Experimental Demonstration of Topological Surface States Protected by Time-Reversal Symmetry
Open accessAbstract
We report direct imaging of standing waves of the nontrivial surface states of topological insulator Bi2Te3 using a scanning tunneling microscope. The interference fringes are caused by the scattering of the topological states off Ag impurities and step edges on the Bi2Te3(111) surface. By studying the voltage-dependent standing wave patterns, we determine the energy dispersion E(k), which confirms the Dirac cone structure of the topological states. We further show that, very different from the conventional surface states, backscattering of the topological states by nonmagnetic impurities is completely suppressed. The absence of backscattering is a spectacular manifestation of the time-reversal symmetry, which offers a direct proof of the topological nature of the surface states.
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TL;DR
Direct low-temperature STM/STS imaging of Bi₂Te₃(111) reveals standing waves from topological surface states scattering off Ag trimers and step edges, confirming a Dirac cone dispersion. The absence of backscattering by nonmagnetic impurities provides a direct demonstration of time-reversal-symmetry protection.
Key findings
- Standing waves appear around Ag trimers and step edges; FFT power spectra show high intensity along Γ–M and vanishing intensity along Γ–K. Only the scattering wave vector q₂ along Γ–M is observed, while q₁ and q₃ are invisible, consistent with forbidden backscattering between k and −k for nonmagnetic impurities due to time-reversal symmetry.
- The measured energy dispersion in the Γ–K direction is linear, giving a Fermi velocity v_F = 4.8 × 10⁵ m/s and an estimated Dirac point energy of 0.25 eV. These agree with first-principles calculations and ARPES data, and the method probes unoccupied states inaccessible to ARPES.
- STS on bare Bi₂Te₃ shows the Fermi level within the bulk gap and a linear dI/dV in the gap attributed to gapless surface states; the Dirac point is invisible because surface states overlap in energy with the bulk valence band. Depositing 0.01 ML Ag shifts the Fermi level upward by 20–30 meV.
- Step-edge standing waves also propagate predominantly along Γ–M and yield the same v_F = 4.8 × 10⁵ m/s as the Ag-trimer standing waves.
Methods
- Experiments used an ultra-high-vacuum low-temperature STM/STS system (down to 0.4 K) with MBE. A stoichiometric Bi₂Te₃ film of ∼100 nm thickness was grown on Si(111); the Te-terminated (111) surface showed steps of 0.94 nm height (one quintuple layer).
- A small amount (0.01 ML) of Ag was deposited, forming trimers. dI/dV maps were recorded at bias voltages from 50 mV to 400 mV with 128 × 128 pixels, then Fourier transformed into q-space.
- The scattering geometry q₂ = √3 k in the Γ–K direction was used to extract E(k); a linear fit gave v_F and the Dirac point energy. First-principles calculations were used for comparison.
Limitations
- Stated: the exact structure of the Ag trimers is not fully determined—top-site adsorption versus substitutional Te model—though the authors state this does not affect the main conclusion and remains for further study.
- Stated: the Dirac point is invisible in STS because the surface states around Γ overlap with the bulk valence band; below 50 meV the interference fringes become obscured due to rapidly increasing wavelength and multiple topological states. The response to magnetic impurities and time-reversal-breaking perturbations remains an open question.
- Evident: the paper has no formal limitations section; the FFT resolution is limited to 2π/28 nm⁻¹ by the STM image size, and the backscattering suppression is demonstrated only for nonmagnetic Ag impurities and step edges, not for magnetic scatterers.
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- Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 4
Using the slope of the linear fitting and the same scattering geometry as for the Ag trimmer, the Fermi velocity from step-edge standing waves is 4.8 × 10^5 m/s, the same as that obtained from Ag impurities.
“Using the slope of the linear fitting together with the same scattering geometry as that for the Ag trimmer, the Fermi velocity is found to be 4 . 8 × 10 5 m/s, the same as that obtained from the standing waves caused by the Ag impurities.”
Quote in the paperSubstantial claim - Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 4
The standing waves produced by steps are predominantly propagating along the Gamma-M direction.
“Similar to the case of Ag trimmers, the standing waves produced by steps are predominantly propagating along the ¯ Γ -¯ M direction.”
Quote in the paperNot a substantial claim - Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 3
The slope of the linear fitting provides a measurement of the Dirac fermion velocity vF = 4.8 × 10^5 m/s.
“The slope of the linear fitting provides a measurement of the Dirac fermion velocity v F , which is 4 . 8 × 10 5 m/s.”
Quote in the paperSubstantial claim - Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
The Fermi level (zero bias) is within the energy gap of the Bi2Te3 film.
“The Fermi level (zero bias) is within the energy gap, indicating that the film is an intrinsic bulk insulator [22].”
Quote in the paperSubstantial claim - Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 3
The resulting k values vary linearly with energy.
