# Phonon-assisted broadening in Bernal boron nitride: A comparison between indirect and direct excitons

Adrien Rousseau, Pierre Valvin, Lianjie Xue, Jiahan Li, James H. Edgar,

Bernard Gil, Guillaume Cassabois

► To cite this version:

Adrien Rousseau, Pierre Valvin, Lianjie Xue, Jiahan Li, James H. Edgar, et al.. Phonon-assisted broadening in Bernal boron nitride: A comparison between indirect and direct excitons. Physical Review B, 2022, 106 (3), pp.035203. 10.1103/PhysRevB.106.035203. hal-03767169

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# Phonon-assisted broadening in Bernal boron nitride: A comparison between indirect and direct excitons

A. Rousseau, P. Valvin, L. Xue, J. Li, J. H. Edgar, B. Gil, and G. Cassabois

Phys. Rev. B **106**, 035203 — Published 12 July 2022

DOI: 10.1103/PhysRevB.106.035203
Phonon-assisted broadening in Bernal boron nitride:
a comparison between indirect and direct excitons

A. Rousseau,¹ P. Valvin,¹ L. Xue,² J. Li,² J. H. Edgar,² B. Gil,¹ G. Cassabois¹,*

¹Laboratoire Charles Coulomb UMR 5221 CNRS-Université de Montpellier, 34095 Montpellier, France

²Tim Taylor Department of Chemical Engineering,
Kansas State University, Manhattan Kansas 66506, USA

(Dated: June 28, 2022)

We study the deep-ultraviolet emission in Bernal boron nitride as a function of temperature. The quasi-degeneracy of indirect and direct excitons in Bernal boron nitride leads to their simultaneous recombination, allowing a comparison of phonon-assisted broadening in the two cases. Temperature-dependent measurements reveal that, below 200 K, the efficiency of phonon-assisted broadening is one order of magnitude lower for the direct transition at 6.035 eV than for the phonon replicas of the indirect exciton. This striking effect results from the inhibition of quasi-elastic acoustic phonon scattering in the strong coupling regime of the light-matter interaction, where the density of final states is reduced by the curvature of the polaritonic dispersion.

# I. INTRODUCTION

Hexagonal boron nitride (hBN) is a wide band gap semiconductor which differs from other nitrides because of its sp² hybridized bonds forming honeycomb atomic planes weakly stacked through the van der Waals interaction [1]. Since the first synthesis of large hBN single crystals by Watanabe et al. in 2004 [2], hBN has become a pivotal material in a wide variety of applications such as photonics, optronics, neutronics and twistronics [3, 4]. hBN corresponds to the so-called AA' stacking of atomic planes, where boron and nitrogen atoms alternate along the c-axis [5]. This is the most common sp²-bonded boron nitride polytype. Other polytypes have different stacking sequences of the basal plane: Bernal boron nitride (bBN) with an AB stacking, rhombohedral boron nitride (rBN) with an ABC stacking, and also turbostratic boron nitride (tBN) which differs from the previous two by the existence of some translational and rotational disorder in the vertical stacking [6–8].

bBN is isostructural to graphite, and like hBN it has a band gap in the range of ~6 eV [9–11]. Still, the different stackings lead to subtle modifications of the electronic and optical properties [10–14]. On the one hand, the independent-particle dispersions of hBN and bBN both exhibit a conduction band minimum at the M point, and the valence band maximum around the K point. This results in free electron-hole pairs with a minimum of their energy dispersion at the T point of the first Brillouin zone in both hBN and bBN. Direct band-to-band transitions occur at higher energies, but with a relative blue-shift that depends on the stacking. In hBN, the direct single-particle gap is ~0.5 eV higher in energy than the fundamental indirect gap, while this value decreases to less than 0.2 eV in bBN [12]. This effect stems from the flat-ter dispersion of the conduction band for the AB stacking [12–14]. The attractive electron-hole interaction further modifies the picture, with different binding energies of direct and indirect excitons [12–14]. In hBN, the direct exciton $dX_h$ lies at 6.125 eV [15], i.e. an energy 0.17

eV higher than the fundamental indirect exciton $iX_h$ at 5.955 eV [9]. On the contrary, in bBN, the direct exciton $dX_b$ is quasi-degenerate with the indirect one $iX_b$, calculations having predicted a marginally direct optical gap [12] while recent experiments have shown it is marginally indirect [11]. The quasi-degeneracy of $dX_b$ and $iX_b$ is the signature of an unusually flat in-plane dispersion of the excitonic states in bBN, that allows their simultaneous observation in luminescence experiments [11].

