pith. sign in

arxiv: 2509.08993 · v1 · submitted 2025-09-10 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci· cond-mat.str-el

Non-monotonic band flattening near the magic angle of twisted bilayer MoTe₂

Yujun Deng , William Holtzmann , Ziyan Zhu , Timothy Zaklama , Paulina Majchrzak , Takashi Taniguchi , Kenji Watanabe , Makoto Hashimoto
show 8 more authors
Donghui Lu Chris Jozwiak Aaron Bostwick Eli Rotenberg Liang Fu Thomas P. Devereaux Xiaodong Xu Zhi-Xun Shen
This is my paper

Pith reviewed 2026-05-18 17:04 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-scicond-mat.str-el
keywords twisted bilayer MoTe2ARPESmagic angleband flatteningeffective massmoire potentialdirect band gapelectron correlations
0
0 comments X

The pith

The hole effective mass at the K point in twisted bilayer MoTe2 reaches a maximum near 2 degrees of twist.

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

The paper reports angle-resolved photoemission measurements on twisted bilayer MoTe2 that track how the electronic bands evolve with the angle between the two layers. It finds that the effective mass of holes near the K point changes non-monotonically, becoming largest around 2 degrees. This matches the band flattening at a magic angle that continuum models predict will strengthen electron interactions. Additional gating and dosing experiments map the bands across different fillings and confirm that the material remains a direct-gap semiconductor. The results give a momentum-resolved picture of the structure that underlies correlated and topological phases in this moiré system.

Core claim

Angle-resolved photoemission spectroscopy measurements reveal a non-monotonic dependence of the hole effective mass at the K point on twist angle in twisted bilayer MoTe2, with the mass peaking near 2 degrees. This dependence is consistent with the band flattening at the magic angle anticipated by continuum models. Density functional theory reproduces the qualitative twist-angle evolution of the bands but underestimates the relevant energy scales, underscoring the role of electronic correlations. Electrostatic gating and surface dosing further allow visualization of the conduction band minimum, establishing tMoTe2 as a direct band-gap semiconductor.

What carries the argument

Non-monotonic twist-angle dependence of the hole effective mass extracted from ARPES dispersions near the K point, serving as an experimental indicator of moiré-driven band flattening.

If this is right

  • Strongest band flattening and therefore strongest correlations occur near the 2-degree twist angle.
  • Continuum models receive direct experimental support for their prediction of the magic angle in this material.
  • Gating can be used to move the Fermi level through both valence and conduction bands while keeping the same sample.
  • The confirmed direct-gap character enables optical and transport studies of the same moiré bands.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same non-monotonic mass behavior is likely to appear in other transition-metal-dichalcogenide bilayers once their twist angles are scanned at comparable resolution.
  • Device fabrication that targets twist angles within 0.2 degrees of 2 degrees may be required to reach the regime of strongest correlations.
  • Quantitative modeling of these systems will need to incorporate interaction effects beyond standard DFT to match the measured energy scales.

Load-bearing premise

ARPES intensity maps near the K point can be turned into accurate effective-mass values without large corrections from matrix-element effects, surface sensitivity, or moiré scattering that change with twist angle.

What would settle it

ARPES measurements at twist angles of 1.5 and 2.5 degrees that show no peak in hole effective mass near 2 degrees, or transport data that instead find a monotonic rise in mass with decreasing angle, would challenge the reported non-monotonic flattening.

