1. Introduction

Two-dimensional (2D) semiconductors, such as monolayer transition metal dichalcogenides (TMDCs) have attracted a lot of attention, due to their unique optical and electronic properties, including direct bandgap and tunability, which make them ideal materials for optoelectronic applications [1,2]. Lead iodide (PbI$_2$) is another type of layered semiconductors from the group of transition metal halides. It has a direct bandgap in its bulk and a hexagonal layered crystalline structure, consisting of close-packed Pb atoms sandwiched between two layers of Iodine atoms [3]. The bandgap structure of PbI$_2$ is strongly dependent on the number of layers, and exhibits a direct-to-indirect band gap transi0tion when the number of layers is reduced from its bulk form to a two-dimensional (2D) monolayer [3,4]. Compared to the widely-studied TMDCs, 2D PbI$_2$ has a relatively large bandgap in the visible ($E_g\sim 2.4$ eV) [5,6], complemented with high refractive index above two in the visible and near-infra-red spectral range, which endows its distinct optical properties.

Despite being studied for decades and being used as a precursor for perovskite materials [7,8], the recently developed PbI$_2$ nanosheets have shown a great promise for miniaturised high-performance optoelectronic devices, such as flexible photodetectors [8–12] and nanolasers [7]. Heterostructures of PbI$_2$ nanosheets with TMDCs and graphene layers have also been employed to realise band structure engineering [12–14]. Similar to TMDC materials [15], PbI$_2$ nanosheets also show a great potential for low-dimensional nonlinear optical devices. However, their nonlinear properties still remain unexplored and all studies to date have been limited to the characterisation of the PbI$_2$ electronic and optoelectronic properties. This lack of nonlinear characterisation is surprising, as the nonlinear properties of perovskite materials [16] and nanosheets [17] have been widely studied and have shown excellent potential for nonlinear and quantum light sources [18]. To define the nonlinear applications of PbI$_2$ nanosheets require careful characterization of their nonlinear properties based on crystalline orientation, thickness, excitation wavelength and influence of the substrate.

Here, we determine the second and third order nonlinear properties of PbI$_2$ nanosheets and demonstrate their enhancement as a function of the crystalline orientation, thickness, pump power, wavelength and influence of the substrate. We utilise a technique based on nonlinear microscopy of the PbI$_2$ nanosheets to realise non-invasive characterisation of their properties. We firstly investigate the thickness dependence of the second harmonic generation (SHG) and third harmonic generation (THG) of solution-grown PbI$_2$ nanosheets, see Fig. 1(a). As a direct band-gap semiconductor, we found resonantly enhanced THG near the excitonic wavelength. Because of their importance as devices on flexible substrate and heterostructures, we further define the influence of substrate on the nonlinear emission, including SiO$_2$, Si and variable air gap. We demonstrate 65 times and 12 times enhancement for THG and SHG, respectively, at a certain air-gap thickness. By using a layered nonlinear model, we perform numerical simulations, which matched perfectly with the experimental thickness and substrate dependence results. Our simulations allow us to estimate the values of the second and third-order nonlinear susceptibility of PbI$_2$ nanosheets, which are found to be $\chi ^{(2)}=2.5$ pm/V and $\chi ^{(3)}=2\times 10^{-20}$ m$^2$/V$^2$. Our results represent an important step to enabling the use of low-dimensional PbI$_2$ nanostructures for nonlinear and quantum light sources.

2. Results and discussion

PbI$_2$ nanosheets of different thickness are fabricated on a silica-on-silicon substrate by saturated solution growth method [19]. In this method, an over-saturated PbI$_2$ aqueous solution was first prepared by dissolving excess PbI$_2$ powder (99.999%, Aldrich) in deionized water. After heating for 2 hours at 90°C, the over-saturated solution was drop-casted on the substrate at room temperature. During the drying and cooling process, the PbI$_2$ nanosheets nucleate on the substrate surface [19]. Such drying and cooling process results in the formation of hexagonal-shaped nanosheets of different thicknesses, ranging from 20 nm to 80 nm, see Fig. 1(a). Similar to other 2D materials, nanosheets of varying thickness display different colors under an optical microscope because of thin-film interference effects [20]. For example, the yellow color sample in the optical image shown in Fig. 1(b), indicates nanosheet thickness of the order of 60 nm. However, the exact thickness or crystalline orientation are not possible to infer from such simple optical observation.

