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Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons
Open accessAbstract
Two-dimensional lattices of coupled micropillars etched in a planar semiconductor microcavity offer a workbench to engineer the band structure of polaritons. We report experimental studies of honeycomb lattices where the polariton low-energy dispersion is analogous to that of electrons in graphene. Using energy-resolved photoluminescence, we directly observe Dirac cones, around which the dynamics of polaritons is described by the Dirac equation for massless particles. At higher energies, we observe p orbital bands, one of them with the nondispersive character of a flatband. The realization of this structure which holds massless, massive, and infinitely massive particles opens the route towards studies of the interplay of dispersion, interactions, and frustration in a novel and controlled environment.
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TL;DR
The paper reports direct observation of Dirac cones and a flatband in a honeycomb lattice of coupled semiconductor micropillars for polaritons. Energy-resolved photoluminescence reveals graphene-like π/π* bands with Dirac points and higher-energy P bands including a nondispersive flatband.
Key findings
- The two lowest S bands mimic graphene’s π and π* bands, with six Dirac cones at the corners of the first Brillouin zone; a tight-binding fit gives first- and second-neighbor tunneling \(t = 0.25\) meV and \(t' = -0.02\) meV, and group velocity \(v = 3at/2\hbar = 1.3 \times 10^6\) m/s around the Dirac points.
- Higher-energy P bands appear above the S bands, separated by \(\Delta E = 3.2\) meV between the two lowest-energy states of individual pillars; the lowest P band is flat, with an infinite effective mass, and the P bands are separated by a gap of about 0.7 meV in Fig. 1(d).
- Increasing excitation intensity produces polariton condensation at the top of the π* band at the Γ point, with a threshold, single emission line, spatial coherence over the pump spot, and a π phase shift between adjacent pillars.
- The flatband real-space mode has intensity maxima between pillars arranged in a kagome geometry; a tight-binding model with \(t_\parallel = -1\) meV and \(t_\perp = 0\) meV gives two flat extreme bands, while \(t_\perp = 0.2\) meV makes them dispersive, and an energy-dependent \(t_\perp\) explains the observed lower flat and upper dispersive bands.
Methods
- Sample: a \(Q = 72000\) λ/2 microcavity with Ga\(_{0.05}\)Al\(_{0.95}\)As layer, 28 top/40 bottom Bragg mirror pairs, twelve 7 nm GaAs quantum wells, 15 meV Rabi splitting, experiments at 10 K and \(-17\) meV cavity-exciton detuning. A honeycomb lattice of coupled micropillars was etched with pillar diameter \(d = 3\) µm and lattice constant \(a = 2.4\) µm; polariton lifetime is 27 ps at the bottom of the lower polariton band, and tunnel coupling is 0.25 meV.
- Measurements: nonresonant Ti:Sapph laser at 730 nm in a 30 µm spot covering about 30 pillars; energy-resolved photoluminescence collected through a NA = 0.65 objective, dispersed in a spectrometer, and imaged in real or momentum space with a CCD. Low-power excitation probes single-particle physics; a chopper was used at high power to avoid heating.
- Analysis: tight-binding fits to the measured dispersion, including first- and second-neighbor tunneling; a P-state tight-binding model with \(t_\parallel\) and \(t_\perp\); supplementary 2D Schrödinger and Gross-Pitaevskii simulations.
Limitations
- The paper states that the highest available excitation density in the experiment prevented observation of the expected modification in the spatial profile when the condensate evolves into a gap soliton.
- Text-evident: the flatband explanation relies on a phenomenological model in which \(t_\perp\) is assumed to increase with energy, supported by numerical simulations rather than direct measurement of the energy-dependent tunneling.
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A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Using energy-resolved photoluminescence we directly observe Dirac cones”
- Also stated in: § Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2 · plain
“We observe the six Dirac points at the corner of the first Bz (yellow points).”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2 · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In this Letter, we report on a honeycomb lattice for polaritons, made of hundreds of coupled micropillars etched in a planar semiconductor microcavity.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The separation between these two groups of bands is ∆ E = 3 . 2 meV, the energy difference between the two lowest-energy states of the individual pillars.”
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“showing spatial coherence extended over the whole excitation spot.”
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIn the paper
“At higher energy, we observe a group of four bands ( P bands) arising from the coupling between the first excited state of the pillars, which is twice degenerate and has two lobes [22]”
What the sources say
- Not matchedcites [22] M. Galbiati, L. Ferrier, D. D. Solnyshkov, D. Tanese, E. Wertz, A. Amo, M. Abbarchi, P. Senellart, I. Sagnes, A. Lemaître, E. Galopin, G. Malpuech, and J. Bloch, Phys. Rev. Lett. 108 , 126403 (2012).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“we report on the presence of higher-energy bands arising from the coupling between higher-energy modes of the pillars.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“we observe a nondispersive band in which polaritons have an infinite effective mass.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Figure 2(a) shows the measured emitted intensity in momentum space at the Dirac points energy [zero energy in Fig. 1(d)]. We observe the six Dirac points at the corner of the first Bz (yellow points).”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For our structure, the tunnel coupling amounts to 0.25 meV.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The lowest two arise from the coupling between the fundamental modes of the micropillars.”
- Also stated in: § Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2 · plain
“The two lowest bands ( S bands) arise from the coupling between the fundamental mode of the pillars ( S modes).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2 · plain
- It may state two separate claims as one. 14
- Probably not a substantial claim. 5
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“By increasing the excitation intensity, we observe polariton condensation, as evidenced by the threshold in the integrated emission intensity [Fig. 3(d)].”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“At higher energies, we observe p orbital bands, one of them with the non-dispersive character of a flatband.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
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In the paper
“If we perform the same measurement along line 3 in Fig. 2(a), we show in Fig. 2(d) that the emission is absent in the upper band (dashed line) within the first Bz, and in the lower band (solid line) within the second Bz.”
