Polariton lasing in Mie-resonant perovskite nanocavity

Deeply subwavelength lasers (or nanolasers) are highly demanded for compact on-chip bioimaging and sensing at the nanoscale. One of the main obstacles for the development of single-particle nanolasers with all three dimensions shorter than the emitting wavelength in the visible range is the high lasing thresholds and the resulting overheating. Here we exploit exciton-polariton condensation and mirror-image Mie modes in a cuboid CsPbBr$_3$ nanoparticle to achieve coherent emission at the visible wavelength of around 0.53~$μ$m from its ultra-small ($\approx$0.007$μ$m$^3$ or $\approxλ^3$/20) semiconductor nanocavity. The polaritonic nature of the emission from the nanocavity localized in all three dimensions is proven by direct comparison with corresponding one-dimensional and two-dimensional waveguiding systems with similar material parameters. Such a deeply subwavelength nanolaser is enabled not only by the high values for exciton binding energy ($\approx$35 meV), refractive index ($&gt;$2.5 at low temperature), and luminescence quantum yield of CsPbBr$_3$, but also by the optimization of polaritons condensation on the Mie resonances. Moreover, the key parameters for optimal lasing conditions are intermode free spectral range and phonons spectrum in CsPbBr$_3$, which govern polaritons condensation path. Such chemically synthesized colloidal CsPbBr$_3$ nanolasers can be easily deposited on arbitrary surfaces, which makes them a versatile tool for integration with various on-chip systems.

section_content: Section 1: Experimental measurement of ASE from perovskite thin film under the light cone

x 0 ≈ As we discuss in the main text, perovskite polycrystalline thin film is able to support waveguide modes, which can be directly measured by angle-resolved spectroscopy with SIL. Angle-resolved emission spectra, measured under femtosecond laser pump at 515 nm at pump fluences around 0. 01 mJ/cm 2 and 1 mJ/cm 2 in the regime of low coupling of SIL (when the waveguide is located far from the SIL). The waveguide mode is shown in Fig. S1 (a) with k /k 1. 8. The low coupling of the SIL provides low radiative optical losses of the waveguide mode. In the case of low pump fluence, we observe a broad intensive PL signal above the light cone and a low-intensive waveguide mode below the light cone. According to our assumptions, in this regime, we measure exciton PL with thermalized polaritons over the branch. At the pump fluences around 1 mJ/cm 2 above the light cone, ASE emission appears in the spectral region around 2. 26 eV. At the same time below the light cone, we also observe an enhancement of the intensity in the same spectral region, corresponding to the ASE. Moreover, the intensity, measured below the light cone strongly exceeds the intensity above. It can be explained as the origin of ASE came from the guided polariton and then it is scattered via defects and film grains, outcoupling to the free space. 

In the experiment we also change the distance between SIL and perovskite thin film, changing the SIL coupling efficiency and hence radiative losses of waveguide mode (Fig. S1(b) ). In the case of the lowest radiative losses, we observe pronounced red broadening of ASE with increasing pump fluence. When we decrease the distance between SIL and film with increased radiative losses the red broadening is suppressed. In the regime of the SIL overcoupling with the highest optical losses we do not observe any ASE. The phenomenon can be explained in terms of the polariton lifetimes. When we increase radiative losses of guided mode, polaritons recombine or scatter to the lower energies. If losses are too high, polariton concentration cannot achieve a nonlinear regime because of fast recombination, which we observe in the case of. 