 

Fig. 1. Nonlinear microscopy of PbI$_2$ nanosheets: (a) Schematic of nonlinear microscopy of PbI$_2$ nanosheets. (b) The optical image of the PbI$_{2}$ nanosheet. (c,d) SHG and THG 2D mapping of PbI$_2$ nanosheets, respectively. The excitation wavelength is 1550 nm. The mapping unit kHz is defined by counts per integration time. The integration time for each point is 1 ms.

Download Full Size | PDF

To get a better insight on the structural properties of the grown PbI$_2$ nanosheets, we therefore perform nonlinear microscopy on samples of different sizes and orientations. For this purpose, we excite the nanosheets with a short-pulse infrared laser, operating at telecommunication wavelengths. The excitation is at normal incidence. We then detect simultaneously the SHG and the THG emitted by the nanosheets in backward direction (reflection geometry, see Fig. 1(a)). Two different experimental configurations have been implemented to acquire the detailed nonlinear information from the PbI$_2$ nanosheets. In the first configuration we use a pulsed laser at a fixed wavelength of 1550 nm, integrated in a scanning confocal microscope to obtain spatial resolution of the PbI$_2$ nanosheets. Due to the fibre coupling with the confocal microscope, we use pulses of picosecond fiber laser (10 ps, transform limited, 20 MHz repetition rate from Pritel). The SHG and THG signal are selected using bandpass filters and the total emission is detected on a silicon avalanche photo-diode. In the second configuration, we use a home-built microscopy setup, free-space coupled to a tunable-wavelength, femtosecond laser (150 fs, 80 MHz repetition rate, Coherent Chameleon Ultra II and Optical Parametric Oscillator). This second configuration allows us to carefully characterise the wavelength dependencies of the emitted harmonics.

Figures 1(c) and (d) show the measured spatial maps of the SHG and THG emission from the sample in Fig. 1(b). Both maps show pronounced harmonic emission from the PbI$_2$ nanosheets with a high-contrast with respect to the substrate. In this way, the nonlinear microscopy allows for precision identification of the samples. In contrast to the optical microscopy in Fig. 1(a), the nonlinear mapping further reveals possible internal strain in the samples, built during the growth process. This is represented in the brighter and darker regions in Figs. 1(c) and (d) and the separation of the hexagonal nanosheets in six triangles. From the nonlinear mapping we further infer that the intensity of the THG is about 1,000 times stronger that the SHG signal. The low SHG intensity is due to the fact that the SHG is intrinsically inhibited in materials with a centrosymmetric crystalline structure and the resonant excitonic enhancement of the THG for the excitation wavelength, as discussed below. Indeed, while second-order dipolar nonlinear interactions are only possible for non-centrosymmetric materials [21], SHG can also be measured in centrosymmetric materials due to surface nonlinearities [22].

PbI$_2$ has similar structures to the TMDCs (MoS$_2$ and WS$_2$), where the metal (Pb) atomic plane is in the middle and the two iodide atomic layers at the top and the bottom. The PbI$_2$ monolayer form in solution growth has 1-H phase, where the two iodide atoms are on top of each other in a trigonal-prismatic configuration [23] (see also X-ray diffraction spectrum in Fig. S2 in Supplement 1). As such, a monolayer of PbI$_2$ should allow for strong SHG, similar to TMDC monolayers [24–26]. The stacking of the layers in the PbI$_2$ nanosheet is 2-H, where each subsequent layer is shifted to result in a fully centrosymmetric structure. The SHG therefore is sensitive to the number of layers of the material. In its bulk form the PbI$_2$ then forms a trigonal P$\bar {3}$m1 symmetry of the nanosheet [27], where at normal incidence the SHG is forbidden. In a nanosheet, therefore, we expect that most SHG will be emitted from the top and bottom layers of the nanosheet, similar to the situation in MoS$_2$ metasurfaces [15]. Overall, the SHG and its properties can reveal accurate information about the crystalline symmetry and orientation of the nanosheets.

We, therefore, study the polarization properties of the emitted SHG and THG from the nanosheet in Fig. 1(b). The exact thickness of this sample is measured at 55 nm by an atomic force microscope (AFM). The excitation beam from our femtosecond laser (second experimental configuration) is linearly polarized and the nonlinear emissions is collected through a polarizer parallel to the incident beam polarization. Figure 2(a) shows the co-polarized intensity component of the THG and SHG, with blue and red dots, respectively. The THG and the SHG exhibit dramatically different polarization behaviour: the THG is always co-polarized to the pump, while the SHG shows a characteristic six-fold symmetry pattern, which is consistent to the 1-H crystal structure [23,28–30]. This six-fold symmetry pattern unambiguously determines the crystalline orientation of this sample, which is rotated at $17^\circ$ with respect to the horizontal axis (Fig. 2(b)).

