Raman scattering from moir\'e phonons

H\'ector Ochoa; Natalia B. Perkins; Rafael M. Fernandes; Vitor Dantas

REVIEW 2 major objections 6 minor 61 references

Raman scattering from moir\'e phonons

T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Twisted bilayer graphene should display a series of low-frequency Raman peaks from moiré phonons, with positions and intensities that encode the twist angle.

desk verdict A useful framework for Raman from moiré phonons with a solid symmetry classification; the computed peak positions and selection rules are robust, but the peak intensities rest on an untested polarizability ansatz. read the letter →

arxiv 2506.17408 v1 pith:Z26M26YZ submitted 2025-06-20 cond-mat.str-el

Vitor Dantas , Héctor Ochoa , Rafael M. Fernandes , Natalia B. Perkins This is my paper · ORCID

classification cond-mat.str-el

keywords twistedbilayergraphenemoiréphononsRamanscatteringstacking-dependentpolarizabilitylatticereconstructionphasonszonefoldingD6pointgroup

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that Raman scattering can directly observe moiré phonons—the collective interlayer-sliding vibrations of a twisted bilayer—and that in twisted bilayer graphene these modes produce a distinctive series of low-frequency peaks that untwisted or decoupled layers do not show. To get there, the authors model the relaxed moiré lattice with continuum elasticity, expand the stacking-dependent optical polarizability in moiré harmonics, and derive explicit Raman tensors and intensities for the $q=0$ modes. A sympathetic reader would care because the predicted peak positions, intensities, and polarization selection rules all carry information about the twist angle and about how strongly the lattice reconstructs, turning Raman into a quantitative probe of moiré structure.

What carries the argument

The load-bearing object is the moiré-phonon Raman tensor derived from a stacking-dependent polarizability ansatz, $\chi_{\mu\nu}[u]\sim\sum_{s,j}[\hat{G}_{sj}]_\mu[\hat{G}_{sj}]_\nu\cos(G_{sj}\cdot r+u\cdot b_{sj})$, whose derivative with respect to the phonon normal coordinates gives $R^{\mu\nu}_n=\sum_{s,G}\zeta_s C^\lambda_n(G)[\hat{b}_{sj}]_\lambda[\hat{G}_{sj}]_\mu[\hat{G}_{sj}]_\nu f_{sj}(G)$. This tensor converts the symmetry and Fourier content of the relaxed stacking texture into quantitative peak intensities and selection rules. It is combined with three ingredients: a continuum elasticity functional for the relaxed lattice (elastic energy plus adhesion potential), the dynamical matrix whose eigenvectors $C_n(G)$ define the phonon branches, and $D_6$ projection operators that select the Raman-active $A_1$ and $E_2$ modes at $q=0$.

What would settle it

Measure the low-frequency Raman spectrum of a high-quality twisted bilayer graphene sample near $\theta=1.08^\circ$ in backscattering geometry at low temperature: the claim fails if no discrete series of peaks below about 80 meV appears, if the peak spacing does not shrink as $\theta$ decreases, or if the $A_1/E_2$ polarization dependence is absent; an ab initio calculation of $\partial\chi/\partial u$ in the relaxed bilayer would directly test the polarizability ansatz.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that the low-frequency Raman response of twisted bilayer graphene is dominated by discrete optical moiré phonons obtained by folding monolayer graphene's acoustic branches into the moiré Brillouin zone, and that these modes are Raman-active through the stacking dependence of the optical polarizability. The Raman intensity is computed as $I(\omega)\propto\sum_{\Gamma,m}|\sum_{\mu\nu}E^\mu_{\mathrm{in}}R^{\mu\nu}_{\Gamma,m}E^\nu_{\mathrm{out}}|^2\,\gamma/((\omega-\omega_{\Gamma,m})^2+\gamma^2)$, with $R^{\mu\nu}_n$ obtained from the phonon eigenvectors and from the polarizability expansion; the Raman-active modes fall into the $A_1$ and $E_2$ representations of the $D_6$ point group. The resulting spectrum below about 80 meV is a ladder of peaks whose number grows and whose spacing shrinks as the twist angle decreases, and the whole ladder approximately collapses when frequencies are scaled by $\omega_0=(2\pi/L_M)\sqrt{\lambda/\rho}$. Even modes within the same representation can have very different intensities, because their eigenvectors sample the moiré harmonics differently, and at small angles lattice relaxation splits the lowest $A_1$ and $E_2$ peaks enough to resolve them separately.

