The history of halide perovskite started in 1893, when Wells  first synthesized cesium lead halide perovskite, while halide perovskites with organic cation (e.g.
Parallel to the progress in the halide perovskites, new “laser era” raised after the first laser was invented by Maiman  in 1960, resulting in the development of semiconductor-based lasers after the demonstration of the first prototype in 1970 operating at room temperature . Further progress of semiconductor lasers developing was size decreasing to overcome diffraction limits and to create ever smaller lasers , , , , , , , , .
In this general trend in laser physics and technologies, the compactization of perovskite-based lasers is one of the most rapidly developing directions. Figure 1B shows the milestones in the progress of development of single-particle perovskite lasers from relatively large microplate (MPL)-based two-dimensional (2D) design  to nanowires (1D design)  and eventually to such 0D designs as nanospheres  and nanocubes . Current record-small perovskite nanolasers have already achieved the volumes less than the volume of emitted photons in free space , . Because further progress in this field would require even stronger light confinement in a single nanoparticle, it is highly important to understand what is the ultimate size limit for perovskite-based single-particle lasers. Figure 2 shows the most important parameters for laser size minimization. In this focused review, we aim to discuss the impact of material and cavity properties on lasing from single-particle lasers based on the halide perovskites.
2 Perovskite light-emitting properties
2.1 Some advantages of perovskites
In this review, we focus on 3D halide perovskites family of materials with formula ABX3, where “A” is the cation (e.g. MA+, FA+, or Cs+), “B” is usually lead (Pb2+), and “X” is the anion (e.g. I−, Br−, or Cl−). Halide lead perovskites can be considered as direct band-gap materials resembling GaAs. But they have some advantages that make them a very perspective material for nanophotonics. One of such advantages is high optical gain being higher than 103 cm−1 at room temperature owing to strong interband transition . Moreover, the halide perovskites have denominated exciton at room temperature with electron–hole binding energy up to 100 meV , , , , , . High defect tolerance allows achieving photoluminescence (PL) even with defect concentration 1017 cm−3 . One of the most important advantages of halide perovskite is capability to tune band-gap size by chemical method in the range 1.7–3 eV , . Another reason why the halide perovskites are so popular materials for photovoltaics and photonics is their cost-efficiency governed by simple wet chemistry methods of fabrication and earth abundance of all components . Regarding additional optical outstanding properties, perovskites possess high nonlinear absorption coefficients. For example, CsPbBr3 perovskite has three-photon and two-photon absorption coefficients 0.14±0.03 cm3 GW−2 at 1200 nm  and 4.57 cm GW−1 at 800 nm , respectively. Comparing GaAs has 26 cm GW−1 at 1060 nm and 0.35±0.05 cm3 GW−2 at 1680 nm , GaN has 6.35×10−3 cm GW−1 at 410 nm and 1.05×10−10 cm3 GW−2 at 820 nm  two- and three-photon absorption coefficients, respectively. These properties enabled demonstrations of multiphoton excitation of various perovskite-based microlasers , , ,  Nevertheless, perovskites have some disadvantages as degradation at atmosphere due to interaction with water and oxygen, light-induced phase segregation, reaction with metals (Au, Ag), and parasitic migration of ions and defects, as well as low thermal conductivity around 0.5 W m−1 K−1 , . These factors limit the realization of perovskite-based electrically pumped lasers . Therefore, material and cavity properties optimization has crucial importance for further development of this direction.
