One (photon), two(-dimensional crystals), a lot (of potential): a quick snapshot of a rapidly evolving field

We present a concise overview of the state of affairs in the development of single-photon sources based on two-dimensional (2D) crystals, focusing in particular on transition-metal dichalcogenides and hexagonal boron nitride. We briefly discuss the current level of advancement (i) in our understanding of the microscopic origin of the quantum emitters (QEs) identified in these two material systems, and (ii) in the characterisation of the optical properties of these emitters; then, we survey the main methods developed to enable the dynamic control of the QEs’ emission energy. Finally, we summarise the main results stemming from the coupling of QEs embedded in 2D materials with photonic and plasmonic structures.


Introduction
Bringing quantum technologies from the lab to our everyday lives will require the engineering and implementation of devices that can meet the rigorous criteria set by mass applications.Of course, the availability of non-classical light sources emitting indistinguishable photons on demand will be of the utmost importance, together with the achievement of a high source efficiency and single-photon purity.As quantum devices inch closer to making the final leap towards the consumer market, however, a few, initially overlooked requirements are coming to the forefront: indeed, commercially viable single photon sources should be produced using manufacturing methods that are cheap, reliable, reproducible, and, preferably, compatible with current photonic integration technologies.
Epitaxial quantum dots (QDs) based on III-V materials are widely regarded as a very promising system within this field: due to the remarkable progresses made in the last decade [1][2][3], indeed, they feature nearly deterministic photon emission, with high efficiency and near-unity indistinguishability. Consequently, III-V QDs are, up to now, the only system that has been exploited commercially (see, e.g.http://quandela.com/and https://sparrowquantum.com/).On the downside, self-assembled QDs notoriously suffer from stochastic site formation.This problem can only be partially mitigated using sophisticated approaches which, however, yield a lower optical quality of the QDs.Among these approaches, we would like to mention the use of lithographically patterned substrates-which leads to the formation of site-controlled pyramidal QDs by metallorganic vapour epitaxy [4] or the fabrication of site-controlled QDs through the exploitation of the effects of spatially controlled H irradiation [5] or removal [6] in dilute nitride semiconductors.In addition, established methods to enhance the QD emission rate and extraction efficiency rely on the spatial and spectral coupling to engineered photonic structures.While single devices fabricated with this method can feature ∼99% indistinguishability and extraction efficiencies up to ∼70% [7], all current approaches to fabricate such sources demand for manually selecting the QD emitters and/or the full devices [8][9][10], hindering scalability and dramatically increasing production time and costs.The combination of manual selection with the stochastic nature of the QD growth processes makes it a superior challenge to produce large numbers of identical sources of single (and, ideally, indistinguishable) photons, as required for quantum information science and technology.Moreover, the reliance of today's commercially available single-photon sources on III-V materials-and on InGaAs, in particular-is inextricably and inherently entangled to the high fabrication costs typical for this platform.Indeed, the price for InGaAs-based devices commonly exceeds various tens of thousands of euros/dollars per single photon source, due both to the high costs of epitaxial growth techniques-crucial for achieving state-of-the-art material quality-and to the low supply potential of gallium and indium [11].The latter point also raises questions about the long-term sustainability of a technological ecosystem entirely based on III-V materials.
Within this context, the quantum emitters (QEs) embedded in two-dimensional (2D) materials, e.g. in transition-metal dichalcogenides (TMDCs) and in hexagonal boron nitride (hBN), have the potential to successfully challenge the status quo.Both TMDCs and hBN belong to the vast family of layered semiconductors, wherein stacks of individual crystal layers-or monolayers (MLs)-characterised by strong in-plane covalent bonds, are held together, in the vertical direction, by much weaker [12] van der Waals forces.As it is well known, research interest around these materials skyrocketed following the successful isolation of atomically thin crystals of graphene, by simple mechanical exfoliation of graphite [13].The application of the same method to a wide variety of materials, including TMDCs and hBN but also transition-metal chalcogenides [14] and layered metal-halide perovskites [15], has led to nearly two decades of uninterrupted scientific discoveries, which would be too long to even summarise here.What is most important, within the context of this work, is the fact that thin layers of TMDCs and hBN consistently display the ability to host bright single-photon sources [16][17][18][19][20][21][22], ranging in energy from the telecom window (MoTe 2 , see [23]) to the visible range (hBN [20]).The costs of these sources are surprisingly modest, as ∼500 euros are currently sufficient to purchase a bulk specimen that can be used to fabricate hundreds of thin-layer flakes that, in turn, can each host hundreds of QEs.Costs aside, these thin flakes, when compared to 3D bulk crystals hosting QEs, feature a lack of internal reflection, leading to an increased light extraction efficiency, while their all-surface nature allows for the external manipulation of the emitters' properties.In addition, 2D MLs can be easily transferred on any substrate-including silicon-based photonic structures-thus ensuring significant advantages in terms of optical losses, manufacturing costs, and compatibility with CMOS electronics [24].
Moreover, 2D materials are particularly well-suited for the implementation of advanced strain engineering protocols.The introduction of controlled amounts of strain in the material indeed represents a well-established method for tuning the electronic properties of semiconductors.In conventional semiconductors, however, the amount of strain that can be introduced is traditionally fixed by the fabrication process [25].While there has recently been significant progress in the development of methods to apply reconfigurable stresses to these materials, the maximal amount of strain that can be introduced in conventional semiconductors is typically capped at ∼1% by their elastic limit.These limitations can be greatly exceeded in the case of 2D materials, wherein sizeable, variable stresses can be easily applied.The reduced dimensionality in one direction-due to the weak van der Waals forces, by which the isolation of atomically thin, dangling bonds-free layers is made possible-allows for a greater deformation of the crystal.This feature, together with the reduced risk of fractures by virtue of the strong in-plane covalent bonds, leads to the possibility of introducing very large isotropic or anisotropic strains (total strains up to 10%-20%), thus tuning the electronic properties of these materials controllably over a very broad range [26][27][28][29][30][31][32].As discussed in the following section, strain plays also an important role for the observation of the single-photon sources hosted by TMDCs [16,18], which can be purposefully induced by localised, site-controlled stressors [33].
In the following two sections, the main features of TMDC-and hBN-based single-photon sources will be described; then, we will discuss the current state of the art with respect to the integration of these sources with photonic structures.Finally, we will conclude, providing our vision of the main breakthroughs to be expected-and of the most likely upcoming challenges to be faced-in this bustling research field.

