Second harmonic NC characterization from HRS

The external reference method can again be applied to determine $<{\beta }_{nc}>$ provided that the suspension concentration is known. In addition, quantitative assessment of the $<d>$ parameter also assumes that the nanocrystal size is independently estimated.

Our HRS experimental set-up [34] is based on a Q-switched Nd:YAG laser (Wedge HB, Bright solutions, 1 mJ/1 ns). The vertically polarized beam is focused by a 20 cm focal length lens into a glass cuvette containing the NCs suspension. Scattered SH signal is collected at 90° from the laser beam axis with a 5 cm focal length lens and measured by a photomultiplier tube. A colored glass filter (short pass) and an interferometric filter (532 nm) are placed before the photomultiplier to remove unwanted signals. The unpolarized HRS signal is finally processed by a boxcar amplifier after integration and averaging over typically 1000 laser shots.

Preparation of NC suspensions

Different types of particles are investigated in this work: agglomerated BaTiO₃ and KNbO₃ nanopowders obtained by ball grinding of bulk crystals (kindly supplied by FEE GmbH, Idar-Oberstein, Germany), KTP nanocrystals extracted from raw powder at the end of the flux-growth process (kindly provided by Cristal Laser, Messein, France) [35]. LiNbO₃ nanocrystals synthesised by a hydrolysis method [36] supplied by the SRSMC laboratory (Nancy, France), and, two commercial batches of ZnO with different average nominal sizes of 20 nm and 90-200 nm purchased from NanoAmor Inc. (Houston, USA).

Colloidal suspensions with low size dispersion and good stability are obtained following a specified protocol [10, 37]. Dry nanopowders are first dispersed in water or ethanol at 0.5 mg/mL concentration and sonicated (Vibra Cell 75043, Bioblock Scientific) during 25 min in pulsed mode (1 s on, 4 s off). Large particles and remaining aggregates are then excluded by leaving the solution settling down during a period of 1 to 7 days depending on the NCs type. As-obtained supernatants have a typical concentration of 0.1 mg/mL. Size and size dispersion are then characterized by Dynamic Light Scattering (Malvern ZetasizerNanoZS). Concentration of solvent-dispersed NCs is measured after evaporation of a precise volume of the obtained suspension and weighing the residual.

Hyper-Rayleigh Scattering Data Analysis

$F_{pNA}$, the Lorentz-Lorenz local field factor correction, is given by $F_{pNA}={\left(\frac{n_{meth}^2+2}{3}\right)}^6$ with $n_{meth}$ the refractive index of methanol. The internal field factor is expressed as $T_{nc}={\left(\frac{3n_s^2}{2n_s^2+n_{nc}^2}\right)}^6$ where $n_s$ and $n_{NC}$ are the refractive indices (considered here without index dispersion i.e., $n\left(\omega \right)~n\left(2\omega \right)$) of solvent and NCs, respectively. The reference value of the pNA molecular hyperpolarizability is ${\beta }_{33}=25.9\times 10^{-30}esu$[38] from which the orientationally averaged hyperpolarisability corresponds to $\sqrt{\frac{6}{35}{\beta }_{33}}$ in our experimental configuration (vertical input polarization and no analyser).

Optical considerations for a correct HRS measurement

Among the optical considerations and possible artefacts, let us first point out that the term HRS is specifically used here because scattering at 2ω is detected with the classical HRS experimental configuration. According to the recent review of Roke et al. [31] Second Harmonic Scattering would be more appropriate for ~100 nm non-centrosymmetric NCs. If an incoherent process is underlying in eq.2 (I_HRS increases linearly with the NC concentration), the SH intensity also depends quadratically on the NC volume as it can be seen after combination of eq.1 with eq.2. The number of induced nonlinear dipoles (at the scale of the crystalline unit cell) being proportional to V_nc, a coherent process thus takes place within each NC between the various radiated SH fields. This bulk response is equivalent to the one observed in single crystals except that phase matching constraints are absent because the NC size is well below the coherent length [5]. Coherent effects are thus to be considered within each NC whereas the incoherent contribution originates from the linear concentration dependence. The Rayleigh regime assumption, at the very basis of HRS theory could be verified for most NC suspensions. The scatterers are assumed to be point-like, i.e. much smaller than the excitation wavelength so that light is scattered isotropically. An experimental verification can be carried out by detecting the normalized HRS intensity of a NC suspension at different collection angles spanning from 90° to 180° with respect to the incident beam direction. Figure 1 shows the curves obtained for ZnO and KTP. The ZnO suspension (130 nm average particle diameter) displays an almost flat angular response leading to conclude on the validity of the Rayleigh regime. The KTP suspension, on the other hand, containing larger NCs of around 200 nm presents higher forward scattering of I_HRS evidencing the limit of the Rayleigh regime assumption in this size range.