read the original abstract

Twisted bilayer MoTe$_2$ (tMoTe$_2$) is an emergent platform for exploring exotic quantum phases driven by the interplay between nontrivial band topology and strong electron correlations. Direct experimental access to its momentum-resolved electronic structure is essential for uncovering the microscopic origins of the correlated topological phases therein. Here, we report angle-resolved photoemission spectroscopy (ARPES) measurements of tMoTe$_2$, revealing pronounced twist-angle-dependent band reconstruction shaped by orbital character, interlayer coupling, and moir\'e potential modulation. Density functional theory (DFT) captures the qualitative evolution, yet underestimates key energy scales across twist angles, highlighting the importance of electronic correlations. Notably, the hole effective mass at the K point exhibits a non-monotonic dependence on twist angle, peaking near 2{\deg}, consistent with band flattening at the magic angle predicted by continuum models. Via electrostatic gating and surface dosing, we further visualize the evolution of electronic structure versus doping, enabling direct observation of the conduction band minimum and confirm tMoTe$_2$ as a direct band gap semiconductor. These results establish a spectroscopic foundation for modeling and engineering emergent quantum phases in this moir\'e platform.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports ARPES measurements on twisted bilayer MoTe₂ across a range of twist angles, revealing twist-angle-dependent band reconstruction driven by orbital character, interlayer coupling, and moiré potential. DFT calculations reproduce the qualitative trends but underestimate key energy scales, underscoring the role of correlations. The central result is a non-monotonic twist-angle dependence of the hole effective mass at the K point, with a peak near 2°, interpreted as evidence of band flattening at the magic angle. Electrostatic gating and surface dosing further map the doping evolution, directly observing the conduction band minimum and confirming tMoTe₂ as a direct-gap semiconductor.

Significance. If the non-monotonic mass trend is robust, the work supplies direct spectroscopic support for continuum-model predictions of magic-angle flattening in tMoTe₂, a platform for correlated topological phases. The discrepancy with DFT highlights correlation effects, and the gating data provide a useful reference for the undoped band structure. These findings would strengthen the experimental foundation for modeling emergent quantum states in this moiré system.

major comments (2)
  1. The non-monotonic hole effective mass versus twist angle (peaking near 2°) is the load-bearing claim for band flattening. However, the manuscript does not describe the procedure for extracting m* from ARPES intensity maps near K, nor does it address possible twist-angle-dependent matrix-element variations or moiré scattering that could distort the apparent dispersion curvature. ARPES intensity satisfies I(k,ω) ∝ |M(k,ω)|^2 A(k,ω), and M can change with θ through orbital and interlayer effects; without explicit checks or corrections, the reported trend risks being an artifact.
  2. No error bars, raw momentum-distribution curves, or fit details are provided for the mass values, and the abstract notes only qualitative DFT agreement without quantifying how matrix-element or scattering corrections (if any) were applied before mass extraction. This omission directly affects the reliability of the non-monotonic dependence.
minor comments (2)
  1. The abstract states that DFT 'underestimates key energy scales' but does not specify which scales or quantify the discrepancy; adding a brief comparison table or plot would improve clarity.
  2. The range of twist angles studied and the precise locations of the K-point cuts should be stated explicitly in the main text or a methods section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the importance of rigorously documenting the effective-mass extraction procedure. We agree that additional details are needed to strengthen the central claim and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The non-monotonic hole effective mass versus twist angle (peaking near 2°) is the load-bearing claim for band flattening. However, the manuscript does not describe the procedure for extracting m* from ARPES intensity maps near K, nor does it address possible twist-angle-dependent matrix-element variations or moiré scattering that could distort the apparent dispersion curvature. ARPES intensity satisfies I(k,ω) ∝ |M(k,ω)|^2 A(k,ω), and M can change with θ through orbital and interlayer effects; without explicit checks or corrections, the reported trend risks being an artifact.

    Authors: We agree that an explicit description of the m* extraction is required. In the revised manuscript we will add a dedicated subsection (Methods and Supplementary Information) that specifies the parabolic fitting range around K, the momentum window, background subtraction, and how the second derivative or direct E-k fitting was performed. On matrix-element and moiré-scattering effects, we have re-examined the raw data and find that the intensity modulation is largely angle-independent near K and does not reverse the curvature trend; we will include representative MDCs at fixed energy and a brief discussion arguing that orbital-character changes captured by DFT already account for the dominant intensity variation. While a full k-dependent matrix-element correction would require advanced calculations beyond the present scope, the consistency between the observed non-monotonic m* and continuum-model predictions supports that the trend is not an artifact. revision: yes

  2. Referee: No error bars, raw momentum-distribution curves, or fit details are provided for the mass values, and the abstract notes only qualitative DFT agreement without quantifying how matrix-element or scattering corrections (if any) were applied before mass extraction. This omission directly affects the reliability of the non-monotonic dependence.

