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Cited by 171Topological Materials and PhenomenaQuantum Mechanics and Non-Hermitian PhysicsStrong Light-Matter Interactions

Type-III and Tilted Dirac Cones Emerging from Flat Bands in Photonic Orbital Graphene

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Milićević, M., Montambaux, G., Ozawa, T., Jamadi, O., Real, B., Sagnes, I., Lemaître, A., Le Gratiet, L. +3 more·2019·Physical Review X
DOI 10.1103/physrevx.9.031010
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Abstract

The extraordinary electronic properties of Dirac materials, the two-dimensional partners of Weyl semimetals, arise from the linear crossings in their band structure.When the dispersion around the Dirac points is tilted, one can predict the emergence of intricate transport phenomena such as modified Klein tunneling, intrinsic anomalous Hall effects, and ferrimagnetism.However, Dirac materials are rare, particularly with tilted Dirac cones.Recently, artificial materials whose building blocks present orbital degrees of freedom have appeared as promising candidates for the engineering of exotic Dirac dispersions.Here we take advantage of the orbital structure of photonic resonators arranged in a honeycomb lattice to implement photonic lattices with semi-Dirac, tilted, and, most interestingly, type-III Dirac cones that combine flat and linear dispersions.Type-III Dirac cones emerge from the touching of a flat and a parabolic band when synthetic photonic strain is introduced in the lattice, and they possess a nontrivial topological charge.This photonic realization provides a recipe for the synthesis of orbital Dirac matter with unconventional transport properties and, in combination with polariton nonlinearities, opens the way to study Dirac superfluids in topological landscapes.

AI snapshot

Written from the paper’s full text.

TL;DR

This paper demonstrates photonic orbital graphene in a honeycomb lattice of polariton micropillars, where strain in the orbital p bands produces semi-Dirac, tilted, and type-III Dirac cones. It reports the first experimental observation of type-III Dirac cones, formed by the touching of a flat band and a linear band, and explains their emergence via topological Lifshitz transitions.

Key findings

  • Type-III Dirac cones were observed for synthetic strain β = 1.5, combining a flat band and a linear band; measured velocities were |v0x| = (0.29 ± 0.03) d|tL| and |vx| = (0.35 ± 0.03) d|tL|, in agreement with tight-binding prediction |v0x| = |vx| = 0.37 d|tL|. Along the perpendicular direction, |v0y| = (0.00 ± 0.04) d|tL| and |vy| = (0.46 ± 0.04) d|tL|, matching predictions of 0 and 0.48 d|tL|.
  • A semi-Dirac dispersion was directly observed at β = 0.5, where two Dirac cones merge, with parabolic touching along ky and linear dispersion along kx.
  • Tilted type-I Dirac cones emerge from the flat-parabolic band touching at Γ for β < 1, with measured tilt v0y agreeing with tight-binding calculations.
  • Topological analysis shows the β = 1 flat-parabolic touching has winding W = 2; for β ≠ 1 it evolves into a pair of Dirac cones with W = 1, while the semi-Dirac merging has W = 0. Type-III cones are robust for 1 < β < 2 and merge at β = 2.

Methods

  • The platform is a honeycomb lattice of coupled micropillars etched from an AlGaAs semiconductor microcavity with twelve GaAs quantum wells, operating in the strong-coupling regime at 10 K to form exciton-polaritons. Micropillar diameter is 2.75 μm (2.60 μm for type-III samples), with center-to-center distances varied between 2.40 and 2.72 μm, or 2.40 and 2.10 μm for type-III.
  • Angle-resolved photoluminescence under nonresonant continuous-wave excitation at 745 nm was used to reconstruct the band dispersion. The p-orbital bands arise from coupling of first excited modes, with longitudinal hopping tL and transverse hopping tT.
  • Synthetic uniaxial strain was implemented by changing the center-to-center distance d′ along one link direction, defining β = t′L/tL. Experimental dispersions were fitted with a nearest-neighbor tight-binding model, and effective Dirac velocities were extracted from linear fits near the Dirac points.