Load-bearing premise

The predicted Raman intensities, which are the main observable, rest on the untested assumption that the stacking-dependent polarizability is well described by the harmonic ansatz in Eq. (4) with harmonic weights that decay as $\zeta_s\sim|G_M|/|G_s|$; if the real polarizability coupling has a different form, the computed peak-height pattern changes.

Editorial extensions

If this is right

A low-frequency Raman measurement on twisted bilayer graphene should reveal a discrete ladder of peaks below about 80 meV, a signature absent in decoupled layers.
The positions of these peaks scale with the moiré length through $\omega_0=(2\pi/L_M)\sqrt{\lambda/\rho}$, so measuring them gives a direct estimate of the twist angle.
Parallel and cross polarizations in backscattering separate $A_1$ from $E_2$ contributions, giving an experimental symmetry filter.
At small twist angles the lowest $A_1$ and $E_2$ peaks split by more than the linewidth, so the two channels can be resolved and the splitting reports on lattice reconstruction.
The same polarizability-expansion method applies to other twisted van der Waals bilayers, making Raman a general probe of moiré phonons.

Reading between the lines

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

If the decay law $\zeta_s\sim|G_M|/|G_s|$ is even approximately right, the computed intensity ratios become a calibration curve: a single low-temperature Raman spectrum could be used to fit the adhesion strength $V_0$, since the $A_1$–$E_2$ splitting and relative peak heights both track lattice reconstruction.
The method should transfer directly to twisted transition-metal dichalcogenides and to twisted trilayers, where the different adhesion energies and elastic constants would shift and reweight the same folded-phonon ladder; the paper notes the broad applicability but does not compute those cases.
Because the linewidth is argued to be set by three-phonon decay, temperature-resolved Raman could serve as a probe of anharmonicity in moiré systems once the full self-energy is evaluated numerically.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a solid theoretical paper that gives the first microscopic framework for Raman scattering by moiré phonons in TBG, with a clean D6 symmetry classification and explicit spectra. The frequency positions and selection rules should be right. The relative intensities, however, hang on an untested polarizability ansatz, so treat Fig. 2's peak heights as plausible illustrations rather than predictions.

What's new: nobody has constructed Raman tensors for the folded sliding modes before. The symmetry projection onto A1/E2 irreps of D6, including the counterintuitive result that the A1 mode is transverse, and the twist-angle dependence of peak spacing, are real additions. The continuum elasticity model for lattice relaxation and phonons follows the established Koshino-Ochoa approach, which is fine. The anharmonic linewidth section is honest: they walk through the three-phonon self-energy and then approximate the width as constant in the low-frequency regime, which is reasonable for a first pass. There are no machine-checked proofs or shipped code, but the derivation is self-contained and reproducible from the equations given.

The soft spot is the one the stress test flags. Eq. (4) is an ansatz for the stacking-dependent polarizability, and the ζ_s weights in Eq. (5) are hand-chosen (1/s decay). All relative intensities in Fig. 2 follow from those choices. If the real electronic polarizability has different Fourier content, the intensity pattern changes while the frequency skeleton remains. The paper says this plainly enough—the constants are 'phenomenological'—but it doesn't offer any independent constraint from DFT or a microscopic model. For a Letter whose headline claim is that the Raman response 'clearly distinguishes' TBG from decoupled layers, that's a meaningful caveat. The 'weak peak near 10 meV' is exactly the kind of feature that would shift or disappear under a different ansatz.