“The resulting k values vary linearly with energy [Fig. 3(b)].”
Quote in the paperSubstantial claim
- Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 1
Backscattering of the topological states by nonmagnetic impurities is completely suppressed.
“We further show that, very different from the conventional surface states, the backscattering of the topological states by nonmagnetic impurities is completely suppressed.”
- Also stated in: § Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 1 · plain
“we find that the backscattering of topological states by a nonmagnetic impurity is forbidden.”
Quote in the paperSubstantial claim - Also stated in: § Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 1 · plain
- Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
The Fermi level shifts upwards by 20-30 meV after Ag deposition.
“the Fermi level shifts upwards in energy by 20 ∼ 30 meV after Ag deposition”
Quote in the paperNot a substantial claim - Findingcitation§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 4
Interference fringes are also found at the step edges on the surface.
“Interference fringes are also found at the step edges on the surface [29] [Figs. 4(a) to 4(h)].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [29] Z. Alpichshev et al. , arXiv:0908.0371 (2009).
Not checked in the free run.
- Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 4
The dispersion curve deduced from step-edge patterns shows a linear relation between the scattering wave vector and the energy.
“The dispersion curve deduced from these patterns again shows a linear relation between the scattering wave vector and the energy [Fig. 4(i)].”
Quote in the paperSubstantial claim - Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 1
Voltage-dependent standing wave patterns determine the energy dispersion E(k).
“By studying the voltage-dependent standing wave patterns, we determine the energy dispersion E ( k )”
Quote in the paperSubstantial claim - Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
Ag atoms deposited at 0.01 ML form trimmers on the Bi2Te3 surface.
“we deposited a small amount (0.01 ML) of Ag atoms, which form trimmers on the surface”
Quote in the paperNot a substantial claim - Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 1
Standing waves of the nontrivial surface states of the topological insulator Bi2Te3 were directly imaged using a low-temperature scanning tunneling microscope.
“We report direct imaging of standing waves of the nontrivial surface states of topological insulator Bi 2 Te 3 by using a low temperature scanning tunneling microscope.”
Quote in the paperSubstantial claim - Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
The differential conductance in the bulk insulating gap linearly depends on the bias.
“The differential conductance in the bulk insulating gap linearly depends on the bias and is attributed to the gapless surface states.”
Quote in the paperSubstantial claim - Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
The spatial modulation of LDOS by an Ag trimmer forms a hexagonal pattern whose edges are perpendicular to the Gamma-M directions in the surface Brillouin zone.
“The spatial modulation of LDOS by an Ag trimmer forms a hexagonal pattern, whose edges are perpendicular to the ¯ Γ -¯ Mdirections in SBZ.”
Quote in the paperNot a substantial claim - Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 3
The energy of the Dirac point is estimated to be 0.25 eV by the intercept of the dispersion with the energy axis.
“In addition, the energy of the Dirac point is estimated to be 0.25 eV by the intercept of the dispersion with the energy axis.”
Quote in the paperNot a substantial claim - Findingown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 3
The dispersion of massless Dirac fermions in the Gamma-K direction can be obtained using the interference patterns and their Fourier transforms.
“We can obtain the dispersion of the massless Dirac fermions in the ¯ Γ -¯ K direction using the interference patterns and their Fourier transforms.”
Quote in the paperSubstantial claim - Findingreasoning§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
Below 50 meV, the fringes become obscured.
“Below 50 meV, the fringes become obscured.”
Quote in the paperNot a substantial claim - Interpretationnone given§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 3
The unoccupied states, which are inaccessible to ARPES, can be probed by the standing waves with STM.
“More importantly, the unoccupied states, which are inaccessible to ARPES, can be probed by the standing waves with STM.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
For fermions, backscattering between k and -k by a time-reversal invariant nonmagnetic impurity potential is quantum-mechanically prohibited.
“Therefore, the backscattering between ⃗ k and -⃗ k is quantum-mechanically prohibited.”
Quote in the paperSubstantial claim- Rests on an assumption: The Ag impurities used in the experiment are nonmagnetic.
- Interpretationown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
After the surface states in the Gamma-K direction merge into the bulk conduction band at about 0.6 eV above the Dirac point, the standing waves fade out again.
“After the surface states in the ¯ Γ -¯ K direction merge into the bulk conduction band at ∼ 0.6 eV above the Dirac point [Fig. 1(d)], the standing waves fade out again.”
Quote in the paperNot a substantial claim - Interpretationcitation§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
The interference pattern is anisotropic as a result of the hexagram constant-energy contour.
“As expected, the interference pattern is anisotropic as a result of the hexagram CEC [10].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [10] source
Not checked in the free run.
- Interpretationown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 1
The determined energy dispersion confirms the Dirac cone structure of the topological states.