TMDCs
TMDCs are layered materials whose single layer is comprised of a three-layer substructure MX 2 , wherein a hexagonal lattice of transition metal atoms (M) is sandwiched between two other hexagonal lattice structures of chalcogen atoms (X).Semiconducting TMDCs, e.g.WS 2 , WSe 2 , MoS 2 , MoSe 2 , and MoTe 2 , display a peculiar thickness-dependent indirect-to-direct bandgap transition [34][35][36].Indeed, TMDCs in their bulk form feature an indirect bandgap and are, therefore, poor light emitters, since the radiative recombination of the electrons promoted in the conduction band with the holes left behind in the valence band requires the contribution of a phonon to conserve momentum.In TMDC MLs, however, the conduction-and valence-band edges can both be found at the K points of the Brillouin zone, thus making the transition direct and dramatically boosting the light-emission efficiency of these materials.The photoluminescence (PL) spectra of TMDC MLs are characterised by the presence of bright, prominent peaks related to excitonic complexes [37].The reduced dimensionality of TMDC MLs leads to an increase in the exciton binding energy [38], making these quasiparticles robust to thermal excitations, and emitting even at room temperature (RT).
Moreover, the all-surface nature of TMDC MLs makes them extremely sensitive to the surrounding environment and, in particular, to crystal lattice deformations (strain).Strain has, thus, been extensively studied as a tuning knob to gain control over the optoelectronic properties of TMDCs [30], as demonstrated by the considerable variations of the emission energy of excitons in deformed MLs.The strain-induced energy modulation is so large, indeed, that for higher strain values (total strain of about 4%-5%) TMDC MLs undergo a remarkable transition back to an indirect bandgap [39].The poor radiative efficiency associated to an indirect bandgap is, in principle, not seen favourably; however, for intermediate strain values (around 5%), the PL peaks assigned to indirect excitons feature an intensity reduced by just a factor of about 10 [39].This relatively modest reduction could be explained by taking into account the effect of the strain-induced hybridisation of direct and indirect excitons in TMDCs [31].In light of this, strain can be exploited to tune on-demand the recombination energy of TMDCs without remarkable signal losses.
The appearance of lines associated to single-photon emitters (SPEs) in the PL spectra of WSe 2 MLs brought to cryogenic temperatures was reported in 2015 by four different groups [16][17][18][19].The purity of these QEs was assessed via second-order autocorrelation measurements employing Hanbury Brown and Twiss setups [40], with the lines appearing to be spatially localised, spectrally narrow (∼100 µeV), at an energy consistently lower than that of the free-exciton transition, linearly polarised, with lifetimes of 1-10 ns, and featuring Zeeman splittings from which large, very spread-out values of g-factor were extracted.The intimate link between the appearance of these emitters and strain appeared evident from the very beginning, since the SPEs were shown to form preferentially at the flakes' edges [41] or at the border of patterned micrometric holes etched in a silicon substrate [42].In the following years, emissions from localised excitons were verified also for WS 2 [43,44], MoSe 2 [45][46][47], MoS 2 [48], for the WSSe alloy [49], and for MoTe 2 in the telecom wavelength range [23].
In the case of WSe 2 in particular, the necessity of a strained configuration for the appearance of the emitters has led to a variety of strategies employing etched substrates featuring micro-and nanostructures acting as stressors, on which the TMDC ML is precisely deposited.To name a few, WSe 2 MLs have been deposited on the nanogap between two single-crystalline gold nanorods [50], on top of SiO 2 nanopillars (figures 1(a) and (b)) [33,51], on platinum nanoparticles-to obtain a high density of emitters [52]-on silver nanoparticles, to generate emitters used in a quantum key distribution experiment to emulate the BB84 protocol using single photons [53], and on several other structures, aiming at enhancing the emission via the Purcell effect (this will be addressed in greater detail in section 4).Strain-localised emitters can also be generated by the surface roughness of glass [54], or by the indentation of an ML, on top of a polymer substrate, via an atomic force microscope (AFM) tip, allowing for the deterministic placement of emitters with nanometric precision [55].Strain can also be used to gain control over the SPEs' polarisation: bright, linearly polarised emitters can indeed be created by placing a WSe 2 ML on the nanogap of a dielectric rod structure, with the polarisation axis abruptly changing depending on the size of the gap [56].Another system saw the use of arrays of dielectric square pillars as stressors for a WSe 2 ML, so that at the very edge of the pillar a quasi-1D strain-induced confinement potential was formed, leading to highly linearly polarised (95%) emitters [57].
The origin of these emitters, especially for what concerns the W-based compounds, is still debated, with one of the main hypotheses being the brightening of dark excitons, via the strain-induced hybridisation with defect states [58].Indeed, due to large spin-orbit coupling, the electronic bands at the K points of the Brillouin zone lift their spin degeneracy, with the valence band experiencing a larger spin splitting than the conduction band [59].The spins of the so-formed bands alternate with different orders, depending on which of the two unequal K points is being considered [60].In WSe 2 , it has been proved that the bottom conduction sub-band and the top valence sub-band have different spins, making the radiative transition of the actual ground-state spin-forbidden, hence the name dark exciton [61].The line associated with the bright exciton (that is, the one that is visible in PL) comes from the transition involving the top conduction sub-band and the top valence sub-band, which have the same spin.Tensile strain leads to the lowering of the conduction band, and to a decrease of the bandgap.For certain values of strain (1%-3%) the lower conduction sub-band could reach the energy of a discrete state-caused by a lattice defect-lying inside the bandgap.The hybridisation of the dark exciton with this defect state would relax the selection rules for radiative recombination, leading to the bright emission reported for TMDC SPEs [58].Although the O-substituted chalcogen vacancy O Se has been proposed as the defect being responsible for this phenomenon [62], the large spread in the values of the g-factor (from −2.02 to −12) extracted from magneto-PL measurements is probably more consistent with the presence of several defect families, all capable of creating QEs [63].
