900(2020)

work page 2020
[12]

, Rohrer, H

Binnig, G. , Rohrer, H. , Gerber, Ch., & Weibel, E . Surface Studies by Scanning Tunneling Microscopy. Phys. Rev. Lett. 49, 57 (1982)

work page 1982
[13]

& Zandvliet, H

Borca, B. & Zandvliet, H. J.W. Image potential states of 2D materials. Applied Materials Today 39, 102304 (2024)

work page 2024
[14]

Ge, J. -F. et al. Probing image potential states on the surface of the topological semimetal antimony, Phys. Rev. B 101, 035152 (2020)

work page 2020
[15]

A., Wang, Y

Kubby, J. A., Wang, Y . R. & Greene, W. J. Fabry-Pérot transmission resonances in tunneling microscopy, Phys. Rev. B 43, 9346(R) (1991)

work page 1991
[16]

Binnig, G. et al. Tunneling spectroscopy and inverse photoemission: image and field states, Phys. Rev. Lett. 55, 991(1985)

work page 1985
[17]

Xiao, S. F. & Liu, Q. H. Geometric momentum and angular momentum for charge monopole system. Modern Physics Letters A. 33,1850125(2018)

work page 2018
[18]

Chen, J. -L. et al. The monopole -hydrogen atom system and its connection with a four - dimensional harmonic oscillator J. Phys. A: Math. Gen. 32, 945(1999)

work page 1999
[19]

Schindler, F. et al. Higher-order topology in bismuth. Nat. Phys. 14, 918–924 (2018)

work page 2018
[20]

Zhang, H. et al. Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nat. Phys. 5, 438–442 (2009)

work page 2009
[21]

Drozdov, I. K. et al. One-dimensional topological edge states of bismuth bilayers. Nat. Phys. 10, 664–669 (2014)

work page 2014
[22]

& Gervais, G

Yu, O., Boivin, F., Silberztein, A. & Gervais, G. Observation of Temperature -Independent Anomalous Hall Effect in Thin Bismuth from Near Absolute Zero to 300 K Temperature. Phys. Rev. Lett. 134, 066603 (2025). 14

work page 2025
[23]

& Gervais, G

Yu, O., Vijayakrishnan, S., Allgayer, R., Szkopek, T. & Gervais, G. Anomalous Hall effect in thin bismuth. Phys. Rev. B 109, L121406 (2024)

work page 2024
[24]

Gou, J. et al. Scanning tunneling microscopy investigations of unoccupied surface states in two-dimensional semiconducting β - 3×3 - Bi/Si(111) surface. Phys. Chem. Chem. Phys. 20, 20188 (2018)

work page 2018
[25]

Lu, Y . et al. Topological Properties Determined by Atomic Buckling in Self -Assembled Ultrathin Bi (110). Nano Lett. 15, 80-87 (2015)

work page 2015
[26]

Liu, C. et al. Tunable topological edge states in black -phosphorus-like Bi(110). Phys. Rev. B 110, 195427 (2024)

work page 2024
[27]

Gallagher, T. F. Rydberg atoms. Cambridge University Press (1994)

work page 1994
[28]

Influence of the tip of the scanning tunneling microscope on surface electron lifetimes, Surf

Crampin, S. Influence of the tip of the scanning tunneling microscope on surface electron lifetimes, Surf. Sci. 600, 4280 (2006)

work page 2006
[29]

Silkin, V . M. et al. Image potential states in graphene. Phys. Rev. B 80, 121408(R) (2009)

work page 2009
[30]

& Fauster, T

Niesner, D. & Fauster, T. Image -potential states and work function of graphene, J. Phys.: Condens. Matter 26, 393001 (2014)

work page 2014
[31]

Sugawara, K. et al. Interaction of Stark-shifted image potential states with quantum well states in ultrathin Ag(111) islands on Si(111)-√3×√3-B substrates. Phys. Rev. B 96, 075444 (2017)

work page 2017
[32]

