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Correlated Insulator Behaviour at Half-Filling in Magic-Angle Graphene Superlattices

Nature, 2018 · paper · Materials

Studies two graphene sheets twisted near the theoretical "magic angle". The resulting flat electronic bands produce insulating states at half-filling without a magnetic field, which the authors identify as Mott-like insulators of electrons localized in the moiré superlattice.

Checked: PDF from arxiv.org, fetched 4 Oct 2026. Page numbers are those of that PDF.

10 findings

Substance Fig. 3 caption says "reduction in ΔC (on p-side only)", but the main text says "capacitance reduction ... for only the n-side HFIP in this device" and Fig. 3a shows the capacitance dip at +ns/2; the caption should read n-side.
  • p. 13 Figure 3 caption, (a)
    reduction in Δ𝐶 (on p-side only)
  • p. 4 Main text, capacitance paragraph
    for only the n-side HFIP in this device
  • p. 13 Figure 3 caption, panel (a)
    (on p-side only)
  • p. 4 Main text, capacitance paragraph
    is weaker and only visible in the dissipation data

Judge's note: Zoomed Fig. 3a from the PDF: the 0.3 K capacitance trace has a sharp dip at n ≈ +1.4e12 cm^-2, between the blue dashed lines labelled ns/2, and no dip at -ns/2, where only two small loss-tangent peaks appear. Page 4 says the state at -ns/2 "is weaker and only visible in the dissipation data" and that the capacitance reduction is seen "for only the n-side HFIP". The paper's convention is p-side = negative n (Fig. 2 caption: (c) p-side at n ≈ -1.8 to -1.3, (d) n-side at positive n; Fig. 4 likewise), so the caption contradicts both the plot and the text and no alternative convention rescues it.

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Substance Twist angle of device D2 is "θ=1.10°" in the Fig. 2a inset (where 1.12° is attached to D3) but "θ = (1.12 ± 0.01)°, compared to θ = (1.12 ± 0.02)°" in SI II.E.2 and "twist angle of device D2 θ = 1.12°" in SI III.B.
  • p. 12 Figure 2a inset (labels in the image)
    D2 θ=1.10°
  • p. 26 SI II.E.2, text below Fig. S7
    θ = (1.12 ± 0.01)◦, compared to θ =
  • p. 29 SI III.B, last paragraph
    slightly larger twist angle of device D2 θ = 1.12◦
  • p. 12 Figure 2a inset, row D2
    θ=1.10°

Only the authors can say what is right here.

Judge's note: Zoomed the Fig. 2a inset from the PDF at 6x: the rows read D1 1.08°, D2 1.10°, D3 1.12°, D4 1.16°. Page 26 gives D2 as 1.12° by two methods (1/B periodicity and ns = 2.9e12 cm^-2) and page 29 uses 1.12° for D2 again. Scaling ns as θ² from D1 (2.7e12 at 1.08°) gives 2.90e12 for 1.12° and 2.80e12 for 1.10°, so the supplement's ns agrees with 1.12°, not with the inset. The quoted uncertainties (±0.01, ±0.02) do not reach 1.10°. No text explains a second value for D2; the supplement gives no angle for D3, so whether the D2 and D3 labels are swapped or the D2 label is stale is for the authors to say. The separate point about the size of the ±0.01° uncertainty is handled in C028.

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Reference or label Page 4: "a Fermi velocity reduced to about 0.15v0 is necessary for a good fit to the data (see Fig. S4 in SI)" — Fig. S4 is the four-probe/two-probe conductance of device D4; the capacitance fit is Fig. S9.
  • p. 4 Main text, flat-band evidence paragraph
    is necessary for a good fit to the data (see Fig. S4 in SI)
  • p. 21 FIG. S4 caption
    FIG. S4. Four-probe (upper panel) and two-probe (lower panel) conductance measured in device
  • p. 28 FIG. S9 caption
    FIG. S9. Capacitance of device D2 near the charge neutrality point, and fitting curves according
  • p. 4 Main text, flat-band evidence paragraph
    (see Fig. S4 in SI)

Judge's note: Page 21: Fig. S4 shows Gxx and G2 of device D4 at 0.3 K, no capacitance data. Page 28: Fig. S9 is "Capacitance of device D2 near the charge neutrality point, and fitting curves" with vF = 0.15v0, 0.5v0 and v0 curves, and the SI text on page 29 itself refers to this fit as Fig. S9. SI figures are numbered consecutively, so no alternative numbering makes the pointer valid.

