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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- Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 1
Dirac cones were directly observed in the honeycomb lattice.
“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).”
Quote in the paperSubstantial claim - Also stated in: § Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2 · plain
- Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 1
A honeycomb lattice for polaritons was realized using hundreds of coupled micropillars etched in a planar semiconductor microcavity.
“In this Letter, we report on a honeycomb lattice for polaritons, made of hundreds of coupled micropillars etched in a planar semiconductor microcavity.”
Quote in the paperSubstantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2
The separation between the S and P groups of bands is ΔE = 3.2 meV, the energy difference between the two lowest-energy states of the individual pillars.
“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.”
Quote in the paperNot a substantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 1
The polariton condensation at the top of the π* band shows spatial coherence extended over the whole excitation spot.
“showing spatial coherence extended over the whole excitation spot.”
Quote in the paperNot a substantial claim - Findingcitation§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2
At higher energy, a group of four bands (P bands) arises from the coupling between the first excited state of the pillars, which is twice degenerate and has two lobes.
“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]”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [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.
- Findingnone given§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 1
Higher-energy bands arise from the coupling between higher-energy modes of the pillars.
“we report on the presence of higher-energy bands arising from the coupling between higher-energy modes of the pillars.”
Quote in the paperSubstantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 1
A nondispersive band is observed in which polaritons have an infinite effective mass.
“we observe a nondispersive band in which polaritons have an infinite effective mass.”
Quote in the paperSubstantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2
The measured emitted intensity in momentum space at the Dirac points energy shows six Dirac points at the corner of the first Bz.
“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).”
Quote in the paperSubstantial claim - Findingnone given§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2
The tunnel coupling between adjacent micropillars in the structure amounts to 0.25 meV.
“For our structure, the tunnel coupling amounts to 0.25 meV.”
Quote in the paperSubstantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 1
The two lowest energy bands (S bands) arise from the coupling between the fundamental modes of the micropillars.
“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).”
Quote in the paperSubstantial claim - Also stated in: § Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2 · plain
- Findingnone given§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2
Under low-power excitation, incoherent relaxation of polaritons results in the population of all the energy bands.
“Under low-power excitation, incoherent relaxation of polaritons results in the population of all the energy bands.”
Quote in the paperSubstantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2
The measured far field photoluminescence contains many groups of bands, separated by energy gaps.
“Figure 1(d) shows the measured far field photoluminescence containing many groups of bands, separated by energy gaps.”
Quote in the paperNot a substantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 1
When increasing excitation intensity, polariton condensation occurs at the top of the π* band.
“When increasing the excitation intensity, we observe polariton condensation occurring at the top of the π ∗ band”
- Also stated in: § Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3 · plain
“we observe that condensation takes place at the top of the π ∗ band”
Quote in the paperSubstantial claim - Also stated in: § Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3 · plain
- Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2
Fitting Eq. (1) to the data yields first-neighbor coupling t = 0.25 meV and second-neighbor coupling t' = -0.02 meV.
“By fitting Eq. (1) to the data in Fig. 2 we extract a value of the coupling between first and second neighbors of t = 0 . 25 and t ′ = -0 . 02 meV, respectively.”
Quote in the paperSubstantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2
The fit yields a group velocity v = 3at/2ħ = 1.3 × 10^6 m/s around the Dirac points.
“The result of the fit is shown in Fig. 2(b), and yields a group velocity v = 3 at/ 2 ℏ = 1 . 3 × 10 6 m.s -1 around the Dirac points.”
Quote in the paperSubstantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3
The lowest P band is flat.
“The P bands are shown in detail in Fig. 4(a) revealing that the lowest one is flat.”
Quote in the paperSubstantial claim - Findingown data§ CONDENSATION IN THE π ∗ BAND: EXPERIMENTS · p. 5
The π and π* bands continuously blueshift when increasing the excitation power due to repulsive interactions between polaritons populating that band and the highly populated exciton reservoir.
“Note that the π and π ∗ bands continuously blueshift when increasing the excitation power due to the repulsive interactions between polaritons populating that band and the highly populated exciton reservoir located at the bare exciton energy (about 20 meV above in energy).”