That said, the caveat is proportionate. The framework is still useful: symmetries, selection rules, peak positions, and the twist-angle scaling are all robust. The intensity variation among same-irrep modes is a qualitative effect that likely survives, even if quantitative heights don't.

Who should read this: experimentalists planning Raman measurements on TBG or other twisted bilayers, and theorists working on moiré phonons. It's a good paper to referee seriously. I'd send it to review with the expectation that the authors add one paragraph acknowledging the sensitivity of intensities to the ansatz and perhaps comparing to any single experimental spectrum if available.

Referee Report

2 major / 6 minor

Summary. This paper develops a theoretical framework for computing the Raman response of moiré phonons in twisted bilayer graphene. The authors first obtain the relaxed stacking texture from a continuum elastic free energy with a sinusoidal adhesion potential, then diagonalize the dynamical matrix for relative layer displacements to obtain moiré phonon dispersions for several twist angles. They construct a stacking-dependent polarizability ansatz periodic in the moiré reciprocal lattice, derive the resulting Raman tensor in the eigenmode basis, classify the q=0 Raman-active modes under the D6 point group, and compute backscattering Raman spectra. They also estimate anharmonic contributions to the linewidth. The central claim is that the low-frequency Raman spectra contain a series of peaks, with twist-angle-dependent frequencies, intensities, and polarization selection rules, that clearly distinguish TBG from decoupled layers.

Significance. The paper addresses an important and timely problem: identifying experimentally accessible signatures of moiré phonons. Its strengths are the explicit coupling of a well-established elasticity model to a symmetry-based Raman tensor formalism, the D6 classification of folded shear modes, and the falsifiable predictions of peak positions and their scaling with twist angle. The normal-mode expansion of the Raman tensor is carried out carefully, and the presentation is clear. However, the quantitative content of the spectra, especially relative peak intensities and the claimed intensity variation within an irrep, is set by an unconstrained polarizability ansatz plus an assumed harmonic decay of the weights. Because the central claim emphasizes the discriminating power of the Raman intensities, the significance of the paper as it stands is contingent on either microscopic validation of that ansatz or demonstration that the qualitative predictions are robust to its variation.

major comments (2)

[Eq. (4) and SM Eq. (S39)] The relative intensities in Eq. (8), including the 'weak peak near 10 meV' in Fig. 2(c), are completely determined by the guessed stacking-dependent polarizability ansatz in Eq. (4)/SM Eq. (S39) and the ad hoc decay law ζ_s ∼ |G_M|/|G_s| stated in SM Sec. IV.A. The SM itself notes that 'we do not know this functional' and that ζ_s are introduced as an assumption. No DFT, microscopic model, or experiment constrains the Fourier content of the polarizability, yet the paper's central claim is that the Raman response 'clearly distinguishes' TBG from decoupled layers and encodes twist-angle information. This is a load-bearing assumption for the primary observable, not merely a detail. I ask the authors to (i) provide an independent constraint on the polarizability Fourier coefficients, or (ii) perform systematic sensitivity tests (e.g., different ζ_s decay exponents, different angular dependence in the cosine) and show that the predicted qualitative features survive, or (iii) explicitly reframe the intensity predictions as illustrative consequences of a phenomenological model. As written, the intensity pattern is presented as a prediction, which overstates the strength of the evidence.
[SM V.A (linewidth)] The frequency-independent broadening γ used in Eq. (8) is justified in SM V.A by assuming that the three-phonon matrix elements vary weakly and that the low-frequency two-phonon phase space is limited. This is reasonable but only qualitative; the 'resolvable splitting' of A1 and E2 modes claimed for θ≲1° in Fig. 2(d)–(e) depends on γ being much smaller than the A1–E2 gap, which is an input rather than a computed result. Please provide a numerical estimate of the low-frequency two-phonon density of states with realistic matrix elements, or at least show the sensitivity of the spectra to the value and frequency dependence of γ, and soften the statement that the linewidth is 'nearly constant' if it is based on a truncated phase-space argument.