 

Fig. 2. (a) Polarization dependence. Intensity of the SHG (red dots and black fitting line) and THG (blue dots), co-polarized to the incident laser, when rotating the excitation by $360^\circ$. The sample thickness is 60 nm (femtosecond excitation at 1550 nm central wavelength). The red arc indicates the angle between crystalline orientation and the lab frame axis. (b) Schematic of the crystals orientation of PbI$_2$. x and y are the lab-frame axes, a is the armchair direction, $\Phi _a$ is the angle between the armchair direction and the lab-frame. (c) Thickness dependence: SHG (red dots) and THG (blue dots) intensity vs. the nanosheets thickness (picosecond excitation at 1550 nm). Insets: optical images of the different samples indicating their thickness. (d) Power dependence of SHG (red dots) and THG (blue dots) for the nanosheet of 60 nm thickness (femtosecond excitation at 1550 nm central wavelength).

Download Full Size | PDF

We also tested the THG and SHG intensities, as a function of the thickness dependence of nanosheets. These measurements were performed using nonlinear mapping with a picosecond laser (first configuration) and the averaged intensities were recorded and plotted for different nanosheet thicknesses. As illustrated in Fig. 2(c), both dependencies exhibit bell-shaped curves with a maximum of the backward emitted third or second harmonic detected for different sample thickness. The insets show optical microscope images of the used samples, which appear as blue, green, yellow, and red color due to Fabry-Perot interference with the SiO$_2$ on Si substrate. These colors indicate sample thickness of 20, 40, 60, and 80 nm, respectively, as also confirmed by AFM measurements. The 40 nm (green) sample generates the strongest backward third harmonic, while strongest backward second harmonic is generated from the 60 nm thick (yellow) nanosheet. This thickness-dependent nonlinear emission is a result of the interference of the harmonics generated by each layer of the PbI$_2$ nanosheet in backward direction. In a process that resembles backward phase-matching, initially the emission increases, reaching maximum enhancement at a certain thickness. For thicker samples, the interference of the generated wave in backward direction is destructive, leading to a decrease of the SHG and THG and larger thickness. Importantly, the ratio between the SHG and THG backward emission can thereby be used to estimate the thickness of the nanosheet, even on substrates where the optical images do not allow for colorimetric identification.

For such estimation of the thickness, it is important to determine the exact power ratio between the SHG and THG for a given input power and nanosheet thickness. This coefficient can be estimated by measuring the SHG and THG power dependencies. Figure 2(d) shows the logarithmic plot of SHG and THG intensity with excitation power. Here we excite the PbI$_2$ nanosheet of thickness of 60 nm with femtosecond pulses (second configuration) through a $20\times$ microscope objective (${\rm NA}=0.4$), resulting in an excitation spot of $4.6~\mu$m. The slope of SHG and THG fittings are determined to be 2.07 and 3.19, respectively. These slopes confirm that the SHG and THG have quadratic and cubic dependence on the incident power, as expected.

As the PbI$_2$ is a direct bandgap semiconductor, the efficiency of the emitted harmonics will be affected by the proximity of the photon energy to the electronic bandgap energy or the excitonic resonances of the material [17]. To test the influence of the excitonic resonance, we next investigated the wavelength dependence of SHG and THG from a 60 nm-thick PbI$_2$ nanosheet. Figures 3(a) and (b) show how SHG and THG intensities vary with different excitation wavelength of the tunable femtosecond laser. The intensities are normalised to the maximum intensity in the experimental spectral range. We can see that the SHG shows a monotonous decrease of the intensity when the pump wavelength varies from 1300 nm to 1580 nm. In this range, the second harmonic photon energy moves away from the band-edge to lower energies, which results in a weak decrease of the nonlinear susceptibility. No material resonances are excited in this spectral range neither at the fundamental nor at the second harmonic waves. The interference of the SHG signal from the top and bottom surface (in backward direction), however, results in a stronger decrease of the SHG with wavelength. This is caused by the dispersion of the phase delay between these two nonlinear sources and is captured by our numerical modeling, see below. The variations from the mean curve (shown with a dashed-blue line in Fig. 3(a)) can be attributed to the fluctuations of the input laser power in the scan wavelength range.

