Pith. sign in

REVIEW 2 major objections 6 minor 1 cited by

Geometrical properties of strained and twisted moir\'e heterostructures

T0 review · 2 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read This review establishes that strain, combined with twist, can continuously control the angle between moiré lattice vectors, making square, hexagonal, and quasi-1D moiré superlattices patternable.

desk verdict A useful, honest review of moiré geometry under strain; the math holds up, but the central design claim is only proven in the rigid limit, and the paper says so. read the letter →

arxiv 2511.16219 v2 pith:JZAVJN6Z submitted 2025-11-20 cond-mat.mes-hall cond-mat.mtrl-sci

Federico Escudero , Francisco Guinea , Zhen Zhan This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-sci
keywords moirésuperlatticesstrainengineeringtwistronicsheterostraingeometricformalismquasi-1Dpatternssquarehexagonalhomobilayers
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 review establishes a unified geometric formalism for moiré patterns formed by two twisted and strained hexagonal layers. It shows that the moiré geometry is governed by a single symmetric tensor F = TᵀT built from rotation and strain matrices, which determines the angle β between moiré vectors. At small twist angles, experimentally accessible strains below ~10% can shift β continuously from 0° to 180°, producing square, hexagonal, or quasi-1D superlattices that are impossible with twist alone. The formalism extends to hexagonal heterobilayers and monoclinic lattices, and explains several strain-induced patterns observed in experiments. A sympathetic reader would care because this turns strain into a design knob for moiré quantum materials.

What carries the argument

The central object is the moiré construction T = (I + E/2)R(−θ/2) − (I − E/2)R(θ/2), a 2×2 matrix mapping undeformed reciprocal vectors bᵢ to moiré vectors Gᵢ = T bᵢ. Its geometry is carried by the symmetric tensor F = TᵀT: the angle between moiré vectors is cosβ = (F b₁·b₂)/√[(F b₁·b₁)(F b₂·b₂)], and the moiré lengths are |Gᵢ| = √(F bᵢ·bᵢ). The strain-dependent part F_ε is non-spherical only for strain that deforms the unit cell, and this is what changes β. A critical condition det T = 0 gives collinear moiré vectors, producing quasi-1D channels; equal-length and perpendicular conditions give square patterns. The formalism assumes small deformations with rigid, homogeneous twist and strain

What would settle it

Measure the moiré angle β in a twisted bilayer with controlled uniaxial heterostrain at a small twist angle (e.g., θ ≈ 1°–2°) and compare with Eq. (38): the prediction is cosβ ≈ −1/2 + (3√3/8)(ν+1) ε_u/θ, and a square pattern should appear near ε_sq ≈ 0.94 tan(θ/2). If no strain value between 0% and 10% produces β = 90°, or if β does not vary continuously with ε/θ, the central claim fails.

Watch

Extended reading notes

Core claim

The paper argues that the geometry of strained and twisted moiré patterns is fully encoded in the symmetric tensor F = TᵀT, where T = (I + E/2)R(−θ/2) − (I − E/2)R(θ/2). The moiré angle β satisfies cosβ = (F b₁·b₂) / √[(F b₁·b₁)(F b₂·b₂)], and the moiré lengths are |Gᵢ| = √(F bᵢ·bᵢ). Because F contains a non-spherical strain-dependent contribution F_ε that mixes twist and strain, β can deviate from the hexagonal 120° value. At low twist angles, the ratio ε/θ controls β, allowing continuous design between 0° and 180° with modest strains. The review catalogs explicit strain parameters for uniaxial, shear, and biaxial strain that yield quasi-1D channels (det T = 0, critical strain ε_c = ±2√ν ta

Load-bearing premise

The load-bearing premise is that both layers deform as rigid lattices under homogeneous in-plane strain with no lattice relaxation; if relaxation is significant, the actual moiré geometry will deviate from these predictions.

Editorial extensions

If this is right

  • With low twist angles and strains ε ≲ 10%, any moiré angle β between 0° and 180° is reachable, so square, hexagonal, and quasi-1D moiré patterns can be designed on demand.
  • Biaxial strain can reproduce the same moiré periodicity as a twist, but the flat-band physics around the magic angle only occurs in the no-strain twist configuration.
  • The critical strain for quasi-1D channels is small at low angles (ε_c ≈ ±θ/√ν; for graphene ≈ ±5θ/2), explaining the ubiquity of strain-induced 1D patterns in experiments.
  • For hexagonal heterobilayers, the lattice mismatch introduces a threshold in twist/strain below which particular geometries (square, hexagonal) do not form.
  • The real-space moiré-vector formula for monoclinic lattices generalizes the hexagonal results, covering rectangular and oblique lattices as well.

Reading between the lines

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

  • In the editor's reading: since the formalism assumes rigid layers with no lattice relaxation, realistic relaxed samples—where AA regions shrink and domain walls form—will likely show the same qualitative design trends but with shifted critical strain values and possibly altered β; relaxation must be added to make quantitative predictions.
  • A practical extension would be to use the closed-form strain parameters from Eqs. (43)–(44) as starting points for atomistic relaxation calculations, which could reveal how much the square and quasi-1D patterns survive when the layer is allowed to deform energetically.
  • The equivalence between uniaxial strain and a combination of shear and biaxial strain (with an effective Poisson ratio) implies that materials with different elastic properties can still reach the same moiré geometry by tuning the strain ratio, potentially extending the design platform to a wide family of 2D materials
  • The mBZ construction for equal-length moiré vectors with arbitrary β could be used to reinterpret transport experiments in strained twisted bilayer graphene, since the Dirac points no longer sit at the mBZ vertices.
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.

Referee Report

2 major / 6 minor

Summary. This manuscript is a review of the geometry of strained and twisted moiré heterostructures. It opens with the linear elasticity formalism for 2D in-plane strain, catalogs uniaxial, shear, and biaxial deformations, and then develops the moiré-vector formalism for hexagonal homobilayers, hexagonal heterobilayers, and monoclinic lattices. The central mathematical object is the transformation T of Eq. (34) and the symmetric tensor F=T^T T of Eq. (35), from which the moiré angle β and vector lengths are derived. The review provides closed-form recipes for quasi-1D, square, and hexagonal moiré patterns, discusses the moiré Brillouin zone reconstruction, acknowledges lattice relaxation as a limitation, and surveys experimental strain-engineering techniques and observed strained moiré patterns. The paper claims that strain plus twist can, in principle, tune β continuously from 0° to 180° in the low-twist, moderate-strain regime.