In regard to the possible optical artefacts, only the nonlinear elastic scattering is to be measured and a monochromator is necessary to spectrally resolve the HRS signal and to detect the presence of any possible inelastic process. This basic verification has been done for all the NCs under consideration. We only observed second harmonic scattering and no fluorescence [39]. Light absorption and scattering can also lead to alteration of the HRS signal. For instance, light absorption at the working wavelengths (1064 and 532 nm) tends to reduce the SH intensity when the species concentration is too high. This results in a deviation from the linear HRS response and a Beer-Lambert correction factor is to be added to correctly estimate the expected proportionality between the HRS intensity and the concentration [40]. Similarly, a deviation has also been observed with non-absorbing NCs (see Figure 2) so that an exponential correction was introduced to account for the very high multiscattering processes of the incident and SH radiations. Note that further dilution of the initial supernatants readily allows the recovery of a linear dependence on the NC concentration.

Influence of the NC size, concentration and possible aggregation state

Size characterization of NCs is of paramount importance for the correct determination of SH efficiencies. As previously described, the averaged nonlinear optical coefficient <d> is retrieved from $<{\beta }_{nc}>$ and the mean NC volume. Combination of eq.1 with eq.2 shows a dependence of HRS intensity with the squared NCs volume $V_{nc}^2$, emphasizing the requirement of a precise size characterization. In addition, assessment of the suspension concentration also relies on the NC size. As detailed hereafter, a mass concentration is indeed first measured and the unit particle concentration then estimated according to the NC volume.

Size characterization can be performed using Dynamic Light scattering (DLS), a technique based on the intensity fluctuations in the scattered light. These fluctuations are caused by Brownian motion and can be related to the particle size. This method provides a particle size distribution in intensity since the detected signal is proportional to a linear elastic scattering. A main issue may arise from size polydispersity because large particles can readily hinder the lower intensity response of smaller scatterers. In Figure 3, two different LiNbO₃ suspensions are characterized by DLS. Sample A was prepared as described in "methods" whereas the sedimentation step was not completed for sample B. A comparison is made with the distributions obtained from a particle-by-particle investigation technique called Nanoparticle Tracking Analysis (NTA) [41]. The good agreement between both results for sample A demonstrates the reliability of DLS measurements in that case. On the contrary, the larger dispersion associated with suspension B results in a less consistent correlation. This again points out the importance of paying a particular attention to suspension preparation.

Other critical issues complicating the interpretation of DLS results are related to the nanoparticles shape and the presence of aggregates. DLS indeed detects the hydrodynamic volume of objects and uses spherical models making the size characterization less relevant for non-spherical particles. Complementary observations of NCs by electron microscopy may help to obtain more information about the NC shape polydispersity. In particular, the discrimination between eventual aggregates and large individual NCs is not possible with DLS. As an exemplary result, ageing of the initial water-based solutions may result in a slow aggregation if electro-steric repulsions are too weak after the initial preparation of NC suspensions. From our experience, the long term stability of water-dispersed NCs is particularly difficult to achieve with BaTiO₃, ZnO and KTP NCs and the use of ethanol as a solvent is generally preferred.

Influence of the aggregation state on the HRS signal is specifically investigated in Figure 4. Simultaneous DLS and HRS measurements for ZnO NCs suspended in ethanol have been carried out for several days. The size increase in DLS measurements is typical of a spontaneous aggregation process (Figure 4a) whereas the almost constant HRS signal attests of the low effect of aggregation on the SH intensity (Figure 4b). This experiment unambiguously demonstrates that the HRS signal only depends on the individual nanocrystal size. Even after aggregation, the incoherent contribution of each individual nanocrystal is preserved indicating the absence of specific orientations within aggregates. On the contrary, the individual orientation and dynamics of spontaneous aggregation were observed to be well different for BaTiO₃ NCs previously dispersed in a nonpolar fluid, like heptane [42, 43].