 

Fig. 3. Wavelength dependence. (a,b) Wavelength dependence of SHG and THG, respectively from 60 nm thick PbI$_{2}$ nanosheet. (c,d) Measured absorbance spectrum and calculated bandgap by Tauc plot, respectively.

Download Full Size | PDF

In contrast, the THG shows a non-monotonic behavior: it increases as the wavelength is tuned above 1300 nm and reaches the maximum intensity for a fundamental wavelength of 1540 nm (THG wavelengths of 513 nm), see Fig. 3(b). This peak of the THG intensity can be explained as a result of a resonant enhancement due to the excitonic resonance near the band-edge, which for the 60 nm PbI$_2$ is at 2.4 eV (516 nm) [5]. To confirm that the observed THG enhancement is indeed due to the excitonic resonance in the material, we further estimate the bandgap energy by performing an optical absorbance measurement. Figure 3(c) shows the absorbance spectrum of the 60 nm-thick PbI$_2$ nanosheet as a function of the excitation wavelength. We can then convert the absorption spectrum to a typical Tauc plot, shown in Fig. 3(d). Here the abscissa shows the photon energy, $h\nu$, while the ordinate shows the quantity $(\alpha h \nu )^2$, where $\alpha$ is the absorption coefficient. By fitting the linear slop of the Tauc plot curve, we can determine that the bandgap of our 60 nm-thick PbI$_2$ nanosheet is indeed 2.4 eV. The experimentally measured bandgap value agrees well with the photon energy at which the THG is maximised. In Figure S3 in Supplement 1, we also give the measured absorbance and bandgap of several other PbI$_2$ nanosheets of different thickness, ranging from 20 nm to 80 nm. These measurements show how the bandgap is increasing with the nanosheet thickness, as well as the transition from indirect bandgap of a monolayer to direct bandgap of the bulk PbI$_2$ [3,4].

To support our experimental results, we next performed numerical simulations on the thickness and wavelength dependencies of the SHG and THG. The calculations were performed by means of a commercial software based on the finite element method (COMSOL Multiphysics). In all simulations the backward generated nonlinear signals where calculated as a weighted average of the nonlinear response assuming an angle of incidence that varies between $0^\circ$ and $40^\circ$. In this way, the simulations account for the variations of the incident excitation angle due to the focusing objective with a numerical aperture ${\rm NA}=0.64$.

To shed light on the second-order nonlinear response of PbI$_2$ nanosheets we modeled our samples as a set of stacked layers with individual nonlinear surface currents. In fact, despite the individual PbI$_2$ layers being non-centrosymmetric, their 2-H packing in the nanosheet, makes it a purely centrosymmetric material. In this case, the only contributions to the SH signal come from the symmetry breaking at each interface. In this layered model, the overall nonlinear emission is a superposition of the contributions of all layers as already successfully demonstrated for other layered materials [15,31]. We note that since PbI$_2$ arranges in the 2-H stacking geometry, the sign of second-order surface contributions must be inverted between the neighboring layers, resulting in an overall configuration where only the top and bottom layers actually contribute to the nonlinear process (see Fig. 4(a)), so that the overall emission depends on the total number of layers, similar to the nonlinearity of TMDCs [25].

 

Fig. 4. Calculated SHG and THG with PbI$_2$ thickness and wavelength. (a) Schematic of the simulation model. For SHG, we assumed each layer as a surface current, alternating between neighboring layers and the top and bottom current are of opposing sign. For the THG, we assumed nonlinear sources of the same sign in each of the PbI$_2$ layers. (b) Calculated thickness dependence of the SHG and THG in backward direction. (c,d) Calculated SHG and THG efficiency vs. excitation wavelength and sample thickness, respectively.

Download Full Size | PDF

To corroborate our findings we compared this model with other two scenarios where: (i) all layers contribute equally to the nonlinear process with nonlinear surface current homogeneously distributed in the stack and (ii) a model where top and bottom nonlinear surface contributions have the same sign. All simulated scenarios are schematically illustrated in Fig. S4(a) in Supplement 1. Their comparison is given by calculating the thickness dependence on the SHG process, as shown in Fig. S4(b) in Supplement 1. We found that, although the homogeneously distributed currents provide a good phenomenological fit of the experimental findings, this model can be justified only for non-centrosymmetric crystalline structures. On the other hand, by assuming top and bottom surface currents with opposite signs we can still fit very well our experimental findings and at the same time confirm what was also previously observed for materials with similar nonlinearities [25]. We note that the current at the bottom surface has been weighted by a factor of 1.34 to account for the contact with the SiO$_2$ surface of the substrate. As shown in Fig. S4 in Supplement 1, our model reproduces a bell shaped curve that peaks at $\approx 55$ nm nanosheet thickness that matches really well the shape of the measured second harmonic signal (see red line in Fig. 4(b)). These results also indicate that the solution growth process results in twin-layer formation, a process which has been observed in binary semiconductors [32].