“which confirms the Dirac cone structure of the topological states.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 1
The absence of backscattering offers a direct proof of the topological nature of the surface states.
“which offers a direct proof of the topological nature of the surface states.”
Quote in the paperSubstantial claim- Rests on an assumption: The Ag impurities used in the experiment are nonmagnetic.
- Interpretationown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 1
The interference fringes are caused by scattering of the topological states off Ag impurities and step edges on the Bi2Te3(111) surface.
“The interference fringes are caused by the scattering of the topological states off Ag impurities and step edges on the Bi 2 Te 3 (111) surface.”
Quote in the paperSubstantial claim - Interpretationown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
The linear differential conductance in the bulk insulating gap is attributed to the gapless surface states.
“The differential conductance in the bulk insulating gap linearly depends on the bias and is attributed to the gapless surface states.”
Quote in the paperSubstantial claim - Interpretationcitation§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
The Bi2Te3 film is an intrinsic bulk insulator.
“The Fermi level (zero bias) is within the energy gap, indicating that the film is an intrinsic bulk insulator [22].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [22] source
Not checked in the free run.
- Interpretationown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 4
The existence of standing wave strongly supports the surface nature of topological states.
“The existence of standing wave strongly supports the surface nature of topological states.”
Quote in the paperSubstantial claim - Interpretationcitation§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
Most observed features in the interference pattern, including the extinction of wave vector q3, have been explained by a full theoretical treatment based on the T-matrix approach for multiband systems.
“Most of the observed features in the interference pattern, including the extinction of wave vector ⃗ q 3 , have been recently well explained by a full theoretical treatment [27] based on the T-matrix approach for multiband systems [28].”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [27] W.-C. Lee, C. J. Wu, D. P. Arovas, and S.-C. Zhang, arXiv:0910.1668 (2009). source
Not checked in the free run.
- cites [28] W.-C. Lee and C. J. Wu, arXiv:0906.1973 (2009).
Not checked in the free run.
- Interpretationown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
The Fermi level shift after Ag deposition suggests electron transfer from the Ag atoms to the substrate.
“suggesting electron transfer from the Ag atoms to the substrate”
Quote in the paperNot a substantial claim - Interpretationreasoning§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
The disappearance of q1 is accounted for by time-reversal invariance.
“There is a simple argument to account for the disappearance of q ⃗ 1 : the time-reversal invariance.”
Quote in the paperSubstantial claim - Interpretationcitation§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 3
These observations are in agreement with the results from first-principles calculation and ARPES data.
“These observations are in agreement with the results from the first-principles calculation and the ARPES data [7, 22].”
Quote in the paperSubstantial claim0 of 2 sources checked - Interpretationown data§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 1
The absence of backscattering is a manifestation of time-reversal symmetry.
“The absence of backscattering is a spectacular manifestation of the time-reversal symmetry”
Quote in the paperSubstantial claim- Rests on an assumption: The Ag impurities used in the experiment are nonmagnetic.
- Premisecitation§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2
The topological states of Bi2Te3 form a single Dirac cone at the center (Gamma point) of the surface Brillouin zone.
“According to the calculations [Fig. 1(d)], the topological states of Bi 2 Te 3 form a single Dirac cone at the center ( ¯ Γ point) of the SBZ [10, 11]”
Quote in the paperSubstantial claim0 of 2 sources checked - Premisereasoning§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 3
For q2, the scattering geometry determines q2 = sqrt(3) k, where k is the wave vector in the Gamma-K direction at a given energy.
“For ⃗ q 2 , the scattering geometry determines q 2 = √ 3 k [see Fig. 3(a)], where k is the wave vector in the ¯ Γ -¯ K direction at a given energy.”
Quote in the paperSubstantial claim
- May combine two claims. 4
- Classified as probably not a substantial claim. 2
May combine two claims.
Findingnone given§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2The spatial period of the standing wave scales inversely with the bias voltage and is determined by the momentum transfer during scattering at a given energy.
“The spatial period of the standing wave scales inversely with the bias voltage and is determined by the momentum transfer during scattering at a given energy.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2Among the three possible scattering wave vectors q1, q2, and q3, only q2 is along the Gamma-M directions and can generate the observed standing waves; q1 and q3 are invisible in the power spectra.
“Among them, however, only q ⃗ 2 is along the ¯ Γ -¯ Mdirections and can generate the observed standing waves. Both ⃗ q 1 and q ⃗ 3 are invisible in the power spectra.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 2In the Fourier-transformed dI/dV maps, the regions with high intensity are always oriented toward the Gamma-M directions, while the intensity in the Gamma-K directions vanishes.
“the regions with high intensity are always oriented toward the ¯ Γ -¯ M directions, while the intensity in the ¯ Γ -¯ K directions vanishes”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Interpretationreasoning§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 4The fringes at step edges are clearly visible even at negative bias voltages, probably owing to the stronger scattering potential compared to that of the Ag trimmers.