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Reference or label Reference 21 (page 9): "Phys. Rev. B 125413, 86 (2012)." — volume and article number are transposed; the SI reference [21] prints the same paper as "Phys. Rev. B 86, 125413 (2012)."
  • p. 9 Reference 21
    Phys. Rev. B 125413, 86 (2012).
  • p. 37 SI reference [21]
    Phys. Rev. B 86, 125413 (2012).
  • p. 9 Reference 21
    Phys. Rev. B 125413, 86 (2012)

Judge's note: Page 9 image shows bold 125413 in the volume position followed by 86. Page 37 image shows the same paper in the SI list as bold 86, 125413. Every other Phys. Rev. B entry in the main list uses the volume, article-number order (e.g. ref. 10 "83, 195127"), except ref. 11 "Phys. Rev. B 094506, 90 (2014)", which shows the same transposition and is not part of this cluster. The correction is established from the paper's own SI list, not outside knowledge.

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Reference or label Reference list: ref. 11 "Phys. Rev. B 094506, 90 (2014)" and ref. 21 "Phys. Rev. B 125413, 86 (2012)" print the article number (in bold) where the volume belongs and the volume after it; ref. 12 title has "Interaction-driver" (for "driven") and ref. 24 title has "Robuts" (for "Robust").
  • p. 9 Reference 11
    Phys. Rev. B 094506, 90 (2014).
  • p. 9 Ref. 24
    Intrinsic and Robuts Nature
  • p. 9 Ref. 12
    Interaction-driver topological
  • p. 9 Ref. 11
    Phys. Rev. B 094506, 90 (2014)

Judge's note: Read page 9. In refs 11 and 21 the six-digit number is set in bold in the volume position, followed by 90 and 86; every other Phys. Rev. B entry on the page gives a bold two- or three-digit volume and then a six-digit article number (e.g. ref. 10 "83, 195127"). The supplement's own list (page 37, ref. [21]) prints the same de Laissardière paper as "Phys. Rev. B 86, 125413 (2012)", which confirms the transposition from inside the paper. "Robuts" and "Interaction-driver" are visible on the page image and are not extraction artefacts.

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Reference or label The same paper (Ashoori et al., Phys. Rev. Lett. 68, 3088) is dated two ways: main ref. 28 "Phys. Rev. Lett. 68, 3088 (1996)" and supplement ref. [13] "Phys. Rev. Lett. 68, 3088 (1992)"; the main-list year 1996 appears to be the wrong one.
  • p. 10 Reference 28
    Rev. Lett. 68, 3088 (1996).
  • p. 37 SI reference [13]
    Lett. 68, 3088 (1992).
  • p. 10 Reference 28
    Rev. Lett. 68, 3088 (1996)
  • p. 10 Reference 28
    Phys. Rev. Lett. 68, 3088 (1996)

Judge's note: Read page 10 (ref. 28) and page 37 (SI ref. [13]). Title, journal, volume and page are identical, so the two entries are one paper with two years. The paper's own reference list indicates which is right: Phys. Rev. Lett. volume 62 is dated 1989 (ref. 13) and the journal runs two volumes a year, which puts volume 68 in 1992, as the supplement has it. The 1996 may have been carried over from the neighbouring Tinkham entry (ref. 7, 1996).

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Reference or label Fig. 4 caption: "The white line cuts show the conductance at the labeled fields." Panels (a) and (b) contain no white line cuts and no field labels.
  • p. 14 Figure 4 caption
    The white line cuts show the conductance at the labeled fields.

Only the authors can say what is right here.

Judge's note: Rendered Fig. 4a-b from the PDF at 4x: both panels are plain colour maps of conductance against n and B with no overlaid white traces and no field labels. Field labels (0T to 8T) appear only in panel (c), the Arrhenius plot, which is described separately in the next caption sentence. The caption sentence describes an element absent from the published figure, probably left from an earlier version; the authors should either remove the sentence or restore the line cuts.