The quality of emitting semiconductor can be assessed from the kinetics of generated charge carriers relaxation. The kinetics of generation and relaxation of free carriers in halide perovskites are usually described with the following general equation:
where G is the photoexcited carriers generation rate; Rrad is the recombination rate caused by spontaneous light emission; and Rnrad is the recombination rate usually associated with trap-assisted recombination that relies on an individual carrier (electron or hole) being captured in a trap. Also, Auger recombination (cubic term of Nfc) should be taken into account at very high densities (Nfc>1018 cm−3). The radiative relaxation term corresponds to intrinsic electron–hole recombination, which depends on both electron and hole densities and thus has quadratic dependence on Nfc. As a result, the internal quantum yield (QY) can be written as follows:
Remarkably, that main negative contribution to QY in real semiconductor nanostructures comes from nonradiative Shockley–Read–Hall recombination mechanism with the rate RSRH=NTNfc(Kn+Kp)/2, where Kn and Kp are kinetic factors that encompass both the trap capture cross section and thermal velocity of the electrons and holes, respectively, and NT is the trap state density. In the halide perovskites, even high trap state density (NT~1014–1017 cm−3) usually yields relatively weak contribution to QY because of the low formation energy defects in perovskites residing near the band edges, which render the lead-halide perovskite semiconductors tolerant to defects. More detailed analysis with taking into account recycling, polaron, and Rashba effects is presented in , .
Shifting PL spectra to blue or red region is possible with mixed-halide CsPbBr3-xClx and CsPbBr3-xIx compositions. However, mixing anions in perovskites, one inevitably sacrifices QY because of the higher trap state density in the forming materials [see Discussion for (2)]  and hence increases Fth as well as decreases the quality factor of lasing modes .
One can also derive from (2) the expression for temperature dependence of QY on taking into account the emission from an excitonic state:
in which U is the coefficient containing the ratio between radiative and nonradiative relaxation rates, which is a function of Nfc; Eb is the binding energy of exciton being dependent on T for the halide perovskites , , and kB is the Boltzmann constant. In order to quantify the lasing threshold properties relatively, the pump threshold is usually fitted by the exponential dependence on temperature relation :
where Fth,0 is the threshold pump fluence at low temperature, and T0 is the characteristic temperature, which is used to signify the strength of PL temperature dependence. For example, single-crystal CsPbBr3 microparticles have relatively high characteristic temperatures T0≈300 K , which is larger than the typical T0 around 100 K of bulk or quantum well-based lasers . Such relatively weak temperature sensitivity opens the way for room-temperature ultracompact lasers. On the other hand, the temperature-dependent PL experiments for polycrystalline films of MAPbI3 were fitted by much lower characteristic temperature, T0≈75 K , which was attributed to the significant role of defects.
First, we should discuss the nature of lasing in lead halide perovskite single-particle cavities. Generally, there are two main mechanisms of lasing discussed in the literature: from electron–hole pair emission  or from exciton–polaritons Bose condensation , . However, it was revealed recently that the threshold carrier density for lasing in the single-particle perovskite designs is usually above the Mott-transition critical density, which most likely results in stimulated emission of electron–hole plasma rather than the Bose condensation of exciton–polaritons .
In order to discuss the main contributions to lasing performance of perovskite-based nanocavities, let us consider basic equations for kinetics of carriers (Nfc) and photons (s) in a laser mode:
where G is the generation rate; Rsp is the recombination rate associated with spontaneous emission; vg is group velocity of light in the medium; s is the number of photons in the mode; g is the material gain; Γ is the coupling factor for the mode with the active media; β is the part of spontaneous emission that couples into the lasing mode; τp=Qcavity/ω is the photon lifetime within the cavity.
We would like to stress two main contributions to the lasing threshold:
- Quality factor (Qcavity) of a cavity, determining the lifetime of photons in the laser mode and governed both by dissipative losses in the material and by radiative losses in the cavity. Thus, assuming that the material losses are compensated by pump, one can derive from (6) the condition for the minimal radiative part of cavity Qcavity factor required for lasing:where gth is the threshold material gain. Keeping in mind that the gain g is a function of Nfc, in the most optimal case, one has to achieve a certain level of gain at the lowest generation rate (G).
- The minimization of G requires high values of QY, which is governed by nonradiative recombination (Rnrad). This contribution is strongly connected with deep-level defects concentration in the material. Thus, one can see from (5) that minimization of Rnrad would give the highest carriers concentration at given G. The minimization of pumping intensities or current densities is crucial for avoiding nanolaser overheating.
Therefore, we will analyze the contributions both from cavity design and material quality in order to discuss the strategy for the creation of ultimately compact perovskite-based lasers.