minor comments (6)

[Results section] In the paragraph before Eq. (6), 'electronic contributions an be neglected' should read 'electronic contributions can be neglected'.
[Fig. 2 caption] The caption states that the number of G shells is indicated on the top right of each panel, but these labels are not visible in the manuscript text; please ensure they appear in the figure.
[Eq. (6) and SM Eq. (S37)] The main-text Eq. (6) omits temperature dependence while SM Eq. (S37) contains the [n_B+1] factor; clarify that the plotted spectra correspond to the low-temperature limit.
[Eq. (5)] The dimensionless weights ζ_s enter without a defined normalization and the intensities are in arbitrary units; state explicitly that no absolute Raman cross-section is predicted and only relative intensities are meaningful.
[SM II] The truncation criterion for the G-shell sum, described as choosing 'a number of G shells such that the Raman spectrum is computed up to ω_n∼80 meV', should be replaced by a convergence test on the phonon frequencies and Raman intensities.
[References] Reference [60] is a duplicate of Ref. [34]; consolidate the two entries.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the computed Raman spectrum is a conditional consequence of an openly stated polarizability ansatz, not a fitted parameter renamed as a prediction.

full rationale

The derivation chain is self-contained. The moiré phonon frequencies and eigenvectors come from diagonalizing the dynamical matrix built from the elasticity energy Eq. (1) and the adhesion potential Eq. (2), with parameters taken from external sources (V0 from Ref. [12]; Lamé coefficients from Refs. [50,51]); the use of Ref. [18], a co-author's earlier paper, is a standard continuum model corroborated by Ref. [17] and does not import the Raman result. The Raman tensor in Eq. (5)/(S41) is obtained by differentiating the stacking-dependent polarizability ansatz (Eq. (4)/(S39)) with respect to the normal-mode coordinates; the paper explicitly calls this "a guess based on symmetry." The star weights ζ_s are assumed to decay as |G_M|/|G_s| and are not fitted to the spectra or to any Raman data, so the peak intensities are conditional predictions of the model rather than outputs defined by their inputs or by self-citation. The confusing phrase that ζ_s are "defined as the relative intensity between the Raman response of a mode n with some reference value" does not create circularity because the subsequent assumed form is independent of n and of the computed intensities. Group-theoretical projection onto irreps of D6 and the backscattering selection rules are standard, leaving no step where an equation reduces to its own input.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper's predictions rest on two kinds of inputs: standard continuum elasticity and adhesion parameters from the literature, and two ad hoc modeling choices specific to this work, namely the polarizability ansatz and the zeta_s decay law. The free parameters are zeta_s and gamma; they govern the intensity pattern and peak widths. No new physical entities are introduced.

free parameters (2)

zeta_s (Raman harmonic weights) = assumed zeta_s ~ |G_M|/|G_s|
Dimensionless amplitudes weighting each star of the polarizability expansion. They set the relative intensities of all Raman peaks and are chosen by hand, with no microscopic derivation or experimental fit.
gamma (phonon linewidth) = 0.1 meV in Fig. 2
Used in Eq. (8) to broaden the Lorentzian peaks; kept frequency-independent after a phase-space approximation. The value is chosen for the plots rather than computed from the self-energy.

assumptions (6)