Limitations

  • The paper states that the β = 2 merging of type-III Dirac cones could not be observed experimentally because engineered hoppings in compressed structures were limited to β = 1.5.
  • The paper notes that tT is not strictly zero in the experiment, producing faint emission between horizontal links; simulations reproduce this with tT = −0.1tL on strained links.
  • The paper states that the uppermost band deviates from a flat band due to coupling to higher-energy bands from photonic d orbitals, and that polarization-related splitting is neglected because it is much smaller than the bandwidth.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevx.9.031010

Cites

58 of 58 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 8 cited sources checked · 0 supported · 0 flagged

  • Findingnone given

    The orbital bands present a flat band that touches a parabolic band.

    “The orbital bands present a flat band that touches a parabolic band.”
    Quote in the paperSubstantial claim
  • Findingown data

    For β=1.5, in the direction k_x parallel to the reduced center-to-center distance, two new Dirac points emerge from the flat-parabolic band touching.

    “The result of the compression in the polariton lattice is shown in Fig. 5(c) for β ¼ 1.5:In the direction k x parallel to that along which the center-to-center distance d 0 is reduced, two new Dirac points emerge from the flat-parabolic band touching.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The system allows observation of semi-Dirac cones, which combine massless and massive dispersions.

    “The richness of this multiband system allows, in addition, the observation of semi-Dirac cones, which combine massless and massive dispersions.”
    Quote in the paperSubstantial claim
  • Findingnone given

    Type-III Dirac cones possess a nontrivial topological charge.

    “they possess a nontrivial topological charge”
    Quote in the paperSubstantial claim
  • Findingnone given

    Type-III Dirac cones emerge from the touching of a flat and a parabolic band when synthetic photonic strain is introduced in the lattice.

    “Type-III Dirac cones emerge from the touching of a flat and a parabolic band when synthetic photonic strain is introduced in the lattice”
    Quote in the paperSubstantial claim
  • May combine two claims. 18
  • Classified as probably not a substantial claim. 4

  • May combine two claims.

    Findingown data

    At β=0.5, a semi-Dirac dispersion is directly observed, with touching of two parabolic bands along k_y and linear dispersion along k_x.

    “At the merging point [β ¼ 0.5, Fig. 3(c)], we provide the direct observation of a semi-Dirac dispersion, with the touching of two parabolic bands along the k y direction and a linear dispersion along k x [shown in the left inset of Fig. 3(c)].”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingcitation

    Four bands are observed: the lowest is flat, while the two central bands present two type-I Dirac points at K and K' and touch the flat band at Γ.

    “Four bands are observed: The lowest one is flat, while the two central bands present two type-I Dirac points at the K and K' points and touch the flat band at the Γ point [36].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [36]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Findingreasoning

    For β > 1, the Dirac points have E_D = E0 - (3/2)|t_L|, cos(K_x) = (2-β^2)/β, and K_y = 0.

    “while for β > 1, they are E D ¼ E 0 - 3 2 jt L j; cosðK x Þ ¼ 2 -β 2 β ; K y ¼ 0:ðB7Þ”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For the type-I Dirac cone at E0 in Fig. 3(a) at positive momenta, measured |v'_y| = 0.07 ± 0.03 and |v_y| = 0.72 ± 0.03 in units of d|t_L|.

    “After converting the slopes to effective Weyl velocities [Eq.( 1)], we obtain jv 0y j ¼ 0.07 Æ 0.03, close to the expected value of 0, and jv y j ¼ 0.72 Æ 0.03 in units of djt L j.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given

    At 10 K, the excitonic and photonic resonances are in the strong coupling regime, with a Rabi splitting of 15 meV.

    “At 10 K, the temperature of the experiments, the excitonic and photonic resonances are in the strong coupling regime, with a Rabi splitting of 15 meV.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    For β < 1, the energy E_D and momentum-space coordinates of the Dirac points emerging at the bottom bands are E_D = E0 - (|t_L|/2) sqrt(3+6β^2), K_x = 0, and cos(K_y) = β.

    “For β < 1, the energy E D and momentum-space coordinates of the Dirac points emerging at the bottom bands are, respectively, E D ¼ E 0 - jt L j 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 þ 6β 2 q ; K x ¼ 0; cosðK y Þ ¼ β;ðB6Þ”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The single band touching at β=1 is described by a winding W=2; when β is greater or less than 1, the band touching evolves into a pair of Dirac cones with winding W=1.