Finally, accuracy of concentration measurements should also be clarified. In our experience, the weighting method used to determine the mass concentration is satisfactory provided that the evaporated volume is sufficient. Typically, a 30 mL suspension is evaporated in several successive steps in light containers of 7 g. The resulting relative uncertainties in the mass concentration are between 1 and 5%. $N_{nc}$, the unit particle concentration, is then calculated from the mass concentration $C_m$ with $N_{nc}=\frac{C_m}{\rho V_{NC}}$ where $\rho$ is the material density and $V_{nc}$ the mean NC volume. As individual NCs are monitored with the NTA analysis (for which the preparation of very diluted suspensions is a prerequisite), comparison between the two approaches is straightforward. Deviations by a factor of 1.5 were noticed in the best cases, which is acceptable considering the inherent size polydispersity that affects the accuracy of the weighing procedure.

Influence of the NC size distribution

Another point of interest is that the sizes used for derivation of the nonlinear optical parameters are actually the mean values of distributions. In other words, HRS intensity is considered as a nonlinear elastic scattering from exactly monodisperse samples. Because of the already mentioned size polydispersity, the HRS signal (that depends quadratically on V_nc) indicates that a mean value is not commensurate with the real scattered SH intensity. In order to evaluate the influence of the actual size distribution width, we have compared the HRS intensities of a virtual (or ideal) monodisperse suspension to an actual polydisperse one. To begin, we only take the volume into consideration. We also assume that both distributions of spherical NCs (with diameter D) have the same mean size, $D_m$, but a non-zero standard deviation for the actual suspension. To that end, DLS distributions by number are fitted by using a normal probability density function ($\rho \left(D\right)$). Ratios of the HRS intensities between polydisperse and monodisperse samples can be then calculated as: $\frac{I_{poly}}{I_{mono}}=\frac{1}{D_m^6}\int D^6\rho \left(D\right)dD$. In the case of the LiNbO₃ suspension A of Figure 3a, the calculated HRS intensity (with a 125 nm mean diameter and a 25 nm standard deviation) is for instance twice higher than a monodisperse suspension containing exclusively 125 nm NCs.

The statistical approach being introduced, our final aim is to evaluate the effect of the NC size distribution on the derived averaged nonlinear optical coefficient $<d>$. Concentration (N_nc) and volume (V_nc) are thus to be considered. Combination of eq.1 with eq.2 allows to write $<d^2>\propto I_{2\omega }/\left(N_{nc}\cdot V_{nc}^2\right)$. As discussed above for the concentration determination, $N_{nc}$ is inversely proportional to the NC volume so that the effect of a size-dispersion on the calculated concentration can be obtained from $\frac{N_{poly}}{N_{mono}}=\frac{D_m^3}{\int D^3\rho \left(D\right)dD}$. Taking into account the squared volume, the final effect of a size dispersion for spherical NCs with diameter D on the averaged $<d>$ coefficient can be expressed as: $\frac{<d_{poly}>}{<d_{mono}>}=\sqrt{D_m^3\frac{{\int }^{D^6}\rho \left(D\right)dD}{{\int }^{D^3}\rho \left(D\right)dD}}.$

Interestingly, the whole effect of the NC-size dispersion for the LiNbO₃ A suspension is relatively weak. A factor of about 0.8 is indeed found between the "polydisperse" and the "monodisperse" averaged coefficient showing that $<d>$ is generally overestimated when the size distribution is ignored.

Finally and after consideration of the previous recommendations and precautions, our experimental results for the 6 different NC type are summarized in table 1. The mean NC size (D_m) is obtained from a DLS distribution by number whereas the effective hyperpolarizabilities $<{\beta }_{nc}>$ are derived from eq.3. The averaged nonlinear optical coefficient calculated with monodisperse $<d_{mono}>$ and polydisperse $<d_{poly}>$ (when available) suspensions are then compared with the literature values $<d_{bulk}>$ of the different material. Note that literal expressions of $<d_{bulk}>$ (corresponding to our experimental polarization configuration) can be found in footnote of table 1. They have been calculated assuming Kleinman conditions and the symmetry class of each material [24, 44].