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Reference or label SI V says "Fig. S11 shows the same data as in Fig. S7a, but plotted versus 1/B instead of B", but Fig. S7a is the derivative dG/dn "measured at 4 K" while Fig. S11 plots G (S) and ΔG; the preceding sentence refers to Fig. S2 (D1 conductance, T ≤ 0.3 K).
  • p. 30 SI V, last paragraph
    Fig. S11 shows the same data as in Fig. S7a, but plotted versus 1/B instead
  • p. 26 FIG. S7 caption
    (a) Derivative of magneto-conductance data of device D1, measured at 4 K
  • p. 30 SI section V, last paragraph
    The quantum oscillations in device D1 is shown in Fig. S2(a).
  • p. 26 Fig. S7 caption
    (a) Derivative of magneto-conductance data of device D1, measured at 4 K.

Only the authors can say what is right here.

Judge's note: Read pages 18, 26, 30 and 31. Fig. S7(a) is dG/dn in arbitrary units at 4 K, plotted against B. Fig. S11 is raw G in S (colour scale up to 1×10^-4 S, the same 0.1 mS scale as the D1 panel of Fig. S2) and background-subtracted ΔG against 1/B, and it shows the half-filling states, which the SI says disappear on warming to 4 K. The sentence before the pointer cites Fig. S2. The pointer to S7a therefore looks like it should be Fig. S2 (D1), but the temperature of Fig. S11 is not stated, so only the authors can say which data set it is. The secondary points in this cluster ("S2(a)" with no panel letters, "oscillations ... is shown") are judged under C033.

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Reference or label Fig. S10 caption reads "Temperature-dependent magneto-conductance of device D1" but the plotted axis is "ΔR (kΩ)"; likewise the text says the Lifshitz-Kosevich formula is "applied to conductance" while Eq. (7) is written for ΔR and the text says a background "of resistance" is removed.
  • p. 30 FIG. S10 caption vs y-axis label 'ΔR (kΩ)'
    Temperature-dependent magneto-conductance of device D1 at gate voltage
  • p. 29 SI V, before Eq. (7)
    applied to conductance
  • p. 30 FIG. S10 caption
    Temperature-dependent magneto-conductance of device D1
  • p. 30 Fig. S10 y-axis
    ΔR (kΩ)

Judge's note: Read pages 29 and 30. The y-axis of Fig. S10 panels a-c is ΔR in kΩ. The caption calls the quantity magneto-conductance. On page 29 the text says "a polynomial background of resistance in B is first removed" and then "the Lifshitz-Kosevich formula applied to conductance", followed by Eq. (7) for ΔR. The plotted and fitted quantity is resistance; the caption and the phrase before Eq. (7) name the wrong quantity. The analysis itself is unaffected.

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Reference or label Fig. S11 caption: "Quantum oscillations in device S1 with twist angle of 1.08°"; no device S1 exists, the 1.08° device is D1.
  • p. 31 FIG. S11 caption
    Quantum oscillations in device S1 with twist angle of 1.08◦
  • p. 31 Fig. S11 caption
    Quantum oscillations in device S1 with twist angle of 1.08◦.
  • p. 31 FIG. S11 caption
    Quantum oscillations in device S1 with twist angle of 1.08
  • p. 30 Section V text
    The quantum oscillations in device D1 is shown in Fig. S2(a)

Judge's note: Read page 31 and the referring paragraph on page 30. The caption prints "device S1". The paper's devices are D1 to D4, 1.08° is D1's twist angle (page 29: "device D1 θ = 1.08°"), and the paragraph introducing Fig. S11 is about device D1. A search of the text finds no other "S1" device.

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20 minor (typos and the like; most checked by the judge only)
Wording Fig. 3 caption (c) says "at different temperatures from 4.5 K to 120 K", while the legend of Fig. 3c labels the lowest-temperature curve "4 K".
  • p. 13 Figure 3 caption, panel (c)
    temperatures from 4.5 K to 120 K
  • p. 13 Figure 3c legend
    4 K
  • p. 13 Fig. 3 caption (c)
    at different temperatures from 4.5 K to 120 K

Judge's note: Zoomed Fig. 3c from the PDF: the legend reads 4 K, 8 K, 15 K, 29 K, 40 K, 50 K, 70 K, 100 K, 120 K. The upper bound agrees with the caption; the lower bound does not as printed. The legend may simply be rounded to whole kelvin (4.5 K shown as 4 K), so either form could be legitimate; noted as a minor consistency point.