The identification of the point defects responsible for the experimentally reported emissions, together with predictions based on the rational engineering of their properties, is the main effort of ab initio computational studies in the field of 2D QEs [64].For all the practical advantages provided by the low dimensionality of these crystals, equal challenges arise in the computation of their properties, with the standard density functional theory approach in the mean-field approximation being unable to accurately capture the physical phenomena related to point defects in a 2D host material.To overcome these challenges, associated, in particular, to the anisotropic dielectric screening and strong many-body interactions characterising 2D materials, complex computational approaches are often employed, such as many-body perturbation theory in the GW approximation and the resolution of the Bethe-Salpeter equation for two-particle absorption spectra [64].To evaluate the formation energy of defects, these are placed in one unit cell at the centre of a larger ensemble of pristine crystal cells, called supercell.In the case of charged defects, the interaction between them and their periodic images leads to large errors, which need to be addressed with a corrective term in the evaluation of the total energy.Such a correction can be obtained by: (i) supercell extrapolation of the analytical expressions relating the supercell size to the electrostatic energies of point charges [65]; (ii) by approximating it to the difference in model charge electrostatic self-energy between isolated and periodic boundary conditions [66][67][68]; or (iii) by employing self-consistent potential corrections, which prevent the appearance of artificial states in vacuum [69,70].For the calculation of the ionisation energy of these defects, it is necessary to take into account electron correlation, which, in the presence of anisotropic screening, requires computationally heavy methods such as GW.To reduce the computational cost of these kinds of evaluations, however, methods based on density functional perturbation theory were successfully employed, in particular for charged defects in hBN [67,68].The promising role of ab initio calculations for the design of point defects was proven in a work describing in detail the steps necessary for the creation of states meeting the criteria necessary for their use in quantum technologies [71].In this work, vacancies are identified as good candidates for single-photon emission in several 2D materials.This conclusion is reached through the analysis of the electronic levels of these defects, evaluated with point symmetry group considerations.In particular, the Jahn-Teller distortions caused by the addition of an electron to the defect are found to originate a two-level system.Ideally, this two-level system needs to be isolated from the band edges of the host crystal, a condition that can be obtained by introducing a paramagnetic impurity next to the vacancy.Gupta et al [71] focuses in particular on the Re Mo V S defect complex in MoS 2 , which is found to feature transitions compatible with the optical telecommunication range.
Coming back to the optical activation of the defects, the hypothesis of the hybridisation with a dark state is compatible with the reported difference in energy between WSe 2 strain-localised SPEs and free excitons, since its lower limit of 42 meV is consistent with the splitting between bright and dark excitons [72].The interplay between strain and defects was also demonstrated in [73], by decoupling these two factors through the fabrication process illustrated in figures 1(c) and (d).Such process involved the deposition of a WSe 2 ML, sandwiched in hBN, on top of silicon nanopillars, followed by its irradiation with an electron beam at 100 keV.The strained, high-quality WSe 2 ML, shielded from substrate effects, showed only a broad defect band prior to irradiation, while the deterministic creation of defects, via the electron beam, led to the appearance of bright lines associated to QEs, which maintained their single-photon nature up to 150 K [73].Similar results were obtained for a WSe 2 ML deposited on CuO nanoparticles, which showed bright narrow lines in its PL spectrum only after the creation of defects via electron-beam irradiation [74].All these findings substantiate the hypothesis of a strain-induced brightening of dark excitons, making it the most convincing picture among those proposed in the literature.
Indeed, other hypotheses on the formation of QEs in TMDCs refer to the possibility of creating bound states via sharp strain gradients [75], or, in the case of softer strain profiles over larger areas, to the funnelling effect [76], i.e. the drifting of excitons towards points of higher strain (and lower energy gap) and their subsequent trapping by lattice defects.These scenarios, however, need to be reconciled with conflicting experimental results.Indeed, the presence of emitters in nanobubbles formed during the deposition process of WSe 2 MLs due to the trapping of contaminants [77] seems to be in agreement with the identification-via near-field PL measurements performed at RT [78]-of localised, deeply bound excitons at the edge of WSe 2 nanobubbles, and with theoretical calculations suggesting the presence of points of high strain at the nanobubble periphery, due to atomic-scale wrinkling [79].Moving to the micrometric scale, however, SPEs were also consistently observed on the edges of the hydrogen-filled bubbles that can be formed, in a controlled manner, on the surface of bulk TMDC crystals irradiated with low energy (∼20 eV) hydrogen ions (see figures 1(e) and (f)) [44].This is partially at odds with the idea that a sharp, nm-scale strain gradient is required for the creation of QEs, given that microbubbles feature a gradual increase of the strain tensor moving from the bubble's edge towards its apex, where the highest strain is reached and where excitons would be expected to funnel [39].This peculiar spatial dependence of the emission suggests a more complex relationship between the strain values obtained with different methods and the appearance of QEs in W-based compounds.
The origin of the QEs in MoS 2 , on the other hand, seems to be mainly connected to the presence of chalcogen-vacancy defects [80], as demonstrated by the deterministic, site-selective creation of emitters via a helium ion beam (a process shown in figure 1(g)) [48], even after encapsulation of the ML in hBN [81,82].Vacancy-related defects in MoS 2 , MoSe 2 , WS 2 and WSe 2 can also be created via proton beam irradiation at an energy of 50 keV [83].Noticeably, proton irradiation can lead to quite opposite effects depending on the beam energy.Low-energy (tens of eV) proton beams were clearly shown not to induce defects in 2D materials [84]; in semiconductor nanostructures, such as self-assembled QDs, proton beams with energies of around a hundred eV were even used to passivate defects and improve their emission efficiency [85]; on the contrary, high-energy (tens of keV) proton-beams can be purposely exploited to create defects acting as QEs in 2D crystal matrices, as shown for different TMDCs [83].Finally, vacancies in MoS 2 were created also by UV light irradiation [86].Interestingly, no QEs appear when the UV irradiation takes place in air, since the sulphur vacancies are passivated by oxygen during the formation process, yet the emitters formed in vacuum can be later exposed to air with no repercussions [86].The site-selective generation of emitters in MoS 2 , via a focused helium ion beam, can also be performed after the assembly of a gate-tuneable device for the application of an electric field, showing the emitters' sensitivity to the charge carrier concentration and how they can be turned on and off depending on the applied voltage [87].
