Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Polaritons

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).”

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.”

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.”

In the paper

“showing spatial coherence extended over the whole excitation spot.”

In 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 matched

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

In the paper

“we report on the presence of higher-energy bands arising from the coupling between higher-energy modes of the pillars.”

In the paper

“we observe a nondispersive band in which polaritons have an infinite effective mass.”

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).”

In the paper

“For our structure, the tunnel coupling amounts to 0.25 meV.”

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).”

In the paper

“Under low-power excitation, incoherent relaxation of polaritons results in the population of all the energy bands.”

In the paper

“Figure 1(d) shows the measured far field photoluminescence containing many groups of bands, separated by energy gaps.”

In the paper

“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”

In the paper

“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.”

In the paper

“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.”

In the paper

“The P bands are shown in detail in Fig. 4(a) revealing that the lowest one is flat.”

In the paper

“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).”

In the paper

“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).”

What the sources say

• ?Not matched

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

  cites [27]

In the paper

“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)].”

In the paper

“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)).”

In the paper

“At the Γ point a brighter point is observed showing efficient relaxation towards that state.”

In the paper

“At low power [Fig. 5(b)], below threshold, all the low energy bands are populated.”

In the paper

“The intensity maxima are centered on the pillars as expected for a state arising from the hybridization of S states.”

In the paper

“We observe spontaneous coherence over the whole size of the pump beam.”

In the paper

“The spatial extension of the condensate coincides with that of the excitation spot that populates the reservoir.”

In the paper

“Note that condensation does not take place in the ground state.”

In the paper

“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.”

In the paper

“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.”

In the paper

“Above threshold it is that particular state the one that becomes macroscopically occupied (Fig. 5(d)).”

In the paper

“However, the higher energy band in Fig. 4(a) is not flat.”

In the paper

“When we approach the threshold for condensation we observe that particles start to accumulate at the top of the π ∗ band”

In the paper

“Even the smallest interactions within the condensate bring its energy up, further into the gap.”

In the paper

“Finally, Fig. 4(b) shows the flatband real space mode for which intensity maxima sit between the pillars, thus arranged in a kagome geometry.”

In the paper

“This phenomenon arises from destructive interference in the far field emission along certain high symmetry directions.”

In the paper

“We report experimental studies of honeycomb lattices where the polariton low-energy dispersion is analogous to that of electrons in graphene.”

In the paper

“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”

What the sources say

• ?Not matched

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

  cites [4] A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81 , 109 (2009).

In the paper

“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.”

In the paper

“They are analogous to the π and π ∗ bands of graphene [4].”

What the sources say

• ?Not matched

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

  cites [4] A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81 , 109 (2009).

In the paper

“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.”

In the paper

“around which the dynamics of polaritons is described by the Dirac equation for massless particles.”

In the paper

“For this set of parameters the simulations reproduce condensation at the Γ point on the top of the π ∗ band, as in the experiment.”

In the paper

“In order to confirm this model, we have performed a numerical simulation of the two-dimensional Schrödinger equation which reproduces the observed dispersion [27].”

What the sources say

• ?Not matched

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

  cites [27]

In the paper

“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).”

In the paper

“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.”

In the paper

“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].”

What the sources say

• ?Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be 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).
• ?Not matched

  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).

In the paper

“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].”

What the sources say

• ?Not matched

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

In the paper

“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.”

In the paper

“For those parameters, this model describes well the lower bands observed in the experiment [Fig. 4(a)].”

In the paper

“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.”

In the paper

“This can be explained by allowing for a weak hopping also for the P states orthogonal to the link.”

In the paper

“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.”

In the paper

“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].”

What the sources say

• ?Not matched

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

  cites [2] I. L. Aleiner, B. L. Altshuler, and Y. G. Rubo, Phys. Rev. B 85 , 121301 (2012).

In the paper

“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.”

In the paper

“We have shown that the two S bands mimic the graphene π and π ∗ bands.”