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In particular, we evidence six Dirac cones at the corners of the first Brillouin zone (Bz), around which the energy dispersion is linear.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Figures 2(b and c) show the measured energy resolved emission along the lines 1 and 2 indicated in Fig. 2(a), passing through four and three Dirac linear intersections respectively.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
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In the paper
“At the intersection between two adjacent pillars [white square in Fig. 3(c)], the fringes are shifted by half a period. Thus there is a π phase shift between adjacent pillars, as expected for the antibonding π ∗ band.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
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In the paper
“Moreover, via resonant excitation of the structure, polariton wave packets can be created with any desired energy and momentum.”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Moreover, the emission spectrum collapses into a single emission line, and extended spatial coherence builds up.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
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In the paper
“The real space emission of the condensate is shown in Fig. 3(b), covering the same area as the pump spot.”
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the limiting case where t ⊥ = 0 meV and t ‖ = -1 meV, the eigenstates of Eq. (3) give rise to four energy bands plotted in Fig. 4(c). The two extreme bands are flat, the two intermediate ones are dispersive.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“The threshold power is similar to that observed in a planar structure [25].”
What the sources say
- Not matchedcites [25] E. Wertz, L. Ferrier, D. D. Solnyshkov, P. Senellart, D. Bajoni, A. Miard, A. Lemaitre, G. Malpuech, and J. Bloch, Appl. Phys. Lett. 95 , 051108 (2009).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The position, shape and size of each lattice site can be controlled at will during fabrication.”
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“Note that for low power excitation polaritonpolariton interactions are negligible so that single particle physics of the honeycomb lattice is probed.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“It occurs in lattices with multiple sites per unit cell [23] and has been observed along the K -Γ -K ′ directions in angle-resolved electron spectroscopy measurements in graphene [24].”
What the sources say
- Not matchedcites [23] E.L. Shirley, L.J. Terminello, A. Santoni, and F.J. Himpsel, Phys. Rev. B 51 , 13614 (1995).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [24] A. Bostwick, T. Ohta, T. Seyller, K. Horn, and E. Rotenberg, Nat. Phys. 3 , 36 (2006).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Indeed, the tunneling probability varies exponentially with the barrier height relative to the state, and thus increases strongly for higher energy states.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“The simulation is in excellent quantitative agreement with the experimental observation: the lowest P band is indeed flat, while the upper band is dispersive.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, the states with negative mass are on the contrary trapped in this potential, and serve as a seed for stimulated scattering.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“The antibonding mode at the Γ point favors condensation due to two features: ( i ) its negative effective mass and positive interaction energy, ( ii ) its longer lifetime, which stems from the antisymmetric character of the state [2], and from the lower nonradiative recombination rate due to the vanishing polariton density at the constrictions between pillars, where the defect density is larger.”
What the sources say
- Not matchedcites [2] I. L. Aleiner, B. L. Altshuler, and Y. G. Rubo, Phys. Rev. B 85 , 121301 (2012).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Not matched
The authors state this as an assumption.
The tunnelling probability of P states is energy dependent, increasing with energy.
“in which we assume that the tunnelling probability is energy dependent”
“The band structure reported in Fig. 4(a) can then be understood assuming that t ⊥ increases with the energy”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The cited source says what the claim attributes to it.
In the citing paper
“The band structure related to Dirac cones has been widely investigated in the energy-momentum space using the bulk measurement with measured density and dispersion pattern [10, 22-25].”
From Observing movement of Dirac cones from single-photon dynamics (2021) · cites this paper as [24]
The check wasn’t confident either way, so it gives no verdict — never disputed.
In the citing paper
“One should note that the opening band gap in graphene requires large strain, even exceeding the order of 20% [24, 29], which is attributed to the deformations and anisotropy by applying strain instead of the inner disorders or fluctuation of the graphene.”
From Observing movement of Dirac cones from single-photon dynamics (2021) · cites this paper as [24]
Experimental — accuracy not yet measuredIn the citing paper
“In the topological photonics context, this technique was used, e.g., to visualize the relativistic Dirac-like dispersion in honeycomb lattices (Jacqmin et al., 2014) and the corresponding edge states (Milićević et al., 2015(Milićević et al., , 2017)).”
From Topological photonics (2019) · cites this paper as Jacqmin et al., 2014
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Metadata
- Authors
- Jacqmin, T., Carusotto, I., Sagnes, I., Abbarchi, M., Solnyshkov, D. D., Malpuech, G., Galopin, E., Lemaître, A., Bloch, J., & Amo, A.
- Affiliations
- Centre National de la Recherche Scientifique (FR)
- University of Trento (IT)
- ORCID
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2014-03-18
- Volume
- 112
- Issue
- 11
- Pages
- 116402-116402
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2043218475
- Cited by
- 509
- References
- 35
- Citations by year
- 2026: 20 · 2025: 24 · 2024: 50 · 2023: 22 · 2022: 44 · 2021: 48 · 2020: 52 · 2019: 35 · 2018: 45 · 2017: 74 · 2016: 39 · 2015: 36 · 2014: 20
- FWCI
- 27.97
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- HAL (Le Centre pour la Communication Scientifique Directe) (submittedVersion)
- INO Open Portal (submittedVersion)
- Iris (University of Trento) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Agence Nationale de la Recherche (ANR-11-BS10-0001, ANR-10-LABX-0035)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Strong Light-Matter Interactions
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Plasmonic and Surface Plasmon Research
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Semiconductor Quantum Structures and Devices
- Keywords
- polaritons, honeycomb lattice, Dirac cones, flat bands, semiconductor microcavities, graphene analogues, massless Dirac equation, band structure engineering
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