The reported emission of the biexciton-exciton cascade in WSe 2 [88] also makes the latter a promising material for quantum technologies based on photon entanglement, even though no entangled photon pairs have been reported yet for SPEs based on 2D materials.Photon entanglement is, indeed, at the basis of several areas in the field of quantum technologies, such as quantum metrology and sensing [89][90][91], quantum communications [92][93][94] and quantum computing [95,96].Unfortunately, the emission lines involved in the cascade are actually doublets characterised by a zero-field fine-structure splitting (FSS) of ∼700 µeV [16,17,19,42], which hinders the degree of entanglement of the emitted photons.Since the FSS is thought to be a consequence of the electron-hole exchange interaction in the presence of an anisotropic potential, the possibility of strain-tuning the emission of these states-using, for example, a piezoelectric substrate [97]-provides us with tools that, in the presence of entangled photons, will enable the reduction of their FSS and the maximisation of their degree of entanglement.A reduction of the FSS by 11% was successfully achieved by employing a device in which the ML was strained by nanopyramid arrays placed on a voltage-controlled cantilever, achieving simultaneous control over the emitters' location, emission energy and splitting [98].A reduction of the FSS was also obtained through the application of an out-of-plane electric field with a gated device [99].More in general, electrical manipulation of TMDC SPEs also led to phenomena such as Coulomb blockade, i.e. the loading of single electrons or holes, one by one, into a QE [100]; to the creation of localised trions, featuring a reduced FSS when compared to the neutral exciton species [101]; or to the simultaneous electrical pumping and energy tuning of SPEs via a lateral p-n junction, interfaced with a WSe 2 ML placed on top of a dielectric nanopillar [102].Moreover, a hybrid electro-optical approach has been recently demonstrated by the appearance of QEs in a WSe 2 ML deposited on a substrate comprised of GaN-based µ-LEDs, operating at low temperatures [103].
Finally, the mechanical manipulation of exfoliated flakes allows for the possibility of stacking layers from different crystals in a predetermined order, easily assembling vertical heterostructures (HSs), which host new and interesting phenomena [104].One of these phenomena is the formation of interlayer excitons (IXs), i.e. excitons in which the electron and the hole are localised in different layers of the HS.IXs are characterised by PL peaks at energies lower than those of the starting materials [105][106][107][108][109], by sublinear power response of their PL intensity, and by longer lifetimes, due to the spatial separation of the two wavefunctions involved in the recombination process [106].The emission intensity from IXs is heavily dependent on the relative alignment of the two layers' crystal axes (twist angle), with maxima near twist angle values of 0 • and 60 • , for which the appropriate band alignment in k-space is achieved [110,111].
The spatial localisation of free IXs can be achieved via strain confinement, as it was the case for a WS 2 /WSe 2 heterobilayer deposited on a pillar array [112], by the application of an electric field [113,114], or by the formation of a moiré superlattice potential landscape [115][116][117].A moiré potential is the periodic modulation of the electronic band structure due to a lattice mismatch and/or twist angle between the two crystals comprising the HS.This potential is characterised by the presence of points of high symmetry acting as traps for the free IX; the now discrete energy levels can give rise to single-photon emission, as reported for a MoSe 2 /WSe 2 heterobilayer, with the moiré-trapped QEs featuring circular polarisation, Zeeman splitting upon application of a magnetic field, energy-tuning via the Stark effect, and long lifetimes (∼12 ns) [117].

hBN
hBN is a layered crystal whose ML is comprised of a single sheet of alternating boron (B) and nitrogen (N) atoms, arranged in a honeycomb structure akin to that of graphene.Its wide bandgap (∼6 eV) makes it completely transparent in the visible range [118].Given its chemical inertness and thermal stability [119,120], hBN is often used as an encapsulating material for other 2D crystals, to improve their electronic and optical properties [121][122][123][124].While the bottom hBN flake provides a smooth surface, free from dangling bonds and charge traps, the top few-layer hBN creates a uniform dielectric environment for the encapsulated crystal [125], protecting the material below from external degradation factors such as oxidation [126] and leading to the narrowing of its PL peaks when compared to an air-exposed sample deposited on a typical SiO 2 substrate [122].
The wide bandgap of hBN is also at the basis of the appearance of colour centres displaying single-photon emission up to RT.While the usage of low-temperature SPEs may still be envisaged for high-end applications-such as quantum cryptography and quantum metrology-requiring the highest brightness and degree of indistinguishability [127], the development of high-quality SPEs working at RT would represent a major breakthrough, allowing for the realisation of scalable, stand-alone photonic circuits with the ability to operate without the aid of external, bulky, and expensive cryogenic systems (either closed-cycle or based on cryogenic liquids).The potential of hBN-based QEs in the field of quantum technologies has been gaining the attention of an increasing number of researchers and is already showing great promise, as confirmed by the use of hBN SPEs for delayed choice experiments [128] and quantum random generation [129], or by the indistinguishability value of 0.56 (corrected by taking into account the non-ideal purity of the single photons), recently measured by Hong-Ou-Mandel interferometry for an hBN SPE [130].While QEs in W-based TMDCs are likely linked to the presence of dark excitons hybridised with defect states through strain (see previous section), colour centres in hBN are discrete states lying in the middle of the bandgap formed by crystal vacancies or impurities.These 'artificial atoms' embedded in a crystalline matrix are therefore optically active even at RT, with their emission line just broadening due to the increasing phonon population [131].QEs in hBN have been found in its ML form [20], as well as in bulk crystals [21,22], with their zero-phonon line (ZPL) spanning the whole visible spectrum [131,132] (as can be seen in the histogram of figure 2(a)) up to the UV region [133], and still maintaining single-photon purity up to 800 K [134], as confirmed by the second-order autocorrelation measurements reported in figure 2(b).Even though emitters can also be found in as-grown crystals [21,135], several strategies have been employed to increase their density and/or efficiency in a controlled manner.