Significance. If read within its explicitly stated rigid-limit scope, this review is a valuable pedagogical synthesis. Its strengths are the explicit closed-form conditions for special geometries—critical strain for 1D collapse in Eq. (69), the square-pattern recipe in Eq. (78), the hexagonal-pattern recipe in Eq. (79), and the generalized mBZ construction in Eq. (82)—and the direct comparison with experiments such as the TBG square pattern at θ≈0.38° in Ref. [87]. The formalism is standard linear elasticity, no parameters are fitted, and the predictive statements are falsifiable in the rigid-limit model. The main weakness is that the practical reach of the central claim is not quantitatively qualified against lattice relaxation, which the review itself states is outside its scope.

major comments (2)
  1. [III A (paragraph before Eq. (38)) and III F] The statement that for low twist angles and ε≲10% one can vary β to any value between 0° and 180° is presented as a practical design capability, but it is a result of the rigid homogeneous-strain model of Eq. (26). Section IIIF explicitly acknowledges that relaxed configurations “cannot be accounted by homogeneous twist and strain profiles,” and cites many works showing that relaxation is strongest at small twist angles—precisely the regime where ε/θ is large. The manuscript should (i) state that this claim is strictly rigid-limit, (ii) provide a quantitative criterion or a discussion of when relaxation corrections can be neglected (e.g., domain-wall width relative to moiré period), and (iii) temper the “accessible experimental platform” language accordingly. As it stands, the abstract and introduction overstate the experimental relevance of a result whose domain of validity is not estab
  2. [III E, Eq. (82)] Eq. (82) defines Q1 and then labels the next two expressions both as Q2; the third expression should define Q3 = Q1 + G1 − λG2, as implied by the reduction to Eq. (81) for β=120°. As printed, the mBZ construction for general β is incomplete. The sentence “In the special case of β=90°, the four points reduce to four because Q1 = −Q3” is also unclear: the six candidate points should reduce to four for a square lattice, but the stated equality does not hold with the definitions given (for β=90°, λ=1, Q1=-(G1+G2)/2 while −Q3=(-G1+3G2)/2). Please correct the definitions and the degeneracy statement.
minor comments (6)
  1. [III D3, Eq. (79) and following text] Typos in the hexagonal-pattern recipe: β=600 should read β=60°, and “ϕsq ≈ −4.38°” should read “ϕhex ≈ −4.38°.” The same paragraph should be checked for other notation slips.
  2. [III A1, text after Eq. (46)] “Possion’s ratio” should be “Poisson’s ratio.”
  3. [Fig. 3 caption] “Wiger-Seitz cell” should be “Wigner-Seitz cell,” and the same typo appears elsewhere.
  4. [IV B3] “unintensional strains” should be “unintentional strains.”
  5. [III E, first paragraph] “In the the case of only a twist” has a duplicated article; also “In the the case” appears earlier in the same section.
  6. [III F] “cannot be accounted by homogeneous twist and strain profiles” should be “cannot be accounted for by ...”; similar grammatical fix in the next sentence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the formal results follow algebraically from the stated rigid-deformation ansatz; self-citations serve as review references, and the relaxation caveat is a scope limitation, not a circular substitution.

full rationale

The paper's derivation chain is self-contained in the sense relevant to circularity. The deformation of each layer is specified by Eq. (26), a_{i,±}=(I+E_±)R(θ_±)a_i, a stated homogeneous-strain assumption; moiré vectors are then defined by the reciprocal-space difference Eq. (33) and the map T in Eq. (34). From T the symmetric tensor F=T^T T is computed and cosβ is obtained in Eq. (35) as an algebraic consequence, not as an imposed value. The 'any angle β between 0 and 180°' claim follows from the leading-order expansion Eq. (38), which is a Taylor limit of Eq. (35) for small θ and ε; it is not a fit to the target β. The special geometries (square, hexagonal, quasi-1D) are likewise derived by imposing defining conditions—equal length and perpendicularity in Eq. (77), or det T=0 in Eq. (68)—and solving the same F equations; they are not inserted as inputs. The review relies on Refs. [27], [28], and [66] for several closed-form expressions, and two of these are authored by members of this review's author team; however, the present text states the underlying model (homogeneous strain, no relaxation), and the cited results are parameter-free algebraic solutions under that stated model, with external experimental comparison, e.g., the square-pattern prediction ε_sq(θ=0.38°)≈0.3% against the measured (0.21±0.12)% from Ref. [87]. No uniqueness theorem or ansatz is imported solely from a self-citation to force the conclusion. The manuscript explicitly flags the limitation that relaxed configurations 'cannot be accounted by homogeneous twist and strain profiles' (Section IIIF) and are beyond scope; this is a stated validity limitation of the rigid model, not a circular step that substitutes the prediction with the input. Accordingly, the correct circularity finding is a non-finding.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The ledger is light because this is a review. No new constants are fitted; material parameters such as Poisson ratios and lattice constants are quoted from prior literature. The burdens are modeling assumptions: linear small-strain elasticity, symmetric strain partitioning, the incommensurate-moiré definition, and neglect of relaxation. The uniaxial closed-form results in Eqs. (38)-(46) and critical conditions (69)-(71) inherit these assumptions rather than being tested against data in this manuscript.

free parameters (1)
  • µ (interlayer strain partitioning)
    Introduced in Sec. III C, Eqs. (61)-(62), as 0<µ<1 to interpolate between heterostrain (µ=0) and homostrain (µ=1). It is not fitted to data and does not enter the closed-form homobilayer results, but the monoclinic generalization depends on it.
assumptions (5)
  • domain assumption Small-strain linear elasticity with homogeneous, time-independent, in-plane deformation.
    Sec. II A linearizes the strain tensor (Eq. 5) and restricts to in-plane displacement. All closed-form results use this, although strains up to ~8.7% are quoted in Fig. 7, where the linearization is less safe.
  • domain assumption Moiré geometry is well defined for incommensurate stacks via G_i = b_{i,-} - b_{i,+}.
    Sec. III A: 'the geometrical properties of the moiré pattern can be well defined and studied even if the system is technically incommensurate', citing Refs. [5,71]. This is required for the continuum geometry formulas but omits commensuration-specific corrections.
  • domain assumption Symmetric configuration θ_±=±θ/2 and E_±=±E/2 captures the relevant physics.
    Eq. (34) and the critical conditions Eqs. (69)-(70) assume this strain and twist partitioning. The text says the analysis is general, but the useful closed forms are derived for the symmetric setup.
  • domain assumption Lattice relaxation is neglected; layers follow externally imposed homogeneous deformations.
    All geometry formulas (Eqs. 26 and 63) are rigid-lattice formulas. Sec. IIIF states that relaxed configurations cannot be accounted by homogeneous twist/strain and are beyond scope, so predictions may deviate for small-twist samples.
  • domain assumption Lattice mismatch in hexagonal heterobilayers is representable as biaxial strain ±E_b/2.
    Sec. III B, Eq. (57), absorbs the lattice mismatch as an effective biaxial strain. This is a modeling choice for the moiré transformation, not an exact mapping for arbitrary mismatches.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Geometrical properties of strained and twisted moir\'e heterostructures." pith.science (2026). https://pith.science/paper/JZAVJN6Z

@misc{pith2026251116219,
  author       = {Pith},
  title        = {Pith review of: Geometrical properties of strained and twisted moir\'e heterostructures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JZAVJN6Z}},
  note         = {Machine review of arXiv:2511.16219}
}
read the original abstract

The experimental observations of many interaction-driven electronic phases in moir\'e superlattices have stimulated intense theoretical and experimental efforts to understand and engineer these correlated physics. Strain is a powerful tool for manipulating and controlling the geometrical and electronic structures of moir\'e superlattices. This review provides a comprehensive introduction to the geometry of strained moir\'e superlattices. First, starting from the linear elasticity theory, we briefly introduce the general formalism of small deformations in two-dimensional materials, and discuss the particular cases of uniaxial, shear and biaxial strain. Then, we apply the theory to twisted and strained moir\'e materials, mainly focusing on the hexagonal homobilayers, hexagonal heterobilayers and monoclinic lattices. Special moir\'e geometries, like the quasi-unidimensional patterns, square patterns and hexagonal, are theoretically predicted by manipulating the strain and twist. Finally, we review recently developed strain techniques and the special moir\'e geometries realized via these approaches. This review aims at equipping the reader with a robust understanding on the description and implementation of strain in moir\'e materials, as well as highlight some major breakthroughs in this active field.