Quote in the paperNot a substantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3
Condensation takes place at the top of the π* band, located at the Γ point.
“By monitoring the energy resolved emitted intensity across the condensation threshold, we observe that condensation takes place at the top of the π ∗ band [arrow in Fig. 2(b), [27]]. This state is located at the Γ point (center of the Bzs) as seen in Fig. 3(a).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [27]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3
In our lattice, the coupling between P modes of the pillars leads to four energy bands which appear above the two S bands, separated by a gap of about 0.7 meV.
“In our lattice, the coupling between P modes of the pillars leads to four energy bands which appear above the two S bands, separated by a gap of about 0.7 meV [see Fig. 1(d)].”
Quote in the paperSubstantial claim - Findingown data§ CONDENSATION IN THE π ∗ BAND: SIMULATIONS · p. 6
The increase of the polariton-polariton interaction term in the simulation leads to the shrinking of the spatial extension of the emission.
“The increase of the polariton-polariton interaction term α | Ψ | 2 in the simulation leads to the shrinking of the spatial extension of the emission (see Fig. 6(b)).”
Quote in the paperNot a substantial claim - Findingown data§ CONDENSATION IN THE π ∗ BAND: EXPERIMENTS · p. 5
At the Γ point a brighter point is observed showing efficient relaxation towards that state.
“At the Γ point a brighter point is observed showing efficient relaxation towards that state.”
Quote in the paperNot a substantial claim - Findingown data§ CONDENSATION IN THE π ∗ BAND: EXPERIMENTS · p. 5
At low power below threshold, all the low energy bands are populated.
“At low power [Fig. 5(b)], below threshold, all the low energy bands are populated.”
Quote in the paperNot a substantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3
The intensity maxima of the condensate are centered on the pillars, as expected for a state arising from the hybridization of S states.
“The intensity maxima are centered on the pillars as expected for a state arising from the hybridization of S states.”
Quote in the paperNot a substantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3
Spontaneous coherence is observed over the whole size of the pump beam.
“We observe spontaneous coherence over the whole size of the pump beam.”
Quote in the paperSubstantial claim - Findingnone given§ CONDENSATION IN THE π ∗ BAND: SIMULATIONS · p. 6
The spatial extension of the condensate coincides with that of the excitation spot that populates the reservoir.
“The spatial extension of the condensate coincides with that of the excitation spot that populates the reservoir.”
Quote in the paperNot a substantial claim - Findingreasoning§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3
Condensation does not take place in the ground state.
“Note that condensation does not take place in the ground state.”
Quote in the paperSubstantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 4
In the case where t⊥ = 0.2 meV and t∥ = -1 meV, the tight-binding result has the two extreme bands no longer flat.
“Indeed in the case where t ⊥ = 0 . 2 meV and t ‖ = -1 meV, the tight-binding result is plotted in Fig. 4(d) where the two extreme bands are no longer flat.”
Quote in the paperSubstantial claim - Findingnone given§ CONDENSATION IN THE π ∗ BAND: SIMULATIONS · p. 6
When varying the position of the pump spot with respect to the center of the lattice, very similar spatial and momentum space patterns are obtained.
“When varying the poition of the pump spot with respect to the center of the lattice, a very similar spatial and momentum space patterns are obtained.”
Quote in the paperNot a substantial claim - Findingown data§ CONDENSATION IN THE π ∗ BAND: EXPERIMENTS · p. 5
Above threshold, the state at the top of the π* band becomes macroscopically occupied.
“Above threshold it is that particular state the one that becomes macroscopically occupied (Fig. 5(d)).”
Quote in the paperSubstantial claim - Findingreasoning§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 4
The higher energy band in Fig. 4(a) is not flat.
“However, the higher energy band in Fig. 4(a) is not flat.”
Quote in the paperSubstantial claim - Findingown data§ CONDENSATION IN THE π ∗ BAND: EXPERIMENTS · p. 5
Approaching the threshold for condensation, particles start to accumulate at the top of the π* band.