Annealing at high temperatures in argon has been found to be the most reliable procedure to optically activate QEs in hBN [20,22], be they naturally occurring or induced.Emitters related to lattice defects can be successfully obtained by chemical etching [136], ion irradiation [137][138][139], electron beam irradiation [132,138], laser ablation [138], plasma treatment [140][141][142], even reaching submicrometre precision in the positioning of the colour centres by employing the electron beam of a scanning electron microscope [143] (figure 2(c)).High energy (MeV) electron irradiation has been shown to lead to the formation of stable SPEs, with no need for annealing in Ar atmosphere, in a variety of exfoliated multilayers (high purity, B enriched, C enriched) as well as chemical vapour deposition (CVD)-grown MLs [144].On the other hand, laser irradiation appears to yield better results in multilayer flakes, with purer emitters displaying brighter and sharper lines when compared to those observed in large MLs, which are affected by a hexagonal-to-cubic lattice structure transition [145].The effects of different types of irradiations can be unveiled by monitoring the emission from the colour centres on samples undergoing layer-by-layer etching with O 2 plasma: by keeping track of the number of layers that need to be etched away before the disappearance of the emitter, it has been found that the centres created by other plasma treatments tend to form on the crystal surface, while electron irradiation leads to emitters located throughout the whole crystal [146].By performing large area PL maps, the different roles played by low-energy electron irradiation and annealing were clearly identified, with the former being responsible for the creation of new emitters and the latter for the brightening of those already present [147].
Strain can be used as a tuning knob for the emission-as it is the case for flakes on bendable polycarbonate beams, where the emitters were formed by He ion irradiation [148]-but it also seems to play a role in the creation of the defects themselves, as demonstrated by the similar spectral features of QEs in  [190], with permission from Springer Nature.untreated BN nanotubes and hBN flakes deposited on diamond nanopillars [149], by the activation yield of 10% of single QEs in flakes deposited on silica nanopillars [150] jumping to 80% when flakes are CVD-grown directly on pillars of 250 nm of diameter [151] (schematic representation in figure 2(d)), or by the appearance of emitters by nanoindentation with an AFM tip [152].A 31% yield of single QEs was also achieved by the deterministically controlled creation of edges via the patterned milling of hBN using a gallium focused beam [153].The link between strained configurations, such as those arising from the wrinkling of the material, and the polarisation axis of the emitters appears evident in samples brought to low temperature (10 K) in which the QEs' polarisation aligns with the wrinkle direction [154].
Given the very broad range of emission of QEs in hBN, several different strategies to gain control over their properties have been developed.An optimised CVD growth technique leads to the formation of multilayer hBN displaying the majority of the emitters at a wavelength of the ZPL of 580 ± 10 nm, greatly reducing the emitters' variability in energy and suggesting that this particular wavelength range is associated to a specific crystal defect related to the growth process [155].Even CVD-grown MLs, featuring large homogeneous domains, display emitters at a wavelength of 575 ± 15 nm, with no need for ion irradiation or annealing in argon [156].The presence of carbon during other growth techniques (metallorganic vapour phase epitaxy and molecular beam epitaxy) appears to be crucial for the appearance of SPEs (as shown in figure 3(a)), with the V B C − N defect as a possible candidate [157], confirming several theoretical studies on the role of carbon [158][159][160][161][162][163][164][165][166][167], which also attributed the UV emission at 4.1 eV to the carbon dimer defect C B C N [168].Control over the emitters wavelength has been partially achieved (i) via the Stark effect, tuning the emission with an out-of-plane electric field applied by graphene gates [169]; (ii) with a four-electrode device controlling both the amplitude and the direction of the electric field, leading to a Stark shift (visible in figure 3(b)) four times larger than the full width at half maximum (FWHM) at RT [170]; or (iii) by placing the emitters between an indium tin oxide coated glass slide and a conductive AFM tip [171].Substrate engineering has also been explored as a possible approach to gain control over the emitters' properties.Even though emitters in suspended thin films, which do not suffer from any substrate-induced effect, still show a large variability in their spectral, temporal and spatial characteristics [172], surface passivation of a SiO 2 substrate with Al 2 O 3 leads to the appearance of QEs with ultranarrow ZPLs (45 µeV), almost no sidebands and a g-factor of just 0.2, making them nonmagnetic [173].A shift of the ZPL wavelength from 600-650 nm to 550-600 nm takes place when nickel is used, instead of quartz, as a supporting substrate for multilayer hBN grown by CVD on copper [174].Controlled localisation of the emitters, on the other hand, is achieved by patterning openings in a graphene layer put on top of three-layer hBN [175].This exploits the graphene-induced suppression of emitters, which takes place via a double transfer mechanism: an energy transfer is responsible for the intensity reduction of all ZPLs, while a charge transfer is responsible for the complete quenching of all the ZPLs beyond 600 nm [176].
The substrate on which hBN flakes are deposited appears to be important also for what concerns the broadening of the emitters' linewidth.Even at cryogenic temperatures, linewidths of 27 µeV have been interpreted as a consequence of ultrafast spectral diffusion [177], i.e. the variation of the electric field around the emitters, leading to small changes in the emission wavelength and, thus, the broadening of the peak.This phenomenon, in the case of hBN, is thought to be linked to two dephasing mechanisms, which according to [178] occur at two different timescales (∼20 µs and ∼500 µs).Similar linewidths were also reported in the case of resonant excitation [179], while reduced linewidths were obtained by performing photoluminescence excitation (PLE) measurements on resonance with the ZPL, while monitoring the intensity of the phonon sideband, at low power excitation [180], by using conductive indium tin oxide as a substrate instead of SiO 2 [181] or by low-power resonant excitation [182].The goal of Fourier transform (FT)-limited linewidths of ∼0.21 µeV was initially achieved at low temperatures by resonant excitation of flakes deposited on a silver mirror and annealed further to eliminate chemical residues [183], or by two-laser resonant repumping [184].FT-limited lines were reported also at RT, suggesting the presence of spectral diffusion but the complete lack of phonon dephasing [185,186].This was interpreted as the consequence of the decoupling of the defect with low-frequency in-plane acoustic phonons, due to an out-of-plane lattice distortion [185,186].