3 Perovskite cavities properties
3.1 Nanoparticles subwavelength in all dimensions: Mie resonances
Optical modes in the dielectric (Re[n]>1 and Im[n]=1) particles with subwavelength dimensions in all three directions are often called Mie resonances , , . Recent works proved that halide perovskite nanoparticles can support Mie resonances in visible and near-infrared (IR) ranges , , . Analytical calculations based on the Mie theory ,  for a sphere with complex refractive index (Re[n]=2.5 and Im[n]=0) are shown in Figure 3A. The Mie theory gives the total scattering efficiencies upon irradiation of an incident plane wave:
where k=2πnh/λ, nh is the refractive index of the host media, λ is the incident wavelength. The coefficients aj and bj correspond to electric and magnetic scattering amplitudes for corresponding Mie modes (multipoles), which were actively studied during the last decade for various high-refractive-index materials (such as Si and GaAs).
The typical calculated scattered cross section for perovskite (n=2.5) spherical nanoparticle is shown in Figure 3A. Black curve shows a total scattering cross section, color curve shows mode decomposition, dotted line shows eclectic-type modes (TE), and solid line shows magnetic-type modes (TM). For example, TM1 corresponds to a magnetic dipole, TE1 corresponds to electric dipole, TM2 corresponds to magnetic quadrupole, TE2 corresponds to electric quadrupole, and so on. In notation for TE, TMi,m, m stands for the azimuthal mode number, and i stands for the radial mode number. This modeling provides important information on spectral width and overlapping between the Mie resonances. More detailed studies of the Mie resonances in halide perovskite nanoparticles were carried out recently both theoretically and experimentally , , , . Remarkably, such spectral overlapping can result in specific modes interference and thus to control the directivity of emitted or scattered light , , , .
More complicated numerical methods are required to solve the Maxwell equations for cylindrical, cubical, and other shapes of the nanoparticles. The Qcavity factors of the eigenmodes in perovskite nanoparticles are calculated numerically by using finite-element method in an eigenmode solver in COMSOL Multiphysics. All calculations are carried out for single particles of three different shapes: cubes, spheres, and cylinders (height of cylinder considered permanent equal to 200 nm). Nanoresonators are considered in homogeneous environment with refractive index equal to one surrounded by perfectly matched layer imitating infinity space. The value for refractive index of particle is 2.5 that close to CsPbBr3 thin film around its excitonic state. Figure 3B shows the dependence of eigenmode Qcavity factors on cavity size. Color ovals show that modes near field distribution satisfy sphere multipole notation from Figure 3A. Insets show near-field distribution of electric field for the first magnetic modes. The considered shapes of perovskite nanoparticles can be fabricated by various methods , , , , , which will be discussed in the following sections in more detail.
3.2 Nanowires: Fabry–Perot resonances
One of the most popular shapes of nanocavities for perovskite single-particle lasers are nanowires (NWs), which are just prolonged nanoparticles in one direction. Owing to high reflectivity on facets, perovskite NW can be considered as a Fabry–Perot (F-P) resonator, whose Qcavity factor is proportional to its length .
where R1 and R2 are the reflection coefficients at each facet, L is the nanowire length, and k|| is the longitudinal projection of wavenumber along the NW axis.
Remarkably, F-P modes can interfere with Mie-like modes in NWs, resulting in Fano-like features in spectral response , . Nevertheless, linear dependence of Qcavity factor on length in NWs might be slower than that for Mie modes increasing in all three dimensions (Figure 3B). Thus, NWs usually do not show record-small volumes of perovskite-based nanolasers, except their integration with metal surfaces  or nanostructures .
In general, there are many approaches to synthesize perovskite 1D structures, but here we focus only on some of them yielding high-quality monocrystalline F-P cavities. The first one is conventional chemical vapor deposition (CVD) technique, which is widely exploited for the growth of inorganic nanowires and microwires from different materials such as ZnO, GaAs, InP, and so on. As CVD is a high-temperature process, it was found to produce various CsPbX3 NWs possessing a cubic crystal phase. These NWs have rectangular or triangular cross section depending on the direction of growth (in plane or bottom-up) and crystal lattice parameters of the substrate material (Si , mica , , Al2O3 ) or perovskite. In the case of lattice parameter inconsistency or utilization of a substrate with an amorphous top layer (for instance Si/SiO2), the area of NW cross section could be different along the crystal growth direction . The second is very similar to CVD, but a bit simpler approach to fabricate inorganic perovskite nanolasers was demonstrated by Wang et al. . The authors showed that the temperature difference between substrate covered with a perovskite thin film and mica substrate triggers controlled growth of triangular NWs on the latter in air.