Figures

Figures reproduced from arXiv: 2511.16219 by the authors.

Figure 1
Figure 1. Schematic effect of three common types of strain in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Strain-dependent construction of moiré vectors. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Examples of moiré patterns in two equal hexagonal layers with relative twist [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Variation of the real space moiré vectors lengths [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Variation of the real-space angle βR between the moiré vectors (top) and the moiré area M = [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Formation of one-dimensional moiré patterns in twisted and strained bilayer graphene. Panels (a) to (c) correspond [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Three ways to generate hexagonal moiré patterns in a hexagonal homobilayer configuration: (a) only twist angle, (b) [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Construction of the moiré Brillouin zone (mBZ) [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 10
Figure 10. Figure 10: The substrate bending technique. Schematic illustration of generating heterostrain in TBG on a flexible substrate [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: Process-induced strain technique. (a) Schematic illustration of generating biaxial heterostrain by fully encapsulating [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: Sliding-based strain technique. (a)-(c) Moiré patterns generated by hexagonal lattices with a lattice mismatch [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: Continuous tuning of twist angle and strain in a graphene–hBN heterostructure. (a) Schematic of the continuous [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: Strain-induced triangular-rectangular moiré pattern evolution in twisted bilayer WSe [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 16
Figure 16. Figure 16: Strain induced giant swirl in bilayer graphene. [PITH_FULL_IMAGE:figures/full_fig_p021_16.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Intertwined chirality and symmetry breaking in moir\'e domain-wall networks

    cond-mat.mes-hall 2026-02 conditional novelty 6.0 of 10

    Domain-wall networks in strained graphene bilayers spontaneously form chiral geometries during relaxation, and this chirality shifts low-energy electrons from junctions to asymmetric wall channels.

Reference graph

Works this paper leans on

147 extracted references · 1 linked inside Pith · cited by 1 Pith paper

  1. [1]

    This re- sult in the formation of quasi-unidimensional moiré chan- nels that have been seen in numerous experiments (see Section IV)

    Quasi-unidimensional patterns The significant deformation of the moiré geometry un- der strain can lead to a critical situation in which the moiré vectors become collinear [23, 27, 28, 66]. This re- sult in the formation of quasi-unidimensional moiré chan- nels that have been seen in numerous experiments (see Section IV). In reciprocal space, as the criti...

    2023

  2. [87]

    Cantele, D

    G. Cantele, D. Alfè, F. Conte, V. Cataudella, D. Ninno, and P. Lucignano, Structural relaxation and low-energy properties of twisted bilayer graphene, Physical Review Research2, 043127 (2020)

    2020

  3. [2]

    In other words, the moiré patterns with both twist and shear strain are not hexagonal [27, 28]

    Shear strain For shear strain the transformationFreads F= 4 sin2 θ 2 +ϵ 2 s cos2 θ 2 I−2ϵ s sin (θ) R (2φ)σz.(47) The last term, which depends on the interplay between twistandstrain, isnotasphericaltensorsoitchangesthe moiré geometry. In other words, the moiré patterns with both twist and shear strain are not hexagonal [27, 28]. The possible strained con...

  4. [3]

    Its effect is to only modify the orientation and length of the moiré vectors

    Biaxial strain For biaxial strain the transformationFreads F= 4 sin2 θ 2 +ϵ 2 b cos2 θ 2 I.(48) As this is, as expected, a spherical tensor, a biaxial strain does not change the moiré geometry. Its effect is to only modify the orientation and length of the moiré vectors. However, both changes are important and can actually significantly influence the geom...

  5. [4]

    Shear and biaxial strain In the presence of both shear and biaxial strains, the lattices change both their shape and size. The corre- sponding strain tensor is given by E= ϵb −ϵ s sin 2φ ϵ s cos 2φ ϵs cos 2φ ϵ b +ϵ s sin 2φ .(53) As noted above, any strain tensor can be generally de- composed as a mixture of biaxial and shear strains, cf. Eq. (27). The co...

    2023

  6. [5]

    These have been predicted theoretically [27, 28] (see Figures 3 and 5), and observed experimentally [85– 87] (see Section IV)

    Square patterns Another interesting case is the formation of square pat- terns. These have been predicted theoretically [27, 28] (see Figures 3 and 5), and observed experimentally [85– 87] (see Section IV). In general there is a family of twist and strain configurations that lead to square moiré pat- terns (see, e.g., label II in Figure 5). Geometrically,...

    2024

  7. [6]

    But one can also have hexagonal moiré patterns with twist and strain [28]

    Hexagonal patterns Hexagonal moiré patterns emerge naturally in the case of only twist without strain (e.g., in twisted bilayer graphene) [2, 3]. But one can also have hexagonal moiré patterns with twist and strain [28]. The simplest case is, ofcourse, withtwistandbiaxialstrain, asthisalwayspre- serves the hexagonal shape or the layers (Section IIIA3). Fo...

    2024

  8. [7]

    In this technique, the active 2D layer can be suspended or directly stacked on the substrate

    Substrate out-of-plane bending One strain technique for moiré materials is based on in-plane or out-of-plane stretching or bending the under- lying substrate or active 2D materials [30, 32, 102, 122]. In this technique, the active 2D layer can be suspended or directly stacked on the substrate. We refer to this method as substrate bending technique. A simi...

Show all 147 references
  1. [8]

    Process-induced strain Another common approach is a process-induced strain technique, which is based on the deposition of stressed thin films onto the active 2D materials [31, 124–127]. In Ref. [31], Peñaet al.utilized this method to gener- ate designable layer-by-layer strain...

  2. [9]

    Sliding-based strain In Ref. [123], Sequeiraet al.presented a sliding- based technique to manipulate the uniaxial heterostrain in graphene-hexagonal boron nitride (hBN) and study the strain effect on moiré geometry. To this end, they fabricated an open-faced graphene device on...

    2024

  3. [10]

    For exam- ple, the superconductivity in TBG is observed around a twist angle 1.1◦, but has a tolerance of only 0.1◦ [129]

    Continuous tuning moiré patterns In the moiré systems, the geometric and electronic properties are often highly dependent on the moiré pe- riodicity and the interlayer atomic registry. For exam- ple, the superconductivity in TBG is observed around a twist angle 1.1◦, but has a...

    2025

  4. [11]

    In this setup, a continuous and homogeneous uniaxial heterostrain is generated by bending a flexible substrate with different radiusR(see Sec

    Rectangular moiré patterns Figure 14 shows a strain-induced moiré pattern ma- nipulation in twisted bilayer WSe2 [32]. In this setup, a continuous and homogeneous uniaxial heterostrain is generated by bending a flexible substrate with different radiusR(see Sec. IVA1). The defo...

  5. [12]

    The critical condition can be, in principle, engineered by extenally induced-strain (Section IVA)

    Quasi-1D pattern As discussed in Section IIID1, in strained moiré mate- rials there can be a critical strain configuration at which the moiré collapse into a quasi-1D pattern. The critical condition can be, in principle, engineered by extenally induced-strain (Section IVA). Bu...