Even though QEs in hBN generally show non-magnetic behaviour [187], the magnetic field-dependent variation of the steady-state, RT PL observed for a small fraction of QEs has been interpreted as a consequence of the presence of optically addressable spin defects [188].This kind of defects could play a major role not only in the field of quantum sensing, given their extreme sensitivity to local magnetic fields and temperature, but also in quantum computing and communication, exploiting the optical initialisation and readout of spin quantum memory [189].An ensemble measurement performed at RT on a PL line attributed to the V − B defect (whose lattice structure is shown in figure 3(c)) featured a triplet groundstate with a zero-field splitting of 14 µeV and optically detected magnetic resonance (ODMR) (whose dependence on the applied magnetic field is reported in figure 3(d)) [190].Going to low temperatures the V − B excited state also showed ODMR, with the state being confirmed as a triplet featuring a longitudinal splitting of 8.93 µeV and a g-factor of 2 [191].The photodarkening of the emitters at 3.5 K, going from an out-of-plane magnetic field of 0 T to 7 T is consistent with a picture in which the spin-dependent non-radiative intersystem crossing transitions happen from the triplet excited state to the lowest-lying spin-singlet metastable, and then from the metastable state to the triplet ground state [192].Carbon-related defect ensembles feature ODMR at RT as well [157,193], which is correlated to a second bunching timescale in the second-order autocorrelation function g 2 (τ ) [193], while a still unidentified paramagnetic defect, which is known to be neither V − B nor carbon related, also features ODMR and has a g-factor of 2 [137].

Integration with plasmonic and photonic structures
Being embedded in a 2D matrix, QEs in TMDCs and hBN are particularly well-suited for the integration in photonic and plasmonic cavities, to which they can be deterministically coupled via simple dry transfer methods.These cavities aim at improving the emitters' performances in terms of brightness and photon collection via the Purcell effect [194].The Purcell effect is the modification of an emitter's radiative rate by controlling its environment, with the two major strategies employed by research groups being the use of plasmonic cavities, which exploit the interaction with plasmons forming at the surface of metallic nanostructures, and photonic cavities, in which the electromagnetic field is confined and amplified.Since 2D crystals lack internal reflection, they can be efficiently interfaced with these cavities by simply depositing the ML on top of the desired structure; moreover, taking into consideration the need for a strained configuration for the activation of the QEs (as far as W-based TMDCs are concerned), it is only natural that the plasmonic/photonic structures, designed for the enhancement of the radiative recombination rate of the emitters, serve also the purpose of acting as stressors for the deposited MLs.As far as hBN is concerned, one basic difference between this system and TMDCs gives rise to an ulterior advantage in using hBN QEs: while emitters in TMDCs are of excitonic nature and are localised by strain-induced potential minima and/or crystal defects, SPEs in hBN are colour centres for which discrete states, caused by crystal defects, lie in the middle of hBN's wide bandgap of 6 eV.This fact decouples almost completely the properties of the QEs found in hBN from the evolution of the electronic bands of the host, which, conversely, plays a major role in TMDCs: in TMDCs, indeed, SPEs can be found almost exclusively in MLs, which are characterised by a direct bandgap in the K points of the Brillouin zone.For this reason, TMDC-based plasmonic and photonic cavities necessarily involve a TMDC ML interfaced with bulk elements made from other materials, i.e. a hybrid platform.hBN, on the other hand, is not constrained by limitations on the flake thickness, with bright emitters forming even in bulk crystals.Even though a thin crystal leads to a reduced total internal reflection and, thus, an increased extraction efficiency, one can deal with hBN flakes that are just thick enough for the creation of monolithic cavities, i.e. a platform in which both the emitters' host and the cavity elements are made with the same crystal, thus removing the photonic losses at the interfaces between different materials.

TMDCs
A plasmonic coupling-induced reduction of the lifetime of the emitters, leading to an enhanced brightness, was observed for WSe 2 MLs deposited on silver nanowires [195], silver-Al 2 /O 3 nanocones [196], gold nanorods [197], Al 2 O 3 /Au nanopillars [198], gold nanocubes coupled to gold mirrors, shown in figure 4(a), [199] (a type of cavity which can maintain single-photon emission up to 160 K, given the right growth conditions for the ML [200]) and gold nanostars [201].Despite their small mode volumes, plasmonic cavities are affected by losses that reduce their Purcell factor, i.e. the number quantifying the emission rate enhancement.To limit these losses, dielectric materials have also been explored as candidates for the creation of cavities.The desired enhancement was achieved for a WSe 2 ML deposited on top of a dielectric nanopillar at the centre of a circular Bragg grating ('bullseye') cavity (figure 4(b)), obtaining lifetime reductions of one order of magnitude [202].An increased quantum efficiency was also achieved for GaP nanoantennas [203], while silicon nitride dimer nanoantennas, with square cross-section, not only led to a reduction of the SPEs' lifetimes, but also provided control over the emitter polarisation by varying the dimer orientation [204].Optical cavities employing distributed Bragg reflectors also led to emission enhancements [205], which become even more drastic when an adjustable top mirror allows for the precise tuning of the Fabry-Perot cavity [206].
The creation of on-chip photonic integrated circuits would greatly improve the production, scalability, and compactness of devices exploiting QEs in TMDCs for quantum technologies, via the miniaturisation of optical quantum circuits.To this end, TMDC QEs need to couple to photonic waveguides, as it was demonstrated for gold [207], SiN [208] and Ti in-diffused lithium niobate waveguides [209].The waveguides not only provide the geometry to strain the MLs that are deposited on top of them, but, as shown for the horseshoe device in figure 4(c), they can also be used to resonantly excite the emitters by shining the laser on one of the waveguide's outputs; while the laser light remains confined in the waveguide, the single photons from the ML are collected with an optical objective on top of the crystal [210].Finally, microring resonators can couple to the chiral emission of TMDC MLs [211].Indeed, the two unequal K points of the Brillouin zone are characterised by emissions of light that is either right-circularly polarised or left-circularly polarised.In the presence of a magnetic field, these two polarisations experience a Zeeman splitting in energy, the QE's σ + peak can then couple to the σ + mode of the outer ring of the resonator, while the scatterers in the inner ring emit spin converted σ − light (figure 4(d) [211].

hBN
An hBN monolithic photonic crystal cavity featuring a high quality (Q-)factor, consisting of a 1D structure (ladder type), was formed via reactive ion etching and electron beam induced etching (EBIE).Reducing the cavity width via EBIE allowed for the tuning of the 1D modes after the identification and characterisation of the emitters [212] (figures 5(a) and (b)).An hBN monolithic bullseye cavity, on the other hand, not only showed an enhancement of the PL signal, but also better contrast in the ODMR measurements of spin defects [213], while a Purcell factor of 15 was achieved for a monolithic crystal cavity for which the coupling with the SPEs was tuned via gas condensation [214].
