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Why do the lowest bands flatten at the first magic angle?

At the first magic angle the interlayer tunnelling and the kinetic scale are comparable, and the Fermi velocity of the lowest bands vanishes.1 The continuum model gives the angle at which this happens main.tex:45-51, the correlated insulator reported at half filling of those bands is the transport signature of the same flattening,2 and superconductivity appears on doping away from it.3

1. Moire bands in twisted double-layer graphene 2. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices 3. Unconventional superconductivity in magic-angle graphene superlattices

Which of those three measures the angle directly?

Only the transport papers do. The continuum model derives the angle rather than measuring it.

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Which of my papers measures the twist angle, rather than deriving it?

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The superconductivity paper states an angle measured on the device.1 The continuum model derives the same angle from the tunnelling and kinetic scales, so it is a prediction rather than a measurement.2

1. Unconventional superconductivity in magic-angle graphene superlattices 2. Moire bands in twisted double-layer graphene

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Compare the twist-angle values across the two transport papers and the continuum model in @

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main.texEditingReviewing

42\section{Flat-band model}43−The flat bands appear near a small twist angle.44The flat bands appear near 1.1 degrees~\cite{cao2018unconventional}.+5−4 wordsAcceptReject45At this angle the moir\'e superlattice quenches the46kinetic energy of the lowest bands, so interaction47rather than dispersion sets the scale.48−Transport measurements report an insulating state at half filling.49Transport measurements report correlated-insulator behaviour at half filling.

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ReferencesAPA 7

Cao, Y., Fatemi, V., Fang, S., Watanabe, K., Taniguchi, T., Kaxiras, E., & Jarillo-Herrero, P. (2018). Unconventional superconductivity in magic-angle graphene superlattices. Nature, 556(7699), 43–50. https://doi.org/10.1038/nature26160

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44and the Fermi velocity of the lowest bands vanishes45at a sequence of values of $\alpha$, the first of46which is $\alpha \approx 0.605$.

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The flat-band condition is derived before either measurement is used, and the two measurements are the right ones for it. Two statements stand on nothing in the library.

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Correlated insulator behaviour at half-filling in magic-angle graphene superlattices reports that result, and the note cites Cao et al., Nature 2018 for it, which is the superconductivity paper paper.tex:71-75.

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No source foundpaper.tex:71

Once the bands are flat, interaction rather than dispersion sets the scale of the problem.

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paper.tex3 findings

69\section{What follows from a flat band}7071Once the bands are flat, interaction rather than dispersion sets the72scale of the problem. Transport measurements on devices near the73first magic angle report correlated-insulator behaviour at half

No source foundpaper.tex:71

The excerpts discuss flat bands and correlated phases but do not establish what sets the scale of the problem once the bands are flat.

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the Fermi velocity of the lowest bands vanishes at a sequence of values of α, the first of which is α ≈ 0.605

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Unconventional superconductivity in magic-angle graphene superlatticesCite
Supports thisIntroduction · page 43
It states the magic angles as the values where the Fermi velocity drops to zero, and gives the first as 1.1 degrees.
Special angles, namely the ‘magic angles’, exist where the Fermi velocity drops to zero, the first of which is about θ_magic = 1.1°.

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the Fermi velocity of the lowest bands vanishes at a sequence ofSearch

Moire bands in twisted double-layer grapheneAdd to library
Bistritzer, R. · MacDonald, A. H. · 2011 · 4,283 citations

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Unconventional superconductivity in magic-angle graphene superlatticesCite
Supports thisIntroduction · page 43
Special angles, namely the ‘magic angles’, exist where the Fermi velocity drops to zero, the first of which is about θ_magic = 1.1°.
This passage credits Bistritzer 2011, Moire bands in twisted double-layer grapheneCite Bistritzer 2011

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Checked againstNature

Length4,180 of 4,300 words
Display items5 of 6
Summary paragraph186 of 200 words
References54 of 50

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Another documentBibTeXZoteroMendeleyMoire transport review (Twisted bilayers, 14)

The marvels of moire materials 2021
Unconventional superconductivity in magic-angle graphene superlattices 2018already here
Graphene bilayer with a twist: electronic structure 2007
Moire bands in twisted double-layer graphene 2011already here
Correlated insulator behaviour at half-filling in magic-angle graphene superlattices 2018

