Ultra-low threshold polariton lasing at room temperature in a GaN membrane microcavity with a zero-dimensional trap

Polariton lasers are coherent light sources based on the condensation of exciton-polaritons in semiconductor microcavities, which occurs either in the kinetic or thermodynamic (Bose-Einstein) regime. Besides their fundamental interest, polariton lasers have the potential of extremely low operating thresholds. Here, we demonstrate ultra-low threshold polariton lasing at room temperature, using an all-dielectric, GaN membrane-based microcavity, with a spontaneously-formed zero-dimensional trap. The microcavity is fabricated using an innovative method, which involves photo-electrochemical etching of an InGaN sacrificial layer and allows for the incorporation of optimally-grown GaN active quantum wells inside a cavity with atomically-smooth surfaces. The resulting structure presents near-theoretical Q-factors and pronounced strong-coupling effects, with a record-high Rabi splitting of 64 meV at room-temperature. Polariton lasing is observed at threshold carrier densities 2.5 orders of magnitude lower than the exciton saturation density. Above threshold, angle-resolved emission spectra reveal an ordered pattern in k-space, attributed to polariton condensation at discrete levels of a single confinement site. This confinement mechanism along with the high material and optical quality of the microcavity, accounts for the enhanced performance of our polariton laser, and pave the way for further developments in the area of robust room temperature polaritonic devices.


Threshold exciton densities
Polariton lasing can be confirmed by estimating the threshold exciton density per QW, per laser pulse ( . The value of is often estimated from the blueshift of the LPB line at = 0 when increasing the pump power up to the threshold value ( , using the following relation 1,2 : (1) where is the matrix element of polariton-polariton interaction given by , with being the exciton fraction at = 0, the exciton binding energy, and the exciton Bohr radius. In our case, however, the blueshift is hardly observable (see Fig. 4a), which is consistent with the much lower exciton densities in our system compared to previous works.
Instead, threshold carrier densities can be estimated based on a simple system of rate equations, coupling the LPB polariton densities at = 0 and at the high-"reservoir" states. Starting from the latter, the variation of the threshold polariton density in the reservoir per coupled QW and per laser pulse ( ) can be written as (2) where is the carrier generation rate per QW per laser pulse at threshold, the relaxation time of polaritons to the bottom of LPB, and the lifetime of polaritons in the reservoir states, which can be approximated by the excitonic lifetime = 275 ps [cf. inset of Supplementary   Fig. S1b]. Taking into consideration that << , at steady state, equation (2) gives: Similarly, the variation of the threshold polariton density at = 0 per coupled QW per laser pulse ( ), can be written as: (4) where is the lifetime of polaritons at = 0. Taking the steady-state limit of equation (4), and using equation (3), we obtain the relation: (5) can be estimated from the threshold power density of  4.5 W/cm 2 , which is the lowest ever reported value for a two-dimensional GaN-based microcavity, taking into account the 5% loss in can be taken as (cf. Supplementary Fig. S4a), which has been independently verified using a two-level coupled oscillator model 3 . Inserting these values into equation (6), we obtain = 0.498 ps. Introducing this value in equation (5), we obtain an upper estimate for of 2.24×10 10 cm -2 . The corresponding exciton density ) at = 0 can be obtained considering the exciton fraction ( ) of LPB 1 at = 0, through the relation: Substituting for and , is estimated to be  7×10 9 cm -2 , which is 2.5 orders of magnitude below the exciton saturation density  2×10 12 cm 2 . 4 The exciton saturation density is also independently estimated as described in references 4,5 , using a QW exciton binding energy = 30 meV 4,6 and a Bohr radius = 2.7 nm. Since at threshold, most of the lasing comes from level "3", as opposed to level "1" (see inset of Fig. 4a), an estimate of is also made for LPB 2 , which turns out to be very similar to the above estimated value, in spite of the smaller detuning.
The above threshold carrier densities at = 0 are consistent with the absence of a measurable blueshift of the LPB line. In fact, assuming that in equation (1) is smaller than the detection limit of our system of 0.2 meV and using = 30 meV, = 2.7 nm and = 0.32, we can obtain an upper bound for ≤ 4.7×10 10 cm -2 , which is more than 40 times lower than the exciton saturation density in these QWs further confirming polariton lasing.
To further check the validity of our estimates, the exciton density in the "uncoupled" QWs ( ), per laser pulse and per QW, can be obtained from the following equation: and are known, as per the discussion above, giving a steady-state estimate for the of approximately 10 12 cm -2 . Alternatively, can be estimated from the blueshift of the PL emission from uncoupled QW excitons, when increasing the pump power up to threshold, which is ≈ 9 meV from data in Fig. 4a. Slightly modifying equation (1), can be written as (9) where is the matrix element of exciton-exciton interaction given by . From this equation, we obtain , which is very close to the value derived from the rate equation model.

x coupled oscillator model
The Hamiltonian defining a 3 x 3 coupled oscillator model can be written as follows: where , and are the energies of the first exciton, second exciton and the cavity mode respectively, as a function of in-plane wave vector . , and are the corresponding linewidths of the first exciton, second exciton and the cavity mode. and are the respective coupling constant"s of the first and second excitons. The coupling constant of an exciton, is given by the following relation: where is the velocity of light in air, is the number of QW"s, is the effective refractive index of the cavity layer, is the effective cavity length including the penetration into the mirrors and finally is given by: (12) where is the charge of an electron, is the vacuum dielectric constant, is the mass of an electron and is the QW oscillator strength per unit area. It should be noted that the expression for coupling constant is an approximate formula, derived under the following assumptions: (1) the cavity layer has a higher refractive index as opposed to its DBR constituents and (2) the QW"s are placed exactly at the antinodes of the field distribution within the cavity layer, estimated for a resonant cavity photon energy .
The Hamiltonian of equation (10) where , and are the Hopfield coefficients. , are the exciton fractions corresponding to the two excitons and represents the photon fraction, for the respective polariton branches. The Hopfield coefficients satisfy the following relation: The energy splitting between the respective branches at the anti-crossing point ( at which the cavity mode energy coincides with the exciton energy), can be referred to as the vacuum field Rabi splitting. It should be noted that, two Rabi splitting"s can be observed in a 3 x 3 system: 1) between the LPB and MPB and 2) between the MPB and UPB. Further discussion about the model can be found in the following references 3,7 .

Comment on the unusual kink between LPB and MPB
The unusual kink at the anticrossing between LPB and MPB dispersions, which can be noticed in the fittings of Figures 2(d), 3(a) and 3(b), is a generic effect that occurs whenever the coupling constant decreases to values just above the transition between weak and strong coupling regimes.
In the 3-level fittings of our system, containing both QW and bulk GaN excitons, the respective coupling constants are 0.031575 eV and 0.01005 eV. In other words, the coupling constant of the bulk GaN excitons is much weaker compared to the QW excitons, in line with the relatively weak Rabi splittings between LPB and MPB.
In order to make our point clear, in a simple way, we show in the Supplementary Figure   S7, the outcome of 2-level calculations, where we only vary the coupling constant, keeping all other parameters intact (e.g. detunings). For relatively large coupling constants, we observe the typical "smooth" anticrossing behaviour, but as we decrease the coupling constant to the limit close to the weak coupling case, we observe a kink-like behaviour, similar to the one observed between LPB and MPB in the 3-level fittings of the manuscript.

Refractive indices
The refractive indices used in the transfer matrix and coupled oscillator model is tabulated