We also study light-light curves for different guided polariton radiative losses shown in Fig. S1(c ). The ASE provides pronounced threshold behavior in the case of low optical losses. Moreover, the ASE threshold is increasing with increasing waveguide optical losses. As there exists a threshold concentration of polaritons for stimulation, increasing losses 2. 20 0 1. 9 1. 8 1. 9 1. 8 1. 9 1. 8 Masharin MA et al. Opto-Electron Adv 7, 230148 (2024) https://doi. org/10. 29026/oea. 2024. 230148

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leads to the requirement of the higher pump fluence to achieve a nonlinear regime. Eventually, the ASE threshold can be larger than the degradation threshold and we will not be able to see any ASE, as we demonstrate in the case of in Fig.
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section_content: Section 2: Interference pattern on angle-resolved emission spectra from perovskite nanowires

Angle-resolved pump-dependent emission measurements from perovskite nanowires are performed in the case of the nanowire axis co-directed with an angle-resolved -axis. To study the interference pattern we normalized angle-resolved spectra from Fig. 2 (g-i) in the main text to the integrated over the k /k spectra to make it more pronounced (Fig. S2 (a)). At the high pump fluence in the multilasing regime, the emission becomes coherent in the nanowire and outcouples through the edges. The pattern appears due to the interference of the coherent emission from the opposite edges of the nanowire on the BFP (Fig. S2 (b)). 

x 0

The number of the interfered maxima over k /k depends on the distance between coherent light sources, or in other words, the physical nanowire length. It should be noticed, that pattern changes the phase over the π with each next order. It can be explained either an even or an odd number of half-length is suited to the nanowire optical length. The parity changes with every next order and therefore the initial phase difference Δ also drops from π + 2π to 2π or vice versa (Fig. S2(c) ). The observed effect helps to distinguish several F-P resonances from each other near the exciton, where it is located very close to each other and when the intensity slightly changes from one resonance to another. By extracting the F-P resonances close to the exciton in the following way, we calculated the free spectral range of the F-P resonances and estimated the group refractive index, shown in Fig. 2 (j) in the main text. Section 3: The radiation pattern of Mie-polariton lasing emission from the nanocubes

Pump-dependent emission measurements shown in the main text were done also with angle resolution. Angle-resolved emission spectrum measured from 0. 02 cuboid upon 6 J/cm 2 is shown in Fig. S3 (a). It is shown two lasing peaks, noted as "Mode 1" and "Mode 2", which are separated on the wavelength. The third mode ("Mode 3") appears at around 2. 3 eV at higher fluence as illustrated in Fig. S3 (b), whereas the first two modes become closer to each other and cannot be clearly distinguished. To make it more clear we plotted the section over the wavelength peak centers in the radiation plot demonstrated in Fig. S2(b ). Polar plots for modes 1 (orange dots), 2 (green dots), and 3 (blue dots) have different radiation patterns (Fig. S3(c-e )). According to numerical eigenmode analysis "Mode 1" (with ) and "Mode2"

) correspond to non-degenerated (because of unequal sides) hybrid mode consisting of mixture of spherical harmonics S1 involving magnetic octupole with azimuthal number m = 1, while the lower energy Mie-mode has with the major contribution from magnetic quadrupole with azimuthal number m = 2 (Fig. S3 ). The spatial electrical field distribution of each eigenmode is presented in Fig. S3 (f-h), while calculated far-field directivity is shown in Fig. S3(c-e ) by dashed curves matching well the experimental results. 