In this paper, we study the deep-ultraviolet photoluminescence (PL) in bBN as a function of temperature. The thermal broadening is extracted for both the direct emission line at ~6.035 eV and the phonon replicas of $iX_b$ below 6 eV. We observe drastically different broadening behaviours. In the first case, for the direct emission line related to $dX_b$, the thermal broadening is linear with the phonon occupation factor $n(T)$. In the second case, for the phonon replicas of $iX_b$, the thermal broadening scales like the square root of $n(T)$. Moreover, below 200 K, the efficiency of phonon-assisted broadening is one order of magnitude lower for the direct transition at ~6.035 eV than for the phonon replicas of $iX_b$. We interpret the different phenomenologies of phonon-assisted dephasing as a result of the strong coupling regime of the light-matter interaction for $dX_b$, where the density of final states is reduced by the curvature of the polaritonic dispersion.

# II. EXPERIMENTAL RESULTS

Our measurements are performed in the experimental conditions of Ref.[11]. hBN and bBN crystals are extracted from a polytypic sp²-bonded carbon-doped BN crystal precipitated from a metal solution, as described in Ref.[10, 16]. **Such a polycrystal contains novel shallow and deep levels emitting at ~5.45 and ~4.14 eV, respectively [16]. The extracted hBN and bBN crystals are presently studied in a scanning confocal cryomicroscope operating in the deep-ultraviolet
<figure><img src="image_2.png" /><figcaption>FIG. 1. PL spectra at 6 K recorded for bBN (top, red line) and hBN (bottom, blue line) crystals. PL excitation energy = 6.39 eV. The bBN spectrum is vertically shifted for clarity.</figcaption></figure>

[17, 18], which was a pivotal tool for identifying bBN crystals [11]. **The PL signal is collected along the c-axis of the crystals.**

A. Quasi-degeneracy of direct and indirect excitons in bBN

The PL spectra of hBN and bBN are displayed in Fig.1 in the spectral range between 5.7 and 6.1 eV. As explained in our recent publication [11], hBN and bBN have different emission spectra in the deep-ultraviolet. In the case of hBN [bottom blue line, Fig.1], the intrinsic emission only results from phonon-assisted recombination of $iX_h$, with is located at the T point of the first Brillouin zone at an energy of 5.955 eV [9, 19–21]. For an electromagnetic field propagating along the c-axis, four different phonon modes are symmetry-allowed in the phonon-assisted processes, namely LO$_3$, TO, LA and TA, as noted on the blue spectrum of Fig.1. Moreover, there is a fine structure of the phonon replicas coming from higher-order processes involving overtones of the interlayer shear mode [22]. There is no signature of the direct exciton $dX_h$ in the PL spectrum of hBN.

The PL spectrum of bBN [top red line, Fig.1] is related to both the direct ($dX_b$) and indirect ($iX_b$) excitons. The series of lines between 5.84 and 6 eV stems from phonon-assisted recombination of $iX_b$, which sits at an energy of 6.029 eV [as indicated in Fig.1]. The bBN phonon replicas are blue-shifted by ~74 meV with respect to the ones of hBN: the LA and TA phonon replicas of bBN occur at ~5.94 and ~5.97 eV, respectively, and the LO$_2$ and TO replicas at ~5.85 and ~5.87 eV [11]. $iX_b$ appears almost resonant with the ~6.035 eV line. This line and its phonon replica at ~5.835 eV are linked to the direct exciton $dX_b$ [10, 11]. Because of the direct nature of $dX_b$, the strong coupling regime of the light-matter interaction notably modifies the excitonic dispersion so

<figure><img src="image_3.png" /><figcaption>FIG. 2. Schematic representation of the dispersion of the upper (UP) and lower (LP) polaritons due to level anti-crossing between excitons and photons in bBN, in the half-space {$k_\parallel > 0$}. The red dots indicate the accumulated populations (i) at the minimum of the excitonic dispersion ($iX_b$) around the T point of the Brillouin zone in the middle of the $\Gamma K$ line (at an energy of 6.029 eV), and (ii) at the relaxation bottleneck on the lower-polariton branch, close to the zone center $\Gamma$ (at an energy ~6.035 eV, slightly below the bare $dX_b$ energy). Inset: irreducible part of the first Brillouin zone, where the red area indicates the landscape of the dispersion plot.</figcaption></figure>

that the ~6.035 eV line only gives a lower bound for the $dX_b$ energy, as detailed in the next part. Still, the direct and indirect excitons are quasi-degenerate in bBN, a peculiarity explaining their simultaneous emission.