For what concerns the Purcell effect caused by plasmonic enhancement of the emitters intensity in hBN crystals, this was reported for flakes wet-transferred on periodic gold and silver nanoparticle arrays [215], for emitters near gold nanospheres which were precisely positioned via an AFM tip [216] and for silver nanopillars, which also acted as stressors [217].Plasmonic effects on the enhancement of the ODMR contrast of V − B defects were also reported for silver nanocubes [218] (figures 5(c)-(e)) and for gold film microwave waveguides [219].The precise ZPL wavelength of this spin defect, which is otherwise characterised by a broad emission, could be determined by tuning the coupling with a high-Q nanobeam cavity, while monitoring the resonance intensity, yielding a value of 773 ± 2 nm [139].
Single photons from hBN have been coupled to AlN waveguides (as demonstrated by the spatial separation between the points of excitation and collection) showing, however, a 1.35% efficiency, when compared to a confocal configuration [220].A monolithic approach can also be applied to waveguides, with the QEs placed in the middle of the structure and the collection taking place from the two side grating couplers [221].A striking 46% coupling efficiency was obtained with a waveguide coupled to a microring resonator made from Si 3 N 4 , with the coupling being deterministic, since the silicon nitride structure is grown after the identification of the emitter's position and of its dipole alignment [222] (a process that is shown in figures 5(f)-(h)).
Other optical cavities, which have been used to increase the radiative efficiency of QEs in hBN, and which are worth mentioning, include plane concave microcavities (leading also to narrower and purer SPEs) [223], CVD-grown hBN multilayer transferred to Si 3 N 4 , where the cavity is then designed with electron-beam lithography (EBL) [224], strain-activated emitters from hBN wrapped around Si 3 N 4 microdisc optical resonators [225] and few-layer CVD-grown hBN coupled to a hybrid open fibre-coupled Fabry-Perot cavity, allowing for strong linewidth reduction, broad spectral tuning range from 565 nm to 635 nm and enhancement of the ZPL up to 50 fold [226].

Conclusions
Since their discovery, less than a decade ago, QEs in 2D materials (2D QEs) have endured a tumultuous development, retracing, if only at an accelerated pace, much of the trail blazed by III-V QDs in the preceding 20 years.As a direct consequence of this research effort, the peculiarities and differences between QEs hosted by TMDCs and hBN have been thoroughly explored and exploited accordingly.To summarise, TMDC QEs, especially in the case of W-based compounds, are thought to be the result of a strain-induced hybridisation of dark excitons and defect states [58], and as such they emerge mainly in the ML form of these crystals, characterised by efficient light emission [34,35].On the other hand, hBN QEs are colour centres, i.e. defect states lying deep in the middle of the large bandgap of the host crystal [118].As a result, not only hBN does not need to be in the ML form to feature QEs, but these QEs are robust to thermal fluctuations, with single-photon emission maintaining its purity even at RT. On the other hand, TMDC QEs appear mainly at cryogenic temperatures, with only hBN-sandwiched high quality WSe 2 samples featuring SPEs up to 150 K [73].QEs hosted by TMDCs emit, for the most part, in the visible range (620-780 nm), with the notable exception of MoTe 2 , which features emitters in the telecom range 1080-1550 nm [23]; hBN QEs, on the other hand, span the whole spectrum from UV to NIR.A plethora of methods have been employed to create defects in the crystalline structure of these materials, while their activation relies on annealing at high temperatures, in the case of hBN [20], and on the creation of strained configurations for W-based TMDCs, whether by deposition on etched substrates acting as stressors [33], by indentation with AFM tips [55], or by pressure-induced bulging of the crystal surface [44].The strategies for the creation/activation of the emitters, as well as their main properties, have been the object of extensive discussions in sections 2 and 3.Even though-as discussed above-some work remains to be done in order to match the performances of these quantum sources to those of the state-of-the-art single-photon sources based on InGaAs et similia, 2D QEs are already at a point where some of the inherent advantages of this material platform can be exploited to push forward the field of quantum technologies as a whole.From a sustainability point of view, for example, TMDCs and hBN crystals are comprised of chemical elements which are relatively available, with MoS 2 being found in nature in the form of the molybdenite mineral.These features are in stark contrast with the availability of gallium and indium and, more in general, with the poor sustainability of the production process of QDs based on III-V materials.As yet another sign of the potential of 2D QEs, it is worth stressing here, for one last time, that the recent discovery of optically addressable spin defects in hBN [190] carries great importance, due to its promise for the realisation of spin-photon interfaces and for quantum sensing.Moreover, as noted in the previous section, the ease with which 2D crystals can be mechanically transferred onto the substrate of choice entails the possibility to deterministically integrate 2D QEs with all kinds of photonic and electronic devices.The potential advantages of the coupling with photonic structures have been thoroughly discussed in section 4, and basically boil down to the possibility of enhancing and optimising the performances of 2D QEs via the Purcell effect-for photonic microcavities-and to the realisation of integrated photonic circuits, for the on-chip implementation of quantum computation protocols.As far as electronic devices are concerned, on the other hand, we would like to briefly note the intriguing possibilities opened by the recent coupling of TMDC-based QEs with GaN micro-LEDs, as detailed in [103].If paired with one of the several methods developed to control the position of 2D QEs, this result could lead to the fabrication of ordered arrays of individually addressable, electrically pumped single-photon sources.Indeed, this system would present several advantages if compared to QDs, such as the fact that QEs in 2D materials are not affected by thermalisation between different emitters, which is instead typical of self-assembled QD ensembles [227].Of course, in order to truly help the development of the complex photonic circuits required for the advancement of quantum technology, the fabricated QE arrays should all emit identical photons.Considering the large spreads almost universally characterising the emission energy of 2D QEs (see, e.g.figure 2(a)), the fulfilment of this strict requirement will likely demand the development of effective post-fabrication methods to control, first of all, the energy of individual QEs, but also their other degrees of freedom, such as polarisation and spatial mode.As noted in the previous sections, strain engineering is one of the most promising tools for the fine-tuning of the emission properties of 2D QEs.The most advanced strain-engineering protocols are currently based on the emitter's integration with micro-machined piezoelectric actuators, whose simultaneous integration with 2D QEs, micro-LEDs, and/or complex photonic circuits poses severe technological challenges, which may well prove too hard to overcome; in this case, the issue may be circumvented by resorting to other methods to tune the emitters, e.g.Stark tuning, which only requires encapsulating the QE-containing 2D layer between graphene electrodes [169].