“The fringes are clearly visible even at the negative bias voltages probably owing to the stronger scattering potential compared to that of the Ag trimmers.”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
Premisecitation§ Experimental demonstration of the topological surface states protected by the time-reversal symmetry · p. 4A time-reversal breaking perturbation, such as magnetic impurities, can induce scattering between the states |k↑> and -|k↓> and open up a local energy gap at the Dirac point.
“Theoretically, it was pointed out [5, 30, 31] that a time-reversal breaking perturbation, such as magnetic impurities, can induce scattering between the states | ⃗ k ↑〉 and -| ⃗ k ↓〉 and open up a local energy gap at the Dirac point.”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [5] source
Not checked in the free run.
- cites [30] Q. Liu, C.-X. Liu, C. Xu, X.-L. Qi, and S.-C. Zhang, Phys. Rev. Lett. 102 , 156603 (2009).
Not checked in the free run.
- cites [31]
Not checked in the free run.
The paper relies on this without support.
The Ag impurities used in the experiment are nonmagnetic.
“the backscattering of the topological states by nonmagnetic impurities is completely suppressed”
“where U is a time-reversal invariant operator and represents the impurity potential of the nonmagnetic Ag impurity.”
The authors state this as an assumption.
The scattering of surface states by defects is elastic.
“During elastic scattering, a defect scatters the incident wave with a wave vector ⃗ k i into ⃗ k f = ⃗ k i + q ⃗ , with ⃗ k i and ⃗ k f being on the same constant-energy contour (CEC).”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The same conclusion has emerged from studies of the electronic interference patterns near defects or steps on the surface that electrons never turn completely around when scattered in other topological insulators.
“The same conclusion has emerged from studies of the electronic interference patterns near defects or steps on the surface that electrons never turn completely around when scattered in other topological insulators (25,26)).”
From Three-Dimensional Topological Insulators (2011) · cites this paper as [25]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Inconsistent panel range for the Fourier spectra in Figure 2
MinorThe main text identifies the Fourier spectra as panels 2(h) through 2(l), but the caption identifies them as 2(h) through 2(k). The caption's range also has four panels for the five dI/dV maps in 2(c) through 2(g).
“(h) to (k) The FFT power spectra of the dI/dV maps in (c) to (g).”
“we performed Fourier transformation of the dI/dV maps into the ⃗ q -space [Figs. 2(h) to 2(l)].”
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Self-citationTransport evidence for Fermi-arc-mediated chirality transfer in the Dirac semimetal Cd3As2
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Isayev, O., Fourches, D., Muratov, E. N., Oses, C., Rasch, K., Tropsha, A., & Curtarolo, S. (2015) · Chemistry of Materials · doi:10.1021/cm503507h · Cited by 300 · see citing works
Quasiparticle interference of the Fermi arcs and surface-bulk connectivity of a Weyl semimetal
Inoue, H., Gyenis, A., Wang, Z., Li, J., Oh, S. W., Jiang, S., Ni, N., Bernevig, B. A., & Yazdani, A. (2016) · Science · doi:10.1126/science.aad8766 · Cited by 190 · see citing works
Zheng, H., Xu, S.-Y., Bian, G., Guo, C., Chang, G., Sanchez, D. S., Belopolski, I., Lee, C.-C., Huang, S.-M., Zhang, X., et al. (2016) · ACS Nano · doi:10.1021/acsnano.5b06807 · Cited by 129 · see citing works
- Authors
- Zhang, T., Cheng, P., Chen, X., Jia, J.-F., Ma, X., He, K., Wang, L., Zhang, H., Dai, X., Fang, Z., Xie, X., & Xue, Q.-K.
- Affiliations
- Chinese Academy of Sciences (CN)
- Tsinghua University (CN)
- ORCID
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2009-12-23
- Volume
- 103
- Issue
- 26
- Pages
- 266803-266803
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2088225002
- Semantic Scholar
- 16e513ba25dcf50a9673b691f72c57c61e648212
- Cited by
- 744
- References
- 27
- Citations by year
- 2026: 11 · 2025: 23 · 2024: 18 · 2023: 20 · 2022: 30 · 2021: 38 · 2020: 45 · 2019: 31 · 2018: 46 · 2017: 42 · 2016: 61 · 2015: 71 · 2014: 62 · 2013: 79 · 2012: 72
- FWCI
- 30.99
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Rare & Special e-Zone (The Hong Kong University of Science and Technology) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Surface and Thin Film Phenomena
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Advanced Condensed Matter Physics
- Keywords
- topological insulators, Bi2Te3, topological surface states, time-reversal symmetry, Dirac cones, standing waves, surface steps, energy dispersion, scanning tunneling microscopy imaging
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-06.
Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-06.
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