B. Strong light-matter interaction

Due to the 3D translational invariance of the system, direct excitons in bulk crystals strongly couple to light [23–25]. The new eigenstates are mixed exciton-photon states, the so-called exciton-polaritons, with the lower polariton (LP) and upper polariton (UP) branches [Fig.2].

Fig.2 is a schematic representation of their energy dispersions in bBN. We plot here the energy levels in a two-dimensional momentum-space defined by the in-plane and out-of-plane wavevectors, $k_\parallel$ and $k_z$ respectively ($k_z$ being parallel to the c-axis), focusing on the $\Gamma K$ line of the first Brillouin zone. The bare excitonic states are weakly dispersed along the c-axis because of the weak van der Waals interaction between adjacent layers. This is also the case for the in-plane dispersion, along the $\Gamma K$ line in bBN, as discussed above and calculated by Sponza et al. [12]. The direct exciton $dX_b$ is located at $k \sim 0$
while the indirect one $iX_b$ is nested **at the T point**, **approximately in the middle of the ΓK line [12]**, **as shown in Fig.2.**

Each excitonic state of a given $k$ interacts with a photon of the same wavevector because of the momentum conservation rule arising from the 3D translational invariance of the system. The polaritonic dispersions significantly deviate from the bare excitonic one around the light cone, i.e. for the direct state $dX_b$. The UP branch is split from the bare exciton state by the so-called longitudinal-transverse splitting $\Delta_{LT}$ [25]. This splitting takes giant values in hBN ($\Delta_{LT}=420$ meV) because of the existence of electronic flat bands in the independent-particle bandstructure [15]. We anticipate an ultralarge value also in bBN because of the similarity of the band-structures of hBN and bBN. On the high-energy side of the reststrahlen band, which width is $\Delta_{LT}$, the polaritons have a photonlike character [UP in Fig.2]. On the low-energy side, for large wavevectors far from the light cone, the LP branch closely follows the excitonic dispersion, with an excitonlike character of the exciton-polaritons. On the contrary, when approaching the anti-crossing, the photonic component of the LP states increases, with a more and more pronounced bending of the LP dispersion [Fig.2]. Such a curvature of the LP dispersion results in the well-documented bottleneck effect for carrier relaxation in direct bulk semiconductors [23–25]. We will show below it also plays a key role in the interpretation of the thermal broadening in bBN. Close to the light cone, the LP branch finally closely follows the photonic dispersion.

Because exciton-polaritons are the eigenstates of bulk semiconductors coupled to light, there is no radiative decay of a direct exciton in an ideal crystal. In other words, there is no luminescence from a direct exciton in a perfect crystal. The luminescence from a bulk semiconductor comes from the polariton recombination which results from the complex interplay between propagation, damping and thermalization processes [25]. Thermalization is very efficient down to the knee of the LP branch, which creates the relaxation bottleneck where polaritons accumulate [as indicated by the red dots around $\Gamma$ in Fig.2]. Therefore the luminescence peak is not at an energy corresponding to the direct exciton but it is slightly below. This is the reason why the ~6.035 eV emission line provides a lower bound for the $dX_b$ energy ($E_{dX_b} > 6.035$ eV).

The overall picture for the population relaxation dynamics in bBN is thus the following: the laser excitation at ~6.39 eV leads to the creation of electron-hole pair states in the absorption continuum; then the photo-generated carriers relax either towards $iX_b$ at the minimum of the excitonic dispersion around the T point of the Brillouin zone along the ΓK line [red dots around T, Fig.2], or along the lower-polariton branch where they accumulate at the relaxation bottleneck around ~6.035 eV, slightly below the $dX_b$ energy [red dots around $\Gamma$, Fig.2]. Given the quasi-degeneracy of the involved states, there

<figure><img src="image_4.png" /><figcaption>FIG. 3. PL spectra in bBN above 6 eV as a function of temperature: data (symbols) and fit following the bimodal distribution of Eq.(A1) (dashed lines). The spectra are vertically shifted for clarity. The dotted lines are guides for the eyes.</figcaption></figure>

is no efficient relaxation channel between them, leading to simultaneous direct and indirect transitions of comparable efficiency in bBN.