    “The single band touching at β ¼ 1 is described by a winding W ¼ 2. When β ≷ 1, the band touching evolves into a pair of Dirac cones (tilted type-I or type-III) with winding W ¼ 1.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    From experimental photoluminescence, |v0x|=(0.29±0.03)d|tL| and |vx|=(0.35±0.03)d|tL|, in good agreement with the tight-binding prediction for β=1.5 (|v0x|=|vx|=0.37 d|tL|).

    “From the experimental photoluminescence of Fig. 5(c), we measure jv 0x j ¼ ð0.29 Æ 0.03Þ djt L j and jv x j ¼ ð0.35 Æ 0.03Þ djt L j [with d ¼ ð2d 0 þ dÞ=3], which is in good agreement with the tight-binding prediction for β ¼ 1.5 (jv 0x j ¼ jv x j ¼ 0.37 djt L j, see Appendix C).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    For 1 < β < 2, the analytical expressions are |v'_x| = |v_x| = (1/4) sqrt((β^2-1)(4-β^2)); |v'_y| = 0; |v_y| = (sqrt(3)/4) sqrt(β^2-1).

    “For 1 < β < 2, we have (same units as above) jv 0x j ¼ jv x j ¼ 1 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðβ 2 -1Þð4 -β 2 Þ q; jv 0y j ¼ 0; jv y j ¼ ffiffi ffi 3 p 4 ffiffiffiffiffiffiffiffiffiffiffiffi ffi β 2 -1 q :”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    Type-III Dirac cones are present for all values of β between 1 and 2, proving their robustness against variations of the strain parameter.

    “The type-III Dirac cones are thus present for all values of β between 1 and 2, proving their robustness against variations of the strain parameter.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    From the tight-binding Hamiltonian, the effective Dirac velocities can be obtained from derivatives of the eigenvalues at the emergent Dirac cones, resulting in analytical expressions for β < 1 (normalized to d|t_L|, assuming d ≈ d').

    “From the tight-binding Hamiltonian, the effective Dirac velocities can be obtained from the derivative along k x and k y of the eigenvalues of Eqs.(B3)-(B5) at the position of the emergent Dirac cones [Eqs.(B6) and (B7)], resulting in the analytical expressions for β < 1 (normalized to djt L j, where we have assumed that, in the experiment, d ≈ d 0 ):”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: In the experiment, d ≈ d'.

  • May combine two claims.

    Findingreasoning

    For β < 1, the analytical expressions for velocities are |v'_x| = 0; |v_x| = |v'_y| = (3β/4) sqrt((1-β^2)/(1+2β^2)); |v_y| = (sqrt(3)/4) sqrt(1-β^2).

    “jv 0x j ¼ 0; jv x j ¼ jv 0y j ¼ 3β 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 -β 2 1 þ 2β 2 s ; jv y j ¼ ffiffi ffi 3 p 4 ffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 -β 2 q :”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    At β=2, the Dirac cones merge at the M point and annihilate, opening a gap for β>2.

    “At β ¼ 2, actually, the Dirac cones merge at the M point and annihilate, opening a gap for β > 2.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingreasoning

    In the etched structures, the lateral defects induced during the microfabrication process reduce the lifetime to about 5-15 ps at the considered photon-exciton detunings.

    “In the etched structures, the lateral defects induced during the microfabrication process reduce the lifetime to about 5-15 ps at the considered photon-exciton detunings.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given

    The polariton lifetime measured in a similar unetched microcavity is of the order of 30 ps.

    “The polariton lifetime measured in a similar unetched microcavity is of the order of 30 ps.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingcitation

    The orbital p bands show the two possible universal scenarios for merging of Dirac cones in two dimensions: two Dirac cones with W=1 emerge from a point with W=2, and two Dirac cones with opposite winding W=±1 merge in a semi-Dirac cone with W=0.

    “Remarkably, the orbital p bands show the two possible universal scenarios for the merging of Dirac cones in two dimensions [50]: (i) Two Dirac cones with W ¼ 1 emerge from a point with W ¼ 2, which we have just discussed, and it is reported in the lower part of the spectra of Fig. 3; (ii) two Dirac cones with opposite winding W ¼ Æ1 merge in a semi-Dirac cone with W ¼ 0, reported in the spectra in”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [50]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    Polaritons are sensitive to external magnetic fields, allowing the implementation of quantum Hall phases.