“When we approach the threshold for condensation we observe that particles start to accumulate at the top of the π ∗ band”
Quote in the paperSubstantial claim - Findingreasoning§ CONDENSATION IN THE π ∗ BAND: SIMULATIONS · p. 6
Even the smallest interactions within the condensate bring its energy up, further into the gap.
“Even the smallest interactions within the condensate bring its energy up, further into the gap.”
Quote in the paperNot a substantial claim - Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 4
The flatband real space mode has intensity maxima sitting between the pillars, arranged in a kagome geometry.
“Finally, Fig. 4(b) shows the flatband real space mode for which intensity maxima sit between the pillars, thus arranged in a kagome geometry.”
Quote in the paperSubstantial claim - Interpretationown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2
This absence of emission arises from destructive interference in the far field emission along certain high symmetry directions.
“This phenomenon arises from destructive interference in the far field emission along certain high symmetry directions.”
Quote in the paperSubstantial claim - Interpretationown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 1
The polariton low-energy dispersion in the honeycomb lattice is analogous to that of electrons in graphene.
“We report experimental studies of honeycomb lattices where the polariton low-energy dispersion is analogous to that of electrons in graphene.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2
The two S bands are expected to present features analogous to the π and π* bands of graphene, including six Dirac points in the first Bz.
“Thus, we expect the two S bands to present features analogous to the π and π ∗ bands of graphene, including six Dirac (contact) points [4] in the first Bz”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [4] A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81 , 109 (2009).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Premisereasoning§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2
The system is well described by the tight-binding approximation because the confinement energy on each site is much larger than the tunneling energy.
“As the confinement energy on each site of the lattice is much larger than the tunneling energy, the system is well described by the tight-binding approximation.”
Quote in the paperSubstantial claim - Interpretationcitation§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 1
The two S bands are analogous to the π and π* bands of graphene.
“They are analogous to the π and π ∗ bands of graphene [4].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [4] A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81 , 109 (2009).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2
The two S bands stem from coupling between micropillar states with cylindrical symmetry similar to carbon Pz orbitals in graphene.
“The two S bands stem from the coupling between micropillar states which have a cylindrical symmetry similar to that of the carbon P z electronic orbitals in graphene.”
Quote in the paperSubstantial claim - Interpretationown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 1
Around the Dirac cones, the dynamics of polaritons is described by the Dirac equation for massless particles.
“around which the dynamics of polaritons is described by the Dirac equation for massless particles.”
Quote in the paperSubstantial claim - Study design and conductown data§ CONDENSATION IN THE π ∗ BAND: SIMULATIONS · p. 6
Simulations reproduce condensation at the Γ point on the top of the π* band, as in the experiment.
“For this set of parameters the simulations reproduce condensation at the Γ point on the top of the π ∗ band, as in the experiment.”
Quote in the paperSubstantial claim - Study design and conductown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 4
A numerical simulation of the two-dimensional Schrödinger equation reproduces the observed dispersion.
“In order to confirm this model, we have performed a numerical simulation of the two-dimensional Schrödinger equation which reproduces the observed dispersion [27].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [27]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning§ CONDENSATION IN THE π ∗ BAND: SIMULATIONS · p. 6
This aspect favors the Γ point of the π* band with respect to other states.
“This aspect favors the Γ point of the π ∗ band with respect to (for example) the Γ points of the non-flat P bands, which might also have negative mass, or with respect to the flat band, which possess much shorter lifetimes due to the location of the wavefunction lobes on the junctions between the pillars (see Fig. 4(b) of the main text).”
Quote in the paperSubstantial claim - Interpretationown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 4
The implemented system allows direct optical access to the basic properties of engineered lattices, as demonstrated by the direct observation of Dirac cones in a honeycomb geometry.
“In summary, we have implemented a system which allows direct optical access to the basic properties of engineered lattices as demonstrated by the direct observation of Dirac cones in a honeycomb geometry.”