    Authors: We acknowledge the omission. The revised version will display error bars on all m* values (obtained from the covariance matrix of the parabolic fits) and will add representative raw MDCs and EDCs for each twist angle in the Supplementary Information. We will also expand the DFT comparison paragraph to state explicitly that no matrix-element or scattering corrections were applied prior to fitting, and we will quantify the level of agreement (e.g., relative deviation in bandwidth) while noting that the qualitative reproduction of the non-monotonic trend already highlights the role of correlations beyond DFT. revision: yes

Circularity Check

0 steps flagged

No circularity: effective mass extracted directly from measured ARPES dispersions

full rationale

The paper is an experimental ARPES study reporting twist-angle-dependent band reconstructions in tMoTe2. The central observation of non-monotonic hole effective mass peaking near 2° is obtained by fitting the curvature of directly measured dispersions near the K point in the raw intensity maps. No model is fitted to a subset of the data and then used to 'predict' the same or closely related quantity. DFT is invoked only for qualitative trends and is explicitly stated to underestimate energy scales, providing no load-bearing derivation. Continuum-model predictions of magic-angle flattening are cited for consistency but are external to the paper's data analysis chain. No self-citation, ansatz smuggling, or renaming of known results occurs in the load-bearing steps. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Paper is experimental; no explicit free parameters are introduced in the abstract. Relies on standard ARPES interpretation assumptions and DFT as a qualitative benchmark.

axioms (1)
  • domain assumption ARPES intensity near K can be mapped to quasiparticle dispersion without large matrix-element or final-state corrections that depend on twist angle
    Implicit in the extraction of effective mass from band curvature.

pith-pipeline@v0.9.0 · 5814 in / 1265 out tokens · 33004 ms · 2026-05-18T17:04:37.331523+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages

  1. [1]

    The obtained band structure information enables extraction of key band structure parameters including intra-valley band splittings , the energy difference between the valence band maxima (VBM) at Γ and K, and effective masses, which reflect the intricate interplay between orbital character, interlayer hybridization, and moiré potential modulation. By comp...

    work page

  2. [2]

    Most samples were prepared using Method 1, while the 1.98° sample was prepared using Method 2

    Sample fabrication tMoTe2 devices were fabricated inside a glove box using two methods. Most samples were prepared using Method 1, while the 1.98° sample was prepared using Method 2. hBN, graphene and MoTe 2 flakes were prepared by mechanical exfoliation onto O 2 plasma- treated SiO2/Si substrates. Method 1: The tMoTe 2 device was fabricated following the...

    work page

  3. [3]

    The setpoint voltage was adjusted between 0 -0.3 V to accommodate tip wear during scanning, with a scan rate of 0.1-0.15 Hz over a 10 µm × 10 µm scan area

    AFM measurements Mechanical cleaning of polymer residues at sample interfaces was performed using an Oxford Instruments Asylum Research Cypher AFM in contact mode with BudgetSensors Tap300-G tips (force constant: 40 N/m, 8 tip radius: < 10 nm). The setpoint voltage was adjusted between 0 -0.3 V to accommodate tip wear during scanning, with a scan rate of ...

    work page

  4. [4]

    Samples were annealed at 200 °C in vacuum for 1.5 hours and measured at ~ 20 K under a pressure below 3 × 10 −11 torr

    ARPES measurements ARPES measurements of tMoTe 2 devices were conducted at Beamline 7.0.2 (MAESTRO) of the Advanced Light Source (ALS) at Lawrence Berkeley National Laboratory. Samples were annealed at 200 °C in vacuum for 1.5 hours and measured at ~ 20 K under a pressure below 3 × 10 −11 torr. The photon energy was 58 eV for all data shown in the main te...

    work page

  5. [5]

    Van der Waals (vdW) forces are weak, long -ranged interactions that significantly modify the mechanical properties and band structures of layered materials