    “they are also sensitive to external magnetic fields, allowing the implementation of quantum Hall phases [54,55]”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [54]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [55]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Premisereasoning

    For type-I Dirac cones, v'_x = v'_y = 0, and the absolute value of the two slopes should be identical and equal to v_y.

    “Note that for type-I Dirac cones, v 0x ¼ v 0y ¼ 0, and the absolute value of the two slopes should be identical and equal to v y .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    Polaritons present significant repulsive interactions in the high-density regime, which can be achieved when using a resonant excitation scheme.

    “Thanks to their excitonic component, they present significant repulsive interactions in the high-density regime, which can be achieved when using a resonant excitation scheme [53]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [53]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The evolution of a touching flat and dispersive band with winding W=2 into two Dirac cones under strain has been predicted for other lattice geometries.

    “This behavior has been predicted for other lattice geometries [45]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [45]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Interpretationreasoning

    For β > 1, |v'_x| = |v_x| with opposite sign (v'_x = -v_x), which is the signature of the type-III Dirac cone and results in a flat band along the k_x direction.

    “Note that the fact that jv 0x j ¼ jv x j (with opposite sign: v 0x ¼ -v x ) for β > 1 is the signature of the type-III Dirac cone, and it results in a flat band along the k x direction.”
    Quote in the paperSubstantial claim
21 of 21

  • The paper relies on this without support.

    Type-III Dirac points have not yet been reported experimentally.

    “Type-III Dirac points have not yet been reported experimentally”

  • The authors state this as an assumption.

    In the nearest-neighbor tight-binding calculation, the coupling t_T is assumed to be zero.

    “assuming t T ¼ 0”

  • The authors state this as an assumption.

    The on-site energy for the p_x, p_y orbitals is 0.

    “where we have assumed that the on-site energy for the p x , p y orbitals is 0,”

  • The authors state this as an assumption.

    In the experiment, d ≈ d'.

    “where we have assumed that, in the experiment, d ≈ d 0”

  • The authors state this as an assumption.

    The hopping of orbitals oriented perpendicular to the links is negligible (t_T = 0).

    “If the hopping of orbitals oriented perpendicular to the links is negligible (t T ¼ 0)”
5 of 5

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Authors
Milićević, M., Montambaux, G., Ozawa, T., Jamadi, O., Real, B., Sagnes, I., Lemaître, A., Le Gratiet, L., Harouri, A., Bloch, J., & Amo, A.
Affiliations
  • Université Paris-Sud (FR)
  • RIKEN (JP)
  • Centre National de la Recherche Scientifique (FR)
ORCID
Venue
Physical Review X
Publisher
American Physical Society (APS)
Published
2019-07-23
Volume
9
Issue
3
ISSN
2160-3308 (ISSN-L)
DOI
10.1103/physrevx.9.031010
OpenAlex
W2884840542
Semantic Scholar
38b2bab9af1d5a17dfd7279a962bf7883f5d587f
Cited by
171
References
55
Citations by year
2026: 8 · 2025: 27 · 2024: 28 · 2023: 22 · 2022: 27 · 2021: 30 · 2020: 25 · 2019: 4
FWCI
9.54
Type
journal article
Language
en
Open access
Licence
Locations
  • Physical Review X (publishedVersion, cc-by)
  • arXiv (Cornell University) (submittedVersion)
  • HAL (Le Centre pour la Communication Scientifique Directe) (submittedVersion, other-oa)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • Figshare (submittedVersion)
  • arXiv (Cornell University)
  • Greater South Information System
Funders
  • H2020 European Research Council
  • Agence Nationale de la Recherche (ANR-16-CE30-0021, ANR-11-LABX-0007, ANR-10-LABX-0035)
  • Japan Society for the Promotion of Science (JP18H05857)
  • RIKEN
  • French RENATECH network
  • CPER Photonics for Society P4S
  • I-Site ULNE
  • Métropole Européenne de Lille
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Strong Light-Matter Interactions
Keywords
tilted Dirac cones, flat bands, photonic lattices, honeycomb lattice, photonic resonators, topological charge, Klein tunneling, anomalous Hall effect, orbital degrees of freedom

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