### C. Phonon-assisted broadening

The strong-coupling regime of the light-matter interaction further impacts the coherence relaxation dynamics in bBN. By temperature-dependent PL measurements we demonstrate that the phonon-assisted broadening has a radically different efficiency for $iX_b$ and for the polariton formed from $dX_b$.

The PL spectra of bBN are recorded from 6 to 250 K in the same conditions as in Fig.1. Fig.3 is a zoom of the direct emission line at ~6.035 eV for different temperatures. The phonon replicas of $iX_b$ below 6 eV display a phenomenology similar to the ones of $iX_h$ in hBN [26]. In Fig. 3, we observe a thermal broadening of the PL line above 50 K, together with a red-shift down to ~6.02 eV at 210 K. The PL intensity decreases by a factor 20 between 10 and 250 K. Interestingly, we note the existence of a weaker peak at higher energy (~14 meV above), showing the same red-shift as a function of temperature as the intense ~6.035 eV line [see the parallel dotted lines in Fig. 3].

In order to extract the thermal broadening, the PL spectra are fitted with the convolution of a line of width $\Gamma$, accounting for homogeneous broadening, with a distribution function $\mathcal{F}$. For the indirect exciton $iX_b$, the homogeneous linewidth is denoted $\Gamma_i$, and the distribution function $\mathcal{F}_i$ is the series of phonon replicas of the $LO_2/TO$ doublet with a detuning given by the energy of the corresponding phonon modes involved in the phonon-
assisted process, as detailed in Ref.[22, 26] for hBN. For the direct emission line at ~6.035 eV, the linewidth is denoted $\Gamma_d$, and $\mathcal{F}_d$ is an ad hoc bimodal function phenomenologically describing the thermalized population of exciton-polaritons at the bottleneck of the lower-polariton branch, as explained in the Appendix. Namely, we use two Lorentzian lines of the same width $\Gamma_d$ to fit the PL spectra above 6 eV.

Fig.4 summarizes the results of our data analysis for both $\Gamma_i$ and $\Gamma_d$. In this figure are displayed the variations of the broadening with respect to 10K ($\Delta\Gamma(T) = \Gamma_\alpha(T) - \Gamma_\alpha(10K)$ where $\alpha = i$ or $d$). The results are drastically different in the two cases. For the indirect exciton $iX_b$ [orange squares in Fig. 4], the phonon-assisted broadening reaches values as high as ~20 meV at 100 K before a **slower increase up to** 50 meV at higher temperatures. In this case, the thermal broadening shows the same unusual curvature as for the indirect exciton $iX_h$ in hBN [dashed line in Fig.4, adapted from Ref.[26]]. On the contrary, for the direct emission at ~6.035 eV related to $dX_b$ [blue circles in Fig.4], phonon dephasing is much less efficient than for $iX_b$, by approximately one order of magnitude below 200 K, before an exponential increase after 200 K. In that case, we recover the standard phenomenology of semiconductor physics [27].

The unusual curvature of the temperature variations of $\Gamma_i$ stems from the so-called slow modulation limit of the phonon-assisted broadening predicted by Toyozawa in his seminal paper [28], and first observed in hBN [26]. In the slow modulation limit, each contribution to the phonon-assisted broadening scales like the square root of the phonon occupation number, so that the total phonon dephasing is written [26]:

$$ \Gamma_i(T) - \Gamma_i(0) = \sqrt{\Delta_A^2 + \Delta_O^2} - \sqrt{S_A E_A} \quad (1) $$

where $\Delta_A$ and $\Delta_O$ are the contributions of acoustic and optical phonons, respectively, given by [28]:

$$ \Delta_A^2 = S_A E_A \coth \left( \frac{E_A}{2k_B T} \right) \quad (2) $$

$$ \Delta_O^2 = S_O E_O \frac{1}{e^{\frac{E_O}{k_B T}} - 1} \quad (3) $$

where $E_A$ and $E_O$ represent the mean energy of the acoustic and optical phonons involved in the dephasing processes, respectively, $S_A$ and $S_O$ are the corresponding coupling strengths, and $k_B$ is the Boltzmann constant. The term $\sqrt{S_A E_A}$ in Eq.1 corresponds to the non-zero limit of $\Delta_A$ at $T=0$, which has to be subtracted to compare with the experimental thermal broadening $\Delta\Gamma$. A fair agreement with our data is obtained by taking $E_A=4$ meV, $E_O=15$ meV, $S_A=4.5$ meV and $S_O=200$ meV [orange line on Fig.4]. The phonon dephasing is in the slow modulation limit in bBN as in hBN, because of the weakly-dispersed excitonic bands in these two BN polytypes, fulfilling Toyozawa's criterion of "flat bands" [28]. Remarkably, an