Section 4: Experimental observation of the Mott transition in the perovskite nanocubes In the previous works, it was suggested, that lasing in the CsPbBr 3 nanowires appears above the Mott transition and originated from the electron-hole plasma and plasmonic resonances S2. Authors sustained their claim by ultrafast emission spectroscopy, where lasing peaks in the spectra shift to the red region with time, which points to the dependence of the peak position with carrier concentration. However, the same result can be explained by the polariton nature, mentioned in the main text. To check the lasing origin in our experiments we pump our nanocubes at the highest pump fluences at 6 K to experimentally achieve the Mott transition (Fig. S4(a) ). As was already discussed in the main text, we observe a few-lasing peak regimes from the nanocube at the low pump. The quality factor of the lasing mode achieves 1680 (Fig. S4(b) ) at 8 J/cm 2. With increasing the pump fluence lasing peaks shift and broad to the blue region. At the fluence around 1000 J/cm 2, broad background PL appears, where stimulated emission still exists. Finally, at fluences above 2500 J/cm 2 lasing stimulated emission disappears and we observe a broad PL spectrum. The spectral center of the broad PL spectrum is shifted from the initial PL center by the value around 32 meV. As the exciton binding energy is estimated to be around 35 meV S3 we attribute the initial PL to the exciton recombination and PL obtained at the high pump fluence to the band-to-band recombination. In other words, we achieve the Mott transition at the pump fluences around 1000 J/cm 2, which is described in a broad PL background, but the Coulomb correlations between electrons and holes still exist and the polariton lasing can still be maintained but with lower intensity. At the higher fluences, even the correlation disappears with the lasing emission and we obtain clear band-to-band PL. It also should be noted, that according to the experiment, the process is reversible, which means, that we are below the degradation threshold for the particular with calculated quality factor. (c) Normalized integrated spectra, obtained at 2 and 2900 μJ/cm 2. TThe first corresponds to exciton PL and the second to band-to-band PL. The difference between these two spectra is close to the exciton binding energy in this material. Masharin MA et al. Opto-Electron Adv 7, 230148 (2024) https://doi. org/10. 29026/oea. 2024. 230148
section_name: S1(c).

section_content: perovskite nanocube. According to the estimation, Mott transition fluences are close to the estimated critical Mott concentration, which fits the known experimental data. For more details see Section S5. This data proves, that the lasing emission in our case originated from the exciton-polariton nature of either electron-hole plasma. 

Section 5: Estimation of the excited exciton concentration in the nanocubes In order to compare experimentally achieved Mott transition with literature data we estimate excited carrier concentration in our experiments. In the simplest estimations absorbed optical power intensity can be estimated as

where is incident power, T is transmittance coefficient, is the imaginary part of the refractive index at the pump wavelength, is incident pump k-vector and is the height of the nanocube. In the pulsed pump regime it can be rewritten in the carrier number notations:

)

The absorbed carrier concentration, where S is the scale area of the incident pulse. And, where is incident fluence and is the energy of the single photon at the pump wavelength. The resulting carrier concentration can be estimated as:

)

According to the previous studies, critical exciton Mott transition density is estimated to be equal. S4 Based on the current approximations in our experiment with a 490 nm pulsed pump, such concentrations are reachable at the fluences around 300 J/cm 2. Deviations from the experimental observation can be explained by overestimation of the excited exciton concentration, which appeared from overestimated absorption coefficient at 6 K, Gaussian distribution of the pump spot profile, and optical effects due to the substrate, and nanocube faces. Also, it should be noted, that estimated Mott concentration does not take into account polaron effects in the material S5, which increases critical Mott concentration S6. Nevertheless, the order of the estimated excited exciton concentration and Mott concentration is well agreed with experimental observations and prove, that observed lasing and ASE regimes originated from exciton-polariton nature. 