The third harmonic process was also simulated assuming nonlinear surface contributions from the individual layers of the PbI$_2$. For the third order nonlinear contribution, which is not affected by the centrosymmetric nature of the material, we assumed identical strength and sign for all layers, as indicated in Fig. 4(a). In Fig. 4(b), we plot the thickness dependencies for the backward SHG and THG, which follow the same trend as in our experimental measurements, Fig. 2(c). We found an excellent agreement with the experimental results, for the maximum of the backward THG being for a nanosheet thickness of 42 nm. This excellent agreement gives us confidence that our model captures the key physics of the observed phenomena. The agreement of the simulated scenarios with our experiments further allows us to estimate the nonlinear susceptibilities of the single PbI$_2$ nanosheet at 1550 nm. By matching the experimental nonlinear efficiencies, we estimate the sheet second and third order susceptibility values. Considering a sheet thickness for the single PbI$_2$ layer equal to 1 nm [33,34], we then obtain the effective bulk second and third-order susceptibility values equal to $\chi ^{(2)}=2.5$ pm/V and $\chi ^{(3)}=2\times 10^{-20}$ m$^2$/V$^2$. Based on these estimates, we also numerically calculate the dependence of the SHG and THG on the incident wavelength. From Fig. 4(c) and (d), we can find that the maximum efficiency shifts to larger thicknesses as the excitation wavelength red-shifts. As for the wavelength dependence, we can infer that the calculations fit the experiment well: THG conversion efficiency dramatically increases with wavelength and it is enhanced around its bandgap at 513 nm while SHG maintains a relatively stable value with wavelength. We note that the slight discrepancy between the calculated and the measured second harmonic signal at lower frequency values can be attributed to the model that has been adopted for the simulation. More specifically, while the actual nonlinear contribution arises from a combination of chromatic and geometrical dispersion of the system, as extensively demonstrated for a variety of materials and nanostructures [35–37], here we simplify our model by reducing the fitting parameters to the effective second-order susceptibility. This approach, which is typically sufficient to model the second-order nonlinear response of 2D materials [38] does not allow to perfectly reproduce the trend of the signal as a function of frequency.

Finally, we investigated the influence of the substrate on the nonlinear emission from the PbI$_2$ nanosheets. We grew a sample on a corrugated substrate of $\approx 300$ nm SiO$_2$ layer with grooves down to the Si layer. The fabrication method is the same as before, except for the newly designed substrate. Figure 5(a) shows the optical image of the sample, which exhibits a yellow-orange color on the top substrate but changes to green over the grooves. The orange color indicates a sample thickness of $\approx 60$ nm. The exact thickness of the nanosheet is measured by an AFM and it is 65 nm (see Figure S5 in Supplement 1). Figure 5(b) shows an AFM measurement of the nanosheet over the corrugated substrate. This measurement confirms the grooves depth of 300 nm $(\pm 5\%)$. We note that the exposed Si in the grooves oxidizes to a depth of 10-30 nm due to the exposure to the ambient atmosphere, however for convenience, we will refer to it as a Si substrate.

 

Fig. 5. Nonlinear microscopy of PbI$_2$ nanosheets on a corrugated substrate: (a) The optical image of the PbI$_{2}$ nanosheet on a corrugated substrate. (b) The AFM image of the red dashed area in (a). (c) The height and enhancement factor of SHG and THG along the white dashed line in (b,d,e). (d,e) 2D mapping of SHG and THG over the corrugated substrate.