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Unconventional superconductivity in magic-angle graphene superlattices

Cao, Y. · Fatemi, V. · Fang, S. +4 · 2018

nature26160.pdf8 pages · indexed

Cite key: cao2018unconventional
Type: Journal article
Publication: Nature
Publisher: Springer Nature
Date: 2018-03-05
Volume: 556
Pages: 43-50
ISSN: 0028-0836
DOI: 10.1038/nature26160
Cited here: 3 times

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Cao et al., Nature 2018Reading aids

Objectivewhether a twist angle alone can flatten the bands · p. 43
Noveltysuperconductivity in a system with no dopant chemistry · p. 43
Resulttwo domes flanking the half-filling insulator · p. 45
Methodtear-and-stack assembly, transport at 70 mK · p. 48

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Near the magic angle the moire superlattice quenches the kinetic energy and the lowest bands become flat bands.

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Cao et al., Nature 2018 · references54 parsed

[18] Cao, Y. et al. Correlated Insulator Behaviour at Half-Filling in Magic Angle Graphene Superlattice. arXiv:1802.00553 (2018).

[19] J. M. B. Lopes dos Santos, N. M. R. Peres, A. H. Castro Neto, Phys. Rev. B 86, 155449 (2012).

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18Correlated insulator behaviour at half-filling in magic-angle graphene superlatticesIn your library
19Continuum model of the twisted graphene bilayerPhys. Rev. B 86 · 2012Add
20van der Waals Heterostructures with High Accuracy Rotational AlignmentNano Lett. 16 · 2016Add

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No PDFNever citedUnverifiedmethodstheoryread laterNew tag

Unconventional superconductivity in magic-angle graphene superlatticesYuan Cao et al., 2018theory
Correlated insulator behaviour at half-filling in magic-angle graphene superlatticesYuan Cao et al., 2018methods
Graphene bilayer with a twist: electronic structureJ. M. B. Lopes dos Santos et al.read later

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Unconventional superconductivity in magic-angle graphene superlatticesCao, Y. · Fatemi, V. · Fang, S. · 2018
Correlated insulator behaviour at half-filling in magic-angle graphene superlatticesCao, Y. · Fatemi, V. · Demir, A. · 2018
Moire bands in twisted double-layer grapheneBistritzer, R. · MacDonald, A. H. · 2011
Graphene bilayer with a twist: electronic structureLopes dos Santos, J. M. B. · Peres, N. M. R. · Castro Neto, A. H. · 2007
The marvels of moire materialsAndrei, E. Y. · Efetov, D. K. · Jarillo-Herrero, P. · 2021

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FileEditInsertViewFormatReviewHelpFlat bands in twisted bilayer graphene

main.tex

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13\section{Introduction}14\label{sec:intro}1516Bilayer graphene admits a flat-band17description~\cite{cao2018unconventional}18near $\theta \approx 1.1^\circ$θ ≈ 1.1°, and the19continuum model of~\cite{bistritzer2011moire}20fixes the angle at which the lowest bands21flatten. The twisted-bilayer band22structure was set out earlier23in~\cite{santos2007graphene}.2425At this angle the moir\'e superlattice

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1 Introduction

Bilayer graphene admits a flat-band description [4] near θ ≈ 1.1°, and the continuum model of [2] fixes the angle at which the lowest bands flatten. The twisted-bilayer band structure was set out earlier in [5].

At this angle the moiré superlattice quenches the kinetic energy of the lowest bands, so interaction rather than dispersion sets the scale of the problem.

2 Flat-band model

The low-energy continuum model treats each graphene layer as a rotated Dirac system coupled by a spatially modulated interlayer tunnelling term [2].

α = w / ℏ v_F k_θ(3)

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FileEditInsertViewFormatReviewHelpFlat bands in twisted bilayer graphene

EditingReviewing

2 Flat-band model

Bilayer graphene admits a flat-band description1 near θ ≈ 1.1°, and the continuum model2 fixes the angle at which the lowest bands flatten.

2.1 The competition of scales

The interlayer tunnelling energy competes with the kinetic scale, and the ratio of the two is what the magic angle is defined by

α = w / ℏ v_F k_θ

Equation 3 · numbered, cross-referenced, and set with the rest

At this angle the moiré superlattice quenches the kinetic energy of the lowest bands.