Craes, F. et al. Mapping Image Potential States on Graphene Quantum Dots, Phys. Rev. Lett. 111, 056804 (2013)

work page 2013
[33]

& Haesendonck, C

Schouteden, K. & Haesendonck, C. V . Quantum Confinement of Hot Image -Potential State Electrons. Phys. Rev. Lett. 103, 266805(2009)

work page 2009
[34]

Fowler, R. H. & Nordheim, L. Electron Emission in Intense Electric Fields. Proceedings of the Royal Society A 119, 173-181(1928). Methods: 15 The experiments were performed in a home -built low temperature STM equipped with molecular beam epitaxial (MBE) chamber, with base pressure of 310-10 mbar. Sample preparation High-purity Bi (99.999%) and Se (99.99...

work page 1928

[1] [1]

Dirac, P. A. M. Quantised singularities in the electromagnetic field. Proc. R. Soc. London A 133, 60–72 (1931)

work page 1931

[2] [2]

& Witten, E

Wen, X.G. & Witten, E. Electric and magnetic charges in superstring models. Nucl. Phys. B, 261, 651–677 (1985)

work page 1985

[3] [3]

Mavromatos, N. E. & Mitsou, V . A. Magnetic monopoles revisited: models and searches at colliders and in the cosmos. Int. J. Mod. Phys. A 35, 2030012 (2020)

work page 2020

[4] [4]

Bramwell, S. T. et al. Measurement of the charge and current of magnetic monopoles in spin ice. Nature 461, 956–959 (2009)

work page 2009

[5] [5]

W., Ruokokoski, E., Kandel, S., Mottonen, M

Ray, M. W., Ruokokoski, E., Kandel, S., Mottonen, M. & Hall, D. S. Observation of Dirac monopoles in a synthetic magnetic field. Nature 505, 657 (2014)

work page 2014

[6] [6]

W., Ruokokoski, E., Kandel, S., Mottonen, M

Ray, M. W., Ruokokoski, E., Kandel, S., Mottonen, M. & Hall, D. S. Observation of isolated monopoles in a quantum field. Science 348, 544 (2015)

work page 2015

[7] [7]

Milton, K. A. Theoretical and experimental status of magnetic monopoles. Reports on Progress in Physics 69, 1637–1711 (2006)

work page 2006

[8] [8]

Qi, X.-L., Hughes, T. L. & Zhang, S. -C. Topological field theory of time -reversal invariant 13 insulators. Phys. Rev. B 78, 195424 (2008)

work page 2008

[9] [9]

& Zhang, S

Qi, X.-L., Li, R., Zang, J. & Zhang, S. -C. Inducing a magnetic monopole with topological surface states, Science 323, 1184 (2009)

work page 2009

[10] [10]

Chang, C.-Z. et al. Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator. Science 340, 167-170(2013)

work page 2013

[11] [11]

Deng, Y . et al. Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4. Science 367, 895-900(2020)

work page 2020

[12] [12]

, Rohrer, H

Binnig, G. , Rohrer, H. , Gerber, Ch., & Weibel, E . Surface Studies by Scanning Tunneling Microscopy. Phys. Rev. Lett. 49, 57 (1982)

work page 1982

[13] [13]

& Zandvliet, H

Borca, B. & Zandvliet, H. J.W. Image potential states of 2D materials. Applied Materials Today 39, 102304 (2024)

work page 2024

[14] [14]

Ge, J. -F. et al. Probing image potential states on the surface of the topological semimetal antimony, Phys. Rev. B 101, 035152 (2020)

work page 2020

[15] [15]

A., Wang, Y

Kubby, J. A., Wang, Y . R. & Greene, W. J. Fabry-Pérot transmission resonances in tunneling microscopy, Phys. Rev. B 43, 9346(R) (1991)

work page 1991

[16] [16]