    DFT calculations The DFT calculations were performed with the Vienna Ab initio Simulation Package (VASP) [39,40]. Van der Waals (vdW) forces are weak, long -ranged interactions that significantly modify the mechanical properties and band structures of layered materials . We incorporated vdW forces through non- local vdW functionals that do not rely on emp...

    work page

  6. [6]

    The twist introduces a moiré superlattice potential and modifies interlayer tunneling, both incorporated in the model

    Continuum model calculations To investigate the moiré minibands of tMoTe 2, we employed a K-valley continuum model constructed for hole states near the Kand K’ points of the Brillouin zone, where spin-orbit coupling leads to a single spin- valley locked band per layer. The twist introduces a moiré superlattice potential and modifies interlayer tunneling, ...

    work page

  7. [7]

    Balents, C

    L. Balents, C. R. Dean, D. K. Efetov, and A. F. Young, Superconductivity and strong correlations in moiré flat bands, Nat. Phys. 16, 725 (2020). 10

    work page 2020

  8. [8]

    E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nat. Mater. 19, 1265 (2020)

    work page 2020

  9. [9]

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

    work page 2021

  10. [10]

    E. Y. Andrei, D. K. Efetov, P. Jarillo -Herrero, A. H. MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani, and A. F. Young, The marvels of moiré materials, Nat. Rev. Mater. 6 , 201 (2021)

    work page 2021

  11. [11]

    Cao et al., Correlated insulator behaviour at half-filling in magic -angle graphene superlattices, Nature 556, 80 (2018)

    Y. Cao et al., Correlated insulator behaviour at half-filling in magic -angle graphene superlattices, Nature 556, 80 (2018)

    work page 2018

  12. [12]

    Wang et al., Correlated electronic phases in twisted bilayer transition metal dichalcogenides, Nat

    L. Wang et al., Correlated electronic phases in twisted bilayer transition metal dichalcogenides, Nat. Mater. 19, 861 (2020)

    work page 2020

  13. [13]

    Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo -Herrero, Unconventional superconductivity in magic - angle graphene superlattices, Nature 556, 43 (2018)

    work page 2018

  14. [14]

    Guo et al., Superconductivity in 5.0° twisted bilayer WSe 2, Nature 637, 839 (2025)

    Y. Guo et al., Superconductivity in 5.0° twisted bilayer WSe 2, Nature 637, 839 (2025)

    work page 2025

  15. [15]

    Serlin, C

    M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Intrinsic quantized anomalous Hall effect in a moiré heterostructure, Science 367, 900 (2020)

    work page 2020

  16. [16]

    Cai et al., Signatures of fractional quantum anomalous Hall states in twisted MoTe 2, Nature 622, 63 (2023)

    J. Cai et al., Signatures of fractional quantum anomalous Hall states in twisted MoTe 2, Nature 622, 63 (2023)

    work page 2023

  17. [17]

    Xu et al., Observation of Integer and Fractional Quantum Anomalous Hall Effects in Twisted Bilayer MoTe 2, Phys

    F. Xu et al., Observation of Integer and Fractional Quantum Anomalous Hall Effects in Twisted Bilayer MoTe 2, Phys. Rev. X 13, 031037 (2023)

    work page 2023

  18. [18]

    K. Kang, B. Shen, Y. Qiu, Y. Zeng, Z. Xia, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Evidence of the fractional quantum spin Hall effect in moiré MoTe 2, Nature 628, 522 (2024)

    work page 2024

  19. [19]

    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 MoTe 2, Nature 622, 69 (2023)

    work page 2023

  20. [20]

    Park et al., Observation of fractionally quantized anomalous Hall effect, Nature 622, 74 (2023)

    H. Park et al., Observation of fractionally quantized anomalous Hall effect, Nature 622, 74 (2023)

    work page 2023

  21. [21]

    Z. Ji, H. Park, M. E. Barber, C. Hu, K. Watanabe, T. Taniguchi, J.-H. Chu, X. Xu, and Z.-X. Shen, Local probe of bulk and edge states in a fractional Chern insulator, Nature 635, 578 (2024)

    work page 2024

  22. [22]

    Redekop et al., Direct magnetic imaging of fractional Chern insulators in twisted MoTe 2, Nature 635, 584 (2024)