The shape and area of the NW cross section affect the confinement factor (η), which is supposed to be maximal for the low threshold lasing and can be expressed as follows :
where nb is the background refractive index, ng is group velocity refractive index, and η0 is a ratio of the energy in the active region to the energy in the entire waveguide and takes values smaller than 1. Therefore, in perovskite NWs, η can be larger than 1 as it is mostly influenced by high group refractive index, which is in the range of 5–17 , , . In turn, the width and height of the most compact laser design based on NW should fall within a specific range to enable efficient propagation of the first fundamental electromagnetic mode along the resonator as well as to avoid competition between this mode and the other ones. For example, the optimal width and height of rectangular MAPbI3 NWs, supporting TE-like fundamental mode (λ=787 nm) having a minor longitudinal component, were calculated to be around 600 and 300 nm, respectively , whereas almost two times smaller dimensions were reported for CsPbX3 nanolasers generating visible light .
In contrast to the vapor deposition techniques, wet chemical approaches employ the processes occurring at moderate temperature and resulting in the formation of tetragonal or orthorhombic perovskite structures. Therefore, NWs have only a rectangular cross section. The early reports on their slow solution growth describe bottom-up crystallization from lead halide or acetate films immersed in alcohol containing methylammonium or cesium halides , . Then, synthesis was expanded to antisolvent diffusion method  and ligand-assisted reactions in which ionic organic species passivate certain crystallographic planes and invoke unidirectional crystallization , . It should be pointed out that Q factors of the F-P resonators obtained by the wet chemical approaches are comparable or even better than those of NWs grown by CVD. In particular, recently, we have developed a few-minute synthesis of CsPbBr3 NWs with truncated pyramidal end facets, which according to numerical calculations trap confined light in the vicinity of the very end of the cavity and decrease the out-coupling efficiency . Moreover, ligand-assisted strategy successfully applied for the production of monodisperse quantum dots  is supposed to accelerate the development of technology, which will afford manufacturing of perovskite NWs having optimized dimensions for high-quality lasing.
3.3 Plates and disks: whispering gallery modes
When two characteristic lateral dimensions of a perovskite particle are larger than the emitted wavelength, and the thickness is subwavelength only, the structures are often called microplates (MPLs) or microdisks (MDs). These types of cavity support so-called whispering gallery modes (WGMs). Qcavity factor of WGM in an MPL resonator with polygon shape is given as follows :
where d is the diameter of the circle that circumscribes the polygon, n is refractive index, N is the number of facets (N>2), and R is reflection on the boundaries. For an ideal spherical resonator, Qcavity factor is given by the following formula:
where i is radial mode number, q=2πd/λ where d is sphere diameter, and χi(q)=qyi(q) is Riccati–Bessel functions, where yi(q) is Neumann functions.
Generally, spherical microparticles or MDs are more preferable as compared with cubic ones when the size is larger than the operating wavelength in free space, i.e. when there is an increase of incidence angle of WGM to the spherical boundary of particle, whereas this angle is fixed for the increasing polygon particles. However, a spherical shape does not correspond to any single-crystal state being likely an amorphous or polycrystalline one, where defect concentration (NT) is usually much higher as compared with single-crystal perovskites.