Section 6: Optical properties of CsPbBr 3 cuboids at room temperature λ c μ Obtained nanocuboids suspension in hexane reveals bright green emission peaked at 524 nm with FWHM ~ 21 nm upon UV lamp excitation, while extinction spectrum contains a sharp dip in the region of PL maximum (Fig. S5(a) ). This dip is attributed to Fano resonance S7. Notably, the suspension consists of nanocuboids of various hydrodynamic sizes from around 200 to 1000 nm which are slightly overestimated based on DLS measurements as compared to the real sizes (Fig. S5(b) ). According to the collected data the majority of nanocuboids have hydrodynamic diameter of about 296 nm. A suspension of perovskite cuboids was drop-casted on a metal-dielectric substrate (Fig. S6(a-b )). Regular shape and smooth facets of cavities with various dimensions were confirmed by SEM (Fig. S7(a-d )). Scattering from single resonators was studied by means of dark-field spectroscopy at room temperature. Scattering spectra were measured by optical microscope (Carl Zeiss Axio Imager. A2m). The sample was illuminated in dark-field mode by a broadband halogen lamp (HAL 100) via 100x objective (EC Epiplan-NEOFLUAR). The scattered signal from single cuboid was collected by the same objective and sent to the spectrometer (Ocean Optics) through 100-m optical fiber. The spectra showed sharp resonances of Mie-type which spectral position and order depends on cavity size. Therefore, small cuboids with size around 200 nm possesses pronounced magnetic dipole and quadrupole Mie-resonances. Increase in cavity size causes shift to low-energy of first-order resonances (purple dashed line) along with appearance of new highorder resonances near the exciton energy (green dashed line) (Fig. S7(e) ). Notably, the resultant scattering spectrum is a superposition of resonator's Mie modes which allows to estimate the approximate geometrical parameters of cuboids. T To solve the inverse scattering problem we applied a relevant numerical model based on the -matrix method and the freely available software "Smuthi" S8-S12. "Smuthi" is a Python package for modeling electromagnetic scattering by one or more wavelength scale objects in layered systems efficiently and accurately. The software integrates the -matrix method of single-particle scattering with the scattering matrix formalism for electromagnetic field propagation through planar interfaces. This method was used to find optimal multilayer systems designs and to obtain an extinction crosssection decomposition for all multipole components (in our case only for the dipole components). To estimate thematrices of nonspherical particles, "Smuthi" uses the advanced NFM-DS ("Null-field Method with Discrete Sources") FORTRAN S13 software. T-matrices computed with NFM-DS are processed in the "Smuthi" software to solve the problem of multiple scattering between such particles as well as between particles and layered media. 

As described earlier in the main text, metal-dielectric substrates are widely used to enhance field localization efficiency and mode selectivity S14,S15. To demonstrate the promising potential of plasmonic substrates, we numerically simulate the scattering behavior of CsPbBr 3 nanocuboids on Al 2 O 3 /Ag/Si substrates (bottom row in Fig. S8 ) and compare with similar nanocuboids on glass substrates (top row in Fig. S8 ).
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section_content: We evaluated a comparison of scattering spectra for a broad range of cube sizes (Fig. S8(a, d ), revealing an order of magnitude higher scattering efficiency on the Al 2 O 3 /Ag/Si substrate, as well as a mode contrast. To validate the -matrix method, the experimental scattering spectra obtained for similar CsPbBr 3 nanocuboids on the glass and Al 2 O 3 /Ag/Si
section_name: T

section_content: substrates were compared with those predicted theoretically. As shown in Fig. S8 (b, e), obtained results demonstrate an excellent agreement between the numerical and experimental data, where the difference in amplitude for the resonance at high energy can be explained by the large absorption in this range. Notably, the scattering spectra were simulated for systems similar to the experimental ones, taking into account all input parameters of the experimental conditions in- cluding geometrical parameters (the difference in width, length and height). Improved scattering efficiency and selectivity of modes on plasmonic substrates is related to the formation of mirror-image Mie modes S14,S15. The multipole decomposition of extinction cross-section of a 242 nm CsPbBr 3 cube excited in TE mode is shown in (Fig. S8(c, f ). TE polarization of the incoming light leads to the efficient excitation of the -oriented magnetic dipole and, in case of plasmonic substrate, formation of a magnetic dipole-image with anti-phase (inset in Fig. S8 (f)). The interaction of antiphase dipole sources leads to an increase in the intensity and -factor of magnetic dipole component in decomposition of the extinction cross-section for a cube on metal-dielectric substrate compared to the cube on glass. Therefore, the integration of compact semiconductor resonators with metal-dielectric substrates is highly promising for lasing application. Moreover, the developed approach is highly useful for fast detection and primary characterization of nanocuboids, promising for lasing study, allowing to avoid premature SEM measurements, which destroys optical properties via increasing nonradiative traps. 