    2021

  6. [13]

    Severo Ochoa

    Giant atomic swirl As noted in Section IIIF, relaxation-induced giant atomic swirls can emerge in the domain wall networks of moiré materials [98, 134]. Figure 16 shows a moiré pattern formed in bilayer graphene with only biaxial het- erostrain, which was produced by hydrogen ...

    2023

  7. [14]

    F. He, Y. Zhou, Z. Ye, S.-H. Cho, J. Jeong, X. Meng, and Y. Wang, Moiré patterns in 2d materials: a review, ACS Nano15, 5944 (2021)

    2021

  8. [15]

    E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yaz- dani, and A. F. Young, The marvels of moiré materials, Nature Reviews Materials6, 201 (2021)

    2021

  9. [16]

    E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nature Materials19, 1265 (2020)

    2020

  10. [17]

    Lopes dos Santos, N

    J. Lopes dos Santos, N. Peres, and A. Castro Neto, Graphene bilayer with a twist: Electronic structure, Physical Review Letters99, 256802 (2007)

    2007

  11. [18]

    Bistritzer and A

    R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proceedings of the Na- tional Academy of Sciences108, 12233 (2011)

    2011

  12. [19]

    Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras,et al., Correlated insulator behaviour at half-filling in magic-angle graphene super- lattices, Nature556, 80 (2018)

    2018

  13. [20]

    Y. Xu, S. Liu, D. A. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, V. Elser, K. F. Mak, and J. Shan, Correlatedinsulatingstatesatfractionalfillingsofmoiré superlattices, Nature587, 214 (2020)

    2020

  14. [21]

    Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional su- perconductivity in magic-angle graphene superlattices, Nature556, 43 (2018)

    2018

  15. [22]

    Y. Xia, Z. Han, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Superconductivity in twisted bilayer wse2, Nature637, 833 (2025)

    2025

  16. [23]

    Y. Guo, J. Pack, J. Swann, L. Holtzman, M. Cothrine, K. Watanabe, T. Taniguchi, D. G. Mandrus, K. Bar- mak, J. Hone,et al., Superconductivity in 5.0 twisted bilayer wse2, Nature637, 839 (2025)

    2025

  17. [24]

    Zheng, Q

    Z. Zheng, Q. Ma, Z. Bi, S. de La Barrera, M.-H. Liu, N. Mao, Y. Zhang, N. Kiper, K. Watanabe, T. Taniguchi,et al., Unconventional ferroelectricity in moiré heterostructures, Nature588, 71 (2020)

    2020

  18. [25]

    A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. Kastner, and D. Goldhaber-Gordon, Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene, Sci- ence365, 605 (2019)

    2019

  19. [26]

    Serlin, C

    M. Serlin, C. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. Young, Intrinsic quantized anomalous hall effect in a moiré het- erostructure, Science367, 900 (2020)

    2020

  20. [27]

    F. Xu, Z. Sun, T. Jia, C. Liu, C. Xu, C. Li, Y. Gu, K.Watanabe, T.Taniguchi, B.Tong,et al.,Observation of integer and fractional quantum anomalous hall effects in twisted bilayer mote2, Physical Review X13, 031037 (2023)

    2023

  21. [28]

    J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe,et al., Signatures of fractional quantum anomalous hall states in twisted mote2, Nature622, 63 (2023)

    2023

  22. [29]

    Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Knüppel, C.Vaswani, K.Watanabe, T.Taniguchi, K.F.Mak,and J. Shan, Thermodynamic evidence of fractional chern insulator in moiré mote2, Nature622, 69 (2023)

    2023

  23. [30]

    H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu,et al., Obser- vation of fractionally quantized anomalous hall effect, Nature622, 74 (2023)

    2023

  24. [31]

    T. Li, S. Jiang, L. Li, Y. Zhang, K. Kang, J. Zhu, K. Watanabe, T. Taniguchi, D. Chowdhury, L. Fu, et al., Continuous mott transition in semiconductor moiré superlattices, Nature597, 350 (2021)

    2021

  25. [32]

    J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Tunable strongly coupled supercon- ductivity in magic-angle twisted trilayer graphene, Na- ture590, 249 (2021)

    2021

  26. [33]

    H. Kim, Y. Choi, C. Lewandowski, A. Thomson, Y. Zhang, R. Polski, K. Watanabe, T. Taniguchi, J. Al- icea, and S. Nadj-Perge, Evidence for unconventional superconductivity in twisted trilayer graphene, Nature 606, 494 (2022)

    2022

  27. [34]

    Zhang, R

    Y. Zhang, R. Polski, C. Lewandowski, A. Thomson, Y. Peng, Y. Choi, H. Kim, K. Watanabe, T. Taniguchi, J. Alicea,et al., Promotion of superconductivity in magic-angle graphene multilayers, Science377, 1538 (2022)

    2022

  28. [35]

    Wang, E.-M

    L. Wang, E.-M. Shih, A. Ghiotto, L. Xian, D. A. Rhodes, C. Tan, M. Claassen, D. M. Kennes, Y. Bai, B. Kim,et al., Correlated electronic phases in twisted bilayer transition metal dichalcogenides, Nature Mate- xxiii rials19, 861 (2020)

    2020

  29. [36]

    D. A. Cosma, J. R. Wallbank, V. Cheianov, and V. I. Fal’Ko, Moiré pattern as a magnifying glass for strain anddislocationsinvanderwaalsheterostructures,Fara- day Discussions173, 137 (2014)

    2014

  30. [37]

    G. G. Naumis, S. Barraza-Lopez, M. Oliva-Leyva, and H. Terrones, Electronic and optical properties of strained graphene and other strained 2d materials: a re- view, Reports on Progress in Physics80, 096501 (2017)

    2017

  31. [38]

    G. G. Naumis, S. A. Herrera, S. P. Poudel, H. Naka- mura, and S. Barraza-Lopez, Mechanical, electronic, optical, piezoelectric and ferroic properties of strained graphene and other strained monolayers and multilay- ers: an update, Reports on Progress in Physics87, 016502 (2023)

    2023

  32. [39]

    Z. Dai, L. Liu, and Z. Zhang, Strain engineering of 2d materials: issues and opportunities at the interface, Ad- vanced Materials31, 1805417 (2019)

    2019

  33. [40]

    M. Kögl, P. Soubelet, M. Brotons-Gisbert, A. Stier, B. Gerardot, and J. Finley, Moiré straintronics: a uni- versal platform for reconfigurable quantum materials, npj 2D Materials and Applications7, 32 (2023)

    2023

  34. [41]

    Escudero, A

    F. Escudero, A. Sinner, Z. Zhan, P. A. Pantaleón, and F. Guinea, Designing moiré patterns by strain, Physical Review Research6, 023203 (2024)

    2024

  35. [42]

    N. P. Kazmierczak, M. Van Winkle, C. Ophus, K. C. Bustillo, S. Carr, H. G. Brown, J. Ciston, T. Taniguchi, K.Watanabe,andD.K.Bediako,Strainfieldsintwisted bilayer graphene, Nature Materials20, 956 (2021)

    2021

  36. [43]