Among the regular shapes of perovskite plates supporting high-quality WGM regime, only three are possible: triangular, rectangular, and hexagonal because of perovskite crystal phases dictating the geometric parameters of the polygons. Rectangular cavities adopting cubic or orthorhombic crystal lattice can be grown by using the same approaches mentioned above for perovskite NWs , , . Triangular and hexagonal plates have been reported only for MAPbI3 and its mixed halide counterparts – MAPbI3−xBrx and MAPbI3−xClx with x=1 . These structures possess a tetragonal phase and can be fabricated by vapor deposition of lead halides crystallizing in the form of thin triangles or hexagons on mica followed by their thermal intercalation with methylammonium iodide. As compared to NWs of similar chemical composition, MAPbI3 hexagonal MPLs exhibit much higher optical gain that allows reducing their lasing cutoff thickness down to 70 nm at room temperature .
Thin WGM cavities with a circular shape cannot be achieved either by CVD or by simple wet chemical approaches. Usually, their manufacturing is a complicated multistep process comprising growth of single-crystal perovskite MPLs or deposition of polycrystalline thin films that are accompanied by top-down lithography , ,  or laser printing procedure , . Thereby, when material losses (1/Qabs) are compensated by pump and mode leakage losses (1/ Qleak) are extremely small, Qcavity suffers mostly from both the sidewall roughness and polycrystallinity involved in the scattering losses (Qscat) owing to technological reasons. Thus, in the following formula :
the first two terms can become almost negligible as compared with the scattering losses. For example, in MD cavities, the losses caused by scattering on sidewall roughness (Qscat) can be expressed as follows :
where d is MD diameter, h is MD thickness, and Lc and ξ are the roughness correlation length and amplitude, respectively.
Among different types of perovskite WGM resonators, the highest Qcavity factors belong to spherical ones manufactured by CVD , ,  and deposition of perovskite precursors or as-prepared quantum dots onto SiO2 microspheres , . Although CVD-grown CsPbX3 spheres adopt an orthorhombic phase, which is supposed to crystallize in the form of cubes or bricks, the activation of surface tension forces at high temperature leads to the formation of spherical cavities , .
4 Experimental demonstrations of single-particle lasers: comparative analysis
Because of high optical losses at emission wavelengths of the halide perovskites, it is difficult to characterize Qcavity factors of the nonexcited cavities. On the other hand, one can easily measure spectral width of the lasing modes above the threshold. The connection between Qcavity factor of cavity and spectral width of the lasing mode was established by Lax  correcting the famous Schawlow–Townes equation:
which can be presented as
where ω is the radial frequency of the resonant mode, Δωlaser and Δωcavity are the full width at half-maximum of the laser line and the resonance mode in cavity, respectively; Pout is the laser output power; and ħ is the Plank constant. This relation shows that laser spectral width above the lasing threshold is always narrower than the spectral width of the cavity mode and additionally depends on gain saturation. Thus, the comparison of laser quality factors (Qlaser) for various perovskite lasers measured from PL spectral above the lasing threshold should take into account properties of cavity and material properties.
Discussing the most optimal shapes of single-particle perovskite lasers, it should be stressed that each of them has some advantages and disadvantages. A spherical cavity has higher quality factor and vanishing contact area with substrate resulting in the reduction of mode leakage. On the contrary, small spherical particle is polycrystalline; i.e. it has high crystal lattice defect concentration. In turn, nanocubes, nanowires, and nanoplates have low defect concentration because these objects correspond to regular shapes, which are natural for single crystals. However, their disadvantage is higher mode leakage to substrate and additional scattering on the sharp edges. Cylindrical particles require lithographic methods of fabrication, which usually generate additional defects in perovskite material owing to overheating, amorphization, or spoiling by etching gas/liquid.
Besides the leakage losses, strong impact on Qlaser has the scattering ones originating from low volume integrity of the cavity and its surface roughness. In particular, it was shown that single-crystal MAPbBr3 MDs having sidewall roughness less than 5 nm support lasing with Qlaser up to 6000 , whereas polycrystalline MAPbI3 cavities with more rough sidewall (>8 nm) fabricated by chemical etching through a mask yield lasing with Qlaser up to 710 . On the other hand, recently, it was demonstrated how the roughness can be exploited to support a single mode lasing in MAPbX3 (X=I, Br0.33I0.66, Br) MDs printed by means of laser ablation technique . When the roughness is close to 50 nm, the WGM cavity with diameter from 2 to 5 μm enters into chaotic operation regime where predominantly only one lasing mode with Qlaser≈5000 survives.