domain assumption Continuum elasticity free energy Eq. (1)/(S7) with the Lamé coefficients of graphene
The moiré phonon spectrum is computed from this standard model following Refs. [17,18]; its validity for small twist angles is prior knowledge.
domain assumption Adhesion potential with first-star Fourier expansion, Eq. (2)/(S6), with V0 = 90 meV/nm^2
The relaxation and the phonon dynamical matrix depend on this form and this value taken from DFT calculations in Ref. [12]; it is an input, not derived here.
domain assumption D6 point group of the relaxed moiré superlattice with rotation axis through a hexagon center
The Raman-active mode classification and the tensor forms in Eq. (S38) depend on this symmetry choice, which is a modeling decision about the twist axis.
ad hoc to paper Stacking-dependent polarizability ansatz, Eq. (4)/(S39)
The Raman coupling is assumed to be a sum of moiré harmonics with unit vectors enforcing symmetry. The paper states it is 'a guess based on symmetry' (SM Sec. IV.A).
ad hoc to paper Decay law for the Raman harmonic weights zeta_s ~ |G_M|/|G_s|
All computed intensities are weighted by this assumed decay, making low harmonics dominate. No microscopic derivation or experimental fit is given for the decay exponent or prefactors.
ad hoc to paper Frequency-independent linewidth gamma at low frequencies
The full three-phonon self-energy evaluation is described as computationally demanding and is replaced by a constant broadening in Fig. 2 (gamma = 0.1 meV).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Raman scattering from moir\'e phonons." pith.science (2026). https://pith.science/paper/Z26M26YZ

@misc{pith2026250617408,
  author       = {Pith},
  title        = {Pith review of: Raman scattering from moir\'e phonons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z26M26YZ}},
  note         = {Machine review of arXiv:2506.17408}
}

read the original abstract

We develop a theoretical framework for probing moir\'e phonon modes using Raman spectroscopy, and illustrate it with the example of twisted bilayer graphene (TBG). These moir\'e phonons arise from interlayer sliding motion in twisted 2D materials and correspond to fluctuations of the stacking order in reconstructed moir\'e superlattices. These include both acoustic-like phason modes and a new set of low-energy optical modes originating from the zone-folding of monolayer graphene's acoustic modes, which are accessible via Raman spectroscopy. We show that the Raman response of TBG exhibits a series of low-frequency peaks that clearly distinguish it from that of decoupled layers. We further examine the role of anharmonic interactions in shaping the phonon linewidths and demonstrate the strong dependence of the Raman spectra on both the twist angle and the polarization of the incident light. Our findings establish Raman spectroscopy as a powerful tool for exploring moir\'e phonons in a broad class of twisted van der Waals systems.

Figures

Figures reproduced from arXiv: 2506.17408 by the authors.

**Figure 2.** FIG. 2. (a)-(c) Raman intensity in arbitrary units. Here we choose the ¯z [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

61 extracted references · 45 canonical work pages

[1]

Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo- Herrero, Nature556, 80 (2018)

2018
[2]

Both arguments are consistent with the linear response regime used to defineR(t). Therefore, we have only the first term, and the position dependent tensor takes the form eRµν λ (r) = X G,n e−iG·r Cλ n(G) ∗ Rµν n ,(S34) while the Fourier component eRµν λ (G) is simply Rµν λ (−G) = X n Cλ n(G) ∗ Rµν n .(S35) Finally, this contribution to the Raman polariza...

work page
[3]

We construct the TBG lattice by taking two overlapping honeycomb lattices with AA stacking

andb 2 = 4π√ 3a(0,1). We construct the TBG lattice by taking two overlapping honeycomb lattices with AA stacking. Then, we rotate each layer by∓θ/2, as depicted in Fig. S1(a). The rotation between the two layers can be performed by theSO(2) transformationR θ. The real and reciprocal lattice vectors in each layer (l= 1,2) are written in terms of the origin...

work page
[4]

Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Nature556, 43 (2018)

2018
[5]

Yankowitz, S

M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watan- abe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Science363, 1059 (2019)

2019
[6]

A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Science365, 605 (2019)

2019
[7]

C. Jin, E. C. Regan, A. Yan, M. Chu, S. Wang, K. Watan- abe, T. Taniguchi, A. Zettl, and F. Wang, Nature567, 76 (2019)

work page 2019
[8]

Y. Tang, L. Li, T. Li, Y. Xu, S. Wang, K. Watanabe, T. Taniguchi, A. Srivastava, J. Hone, and X. Xu, Nature 579, 353 (2020)

work page 2020

Show all 61 references

[9]

Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Kn¨ uppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, Nature622, 69 (2023)

2023
[10]

Y. Xia, Z. Han, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Nature637, 833 (2025)

2025
[11]

Y. Guo, J. Pack, J. Swann, L. Holtzman, M. Cothrine, K. Watanabe, T. Taniguchi, D. G. Mandrus, K. Barmak, J. Hone, A. J. Millis, A. Pasupathy, and C. R. Dean, Nature637, 839 (2025)

2025
[12]

H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, 14 C. Wang, W. Holtzmann, C. Hu, Z. Liu, T. Taniguchi, K. Watanabe, J.-H. Chu, T. Cao, L. Fu, W. Yao, C.-Z. Chang, D. Cobden, D. Xiao, and X. Xu, Nature622, 74 (2023)

2023
[13]

N. N. T. Nam and M. Koshino, Phys. Rev. B96, 075311 (2017)

2017
[14]

S. Carr, D. Massatt, S. B. Torrisi, P. Cazeaux, M. Luskin, and E. Kaxiras, Phys. Rev. B98, 224102 (2018)

2018
[15]

Guinea and N

F. Guinea and N. R. Walet, Phys. Rev. B99, 205134 (2019)

2019
[16]

Wang, X.-W

C. Wang, X.-W. Zhang, X. Liu1, Y. He, X. Xu, Y. Ran, T. Cao, and D. Xiao, Phys. Rev. Lett.132, 036501 (2024)

2024
[17]

Kang and O

J. Kang and O. Vafek, arXiv:2502.15154
[18]

De Beule, G

C. De Beule, G. N. Pallewela, M. M. Al Ezzi, L. Pend, E. J. Mele, and S. Adam, arXiv:2503.19162
[19]

Koshino and Y.-W

M. Koshino and Y.-W. Son, Phys. Rev. B100, 075416 (2019)

2019
[20]

Ochoa, Phys

H. Ochoa, Phys. Rev. B100, 155426 (2019)

2019
[21]

B. Lian, Z. Wang, and B. A. Bernevig, Phys. Rev. Lett. 122, 257002 (2019)

2019
[22]

F. Wu, E. Hwang, and S. Das Sarma, Phys. Rev. B99, 165112 (2019)

2019
[23]

Maity, M

I. Maity, M. H. Naik, P. K. Maiti, H. R. Krishnamurthy, and M. Jain, Phys. Rev. Res.2, 013335 (2020)

2020
[24]

N. Suri, C. Wang, Y. Zhang, and D. Xiao, Nano Lett. 21, 10026 (2021)

2021
[25]

Samajdar, Y

R. Samajdar, Y. Teng, and M. S. Scheurer, Phys. Rev. B 106, L201403 (2022)

2022
[26]

J. Z. Lu, Z. Zhu, M. Angeli, D. T. Larson, and E. Kaxiras, Phys. Rev. B106, 144305 (2022)

2022
[27]

Gao and E

Q. Gao and E. Khalaf, Phys. Rev. B106, 075420 (2022)

2022
[28]

Birkbeck, J

J. Birkbeck, J. Xiao, A. Inbar, T. Taniguchi, K. Watan- abe, E. Berg, L. Glazman, F. Guinea, F. von Oppen, and S. Ilani, Nature641, 345 (2025)

2025
[29]

Xiao, Jiewen and Berg, Erez and Glazman, Leonid I and Guinea, Francisco and Ilani, Shahal and von Oppen, Fe- lix, Phys. Rev. B110, 205407 (2024)

2024
[30]

Ochoa and R

H. Ochoa and R. M. Fernandes, Phys. Rev. B108, 075168 (2023)

2023
[31]

Girotto, L

N. Girotto, L. Linhart, and F. Libisch, Phys. Rev. B108, 155415 (2023)

2023
[32]

Cappelluti, J

E. Cappelluti, J. A. Silva-Guill´ en, H. Rostami, and F. Guinea, Phys. Rev. B108, 125401 (2023)

2023
[33]