    Kapfer, B

    M. Kapfer, B. S. Jessen, M. E. Eisele, M. Fu, D. R. Danielsen, T. P. Darlington, S. L. Moore, N. R. Finney, A. Marchese, V. Hsieh,et al., Programming twist an- gle and strain profiles in 2d materials, Science381, 677 (2023)

    2023

  37. [44]

    T. Peña, A. Dey, S. A. Chowdhury, A. Azizimanesh, W. Hou, A. Sewaket, C. Watson, H. Askari, and S. M. Wu, Moiré engineering in 2d heterostructures with process-induced strain, Applied Physics Letters122 (2023)

    2023

  38. [45]

    H. Ou, K. Oi, R. Usami, T. Endo, K. Shinokita, R. Ki- taura, K. Matsuda, Y. Miyata, J. Pu, and T. Takenobu, Continuous strain modulation of moiré superlattice symmetry from triangle to rectangle, Small , 2407316 (2025)

    2025

  39. [46]

    Huder, A

    L. Huder, A. Artaud, T. Le Quang, G. T. De Lais- sardière, A. G. Jansen, G. Lapertot, C. Chapelier, and V. T. Renard, Electronic spectrum of twisted graphene layers under heterostrain, Physical Review Letters120, 156405 (2018)

    2018

  40. [47]

    Kerelsky, L

    A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian, M. Yankowitz, S. Chen, K. Watanabe, T. Taniguchi, J. Hone, C. Dean,et al., Maximized electron interac- tions at the magic angle in twisted bilayer graphene, Nature572, 95 (2019)

    2019

  41. [48]

    Y. Xie, B. Lian, B. Jäck, X. Liu, C.-L. Chiu, K. Watan- abe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene, Nature572, 101 (2019)

    2019

  42. [49]

    Mesple, A

    F. Mesple, A. Missaoui, T. Cea, L. Huder, F. Guinea, G. Trambly de Laissardière, C. Chapelier, and V. T. Renard, Heterostrain determines flat bands in magic- angle twisted graphene layers, Physical Review Letters 127, 126405 (2021)

    2021

  43. [50]

    X. Wang, J. Finney, A. L. Sharpe, L. K. Rodenbach, C. L. Hsueh, K. Watanabe, T. Taniguchi, M. Kast- ner, O. Vafek, and D. Goldhaber-Gordon, Unusual mag- netotransport in twisted bilayer graphene from strain- induced open fermi surfaces, Proceedings of the Na- tional Academy of...

    2023

  44. [51]

    J. Yu, G. Jia, Q. Li, Y. Wang, K. Xiao, Y. Ju, H. Zhang, Z. Hu, Y. Guo, B. Lian,et al., Twist angle driven elec- tronic structure evolution of twisted bilayer graphene, arXiv preprint arXiv:2406.20040 (2024)

    2024 arXiv

  45. [52]

    Carrasco, F

    R. Carrasco, F. Escudero, Z. Zhan, E. Cortés, B. Viña- Bausá, Y. Maximenko, P. A. Pantaleón, F. Guinea, and I. Brihuega, Twistraintronics in square moiré superlat- ticesofstackedgraphenelayers,Tobepublished (2025)

    2025

  46. [53]

    Y. Hou, J. Zhou, M. Xue, M. Yu, Y. Han, Z. Zhang, and Y. Lu, Strain engineering of twisted bilayer graphene: The rise of strain-twistronics, Small21, 2311185 (2025)

    2025

  47. [54]

    D. Zhai, H. Yu, and W. Yao, Twistronics and moiré superlattice physics in 2d transition metal dichalco- genides, Reports on Progress in Physics88, 084501 (2025)

    2025

  48. [55]

    Trambly de Laissardière, S

    G. Trambly de Laissardière, S. Venkateswarlu, A. Miss- saoui, G. Jemaï, K. Chika, J. Vahedi, O. Faizy Na- marvar, J.-P. Julien, A. Honecker, L. Magaud,et al., Electronic structure and transport in materials with flat bands: 2d materials and quasicrystals, Physica E Low- Dimens...

    2026

  49. [56]

    V. M. Pereira, A. Castro Neto, and N. Peres, Tight- binding approach to uniaxial strain in graphene, Phys- ical Review B80, 045401 (2009)

    2009

  50. [57]

    J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J. Booth, and S. Roth, The structure of suspended graphene sheets, Nature446, 60 (2007)

    2007

  51. [58]

    Fasolino, J

    A. Fasolino, J. Los, and M. I. Katsnelson, Intrinsic rip- ples in graphene, Nature Materials6, 858 (2007)

    2007

  52. [59]

    Uchida, S

    K. Uchida, S. Furuya, J.-I. Iwata, and A. Oshiyama, Atomic corrugation and electron localization due to moiré patterns in twisted bilayer graphenes, Physical Review B90, 155451 (2014)

    2014

  53. [60]

    N. N. Nam and M. Koshino, Lattice relaxation and energy band modulation in twisted bilayer graphene, Physical Review B96, 075311 (2017)

    2017

  54. [61]

    Koshino, N

    M. Koshino, N. F. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Maximally localized wannier or- bitals and the extended hubbard model for twisted bi- layer graphene, Physical Review X8, 031087 (2018)

    2018

  55. [62]

    Cazeaux, D

    P. Cazeaux, D. Clark, R. Engelke, P. Kim, and M. Luskin, Relaxation and domain wall structure of bilayer moiré systems, Journal of Elasticity154, 443 (2023)

    2023

  56. [63]

    L. D. Landau, L. Pitaevskii, A. M. Kosevich, and E. M. Lifshitz,Theory of elasticity: volume 7, Vol. 7 (Elsevier, Amsterdam Heidelberg, 2012)

    2012

  57. [64]

    M. E. Gurtin,An introduction to continuum mechanics, Vol. 158 (Academic press, San Diego, Calif., 1982)

    1982

  58. [65]

    M. E. Gurtin, E. Fried, and L. Anand,The mechanics and thermodynamics of continua(Cambridge university press, Cambridge, 2010)

    2010

  59. [66]

    M. A. Vozmediano, M. Katsnelson, and F. Guinea, Gauge fields in graphene, Physics Reports496, 109 (2010)

    2010

  60. [67]

    Guinea, Strain engineering in graphene, Solid State Communications152, 1437 (2012)

    F. Guinea, Strain engineering in graphene, Solid State Communications152, 1437 (2012). xxiv

    2012

  61. [68]

    Amorim, A

    B. Amorim, A. Cortijo, F. De Juan, A. G. Grushin, F. Guinea, A. Gutiérrez-Rubio, H. Ochoa, V. Parente, R. Roldán, P. San-Jose,et al., Novel effects of strains in graphene and other two dimensional materials, Physics Reports617, 1 (2016)

    2016

  62. [69]

    C. Si, Z. Sun, and F. Liu, Strain engineering of graphene: a review, Nanoscale8, 3207 (2016)

    2016

  63. [70]

    C. Lee, X. Wei, J. W. Kysar, and J. Hone, Measure- ment of the elastic properties and intrinsic strength of monolayer graphene, Science321, 385 (2008)

    2008

  64. [71]

    D. G. Papageorgiou, I. A. Kinloch, and R. J. Young, Mechanical properties of graphene and graphene-based nanocomposites, Progress in Materials Science90, 75 (2017)

    2017

  65. [72]

    Frisenda, M

    R. Frisenda, M. Drüppel, R. Schmidt, S. Michaelis de Vasconcellos, D. Perez de Lara, R. Bratschitsch, M. Rohlfing, and A. Castellanos-Gomez, Biaxial strain tuning of the optical properties of single-layer transition metal dichalcogenides, npj 2D Materials and Applica- tions1, ...