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Wording SI II.E.2: "A = (1.37 ± 0.04) × 10^-12 cm^2 and θ = (1.12 ± 0.01)°"; propagating the stated uncertainty on A (θ ∝ A^-1/2) gives about ±0.016-0.017°, which rounds to ±0.02°, not ±0.01°.
  • p. 26 SI Sec. II.E.2
    θ = (1.12 ± 0.01)◦
  • p. 26 SI text after Fig. S7
    A = (1.37 ± 0.04) × 10−12 cm2 and θ = (1.12 ± 0.01)◦

Judge's note: Read page 26. A carries a 2.9% relative uncertainty (and the periodicity 0.033 ± 0.001 T^-1 a 3.0% one); half of that on θ = 1.12° is 0.016-0.017°. The companion estimate from ns, with a similar relative uncertainty (0.1/2.9 = 3.4%), is quoted as ±0.02°, which is consistent with the same propagation. The paper does not describe how the ±0.01° was obtained, so truncation or a different error estimate cannot be excluded; the central value is unaffected. A minor consistency point.

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Wording SI Eq. (8): "vF(θ) = (1 − 3α²)/(1 + 6α²) ≈ 1 − 9α² (α ≤ 1)"; the right-hand side is the dimensionless ratio vF/v0 (v0 is omitted), and the expansion holds for α ≪ 1, not α ≤ 1.
  • p. 31 SI Eq. (8)
    ≈1 −9α2(α ≤1)

Judge's note: Read page 31. The image shows "≤", not "≪". The text defines v0 as the original Fermi velocity yet the equation sets vF equal to a pure number, so a factor v0 (or a statement that vF is in units of v0) is missing. The expansion 1 − 9α² is the small-α limit: at α = 1 the exact expression is about −0.29 and the approximation −8; at α² = 1/3, where the text says vF passes through zero, the approximation gives −2. Notational looseness; no result depends on it.

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Typo Abstract, last sentence: "the bandwidth tunability though twist angle may pave the way" — "though" should read "through".
  • p. 1 Abstract, last sentence
    tunability though twist angle may pave the way
  • p. 1 Abstract, last sentence
    bandwidth tunability though twist angle

Judge's note: Read the abstract on the page 1 image; the word printed is "though" and the sentence only parses with "through". Not an extraction artefact.

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Typo Page 2, first paragraph: "the fractional quantum hall effect" — "Hall" is printed in lower case.
  • p. 2 Introduction, paragraph 1
    fractional quantum hall effect

Judge's note: Page 2 image shows "quantum hall effect 8". The paper capitalises it elsewhere ("Hall measurements" and "Hall densities" on page 6; references 3, 8 and 19 "quantum Hall effect").

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Typo Page 5: "when the thermal energy kT becomes comparable to vF kθ/2 ≈ 4 meV, the energy scale spanning the Dirac-like portion of the band" — the factor ħ is missing; elsewhere the quantity is written ħv0kθ (page 3, Fig. 1e).
  • p. 5 Main text, first paragraph (inline fraction)
    becomes comparable to
  • p. 11 Figure 1e label
    ℏ𝑣0𝑘𝜃

Judge's note: Page 5 image shows the inline fraction as vF kθ over 2 with no ħ, equated to an energy in meV. Page 3 writes "ħv0kθ" and Fig. 1e on page 11 is labelled ħv0kθ/2, so the main text does not use ħ=1 units. With ħ included, vF = 4×10^4 m/s and kθ ≈ Kθ at 1.08° give about 4.2 meV, matching the printed value. Notation omission only; the number is fine.

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Typo Page 5: "results in half-filling Mott-like gaps. 32The realistic scenario" — no space between the superscript citation 32 and the next sentence.
  • p. 5 paragraph on Hubbard model
    Mott-like gaps. 32The realistic

Judge's note: Page 5 image shows the superscript 32 directly abutting "The" with no space; other citations in the paper are followed by a space (e.g. "electrons.32, 33 To this end" in the same paragraph). Visible in the image, not an extraction artefact. Purely typographic.