Y. H. Kwan, G. Wagner, N. Bultinck, S. H. Simon, E. Berg, and S. A. Parameswaran, Phys. Rev. B110, 085160 (2024)

2024
[34]

Maity, A

I. Maity, A. A. Mostofi, and J. Lischner, Nano Letters 23, 4870 (2023)

2023
[35]

Zhang, B

Y. Zhang, B. Ahammed, S. H. Bae, C.-H. Lee, J. Huang, M. A. Hossain, T. Rakib, A. van der Zande, E. Ertekin, and P. Y. Huang, arXiv:2505.03060 (2025)

2025 arXiv
[37]

Ochoa and R

H. Ochoa and R. M. Fernandes, Phys. Rev. Lett.128, 065901 (2022)

2022
[38]

Carozo, C

V. Carozo, C. M. Almeida, E. H. Ferreira, L. G. Cancado, C. A. Achete, and A. Jorio, Nano letters11, 4527 (2011)

2011
[39]

K. Kim, S. Coh, L. Z. Tan, W. Regan, J. M. Yuk, E. Chatterjee, M. F. Crommie, M. L. Cohen, S. G. Louie, and A. Zettl, Phys. Rev. Lett.108, 246103 (2012)

2012
[40]

K. Sato, R. Saito, C. Cong, T. Yu, and M. S. Dresselhaus, Phys. Rev. B86, 125414 (2012)

2012
[41]

Carozo, C

V. Carozo, C. M. Almeida, B. Fragneaud, P. M. Bedˆ e, M. V. O. Moutinho, J. Ribeiro-Soares, N. F. Andrade, A. G. Souza Filho, M. J. S. Matos, B. Wang, M. Terrones, R. B. Capaz, A. Jorio, C. A. Achete, and L. G. Can¸ cado, Phys. Rev. B88, 085401 (2013)

2013
[42]

R. E, T.-F. Chung, C. Delaney, C. Keiser, L. A. Jauregui, P. M. Shand, C. C. Chancey, Y. Wang, J. Bao, and Y. P. Chen, Nano Letters13, 3594 (2013)

2013
[43]

Campos-Delgado, L

J. Campos-Delgado, L. G. Can¸ cado, C. A. Achete, A. Jo- rio, and J.-P. Raskin, Nano Research6, 269 (2013)

2013
[44]

Sch¨ apers, J

A. Sch¨ apers, J. Sonntag, L. Valerius, B. Pestka, J. Stras- das, K. Watanabe, T. Taniguchi, L. Wirtz, M. Morgen- stern, B. Beschoten, R. J. Dolleman, and C. Stampfer, 2D Materials9, 045009 (2022)

2022
[45]

A. C. Gadelha, D. A. Ohlberg, C. Rabelo, E. G. Neto, T. L. Vasconcelos, J. L. Campos, J. S. Lemos, V. Ornelas, D. Miranda, R. Nadas,et al., Nature590, 405 (2021)

2021
[46]

T. C. Barbosa, A. C. Gadelha, D. A. A. Ohlberg, K. Watanabe, T. Taniguchi, G. Medeiros-Ribeiro, A. Jo- rio, and L. C. Campos, 2D Materials9, 025007 (2022)

2022
[47]

Pandey, S

V. Pandey, S. Mishra, N. Maity, S. Paul, A. M. B, A. K. Roy, N. R. Glavin, K. Watanabe, T. Taniguchi, A. K. Singh, and V. Kochat, ACS Nano18, 4756 (2024), epub 2024 Jan 31

2024
[48]

K.-Q. Lin, J. Holler, J. M. Bauer, P. Parzefall, M. Scheuck, B. Peng, T. Korn, S. Bange, J. M. Lup- ton, and C. Sch¨ uller, Advanced Materials33, e2008333 (2021)

2021
[49]