    2017

  66. [73]

    Qiao, L.-J

    J.-B. Qiao, L.-J. Yin, and L. He, Twisted graphene bi- layer around the first magic angle engineered by het- erostrain, Physical Review B98, 235402 (2018)

    2018

  67. [74]

    Blakslee, D

    O. Blakslee, D. Proctor, E. Seldin, G. Spence, and T. Weng, Elastic constants of compression-annealed py- rolytic graphite, Journal of Applied Physics41, 3373 (1970)

    1970

  68. [75]

    Z. Bi, N. F. Yuan, and L. Fu, Designing flat bands by strain, Physical Review B100, 035448 (2019)

    2019

  69. [76]

    Moon and M

    P. Moon and M. Koshino, Optical absorption in twisted bilayer graphene, Physical Review B87, 205404 (2013)

    2013

  70. [77]

    D. M. Kennes, M. Claassen, L. Xian, A. Georges, A. J. Millis, J. Hone, C. R. Dean, D. Basov, A. N. Pa- supathy, and A. Rubio, Moiré heterostructures as a condensed-matter quantum simulator, Nature Physics 17, 155 (2021)

    2021

  71. [78]

    A. H. Castro Neto, F. Guinea, N. M. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Reviews of Modern Physics81, 109 (2009)

    2009

  72. [79]

    Sinner, P

    A. Sinner, P. A. Pantaleon, and F. Guinea, Strain- induced quasi-1d channels in twisted moiré lattices, Physical Review Letters131, 166402 (2023)

    2023

  73. [80]

    Halbertal, N

    D. Halbertal, N. R. Finney, S. S. Sunku, A. Kerelsky, C. Rubio-Verdú, S. Shabani, L. Xian, S. Carr, S. Chen, C. Zhang,et al., Moiré metrology of energy landscapes in van der waals heterostructures, Nature Communica- tions12, 242 (2021)

    2021

  74. [81]

    Shallcross, S

    S. Shallcross, S. Sharma, E. Kandelaki, and O. Pankra- tov,Electronicstructureofturbostraticgraphene,Phys- ical Review B81, 165105 (2010)

    2010

  75. [82]

    H. Shi, Z. Zhan, Z. Qi, K. Huang, E. v. Veen, J. Á. Silva- Guillén, R.Zhang, P.Li, K.Xie, H.Ji,et al.,Large-area, periodic, and tunable intrinsic pseudo-magnetic fields in low-angletwistedbilayergraphene,NatureCommunica- tions11, 371 (2020)

    2020

  76. [83]

    D. Wang, F. Escudero, P. A. Pantaleón, S. Yuan, F. Guinea, and Z. Zhan, Straintronics and twistronics in bilayer graphene, To be published (2025)

    2025

  77. [84]

    Koshino, Interlayer interaction in general incommen- surate atomic layers, New Journal of Physics17, 015014 (2015)

    M. Koshino, Interlayer interaction in general incommen- surate atomic layers, New Journal of Physics17, 015014 (2015)

    2015

  78. [85]

    Suárez Morell, J

    E. Suárez Morell, J. Correa, P. Vargas, M. Pacheco, and Z. Barticevic, Flat bands in slightly twisted bilayer graphene: Tight-binding calculations, Physical Review B82, 121407 (2010)

    2010

  79. [86]

    M. Long, P. A. Pantaleón, Z. Zhan, F. Guinea, J. Á. Silva-Guillén, and S. Yuan, An atomistic approach for the structural and electronic properties of twisted bi- layergraphene-boronnitrideheterostructures,npjCom- putational Materials8, 73 (2022)

    2022

  80. [88]

    Leconte, S

    N. Leconte, S. Javvaji, J. An, A. Samudrala, and J. Jung, Relaxation effects in twisted bilayer graphene: A multiscale approach, Physical Review B106, 115410 (2022)

    2022

  81. [89]

    San-Jose, J

    P. San-Jose, J. González, and F. Guinea, Non-abelian gauge potentials in graphene bilayers, Physical Review Letters108, 216802 (2012)

    2012

  82. [90]

    J. S. Alden, A. W. Tsen, P. Y. Huang, R. Hov- den, L. Brown, J. Park, D. A. Muller, and P. L. McEuen, Strain solitons and topological defects in bi- layer graphene, Proceedings of the National Academy of Sciences110, 11256 (2013)

    2013

  83. [91]

    Y. Bai, L. Zhou, J. Wang, W. Wu, L. J. McGilly, D. Hal- bertal, C. F. B. Lo, F. Liu, J. Ardelean, P. Rivera,et al., Excitons in strain-induced one-dimensional moiré po- tentials at transition metal dichalcogenide heterojunc- tions, Nature Materials19, 1068 (2020)

    2020

  84. [92]

    N. C. Hesp, S. Batlle-Porro, R. Krishna Kumar, H. Agarwal, D. Barcons Ruiz, H. Herzig Shein- fux, K. Watanabe, T. Taniguchi, P. Stepanov, and F. H. Koppens, Cryogenic nano-imaging of second-order moiré superlattices, Nature Materials23, 1664 (2024)

    2024

  85. [93]

    Yankowitz, J

    M. Yankowitz, J. Xue, D. Cormode, J. D. Sanchez- Yamagishi, K. Watanabe, T. Taniguchi, P. Jarillo- Herrero, P. Jacquod, and B. J. LeRoy, Emergence of su- perlattice dirac points in graphene on hexagonal boron nitride, Nature Physics8, 382 (2012)

    2012

  86. [94]

    J. Xue, J. Sanchez-Yamagishi, D. Bulmash, P. Jacquod, A. Deshpande, K. Watanabe, T. Taniguchi, P. Jarillo- Herrero, and B. J. LeRoy, Scanning tunnelling mi- croscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride, Nature Materials10, 282 (2011)

    2011

  87. [95]

    Decker, Y

    R. Decker, Y. Wang, V. W. Brar, W. Regan, H.-Z. Tsai, Q.Wu, W.Gannett, A.Zettl,andM.F.Crommie,Local electronic properties of graphene on a bn substrate via scanning tunneling microscopy, Nano Letters11, 2291 (2011)

    2011

  88. [96]

    T. Cea, P. A. Pantaleón, and F. Guinea, Band structure of twisted bilayer graphene on hexagonal boron nitride, Physical Review B102, 155136 (2020)

    2020

  89. [97]

    M. Long, Z. Zhan, P. A. Pantaleón, J. Á. Silva- Guillén, F. Guinea, and S. Yuan, Electronic properties of twisted bilayer graphene suspended and encapsulated with hexagonal boron nitride, Physical Review B107, 115140 (2023)

    2023

  90. [98]

    C. D. Mendoza, I. J. Califrer, and F. L. Freire Jr, Strain in twisted bilayer graphene grown by chemical vapour deposition on ni surfaces, Applied Surface Science544, 148884 (2021)

    2021

  91. [99]