<figure><img src="image_5.png" /><figcaption>FIG. 4. Thermal broadening of the indirect exciton $iX_b$ (orange squares), and of the lower exciton-polariton related to $dX_b$ (blue circles) in bBN. $\Delta\Gamma(T) = \Gamma_\alpha(T) - \Gamma_\alpha(10K)$ where $\alpha = i$ for the broadening of the indirect exciton $iX_b$, or $\alpha = d$ for the broadening of the direct emission at ~6.035 eV. The dashed line is the thermal broadening of the indirect exciton $iX_h$ in hBN, adapted from Ref.[26]. The orange solid line is a fit with Eq.1 with the fitting parameters $E_A=4$ meV, $E_O=15$ meV, $S_A=4.5$ meV and $S_O=200$ meV. The blue solid line is a fit with Eq.4 with the fitting parameters $a=0.03$ meV.K$^{-1}$, $b=600$ meV, and $\Lambda=90$ meV.</figcaption></figure>

enhancement by a factor ~2 is observed in bBN in comparison to the thermal broadening of the indirect exciton $iX_h$ in hBN [black dashed line on Fig. 4]. We interpret this effect as a result of the weak excitonic dispersion, not only along the c-axis, but also in the plane of the layers in bBN, as discussed above [Fig.2], and witnessed by the quasi-degeneracy of $iX_b$ and $dX_b$.

The thermal broadening of the direct emission at ~6.035 eV related to $dX_b$ [blue circles in Fig.4] can not be explained by the slow modulation of phonon dephasing, but on the contrary by the fast modulation **limit for which** the linewidth increases linearly with the phonon occupation number $n(T)$. Such a difference comes from the motional narrowing regime of the exciton-phonon interaction that is quasi-universal in semiconductor physics [26-28], leading to the well-known phonon dephasing:

$$ \Gamma_d(T) - \Gamma_d(0) = \Xi_A + \Xi_O \quad (4) $$

where $\Xi_A$ and $\Xi_O$ are the contributions of acoustic and optical phonons, respectively, given by [27]:

$$ \Xi_A = aT \quad (5) $$

$$ \Xi_O = b \frac{1}{e^{\frac{\Lambda}{k_B T}} - 1} \quad (6) $$

where $a$ and $b$ characterize the coupling to acoustic and optical phonons, respectively, and $\Lambda$ is the typical optical
phonon energy. An excellent agreement with our data is obtained with a=0.03 meV.K⁻¹, b=600 meV and Λ=90 meV. The dramatic modification of the thermal broadening with respect to the indirect exciton $iX_b$ originates from the strong coupling regime of the light-matter interaction in the case of $dX_b$. Because one has to consider the density of polaritonic states, and not the density of excitonic states, in the calculations of phonon dephasing within the Fermi's golden rule [27], the curvature of the polaritonic dispersions leads to a reduction of the density of final states. This is particularly important for polaritons at the relaxation bottleneck [Fig.2] that are quasi-elastically scattered at high k by acoustic phonons [27]. Here, the reduced density of final states modifies the phonon dephasing efficiency both qualitatively and quantitatively. First, there is a transition from the slow modulation limit [Eq.1] to the fast one [Eq.4], with different curvatures of the broadening as a function of temperature [Fig. 4], because Toyozawa's criterion of "flat bands" is no longer fulfilled. Second, there is an overall decrease of the phonon dephasing efficiency, by approximately one order of magnitude below 200 K [Fig.4], when quasi-elastic acoustic phonon scattering dominates.

Although the impact of the polariton formation on the phonon dephasing was early pointed out and calculated in Ref. [27], we are not aware of any demonstration for exciton-polaritons, even though an analogous and moderate effect was reported for cavity-polaritons in two-dimensional systems [29]. bBN appears as a model system for demonstrating the influence of the reduced density of polaritonic states in quasi-elastic acoustic phonon dephasing, since the quasi-degeneracy of $iX_b$ and $dX_b$ allows a direct comparison of the corresponding thermal broadening below 200 K, as seen in Fig.4.

Above 200 K, the broadening processes assisted by optical phonons become more and more important. Such scattering mechanisms are no longer quasi-elastic so that they involve other final states than the polaritons of the lower branch. As a consequence, the thermal broadening starts to converge to similar values above 200 K [Fig.4]. Although the signal-to-noise ratio was too small to accurately estimate the homogeneous linewidth above 250 K up to room temperature, we note the thermal broadening of $iX_b$ and $dX_b$ already differs by only a factor ~3 around 250 K, while it was by one order of magnitude at ~200 K.