Parzefall, J

P. Parzefall, J. Holler, M. Scheuck, A. Beer, K.-Q. Lin, B. Peng, B. Monserrat, P. Nagler, M. Kempf, T. Korn, and C. Sch¨ uller, 2D Materials8, 035030 (2021)

2021
[50]

Y. Wu, S. Wang, C. Chen, C. Zhou, K. Chen, J. Zhou, Z. Wang, Y. Bie, and S. Deng, Advanced Optical Mate- rials12, 10.1002/adom.202301747 (2023)

2023 doi
[51]

See Supplemental Material at XXX for further informa- tion on the numerical and analytical calculations regard- ing the lattice relaxation, moir´ e phonon dispersion, Ra- man intensity and anharmonicity effects
[52]

K. V. Zakharchenko, M. I. Katsnelson, and A. Fasolino, Phys. Rev. Lett.102, 046808 (2009)

2009
[53]

C. Lee, X. Wei, J. W. Kysar, and J. Hone, Science321, 385 (2008)

2008
[54]

Koshino and N

M. Koshino and N. N. T. Nam, Phys. Rev. B101, 195425 (2020)

2020
[55]

Poulet and J

H. Poulet and J. P. Mathieu,Spectres de vibration et sym´ etrie des cristaux(CRC Press, 1970)

1970
[56]

Hayes and R

W. Hayes and R. Loudon,Scattering of light by crystals (Dover, 1978)

1978
[57]

M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov, and H. Won- dratschek, Zeitschrift f¨ ur Kristallographie-Crystalline Materials221, 15 (2006)

2006
[58]

M. S. Dresselhaus, G. Dresselhaus, and A. Jorio,Group theory: application to the physics of condensed matter (Springer Science & Business Media, 2007)

2007
[59]

A. C. Ferrari and D. M. Basko, Nature nanotechnology 8, 235 (2013)

2013
[60]

S. Zhou, J. Han, S. Dai, J. Sun, and D. J. Srolovitz, Phys. Rev. B92, 155438 (2015)

2015
[61]

S. Dai, Y. Xiang, and D. J. Srolovitz, Nano letters16, 5923 (2016)

2016
[62]

Ramos-Alonso, B

A. Ramos-Alonso, B. Remez, D. Bennett, R. M. Fernan- des, and H. Ochoa, Phys. Rev. Lett.134, 026501 (2025)

2025

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo- Herrero, Nature556, 80 (2018)

2018

[2] [2]

Both arguments are consistent with the linear response regime used to defineR(t). Therefore, we have only the first term, and the position dependent tensor takes the form eRµν λ (r) = X G,n e−iG·r Cλ n(G) ∗ Rµν n ,(S34) while the Fourier component eRµν λ (G) is simply Rµν λ (−G) = X n Cλ n(G) ∗ Rµν n .(S35) Finally, this contribution to the Raman polariza...

work page

[3] [3]

We construct the TBG lattice by taking two overlapping honeycomb lattices with AA stacking

andb 2 = 4π√ 3a(0,1). We construct the TBG lattice by taking two overlapping honeycomb lattices with AA stacking. Then, we rotate each layer by∓θ/2, as depicted in Fig. S1(a). The rotation between the two layers can be performed by theSO(2) transformationR θ. The real and reciprocal lattice vectors in each layer (l= 1,2) are written in terms of the origin...

work page

[4] [4]

Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Nature556, 43 (2018)

2018

[5] [5]

Yankowitz, S

M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watan- abe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Science363, 1059 (2019)

2019

[6] [6]

A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Science365, 605 (2019)

2019

[7] [7]

C. Jin, E. C. Regan, A. Yan, M. Chu, S. Wang, K. Watan- abe, T. Taniguchi, A. Zettl, and F. Wang, Nature567, 76 (2019)

work page 2019

[8] [8]

Y. Tang, L. Li, T. Li, Y. Xu, S. Wang, K. Watanabe, T. Taniguchi, A. Srivastava, J. Hone, and X. Xu, Nature 579, 353 (2020)

work page 2020