Quote in the paperSubstantial claim - Interpretationcitation§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3
This feature arises from the out of equilibrium nature of polaritons in which the steady state is fixed by the interplay between pump, relaxation and decay.
“This feature arises from the out of equilibrium nature of polaritons in which the steady state is fixed by the interplay between pump, relaxation and decay [29, 30].”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [29] C. W. Lai, N. Y. Kim, S. Utsunomiya, G. Roumpos, H. Deng, M. D. Fraser, T. Byrnes, P. Recher, N. Kumada, T. Fujisawa, and Y. Yamamoto, Nature (London) 450 , 529 (2007).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [30] D. Tanese, H. Flayac, D. Solnyshkov, A. Amo, A. Lemaître, E. Galopin, R. Braive, P. Senellart, I. Sagnes, G. Malpuech, and J. Bloch, Nat. Comm. 4 , 1749 (2013).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Study design and conductcitation§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3
The low value of the measured emission blueshift certifies that the system remains in the strong coupling regime across the threshold.
“The low value of the measured emission blueshift, due to interactions between polaritons and uncondensed excitons [see Fig. 3(d)], certifies that the system remains in the strong coupling regime across the threshold [26].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [26] L. Ferrier, E. Wertz, R. Johne, D. D. Solnyshkov, P. Senellart, I. Sagnes, A. Lemaître, G. Malpuech, and J. Bloch, Phys. Rev. Lett. 106 , 126401 (2011).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning§ CONDENSATION IN THE π ∗ BAND: SIMULATIONS · p. 6
The reservoir of excitons created by the nonresonant pump creates a repulsive potential for polaritons, which pushes away particles created by spontaneous scattering, preventing the formation of the condensate in the states with positive mass.
“First, the reservoir of excitons created by the nonresonant pump creates a repulsive potential for polaritons, which pushes away particles created by spontaneous scattering, preventing the formation of the condensate in the states with positive mass.”
Quote in the paperSubstantial claim - Interpretationown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 4
For those parameters, this model describes well the lower bands observed in the experiment.
“For those parameters, this model describes well the lower bands observed in the experiment [Fig. 4(a)].”
Quote in the paperSubstantial claim - Interpretationreasoning§ CONDENSATION IN THE π ∗ BAND: SIMULATIONS · p. 6
This longer lifetime is due to the reduced evanescent fraction of the mode outside the pillars for these modes, due to zeroes of the wavefunction at all junctions between the pillars, where there is a larger density of non-radiative centers.
“This is due to the fact that the evanescent fraction of the mode outside the pillars is reduced for these modes due to the presence of the zeroes of the wavefunction at all junctions between the pillars, where there is a larger density of non-radiative centers that contribute to the lifetime reduction.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 4
Allowing a weak hopping for the P states orthogonal to the link explains the non-flat higher energy band.
“This can be explained by allowing for a weak hopping also for the P states orthogonal to the link.”
Quote in the paperSubstantial claim - Interpretationreasoning§ P BANDS: 2D SCHRÖDINGER EQUATION SIMULATION · p. 7
The full 2D model reproduces this behavior correctly because it automatically takes into account the exponential increase with energy of the tunneling rate of the P states of the individual pillars.
“The full 2D model reproduces this behavior correctly, because it automatically takes into account the exponential increase with energy of the tunneling rate of the P states of the individual pillars, as explained in the main text and illustrated by the tight-binding model calculations.”
Quote in the paperSubstantial claim - Premisecitation§ CONDENSATION IN THE π ∗ BAND: SIMULATIONS · p. 6
The lifetime of anti-symmetric states is in general longer than that of the symmetric one.
“A second reason for the condensation of polaritons on top of the π ∗ band is that the lifetime of anti-symmetric states is in general longer than that of the symmetric one [2].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [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.
- Interpretationreasoning§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 4
The band structure in Fig. 4(a) can be understood assuming that t⊥ increases with energy, resulting in a flat band at low energy and a dispersive band at higher energy.