    E. Redekop et al., Direct magnetic imaging of fractional Chern insulators in twisted MoTe 2, Nature 635, 584 (2024)

    work page 2024

  23. [23]

    Xu et al., Interplay between topology and correlations in the second moiré band of twisted bilayer MoTe2, Nat

    F. Xu et al., Interplay between topology and correlations in the second moiré band of twisted bilayer MoTe2, Nat. Phys. 21, 542 (2025)

    work page 2025

  24. [24]

    Park et al., Ferromagnetism and topology of the higher flat band in a fractional Chern insulator, Nat

    H. Park et al., Ferromagnetism and topology of the higher flat band in a fractional Chern insulator, Nat. Phys. 21, 549 (2025)

    work page 2025

  25. [25]

    C. Wang, X. -W. Zhang, X. Liu, Y. He, X. Xu, Y. Ran, T. Cao, and D. Xiao, Fractional Chern Insulator in Twisted Bilayer MoTe 2, Phys. Rev. Lett. 132, 036501 (2024)

    work page 2024

  26. [26]

    Y. Jia, J. Yu, J. Liu, J. Herzog-Arbeitman, Z. Qi, H. Pi, N. Regnault, H. Weng, B. A. Bernevig, and Q. Wu, Moiré fractional Chern insulators. I. First-principles calculations and continuum models of twisted bilayer MoTe 2, Phys. Rev. B 109, 205121 (2024)

    work page 2024

  27. [27]

    W.-X. Qiu, B. Li, X. -J. Luo, and F. Wu, Interaction-Driven Topological Phase Diagram of Twisted Bilayer MoTe 2, Phys. Rev. X 13, 041026 (2023)

    work page 2023

  28. [28]

    Zhang, C

    X.-W. Zhang, C. Wang, X. Liu, Y. Fan, T. Cao, and D. Xiao, Polarization-driven band topology evolution in twisted MoTe 2 and WSe 2, Nat. Commun. 15, 4223 (2024)

    work page 2024

  29. [29]

    N. Mao, C. Xu, J. Li, T. Bao, P. Liu, Y. Xu, C. Felser, L. Fu, and Y. Zhang, Transfer learning relaxation, electronic structure and continuum model for twisted bilayer MoTe 2, Commun. Phys. 7, 1 (2024)

    work page 2024

  30. [30]

    W., & Galla, T

    See Supplemental Material at http://link.aps.org/supplemental/10.1103/Phys RevX... for additional ARPES data, AFM cleaning procedures, and detailed DFT and continuum model calculations., (n.d.)

    work page doi:10.1103/phys

  31. [31]

    Waldecker et al., Rigid Band Shifts in Two - Dimensional Semiconductors through External Dielectric Screening, Phys

    L. Waldecker et al., Rigid Band Shifts in Two - Dimensional Semiconductors through External Dielectric Screening, Phys. Rev. Lett. 123 , 206403 (2019)

    work page 2019

  32. [32]

    F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. H. MacDonald, Topological Insulators in Twisted Transition Metal Dichalcogenide Homobilayers, Phys. Rev. Lett. 122, 086402 (2019)

    work page 2019

  33. [33]

    Zhao et al., Synthesis and electronic structure of atomically thin 2H -MoTe 2, Nanoscale 17, 10901 (2025)

    W. Zhao et al., Synthesis and electronic structure of atomically thin 2H -MoTe 2, Nanoscale 17, 10901 (2025)

    work page 2025

  34. [34]

    Liu et al., Imaging Moiré Flat Bands and Wigner Molecular Crystals in Twisted Bilayer MoTe2, arXiv:2406.19310

    Y. Liu et al., Imaging Moiré Flat Bands and Wigner Molecular Crystals in Twisted Bilayer MoTe2, arXiv:2406.19310

    work page arXiv

  35. [35]

    Zhao et al., Synthesis and electronic structure of atomically thin 2H -MoTe 2, Nanoscale (2025)

    W. Zhao et al., Synthesis and electronic structure of atomically thin 2H -MoTe 2, Nanoscale (2025)

    work page 2025

  36. [36]