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Undefined control sequencemain.tex, 44

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! Undefined control sequence.
l.44 The flat bands appear near \parencite

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main.tex

72dispersion sets the scale of the problem. Transport73measurements near the first magic angle74reports correlated-insulator behaviour atAI · grammarThe plural subject “measurements” requires the plural verb “report”.Replace with “report”Dismiss75half filling, and superconducitvity on76doping away from it, in the same device~\cite{cao2018unconventional}.7778\bibliographystyle{unsrt}79\bibliography{paper}8081\end{document}

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main.texROAFEditing now

44The flat bands appear near 1.1 degrees~\cite{cao2018unconventional}.45Devices were assembled by tear and stackA. Fischer46and measured at 70 mK.47−The twist angle was measured from the opticalalignment.48The twist angle was extracted from the superlattice49density rather than from the optical alignment.R. Okonkwo

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AFSay which device this angle belongs to.line 45 · 20 minutes ago
ROAdding it with the density in the next sentence.line 49 · just now

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Sent to co-authors14:20 · R. Okonkwo

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51%52and the Fermi velocity vanishes at a53sequence of values of $\alpha$.−and the lowest bands flatten at a−discrete set of twist angles.54%

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Cao et al., Nature 20186/ 8Fit width

Methods

The separated graphene pieces are manually rotated by a twist angle θ about 1.2° ∼ 1.3° and stacked together again, resulting in a precisely controlled TBG structure.

Transport measurements are performed in a dilution refrigerator with a base temperature of ∼ 70 mK except for the temperature-dependent quantum oscillations which are measured in a He-3 fridge.

A rough estimate of the twist angle can be given by the carrier density of the superlattice gaps at ±n_s which exhibit as strongly insulating states.

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What temperature were the devices measured at?

Down to a base temperature of 70 mK, in a dilution refrigerator.48

Page 48, paragraph 2

And how was the twist angle established?

From the carrier density of the superlattice gaps, which show as strongly insulating states.48

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Cao et al., Nature 20184 highlights · 2 notes

Unconventional superconductivity in magic-angle graphene superlattices

At low twist angles, each electronic band in the MBZ has a four-fold degeneracy of spins and valleys, the latter of which are inherited from the original graphene electronic structure. Special angles, namely the ‘magic angles’, exist where the Fermi velocity drops to zero, the first of which is about θ_magic = 1.1°.

Near this twist angle, the energy bands near charge neutrality, which are separated from other bands by single-particle gaps, become remarkably flat.

ROThis is the number to cite in the introduction.p. 43 · 2 days ago
AFWorth contrasting with the half-filling paper.p. 43 · yesterday

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Cao et al., Nature 2018Figure 1 · p. 44

Figure 1 | 2D superconductivity in a graphene superlattice.

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supervision-14-03.m4atranscribed

12:0448:12

11:52: The tear-and-stack step is where the angle is actually set, and it drifts while the stack is annealed.
12:04: Extract the twist angle from the superlattice density, not from the optical alignment, or the two devices cannot be compared at all.
12:31: Anything more than a tenth of a degree off and the flat bands are gone, so put both angles in the caption.

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nature.com/articles/nature26160

Article

Unconventional superconductivity in magic-angle graphene superlattices

Cao, Y., Fatemi, V., Fang, S., Watanabe, K., Taniguchi, T., Kaxiras, E. & Jarillo-Herrero, P.

Nature 556, 43-50 (2018)

doi.org/10.1038/nature26160

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Unconventional superconductivity in magic-angle graphene superlattices

Cao, Y. et al. · Nature 556 · 2018

Flat bands in twisted bilayer

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Flat bands and correlated phases in twisted bilayer graphene

2 Flat-band model

Superconductivity appears near 1.1 degrees [1] and the correlated insulator sits at half filling of the same bands [2].

At this angle the moiré superlattice quenches the kinetic energy of the lowest bands.

Flat bands in twisted bilayer

Unconventional superconductivity in magic-angle graphene superlatticesYuan Cao et al., 2018
Correlated insulator behaviour at half-filling in magic-angle graphene superlatticesYuan Cao et al., 2018

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Results rewritten after reviewa3f19c2 · from the editor · 2h ago+128−64
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