Binnig, G. et al. Tunneling spectroscopy and inverse photoemission: image and field states, Phys. Rev. Lett. 55, 991(1985)

work page 1985

[17] [17]

Xiao, S. F. & Liu, Q. H. Geometric momentum and angular momentum for charge monopole system. Modern Physics Letters A. 33,1850125(2018)

work page 2018

[18] [18]

Chen, J. -L. et al. The monopole -hydrogen atom system and its connection with a four - dimensional harmonic oscillator J. Phys. A: Math. Gen. 32, 945(1999)

work page 1999

[19] [19]

Schindler, F. et al. Higher-order topology in bismuth. Nat. Phys. 14, 918–924 (2018)

work page 2018

[20] [20]

Zhang, H. et al. Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nat. Phys. 5, 438–442 (2009)

work page 2009

[21] [21]

Drozdov, I. K. et al. One-dimensional topological edge states of bismuth bilayers. Nat. Phys. 10, 664–669 (2014)

work page 2014

[22] [22]

& Gervais, G

Yu, O., Boivin, F., Silberztein, A. & Gervais, G. Observation of Temperature -Independent Anomalous Hall Effect in Thin Bismuth from Near Absolute Zero to 300 K Temperature. Phys. Rev. Lett. 134, 066603 (2025). 14

work page 2025

[23] [23]

& Gervais, G

Yu, O., Vijayakrishnan, S., Allgayer, R., Szkopek, T. & Gervais, G. Anomalous Hall effect in thin bismuth. Phys. Rev. B 109, L121406 (2024)

work page 2024

[24] [24]

Gou, J. et al. Scanning tunneling microscopy investigations of unoccupied surface states in two-dimensional semiconducting β - 3×3 - Bi/Si(111) surface. Phys. Chem. Chem. Phys. 20, 20188 (2018)

work page 2018

[25] [25]

Lu, Y . et al. Topological Properties Determined by Atomic Buckling in Self -Assembled Ultrathin Bi (110). Nano Lett. 15, 80-87 (2015)

work page 2015

[26] [26]

Liu, C. et al. Tunable topological edge states in black -phosphorus-like Bi(110). Phys. Rev. B 110, 195427 (2024)

work page 2024

[27] [27]

Gallagher, T. F. Rydberg atoms. Cambridge University Press (1994)

work page 1994

[28] [28]

Influence of the tip of the scanning tunneling microscope on surface electron lifetimes, Surf

Crampin, S. Influence of the tip of the scanning tunneling microscope on surface electron lifetimes, Surf. Sci. 600, 4280 (2006)

work page 2006

[29] [29]

Silkin, V . M. et al. Image potential states in graphene. Phys. Rev. B 80, 121408(R) (2009)

work page 2009

[30] [30]

& Fauster, T

Niesner, D. & Fauster, T. Image -potential states and work function of graphene, J. Phys.: Condens. Matter 26, 393001 (2014)

work page 2014

[31] [31]

Sugawara, K. et al. Interaction of Stark-shifted image potential states with quantum well states in ultrathin Ag(111) islands on Si(111)-√3×√3-B substrates. Phys. Rev. B 96, 075444 (2017)

work page 2017

[32] [32]

Craes, F. et al. Mapping Image Potential States on Graphene Quantum Dots, Phys. Rev. Lett. 111, 056804 (2013)

work page 2013

[33] [33]

& Haesendonck, C

Schouteden, K. & Haesendonck, C. V . Quantum Confinement of Hot Image -Potential State Electrons. Phys. Rev. Lett. 103, 266805(2009)

work page 2009

[34] [34]

Fowler, R. H. & Nordheim, L. Electron Emission in Intense Electric Fields. Proceedings of the Royal Society A 119, 173-181(1928). Methods: 15 The experiments were performed in a home -built low temperature STM equipped with molecular beam epitaxial (MBE) chamber, with base pressure of 310-10 mbar. Sample preparation High-purity Bi (99.999%) and Se (99.99...

work page 1928