“The band structure reported in Fig. 4(a) can then be understood assuming that t ⊥ increases with the energy, resulting in a flat band ( t ⊥ ≃ 0 ) at low energy and a dispersive band ( t ⊥ ≃ 0 . 2 meV) at higher energy.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The tunnelling probability of P states is energy dependent, increasing with energy.
- Interpretationnone given§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3
The two S bands mimic the graphene π and π* bands.
“We have shown that the two S bands mimic the graphene π and π ∗ bands.”
Quote in the paperSubstantial claim
- May combine two claims. 14
- Classified as probably not a substantial claim. 5
May combine two claims.
Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3Increasing excitation intensity leads to polariton condensation, evidenced by a threshold in the integrated emission intensity.
“By increasing the excitation intensity, we observe polariton condensation, as evidenced by the threshold in the integrated emission intensity [Fig. 3(d)].”
Quote in the paperSubstantial claimMay combine two claims.
Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 1At higher energies, p orbital bands were observed, one of them with the non-dispersive character of a flatband.
“At higher energies, we observe p orbital bands, one of them with the non-dispersive character of a flatband.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2Along line 3, emission is absent in the upper band within the first Bz, and in the lower band within the second Bz.
“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.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 1Six Dirac cones are evidenced at the corners of the first Brillouin zone, around which the energy dispersion is linear.
“In particular, we evidence six Dirac cones at the corners of the first Brillouin zone (Bz), around which the energy dispersion is linear.”
Quote in the paperSubstantial claimMay combine two claims.
Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2The measured energy resolved emission along lines 1 and 2 passes through four and three Dirac linear intersections respectively.
“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.”
Quote in the paperSubstantial claimMay combine two claims.
Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3At the intersection between two adjacent pillars, the fringes are shifted by half a period, indicating a π phase shift between adjacent pillars, as expected for the antibonding π* band.
“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.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingnone given§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 4Via resonant excitation of the structure, polariton wave packets can be created with any desired energy and momentum.
“Moreover, via resonant excitation of the structure, polariton wave packets can be created with any desired energy and momentum.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3The emission spectrum collapses into a single emission line, and extended spatial coherence builds up.
“Moreover, the emission spectrum collapses into a single emission line, and extended spatial coherence builds up.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3The real space emission of the condensate covers the same area as the pump spot.
“The real space emission of the condensate is shown in Fig. 3(b), covering the same area as the pump spot.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingown data§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 4In the limiting case where t⊥ = 0 meV and t∥ = -1 meV, the eigenstates of Eq. (3) give rise to four energy bands, with the two extreme bands flat and the two intermediate ones dispersive.
“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.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingcitation§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3The threshold power is similar to that observed in a planar structure.
“The threshold power is similar to that observed in a planar structure [25].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [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.
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Findingnone given§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 4The position, shape and size of each lattice site can be controlled at will during fabrication.
“The position, shape and size of each lattice site can be controlled at will during fabrication.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based). May combine two claims.
Study design and conductnone given§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2For low power excitation, polariton-polariton interactions are negligible so that single particle physics of the honeycomb lattice is probed.
“Note that for low power excitation polaritonpolariton interactions are negligible so that single particle physics of the honeycomb lattice is probed.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisecitation§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 2This destructive interference occurs in lattices with multiple sites per unit cell and has been observed along the K-Γ-K' directions in angle-resolved electron spectroscopy measurements in graphene.
“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].”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [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.
- cites [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.
May combine two claims.
Premisenone given§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 4The tunneling probability varies exponentially with the barrier height relative to the state, and thus increases strongly for higher energy states.
“Indeed, the tunneling probability varies exponentially with the barrier height relative to the state, and thus increases strongly for higher energy states.”
Quote in the paperSubstantial claimMay combine two claims.
Study design and conductreasoning§ P BANDS: 2D SCHRÖDINGER EQUATION SIMULATION · p. 7The 2D Schrödinger equation simulation is in excellent quantitative agreement with the experimental observation: the lowest P band is indeed flat, while the upper band is dispersive.