### III. CONCLUSION

We have studied the thermal broadening in bBN crystals by means of temperature-dependant PL measurements in the deep-ultraviolet. The quasi-degeneracy of the indirect and direct excitons in bBN allows a unique comparison of the phonon dephasing efficiency in the two cases. For the indirect exciton $iX_b$, the phonon-assisted broadening is qualitatively identical to the indirect exciton in hBN, i.e. in the slow modulation of phonon de-

phasing, but with an enhancement by a factor ~2 due to the weak in-plane excitonic dispersion in bBN. For the direct emission at ~6.035 eV related to $dX_b$, the thermal broadening switches to the standard fast modulation limit, with a reduction by approximately one order of magnitude of the phonon dephasing efficiency below 200 K. This striking effect results from the inhibition of quasi-elastic acoustic phonon scattering due to the reduced density of final states along the polaritonic dispersion. The thermal broadening in bBN provides a textbook example for the diversity of phonon dephasing in excitons, and the influence of the strong coupling regime of the light-matter interaction in the case of direct excitons. Our work also illustrates the wealth of original effects to be explored in the recently identified Bernal polytype of boron nitride.

**Acknowledgments** We gratefully acknowledge C. L'Henoret and T. Cohen for their technical support at the mechanics workshop. This work was financially supported by the network GaNeX (ANR-11-LABX-0014) and the BONASPES project (ANR-19-CE30-0007). JL and JHE are grateful for support from the Office of Naval Research, award number N00014-20-1-2474, for the BN crystal growth.

*e-mail: guillaume.cassabois@umontpellier.fr

### Appendix A: Lineshape fitting of the direct emission at ~6.035 eV

A fair agreement with the experimental data at ~6.035 eV is reached by taking for the PL signal intensity $I(E)$ the sum of two Lorentzian functions of the same width $Γ$ and of amplitudes $A_j$:

$$I(E) = \sum_{j=1}^{2} A_j L_j(E) \quad (A1)$$

where $L_j(E)$ is a normalized Lorentzian function:

$$L_j(E) = \frac{\frac{\Gamma}{2}}{\pi (E - \Omega_j)^2 + \left(\frac{\Gamma}{2}\right)^2} \quad (A2)$$

with $\Omega_j$ the central energy, and $\Omega_2 > \Omega_1$.

$\{\Gamma, A_j, \Omega_j\}$ are free parameters, which are adjusted for each temperature to reproduce our experimental data [Fig.4]. The temperature variations of the energy $\Omega_1$ and of the energy splitting $\Omega_2 - \Omega_1$ are displayed in Fig.5(a) and (b), respectively. The thermal redshift of $L_1(E)$ centered at $\Omega_1$ comes from the temperature dependence of the bBN bandgap, with a decrease of ~20 meV from 10 to 250 K [Fig.5(a)] similar to the one observed in hBN [19]. The energy splitting $\Omega_2 - \Omega_1$ of about ~14 meV is approximately constant with temperature, within our experimental error [Fig.5(b)]. The discontinuity of the energy splitting $\Omega_2 - \Omega_1$ at ~130 K is attributed to a spatial drift of the sample when changing the temperature.
<figure><img src="image_6.png" /><figcaption>FIG. 5. Temperature variations of the central energy $\Omega_1$ (a) and energy splitting $\Omega_2 - \Omega_1$ (b).</figcaption></figure>

Since the high-energy component of the PL spectrum centered at $\Omega_2$ is more and more intense when raising the temperature [Fig. 5], the ratio $A_2/A_1$ increases with temperature, suggesting a thermal activation. We find that $A_2/A_1$ follows a Boltzmann law $\exp(-E_0/k_B T)$, with $E_0$ of the order of the energy splitting $\Omega_2 - \Omega_1$. More precisely, the best fit to the experimental data is obtained by taking for $E_0$ the energy splitting $\Omega_2 - \Omega_1$ but with a small correction of -2 meV.

From this quantitative interpretation of the direct emission at ~6.035 eV, we thus conclude that the bimodal PL lineshape observed in Fig. 5 originates from a two-component distribution of thermalized exciton-polaritons at the bottleneck of the lower-polariton branch. At this stage, we have no clear interpretation for the origin of these two peaks, which will require further investigation beyond the scope of this work.

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