In summary, one could say that the defining trait of QEs embedded in 2D crystals-as, we hope, is clearly evidenced by many of the examples provided in the previous paragraphs-is represented by their flexibility, to be intended not only literally, but also in the sense of their extreme adaptability.This feature, which makes it comparatively simple to find creative solutions to any of the issues limiting the performance of devices based on 2D QEs, will represent an invaluable tool in our journey towards the realisation of optimised single-photon sources, as required by future quantum-technology applications.

Figure 1 .
Figure 1.Creation of SPEs in TMDC MLs via strain and defect engineering.(a) AFM image of a WSe2 ML deposited on top of an ordered array of SiO2 nanopillars.(b) PL map of the strain-activated QEs, showing the correlation between the bright emission and the nanopillars' location.(a) and (b) Reproduced from [33].CC BY 4.0.(c) Schematic representation of the creation process of SPEs in WSe2 via the deposition of an hBN-sandwiched ML on top of an array of dielectric nanopillars, and its subsequent irradiation with an electron beam.(d) PL map showing the appearance of bright emitters at the nanopillars sites, after electron irradiation, demonstrating that the appearance of QEs depends on the presence of both defects and strain in the 2D lattice.On the right, a zoomed-in picture highlights the localised nature of the emission.(c) and (d) Reproduced from [73].CC BY 4.0.(e) AFM image of an ordered array of hydrogen-filled WS2 bubbles formed on the surface of a bulk crystal via low-energy proton irradiation.The flake is partially capped with few-layer hBN to prevent the bubbles' deflation at low T. (f) Second order autocorrelation function, showing single-photon emission from a bright narrow line appearing at the edge of a capped WS2 bubble, as shown in the optical image in the inset.Scale bar, 2 µm.(e) and (f) Adapted from [44].CC BY 4.0.© 2023 The Authors.Advanced Optical Materials published by Wiley-VCH GmbH.(g) Schematic representation of the formation of QEs in a MoS2 ML via the creation of chalcogen vacancies with a focused helium ion beam.The ML is then capped with hBN to stabilise the emitters.Reproduced from [48].CC BY 4.0.

Figure 2 .
Figure 2. Quantum emitters in hBN.(a) Histogram of the QE energy of 90 different multilayer hBN flakes, spanning a wide energy range.The green hexagons indicate the lines for which single-photon emission has been verified via autocorrelation measurements.Reprinted with permission from [131].Copyright (2016) American Chemical Society.(b) Second-order autocorrelation function g 2 (τ ), showing robust single-photon emission from a line in the spectrum of an hBN flake, measured from 300 K to 800 K with steps of 100 K. Reprinted with permission from [134].Copyright (2017) American Chemical Society.(c) Low temperature, resolution-limited PL line from a colour centre created with submicrometric precision via the electron beam of a scanning electron microscope.Reproduced from [143].CC BY 4.0.(d) Schematic representation of the direct CVD-growth of hBN flakes on top of SiO2 nanopillars and wide-field PL image showing the strain-activated quantum emitters.Reprinted with permission from [151].Copyright (2021) American Chemical Society.

Figure 3 .
Figure 3.Control over the emission properties of hBN QEs and spin defects.(a) PL spectra showing carbon-related defects' ZPLs and their phonon side bands (PSBs) for increasing concentrations of carbon during the growth process.Reproduced from [157], with permission from Springer Nature.(b) Stark shift of an emitter's ZPL as a function of the voltages applied to electrodes A and B of the device shown in the inset.The Stark shift of up to 31 meV is four-fold the FWHM of the ZPL at RT. Reprinted with permission from [170].Copyright (2019) American Chemical Society.(c) Lattice representation of the spin defect V − B .Boron (nitrogen) atoms are represented by red (blue) spheres.(d) ODMR contrast shift of the V − B defect for increasing magnetic field.(c) and (d) Reproduced from[190], with permission from Springer Nature.

Figure 4 .
Figure 4. Integration of TMDC QEs in cavities for the enhancement of their radiative emission.(a) Plasmonic structure comprised of a WSe2 ML placed in the nanogap between a gold nanocube, acting also as a stressor, and a gold mirror.Reproduced from[200].© IOP Publishing Ltd.All rights reserved.(b) WSe2 ML placed on top of a dielectric nanopillar at the centre of a circular Bragg grating (bullseye) cavity.Reprinted with permission from[202].Copyright (2021) American Chemical Society.(c) WSe2 ML deposited on top of a silicon nitride waveguide.The waveguide's output can also be employed to resonantly excite the emitters.Reproduced from[210].CC BY 4.0.(d) Optical image of a WSe2 ML coupled to a microring resonator.The σ + polarised emission from the QEs couples to the mode of the outer ring.Reprinted with permission from[211] © The Optical Society of America.

Figure 5 .
Figure 5. Integration of hBN QEs with plasmonic and photonic cavities.(a) Monolithic 1D photonic cavity of the ladder type.The cavity resonance can be tuned by scanning a focused electron beam along the outer walls of the nanocavity in the presence of water molecules, in order to activate the etch reaction.(b) PL spectra of emitters before (black) the tuning process and after the first (red) and second (blue) tuning steps.(a) and (b) Reproduced from [212].CC BY 4.0.(c) Schematic representation of a plasmonic cavity comprised of an hBN flake deposited on a Al2O3 film on top of a gold surface, and topped by a silver nanocube.(d) PL intensity map of an hBN flake in the plasmonic cavity.The white circle indicates an area in which the V − B spin defect coupled to the plasmonic resonance.(e) ODMR spectrum of a V − B spin defect uncoupled (red) and coupled (blue) to the cavity, showing an enhancement of the contrast.(c)-(e) [218] John Wiley & Sons.© 2021 Wiley-VCH GmbH.(f) Characterisation of the polarisation axis of an emitter in an hBN flake near positioning markers.(g) Optical image of a completed device in which the emitter is integrated and dipole-aligned with the microring resonator cavity.The inset shows the number of occurrences for different values of misalignment between the QEs and the resonators.(h) Characterisation of a microring resonator.A broadband diode (dashed line) is coupled to the output port of the waveguide, while the scattered light from the ring (solid line) is collected with an objective.A Q-factor of 1512 is measured.(f)-(h) Reproduced from [222].CC BY 4.0.