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Typo Page 6: "Figures 4a-b shows that both insulating phases start to conduct" — plural subject with singular verb; should be "show".
  • p. 6 paragraph 2
    Figures 4a-b shows that both insulating phases

Judge's note: Read on the page 6 image: "Figures 4a-b" at the end of one line, "shows that" at the start of the next.

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Typo Page 8: Acknowledgements have "Capacitance work by R.A., A.D., and S.L.T." while Author Contributions have "S.L.T., A.D. and R.C.A. measured capacitance data." — the same author (R.C. Ashoori) abbreviated two ways.
  • p. 8 Acknowledgements
    Capacitance work by R.A., A.D., and S.L.T.
  • p. 8 Author Contributions
    S.L.T., A.D. and R.C.A. measured capacitance data.

Judge's note: Both read on the page 8 image. The author list on page 1 gives "R.C. Ashoori", and no other author has initials R.A., so "R.A." should be "R.C.A.".

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Typo SI reference [12] title reads "Hofstadters butterfly and the fractal quantum Hall effect", without the apostrophe that main-text reference 3 has ("Hofstadter's butterfly").
  • p. 37 SI reference [12]
    Hofstadters butterfly and the fractal quantum Hall effect
  • p. 8 Reference 3
    Hofstadter's butterfly and the fractal quantum Hall effect

Judge's note: Read page 37: the image shows "Hofstadters butterfly" with no apostrophe, so it is not an extraction loss. The main reference list (text of page 8, reference 3) prints the same title as "Hofstadter's butterfly". Cosmetic.

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Typo The same journal is abbreviated two ways in the main reference list, one with a misspelling: ref. 6 "Proc. Natl. Acad. Scie. U. S. A." and ref. 20 "Proc. National Acad. Sci. U. S. A."; the supplement uses "Proc. Natl. Acad. Sci. U. S. A." for both papers.
  • p. 9 Reference 6
    Acad. Scie. U. S. A. 108(30)
  • p. 9 Reference 20
    Proc. National Acad. Sci. U. S. A. 114, 3364 (2017).

Judge's note: Read page 9 (refs 6 and 20) and the supplement list on page 37 (ref. [18], the same Bistritzer and MacDonald paper, printed "Proc. Natl. Acad. Sci. U. S. A."); the text layer of the supplement list gives the same form for the Kim et al. paper. "Scie." is a misspelling and "National" is an unabbreviated variant. Cosmetic.

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Typo Ref. 11 prints two author names with grave accents: "Hèbert, F., Grèmaud, B."; the names are probably Hébert and Grémaud (acute accents).
  • p. 9 Reference 11
    Hèbert, F., Grèmaud, B.
  • p. 9 Reference 11
    Phys. Rev. B 094506, 90 (2014)

Only the authors can say what is right here.

Judge's note: Rests on outside knowledge: that these authors spell their names Hébert and Grémaud with acute accents. Page 9 does show grave accents on both names, so the quote is accurate, but the correct spelling cannot be checked from the files. The second half of this cluster (volume and article number transposed in ref. 11) is real and is already covered by C016; it is not counted again here.

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Typo SI V cites "Fig. S2(a)" although Fig. S2's panels are labelled only D1 and D3; and subject-verb disagreements: "The quantum oscillations in device D1 is shown", "The dashed fans labels" (Fig. S7 caption), "the thick lines denotes" (Fig. S13 caption).
  • p. 30 SI V, last paragraph
    The quantum oscillations in device D1 is shown in Fig. S2(a).
  • p. 18 FIG. S2
    FIG. S2. Magneto-conductance of two different samples D1 (also shown in main text) and D3.
  • p. 30 Section V
    The quantum oscillations in device D1 is shown
  • p. 26 FIG. S7 caption
    The dashed fans labels the main (green)

Judge's note: Read pages 18, 26, 30 and 34. Fig. S2 has two panels marked D1 and D3 with no (a)/(b) letters; the reader can infer the upper D1 panel is meant. The three grammar slips are printed as quoted. All cosmetic. The separate question of the Fig. S7a pointer is in C032.