Section 7: Calculation of cold dispersion Analytical dispersion of CsPbBr 3 at room and 6 K temperatures have been obtained from measurements of reflection coefficient of CsPbBr 3 microplates (MPs) by using of transfer matrix method (TMM). 

μ CsPbBr 3 MPs with various thicknesses (from 145 nm up to 1485 nm) were obtained on glass substrate according to the procedure reported by Zhizhchenko et al. S16 (Fig. S9(a) ). Reflection spectra were measured from 18 MPs at room and cryogenic temperatures upon broadband halogen lamp excitation at almost normal incidence. Reflected signal was collected from the center of MP through 100 m-pinhole to avoid scattering from edges. The collected spectra were normalized to the incoming spectrum of source. Resulting experimental dependencies of reflection coefficient on photon energy obtained for selected five MPs with different thicknesses at room and cryogenic temperatures are shown in Fig. S9(b, c ). 

To obtain the refractive index and the extinction coefficient of the perovskite MP on a glass substrate TMM has been employed. By treating the substrate as a semi-infinite space, the following analytical formula for the reflection coefficient can be derived:

where is the intensity reflection coefficient depending on the wavelength and temperature, is the thickness of perovskite MP, is the complex refractive index of perovskite MP, is the refractive index of the surrounding atmosphere (we consider ), is the substrate refractive index (we consider ), and

θ θ = 0 where is the incidence angle (we consider ). 

In order to parametrize the dispersion, we define the function by its values at fixed wavelengths and use shape-preserving piecewise cubic interpolation to interconnect these points. Namely, we fix the refractive index at 20 uniformly distributed wavelengths in the 500 nm 550 nm range, and the refractive index at 5 uniformly distributed points in the 550 nm 750 nm range. The extinction coefficient is taken from direct measurements. Therefore we obtain a complex function dependent on 24 fitting parameters (for a single temperature). 

Additionally, due to the experimental errors, we introduce the following fitting parameters. First, due to the MP's surface corrugations, the measurement of the plate thicknesses with a profilometer has an experimental error of, so we introduce the thickness correction for each plate, which is varied between and. Second, due to the experimental errors in the reflection measurement, the reflection spectra were multiplied by the correction coefficients (for room temperature) and (for 6 K), which became fitting parameters. The typical values of the correction coefficients lie between 0. 9 and 1. 1. The spectra are fitted together by minimizing the function by varying the total 102 fitting parameters: S15,S18. This enhancement related to Purcell effect and can be defined by Purcell factor proportional to value and expressed as follows:

, where is a wavelength inside cavity, Q and V are quality factor and volume of mode, respectively. Experimentally, Purcell effect can be observed by shortening of the decay (lifetime) of spontaneous emission:

, where and -are radiative decay rates of dipole source in cavity or bulk, respectively. Therefore, PL decay in single cuboids with similar size on glass and plasmonic substrates (Fig. S11(a, b )) was studied in the same excitation conditions (pump fluence around 0. 1 μJcm -2, pulse duration around 220 fs, repetition rate equaled to 100 kHz and pump wavelength is centred to 490 nm) at room and cryogenic temperatures. At room temperature, PL lifetime equals to = 1. 9 ns and = 0. 5 ns for nanocuboids on glass and metal-dielectric substrates, respectively (Fig. S11(c, d )). At cryogenic temperature, lifetimes are getting nearly 3 times shorter for the glass substrate and at least more than 5 times shorter for plasmonic one that cannot be precisely resolved because of instrumental response (black curve in Fig. S11(b-c )) of the detector. Thus, the positive influence of metal-dielectric substrate on optical properties of perovskite nanocuboids was confirmed by time-resolved measurements. Masharin MA et al. Opto-Electron Adv 7, 230148 (2024) https://doi. org/10. 29026/oea. 2024. 230148
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