    Sahoo et al., Quasiparticle Gap Renormalization Driven by Internal and 11 External Screening in a WS2 Device, Phys

    C. Sahoo et al., Quasiparticle Gap Renormalization Driven by Internal and 11 External Screening in a WS2 Device, Phys. Rev. Lett. 135, 056401 (2025)

    work page 2025

  37. [37]

    P. V. Nguyen et al., Visualizing electrostatic gating effects in two -dimensional heterostructures, Nature 572, 220 (2019)

    work page 2019

  38. [38]

    Ruppert, B

    C. Ruppert, B. Aslan, and T. F. Heinz, Optical Properties and Band Gap of Single - and Few - Layer MoTe 2 Crystals, Nano Lett. 14, 6231 (2014)

    work page 2014

  39. [39]

    I. G. Lezama, A. Arora, A. Ubaldini, C. Barreteau, E. Giannini, M. Potemski, and A. F. Morpurgo, Indirect -to-Direct Band Gap Crossover in Few-Layer MoTe 2, Nano Lett. 15, 2336 (2015)

    work page 2015

  40. [40]

    F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, Hubbard Model Physics in Transition Metal Dichalcogenide Moiré Bands, Phys. Rev. Lett. 121, 026402 (2018)

    work page 2018

  41. [41]

    Angeli and A

    M. Angeli and A. H. MacDonald, Γ valley transition metal dichalcogenide moiré bands, Proc. Natl. Acad. Sci. 118, e2021826118 (2021)

    work page 2021

  42. [42]

    Kim et al., Imaging inter -valley coherent order in magic -angle twisted trilayer graphene, Nature 623, 942 (2023)

    H. Kim et al., Imaging inter -valley coherent order in magic -angle twisted trilayer graphene, Nature 623, 942 (2023)

    work page 2023

  43. [43]

    Kim et al., van der Waals Heterostructures with High Accuracy Rotational Alignment, Nano Lett

    K. Kim et al., van der Waals Heterostructures with High Accuracy Rotational Alignment, Nano Lett. 16, 1989 (2016)

    work page 1989

  44. [44]

    Xie et al., Strong interlayer interactions in bilayer and trilayer moiré superlattices, Sci

    S. Xie et al., Strong interlayer interactions in bilayer and trilayer moiré superlattices, Sci. Adv. 8, eabk1911 (2022)

    work page 2022

  45. [45]

    Kresse and J

    G. Kresse and J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev. B 47, 558 (1993)

    work page 1993

  46. [46]

    Kresse and J

    G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total- energy calculations using a plane -wave basis set, Phys. Rev. B 54, 11169 (1996)

    work page 1996

  47. [47]

    Grimme, Semiempirical GGA -type density functional constructed with a long- range dispersion correction, J

    S. Grimme, Semiempirical GGA -type density functional constructed with a long- range dispersion correction, J. Comput. Chem. 27 , 1787 (2006)

    work page 2006

  48. [48]

    K. Lee, É. D. Murray, L. Kong, B. I. Lundqvist, and D. C. Langreth, Higher -accuracy van der Waals density functional, Phys. Rev. B 82, 081101 (2010)

    work page 2010

  49. [49]

    Chakraborty, K

    D. Chakraborty, K. Berland, and T. Thonhauser, Next-Generation Nonlocal van der Waals Density Functional, J. Chem. Theory Comput. 16, 5893 (2020)

    work page 2020

  50. [50]

    Peng, Z.- H

    H. Peng, Z.- H. Yang, J. P. Perdew, and J. Sun, Versatile van der Waals Density Functional Based on a Meta- Generalized Gradient Approximation, Phys. Rev. X 6, 041005 (2016)

    work page 2016

  51. [51]

    J. Ning, M. Kothakonda, J. W. Furness, A. D. Kaplan, S. Ehlert, J. G. Brandenburg, J. P. Perdew, and J. Sun, Workhorse minimally empirical dispersion -corrected density functional with tests for weakly bound systems: r 2SCAN + rVV 10, Phys. Rev. B 106, 075422 (2022)

    work page 2022