    Tilak, G

    N. Tilak, G. Li, T. Taniguchi, K. Watanabe, and E. Y. Andrei, Moiré potential, lattice relaxation, and layer polarization in marginally twisted mos2 bilayers, Nano Letters23, 73 (2022). xxv

    2022

  92. [100]

    Y. Yu, K. Zhang, H. Parks, M. Babar, S. Carr, I. M. Craig, M. Van Winkle, A. Lyssenko, T. Taniguchi, K. Watanabe,et al., Tunable angle-dependent electro- chemistry at twisted bilayer graphene with moiré flat bands, Nature Chemistry14, 267 (2022)

    2022

  93. [101]

    Van Wijk, A

    M. Van Wijk, A. Schuring, M. Katsnelson, and A. Fa- solino, Relaxation of moiré patterns for slightly mis- aligned identical lattices: graphene on graphite, 2D Ma- terials2, 034010 (2015)

    2015

  94. [102]

    Zhang and E

    K. Zhang and E. B. Tadmor, Structural and electron diffraction scaling of twisted graphene bilayers, Journal of the Mechanics and Physics of Solids112, 225 (2018)

    2018

  95. [103]

    S. Carr, D. Massatt, S. B. Torrisi, P. Cazeaux, M. Luskin, and E. Kaxiras, Relaxation and domain formation in incommensurate two-dimensional het- erostructures, Physical Review B98, 224102 (2018)

    2018

  96. [104]

    Guinea and N

    F. Guinea and N. R. Walet, Continuum models for twisted bilayer graphene: Effect of lattice deformation and hopping parameters, Physical Review B99, 205134 (2019)

    2019

  97. [105]

    Lucignano, D

    P. Lucignano, D. Alfè, V. Cataudella, D. Ninno, and G.Cantele,Crucialroleofatomiccorrugationontheflat bands and energy gaps of twisted bilayer graphene at the magic angleθ1. 08, Physical Review B99, 195419 (2019)

    2019

  98. [106]

    M. M. A. Ezzi, G. N. Pallewela, C. De Beule, E. Mele, and S. Adam, Analytical model for atomic relaxation in twisted moiré materials, Physical Review Letters133, 266201 (2024)

    2024

  99. [107]

    Kang and O

    J. Kang and O. Vafek, Analytical solution for the re- laxed atomic configuration of twisted bilayer graphene including heterostrain, Physical Review B112, 125138 (2025)

    2025

  100. [108]

    Kang and O

    J. Kang and O. Vafek, Pseudomagnetic fields, particle- hole asymmetry, and microscopic effective continuum hamiltonians of twisted bilayer graphene, Physical Re- view B107, 075408 (2023)

    2023

  101. [109]

    Ceferino and F

    A. Ceferino and F. Guinea, Pseudomagnetic fields in fully relaxed twisted bilayer and trilayer graphene, 2D Materials11, 035015 (2024)

    2024

  102. [110]

    Kundu, I

    S. Kundu, I. Maity, R. Bajaj, H. Krishnamurthy, and M. Jain, Atomic relaxation and flat bands in strain- engineeredtransitionmetaldichalcogenidebilayermoiré systems, Physical Review B112, 155412 (2025)

    2025

  103. [111]

    Mesple, N

    F. Mesple, N. R. Walet, G. Trambly de Laissardière, F. Guinea, D. Došenović, H. Okuno, C. Paillet, A. Mi- chon, C. Chapelier, and V. T. Renard, Giant atomic swirl in graphene bilayers with biaxial heterostrain, Ad- vanced Materials35, 2306312 (2023)

    2023

  104. [112]

    Lalmi, J

    B. Lalmi, J. Girard, E. Pallecchi, M. Silly, C. David, S. Latil, F. Sirotti, and A. Ouerghi, Flower-shaped do- mains and wrinkles in trilayer epitaxial graphene on sil- icon carbide, Scientific Reports4, 4066 (2014)

    2014

  105. [113]

    Cazeaux, M

    P. Cazeaux, M. Luskin, and D. Massatt, Energy min- imization of two dimensional incommensurate het- erostructures,ArchiveforRationalMechanicsandAnal- ysis235, 1289 (2020)

    2020

  106. [114]

    Vafek and J

    O. Vafek and J. Kang, Continuum effective hamiltonian for graphene bilayers for an arbitrary smooth lattice de- formation from microscopic theories, Physical Review B 107, 075123 (2023)

    2023

  107. [115]

    X. Gao, H. Sun, D.-H. Kang, C. Wang, Q. J. Wang, and D. Nam, Heterostrain-enabled dynamically tunable moiré superlattice in twisted bilayer graphene, Scientific Reports11, 21402 (2021)

    2021

  108. [116]

    G.-H. Lee, R. C. Cooper, S. J. An, S. Lee, A. Van Der Zande, N. Petrone, A. G. Hammerberg, C. Lee, B. Crawford, W. Oliver,et al., High-strength chemical- vapor–deposited graphene and grain boundaries, Sci- ence340, 1073 (2013)

    2013

  109. [117]

    H. Li, A. W. Contryman, X. Qian, S. M. Ardakani, Y. Gong, X. Wang, J. M. Weisse, C. H. Lee, J. Zhao, P. M. Ajayan,et al., Optoelectronic crystal of artificial atoms in strain-textured molybdenum disulphide, Na- ture Communications6, 7381 (2015)

    2015

  110. [118]

    Y.Zhang, M.Heiranian, B.Janicek, Z.Budrikis, S.Zap- peri, P. Y. Huang, H. T. Johnson, N. R. Aluru, J. W. Lyding, and N. Mason, Strain modulation of graphene by nanoscale substrate curvatures: a molecular view, Nano Letters18, 2098 (2018)

  111. [119]

    So, K.-Y

    J.-P. So, K.-Y. Jeong, J. M. Lee, K.-H. Kim, S.-J. Lee, W. Huh, H.-R. Kim, J.-H. Choi, J. M. Kim, Y. S. Kim,et al., Polarization control of deterministic single- photon emitters in monolayer wse2, Nano Letters21, 1546 (2021)

    2021

  112. [120]

    T. P. Darlington, C. Carmesin, M. Florian, E. Yanev, O. Ajayi, J. Ardelean, D. A. Rhodes, A. Ghiotto, A.Krayev, K.Watanabe,et al.,Imagingstrain-localized excitonsinnanoscalebubblesofmonolayerwse 2 atroom temperature, Nature Nanotechnology15, 854 (2020)

    2020

  113. [121]

    N. Levy, S. Burke, K. Meaker, M. Panlasigui, A. Zettl, F. Guinea, A. C. Neto, and M. F. Crommie, Strain- induced pseudo–magnetic fields greater than 300 tesla in graphene nanobubbles, Science329, 544 (2010)

    2010

  114. [122]

    J. S. Bunch, S. S. Verbridge, J. S. Alden, A. M. Van Der Zande, J. M. Parpia, H. G. Craighead, and P. L. McEuen, Impermeable atomic membranes from graphene sheets, Nano Letters8, 2458 (2008)

    2008

  115. [123]

    Lloyd, X

    D. Lloyd, X. Liu, J. W. Christopher, L. Cantley, A. Wadehra, B. L. Kim, B. B. Goldberg, A. K. Swan, and J. S. Bunch, Band gap engineering with ultralarge biaxial strains in suspended monolayer mos2, Nano Let- ters16, 5836 (2016)