Download Full Size | PDF

Next, we map the SHG and THG from this sample. The mapping is shown in Fig. 5(d) and (e). Both SHG and THG are uniform on the flat areas, however the emission from the top SiO$_2$ surface is stronger than from the bottom Si substrate. Interestingly, on the edge of the grooves and holes, the SHG and THG are strongly enhanced. To quantitatively estimate this nonlinear enhancement, we plot their intensity profiles along the groove, as indicated by the white-dashed line in the mappings of Fig. 5(b), (d), and (e). The corresponding topography profile with the THG and SHG intensity profiles across the grove are shown in Fig. 5(c). The enhancement factor are calculated by dividing the SHG and THG intensity at each point by the average intensity from the bottom Si surface. The plots show that THG and SHG are 5.3 and 3.8 times stronger on the top SiO$_2$ than on the bottom Si. Importantly, the suspended nanosheet in the transition region can achieve 65 and 12 times enhancement, seen as sharp peaks in the intensity profiles. This transition area between the top and bottom surface can be considered as a variable air gap. The possible reasons for the enhancement then could be Fabry-Pérot interference of the harmonics emitted in forward direction and reflected by the Si substrate with the backward nonlinear emission. Additionally, the suspended nanosheets exhibits a reduced dielectric screening effect, which could further enhance the nonlinear emission [39]. We note that the strain can also play a role in the transition region [40]. However, the bending of our sample is weak and does not play a significant role in our experiments. We also noticed that position of the THG and SHG maximum is at different locations along the groove, i.e., for different air gap values under the PbI$_{2}$ nanosheet: THG peaks are near the top edge, while the SHG peaks are closer to the middle of the slope. This difference could be due to the wavelength dispersion of the Fabry-Pérot interference but requires further numerical investigation.

To support our assumption, we perform numerical simulations of the substrate dependence on the THG and SHG. Figure 6(a) shows the schematic of the PbI$_2$ nanosheet over the groove. The groove is 10 $\mu$m wide and 300 nm high, and air gap is about 2 $\mu$m, estimated from the AFM plot in Fig. 5(c). As such, the tilt angle of the nanosheets is estimated to be about $8.5^\circ$, as shown in Fig. 6(a). The bottom substrate is modeled as weakly oxidized Si, consisting of a very thin SiO$_2$ layer (30 nm) on top of this Si substrate, labelled as oxSi. Figures 6(b) and (c) show the plot of the calculated THG and SHG conversion efficiencies, as a function of the nanosheet thickness. Four curves are given for different surrounding environment: green - SiO$_2$ on Si (corresponding to the top surface); grey - air suspension; red - oxSi (corresponding to the bottom surface); and blue - pure Si substrate. The nanosheet thickness of 65 nm, corresponding to the experimental measurement, is indicated with a vertical dashed line. The ratio between the top (SiO$_2$/Si) and the bottom (oxSi) surface of THG and SHG are 12 and 2.1, respectively. These numbers represent the trend observed in experiment (Fig. 5(c)) and demonstrate that our model qualitatively explains the observed physics. The quantitative differences, can be explained by experimental noise error and the estimate of the oxidation layer in our simulations.

 

Fig. 6. Calculated efficiency of SHG and THG on different substrate conditions: (a) Schematic of cross-section of PbI$_2$ nanosheet on the groove. The schematic is not in real scale and the sample on the bottom part is skipped by the break symbol. (b,c) THG and SHG efficiency vs sample thickness and different substrate. (d,e) THG and SHG efficiency vs variable air gap distance.

Download Full Size | PDF

Figures 6(d) and (e) also show how the THG and SHG conversion efficiency change by varying the air gap height. We can consider the air gap as a Fabry-Pérot cavity, so that THG and SHG waves are resonant at different thicknesses. The THG is maximised at a thickness of 260 nm, while the SHG has a maximum around 173 nm. By converting the air gap thickness to the spatial position along the groove, we can also reproduce the observed peaks of the THG and SHG near the edges of the grooves (Fig. 5(c)). The normalised THG and SHG conversion efficiencies along the groove are shown in Fig. 6(f). This figures, shows how the two harmonics are maximised at different spatial positions near the edge of the grooves. The spacial separation of two peaks is then estimated to be $0.6~\mu$m, which is consistent with the experimental results in Fig. 5(c). The calculated enhancement values for the THG and the SHG are 65 and 12 times, which is in excellent agreement with the experimental measurements.

3. Conclusion

In summary, we have employed nonlinear microscopy of PbI$_2$ nanosheets to characterise their nonlinear properties. Our technique allows us to determine the second and third-order nonlinear susceptibilities of the nanosheets, as well as to explain the nonlinear enhancement as a function of crystalline orientation, nanosheet thickness, incident wavelength and power. We further demonstrate the enhancement of the nonlinear emission as a result of the substrate and demonstrate 65 and 12 times enhancement of the backward THG and SHG by employing an air-gap substrate. Our results open new possibilities for non-invasive optical characterization of nonlinear PbI$_2$ devices, including coherent and quantum light sources.