Among the numerous results on lasing in perovskite single-particle structures presented in literature since 2014, we would like to point out the outstanding ones in our opinion: (i) the smallest NW supporting lasing at F-P resonances with Qlaser≈6000 at temperature 4 K (Figure 4A) ; (ii) the smallest perovskite laser ever – Mie-resonant nanocube with ≈420-nm edge size and showing Qlaser=670 (Figure 4B) ; (iii) spherical WGM microcavity demonstrating the highest Qlaser ever – 15,000 (Figure 4C) among perovskite lasers .
As many factors affect the lasing properties of nanocavities and microcavities, some figure of merit is required to compare them to each other. As a reasonable criterion, we propose a dependence of Qlaser on volume (V) of a single-particle perovskite laser (Figure 4D), resembling the calculated dependence of Qcavity on cavity size. Indeed, the results in Figure 3B help to predict a general trend for the experimentally observed Qlaser(V) values. In particular, it seems to be challenging to achieve Qlaser>10,000 for the volumes V<1 μm3, because high-quality laser has to compromise on compact dimensions.
Achievement of high Qlaser from single-particle resonators makes it possible to design optical sensors capable of detection of various analytes at ultralow concentrations. Usually, the higher the Q factor, the higher the sensitivity of a laser to changes of any external parameters . Another important application is photonic integrated circuit consisting of high-quality ultracompact nanolasers integrated with complementary metal-oxide-semiconductor technology compatible waveguides. In particular, it can be applied for a high-speed multichannel coherent transmitter  operating at extremely low spacing between channels. In this regard, perovskites are promising materials because they are tunable, cheap, and easily compatible with silicon-based materials and designs. Also, the lasers with higher Qcavity factors operate at lower pump fluence, which should help to substantially increase their long-term stability.
5 Conclusion and outlook
To summarize, the application of the halide perovskites has evolved from a simple synthesis optimization (XIX century) to the fabrication of advanced nanophotonic designs allowing for lasing at the deeply subwavelength scale (XXI century). By now, current understanding of nanocavity and material properties, as well as available advanced methods of nanofabrication of perovskite-based designs, allowed to realize the compact and efficient subwavelength lasers. For example, recent experimental results demonstrated that Mie-resonant nanocubes CsPbBr3 nanoparticles with high crystalline quality are the smallest designs among those reported so far. Moreover, these particles have sizes comparable to or smaller than the cavities based on GaN , , InGaP , and CdSe . Also, nanolasers made of halide perovskites with high multiphoton photoexcitation efficiency would allow working in deeply subwavelength regime relative to the pumping wavelengths, being useful for various up-conversion applications .
Nevertheless, there are many challenges to solve. First, electrically driven perovskite nanolasers are still not realized, owing to low thermal conductivity (<10 W/m·K)  of the halide perovskites resulting in their overheating at high current densities. This problem requires further lowering of lasing threshold via optimization of the material properties (e.g. defects concentration) and maximizing Q factor of the nanocavity. Importantly, both improvement of thermal sink and Q factor can be achieved via proper substrate engineering, which is definitely a topic of further studies. The second problem is the control of wavelength of stimulated emission from perovskite nanolasers. In principle, anion exchange reaction allowed to demonstrate a broad range of colors from blue to near-IR range. However, stability and QY of the mixed-anion nanolasers always get worse. To solve this problem, smart postprocessing is required for defect healing and phase stabilization. The third problem is the integration of the nanolasers with advanced optical elements for further waveguiding and signal processing. In this way, nanoantenna-like effects related to interference of resonances in subwavelength cavities could improve out-coupling efficiency and directivity for the emitted light.
We are thankful to Kirill Koshelev, Prof. Sergei Kulinich, and Prof. Yuri Kivshar for the useful discussions. This work was partially supported by Russian Science Foundation (17-73-20336). E.Y.T. acknowledges the Russian Science Foundation (19-19-00177, Funder Id: http://dx.doi.org/10.13039/501100006769) supporting numerical simulations and the Russian Federation President Scholarship for Young Scientists.
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