    2016

  116. [124]

    K. Y. Thai, I. Park, B. J. Kim, A. T. Hoang, Y. Na, C. U. Park, Y. Chae, and J.-H. Ahn, Mos2/graphene photodetector array with strain-modulated photore- sponse up to the near-infrared regime, ACS Nano15, 12836 (2021)

    2021

  117. [125]

    H. J. Conley, B. Wang, J. I. Ziegler, R. F. Haglund Jr, S. T. Pantelides, and K. I. Bolotin, Bandgap engineering of strained monolayer and bilayer mos2, Nano Letters 13, 3626 (2013)

    2013

  118. [126]

    I. M. Datye, A. Daus, R. W. Grady, K. Brenner, S. Vaziri, and E. Pop, Strain-enhanced mobility of monolayer mos2, Nano Letters22, 8052 (2022)

    2022

  119. [127]

    Castellanos-Gomez, R

    A. Castellanos-Gomez, R. Roldán, E. Cappelluti, M. Buscema, F. Guinea, H. S. Van Der Zant, and G. A. Steele, Local strain engineering in atomically thin mos2, Nano Letters13, 5361 (2013)

    2013

  120. [128]

    Deng and V

    S. Deng and V. Berry, Wrinkled, rippled and crumpled graphene: an overview of formation mechanism, elec- tronic properties, and applications, Materials Today19, 197 (2016)

    2016

  121. [129]

    J. Lee, S. J. Yun, C. Seo, K. Cho, T. S. Kim, G. H. An, K. Kang, H. S. Lee, and J. Kim, Switchable, tunable, anddirectableexcitonfunnelinginperiodicallywrinkled ws2, Nano Letters21, 43 (2020). xxvi

    2020

  122. [130]

    H. H. Pérez Garza, E. W. Kievit, G. F. Schneider, and U. Staufer, Controlled, reversible, and nondestruc- tive generation of uniaxial extreme strains (> 10%) in graphene, Nano Letters14, 4107 (2014)

    2014

  123. [131]

    D. Ziss, J. Martín-Sánchez, T. Lettner, A. Halilovic, G. Trevisi, R. Trotta, A. Rastelli, and J. Stangl, Com- parison of different bonding techniques for efficient strain transfer using piezoelectric actuators, Journal of Applied Physics121(2017)

    2017

  124. [132]

    Antoniazzi, T

    I. Antoniazzi, T. Woźniak, A. Pawbake, N. Zawadzka, M. Grzeszczyk, Z. Muhammad, W. Zhao, J. Ibáñez, C. Faugeras, M. R. Molas,et al., Pressure-induced op- tical anisotropy of hfs2, Journal of Applied Physics136 (2024)

    2024

  125. [133]

    Jiang, J

    Y. Jiang, J. Mao, J. Duan, X. Lai, K. Watanabe, T. Taniguchi, and E. Y. Andrei, Visualizing strain- induced pseudomagnetic fields in graphene through an hbn magnifying glass, Nano Letters17, 2839 (2017)

    2017

  126. [134]

    Z. Liu, X. Ma, J. Cenker, J. Cai, Z. Fei, P. Malinowski, J. Mutch, Y. Zhao, K. Hwangbo, Z. Lin,et al., Contin- uously tunable uniaxial strain control of van der waals heterostructure devices, Journal of Applied Physics135 (2024)

    2024

  127. [135]

    L. Wang, S. Zihlmann, A. Baumgartner, J. Overbeck, K. Watanabe, T. Taniguchi, P. Makk, and C. Scho- nenberger, In situ strain tuning in hbn-encapsulated graphene electronic devices, Nano Letters19, 4097 (2019)

    2019

  128. [136]

    Sequeira, A

    I. Sequeira, A. Z. Barabas, A. H. Barajas-Aguilar, M. G. Bacani, N. Nakatsuji, M. Koshino, T. Taniguichi, K. Watanabe, and J. D. Sanchez-Yamagishi, Manipulat- ing moires by controlling heterostrain in van der waals devices, Nano Letters24, 15662 (2024)

    2024

  129. [137]

    T. Peña, S. A. Chowdhury, A. Azizimanesh, A. Sewaket, H. Askari, and S. M. Wu, Strain engineering 2d mos2 with thin film stress capping layers, 2D Materials8, 045001 (2021)

    2021

  130. [138]

    Zhang, H

    Y. Zhang, H. L. Zhao, S. Huang, M. A. Hossain, and A. M. van der Zande, Enhancing carrier mobility in monolayer mos2 transistors with process-induced strain, ACS Nano18, 12377 (2024)

    2024

  131. [139]

    Zhang, M

    Y. Zhang, M. A. Hossain, K. J. Hwang, P. F. Ferrari, J. Maduzia, T. Peña, S. M. Wu, E. Ertekin, and A. M. van der Zande, Patternable process-induced strain in 2d monolayers and heterobilayers, ACS Nano18, 4205 (2024)

    2024

  132. [140]

    Q. Liu, Y. Zhang, H. Dong, N. Mason, A. M. van der Zande, and H. T. Johnson, Programmable strainscapes in a two-dimensional (2d) material monolayer, ACS Nano (2025)

    2025

  133. [141]

    Shabani, D

    S. Shabani, D. Halbertal, W. Wu, M. Chen, S. Liu, J. Hone, W. Yao, D. N. Basov, X. Zhu, and A. N. Pasu- pathy, Deep moiré potentials in twisted transition metal dichalcogenide bilayers, Nature Physics17, 720 (2021)

    2021

  134. [142]

    Y. Cao, D. Rodan-Legrain, J. M. Park, N. F. Yuan, K. Watanabe, T. Taniguchi, R. M. Fernandes, L. Fu, andP.Jarillo-Herrero,Nematicityandcompetingorders in superconducting magic-angle graphene, Science372, 264 (2021)

    2021

  135. [143]

    P. A. Pantaleón, H. Sainz-Cruz, and F. Guinea, Design- ing moiré patterns by shearing, ACS Nano18, 28575 (2024)

    2024

  136. [144]

    Halbertal, S

    D. Halbertal, S. Turkel, C. J. Ciccarino, J. B. Profe, N. Finney, V. Hsieh, K. Watanabe, T. Taniguchi, J. Hone, C. Dean,et al., Unconventional non-local re- laxation dynamics in a twisted trilayer graphene moiré superlattice, Nature Communications13, 7587 (2022)

    2022

  137. [145]

    X. Yang, Y. Zhang, L. Chen, K. Aso, W. Yamamori, R. Moriya, K. Watanabe, T. Taniguchi, T. Sasagawa, N. Nakatsuji,et al., Intrinsic one-dimensional moire su- perlattice in large-angle twisted bilayer wte2, ACS Nano 19, 13007 (2025)

    2025

  138. [146]

    F. S. Boi, O. Odunmbaku, A. Taallah, and S. Wang, Quasi-1d, rectangular-like and hexagonal moiré super- lattices in exfoliated highly oriented pyrolytic graphite, Diamond and Related Materials151, 111843 (2025)

    2025

  139. [147]

    J. Shi, G. Chaudhary, A. H. MacDonald, and I. Martin, Spontaneous twirls and structural frustration in moir\’e materials, arXiv preprint arXiv:2509.10907 (2025)

    2025

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.