“The simulation is in excellent quantitative agreement with the experimental observation: the lowest P band is indeed flat, while the upper band is dispersive.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ CONDENSATION IN THE π ∗ BAND: SIMULATIONS · p. 6The states with negative mass are trapped in this potential and serve as a seed for stimulated scattering.
“However, the states with negative mass are on the contrary trapped in this potential, and serve as a seed for stimulated scattering.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationcitation§ Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons · p. 3The antibonding mode at the Γ point favors condensation due to its negative effective mass and positive interaction energy, and its longer lifetime.
“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.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [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.
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 band structure related to Dirac cones has been widely investigated in the energy-momentum space using bulk measurement with measured density and dispersion pattern.
“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 opening band gap in graphene requires large strain, even exceeding the order of 20%, attributed to deformations and anisotropy by applying strain instead of inner disorders or fluctuation.
“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]
The paper describes Dirac points in a polariton honeycomb lattice and compares its bands with graphene (pp. 1–2). Searches for graphene strain, a 20% threshold, and strain-induced anisotropy found no passage addressing the claimed requirement for opening a band gap or its proposed cause. (LLM, conf=0.76)
Experimental — accuracy not yet measured
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“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].”
BackgroundMethodInfluentialSelf-citationSpin-Orbit Coupling and the Optical Spin Hall Effect in Photonic Graphene
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Self-citationModulational instability and solitary waves in polariton topological insulators
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Self-citationScalable single-mode surface-emitting laser via open-Dirac singularities
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Ultralow Threshold Polariton Condensate in a Monolayer Semiconductor Microcavity at Room Temperature
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Self-citationMany-body quantum electrodynamics networks: Non-equilibrium condensed matter physics with light
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Defect States Emerging from a Non-Hermitian Flatband of Photonic Zero Modes
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Kibble-Zurek Mechanism in Topologically Nontrivial Zigzag Chains of Polariton Micropillars
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Self-citationTwo-Dimensional Topological Polariton Laser
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Non-equilibrium Bose–Einstein condensation in photonic systems
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Self-citationA polariton condensate in a photonic crystal potential landscape
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Interaction-induced hopping phase in driven-dissipative coupled photonic microcavities
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Self-citationMapping plasmonic topological states at the nanoscale
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Non-Abelian physics in light and sound
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Synthetic Unruh effect in cold atoms
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Tuning of the Berry curvature in 2D perovskite polaritons
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Self-citationEdge states in polariton honeycomb lattices
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Self-citationDirect observation of photonic Landau levels and helical edge states in strained honeycomb lattices
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Self-citationPlexciton Dirac points and topological modes
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Edge-emitting polariton laser and amplifier based on a ZnO waveguide
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Self-citationOptical analogue of Dresselhaus spin–orbit interaction in photonic graphene
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Interacting quantum fluid in a polariton Chern insulator
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Self-citationMicrocavity polaritons for topological photonics [Invited]
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Self-citationFerromagnetism and Wigner crystallization in kagome graphene and related structures
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Micropillar Resonators for Optomechanics in the Extremely High 19–95-GHz Frequency Range
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Self-citationObservation of Dirac bands in artificial graphene in small-period nanopatterned GaAs quantum wells
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Unstable and stable regimes of polariton condensation
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Self-citationMesoscale Graphene-like Honeycomb Mono- and Multilayers Constructed via Self-Assembly of Coclusters
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Direct observation of photonic Landau levels and helical edge states in strained honeycomb lattices
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Self-citationVariable potentials for thermalized light and coupled condensates
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Parity-time symmetry in a flat-band system
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Spin–orbit coupling in photonic graphene
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Self-citationRealization of photonic p-orbital higher-order topological insulators
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Exotic Attractors of the Nonequilibrium Rabi-Hubbard Model
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Self-citationSynthetic band-structure engineering in polariton crystals with non-Hermitian topological phases
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Self-citationElectrically tunable artificial gauge potential for polaritons
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Stabilizing strongly correlated photon fluids with non-Markovian reservoirs
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Self-citation
- 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 › Engineering › Biomedical Engineering › 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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