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Typo Subject-verb disagreement in three SI captions: Fig. S7 "The dashed fans labels the main (green) and satellite (blue) Landau fans"; Fig. S13 "the thick lines denotes the Ks – Ms – K's direction"; Fig. S14 "The red dashed lines denotes the energy".
  • p. 26 FIG. S7 caption
    The dashed fans labels the main (green) and satellite (blue)
  • p. 35 FIG. S14 caption
    The red dashed lines denotes the energy
  • p. 34 FIG. S13 caption
    the thick lines denotes the Ks

Judge's note: All three read as quoted on pages 26, 34 and 35. Plural subjects with singular verbs; should be "label" and "denote". Cosmetic.

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Typo SI III.B: "the analysis present here suffices to demonstrate" should read "the analysis presented here".
  • p. 29 Section III.B, last paragraph
    the analysis present here suffices to demonstrate

Judge's note: Reads as quoted on page 29. Wrong word form; cosmetic.

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Typo SI reference "[2] Sanchez-Yamagishi, J. D., Taychatanapat, T., Watanabe, K., Taniguchi, T., Yacoby, A. & ..." is listed but never cited in the supplement text.
  • p. 35 SI reference [2]
    [2] Sanchez-Yamagishi, J. D., Taychatanapat, T., Watanabe, K., Taniguchi, T., Yacoby, A. &

Judge's note: Listed every bracketed citation in the supplement text: [1], [1, 7], [1, 19], [1, 3, 19], [3, 4], [4], [5], [6], [8], [9], [10–12], [13], [14], [14–16], [17–22], [18], [18, 22], [24–26], [27], [28]. None includes 2, alone or in a range. (By the same list [23] is also uncited.) The entry itself is on pages 35-36. Cosmetic.

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Typo SI reference [3] prints "Huang, S., Larentiz, S.," while SI reference [4] prints "Fallahazad, B., Larentis, S.": the same co-author's surname is spelled two ways.
  • p. 36 SI reference [3]
    Huang, S., Larentiz, S.,
  • p. 36 SI reference [4]
    Fallahazad, B., Larentis, S., Taniguchi, T.
  • p. 36 SI reference [3]
    Huang, S., Larentiz, S.
  • p. 36 SI reference [4]
    Huang, S., Fallahazad, B., Larentis, S.

Judge's note: Read page 36. Reference [3] has "Larentiz, S." and reference [4] has "Larentis, S.", in two consecutive papers from the same group with overlapping author lists. The inconsistency is internal to the page; that "Larentis" is the right form rests on outside knowledge of the cited papers.

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Typo SI reference [11] prints the author as "Mischenko, A."; the reviewer says the cited Nature paper spells it Mishchenko.
  • p. 36 SI reference [11]
    Mischenko, A. S.

Only the authors can say what is right here.

Judge's note: Rests on outside knowledge: that the author of Ponomarenko et al., Nature 497, 594 (2013) is spelled "Mishchenko". Read page 36: the entry prints "Mischenko, A., Mayorov, A. S." (the reviewer's quote "Mischenko, A. S." ran two names together, which is why it was not found in the text layer). The spelling on the page is as claimed; the correct form cannot be checked from the files.

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Typo SI reference [15] prints "Fang, T., Aniruddha, K., Xing, H. & Jena, D."; the reviewer says the second author is Aniruddha Konar, so given name and surname are inverted.
  • p. 37 SI reference [15]
    Fang, T., Aniruddha, K., Xing, H. & Jena, D.

Only the authors can say what is right here.

Judge's note: Rests on outside knowledge: that the second author of Appl. Phys. Lett. 91, 092109 (2007) is Aniruddha Konar. Read page 37: the entry prints "Aniruddha, K." as quoted. Nothing in the files gives the cited paper's author list, so the correct form is for the authors to confirm.

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Typo SI reference [28] prints "Fukui, T., Hatsugi, Y. & Suzuki, H."; the reviewer says the second author is Hatsugai.
  • p. 38 SI reference [28]
    Fukui, T., Hatsugi, Y. & Suzuki, H.

Only the authors can say what is right here.

Judge's note: Rests on outside knowledge: that the second author of J. Phys. Soc. Jpn. 74, 1674 (2005) is Y. Hatsugai. Read page 38: the entry prints "Hatsugi, Y." as quoted. The name appears nowhere else in the files, so the correct spelling cannot be checked here.

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