Dielectric nanophotonics has recently emerged as a very active field due to the strong electric and magnetic multipole resonances combined with low losses of metal-less nanoparticles in the visible and near-infrared spectral ranges , , . Semiconductor nanostructures are particularly attractive for on-chip nonlinear optical applications because the electromagnetic field enhancement is not limited to interfaces; thus, higher conversion efficiencies may be achieved compared to plasmonic nanostructures. Although third harmonic generation has been achieved in silicon , ,  and germanium ,  nanostructures, the central symmetry of these group IV elements forbids bulk second-order nonlinear effects. The latter can be obtained in the optical telecom band with a large-gap III to V optoelectronic material, such as AlGaAs, thanks to its high χ(2) and negligible linear and two-photon absorption (TPA).
Recently, second harmonic generation (SHG) was demonstrated from high-contrast AlGaAs nanoantennas on low-index insulator substrates , , , , such as aluminum oxide (AlOx) and transparent benzocyclobutene (BCB). These all-dielectric approaches allowed to considerably boost the SHG efficiency, achieving a nonlinear conversion coefficient
In the case of AlGaAs-on-AlOx nanoantennas, the oxidation process of such a substrate could in principle produce mechanical stress in the nanocylinder. This effect was recently exploited to achieve controlled deformation of the nonlinear susceptibility tensor , . This motivates a detailed study on the role of the substrate in the AlGaAs-on-AlOx platform, with particular attention to fabrication tolerances and their effect on SHG measurements. Hereafter, based on transmission electron microscopy (TEM) and electron diffraction imaging, we exclude that oxidation-induced stress can modify the χ(2) tensor of the nanostructures and only results in a slight change of the SHG polarization. These results confirm the robustness of our monolithic fabrication technique and strengthen the claim that SHG in our nanostructures is purely due to bulk χ(2) .
2 Fabrication and experiments
Our samples were fabricated on a (100) nonintentionally doped GaAs wafer, with a 400nm layer of Al0.18Ga0.82As on top of an aluminum-rich substrate. We first used electron beam lithography to pattern a rectangular array of circles with nominal radii varying from 175 to 225 nm along the  direction and e-beam dose varying from 108 to 228 μC/cm2 along the [11̅0] direction followed by ICP-RIE dry etching with SiCl4. Lastly, we performed a selective oxidation of the Al-rich substrate with an H2:N2 gas carrier at 390°C for 30 min. The result is an array of nanocylinders with variable radius, equally spaced by 3 μm, as sketched in Figure 1.
To experimentally probe the nonlinear properties of AlGaAs nanoantennas, we employed the setup described in Ref. , which consists of a confocal microscope coupled with a linearly polarized (extinction ratio >100:1) ultrafast laser (150 fs pulse width) centered at 1.55 μm. To access the various multipole resonances at the SH frequency and gain spectral information without varying the excitation wavelength, we analyzed nanoantennas with different radii . Figure 2A reports the SHG intensity map on a portion of the array of AlGaAs nanoantennas, with variable nominal radius (192–203 nm) and e-beam dose. Extracting the data from the second row, we plotted in Figure 2B the SHG intensity counts as a function of position, finding a signal-to-noise ratio of more than 104 .
Figure 3 shows the experimental polar plots retrieved from a series of SHG confocal maps on individual, virtually identical nanoantennas located in six different arrays fabricated using identical parameters. The measurements show different tilt angles ranging from −9° to 6° in the SHG radiation diagram for each nanocylinder. Very recently, the analysis of SHG polarization properties in the same nanoantennas showed its pure bulk χ(2) origin  as opposed to the hypothesis of surface χ(2) contribution in GaAs nanoantennas affected by TPA . To understand if the observed tilts in the radiation diagrams can be explained by a lattice deformation in AlGaAs pillars resulting in off-diagonal terms in the χ(2) tensor, TEM and electron diffraction characterization of our nanostructures were combined with finite-element simulations.
Before TEM observation, a platinum layer was deposited on the sample for metallization, and a cross-section thin film was extracted from the whole sample using Zeiss crossbeam Ga focused ion beam (FIB). The plane of this thin film is (001) and contains three virtually identical pillars, with their axis nominally oriented perpendicular to the  direction, as sketched in Figure 4A. We performed TEM analysis on a JEOL ARM 200F electron microscope equipped with a spherical aberration corrector. High-resolution TEM images show that the three AlGaAs pillars are single crystals without any alteration of the crystalline matrix (see Figure 4B and C). However, according to electron diffraction experiments on selected areas, their crystallographic directions are neither strictly parallel to the ones of the GaAs substrate nor to each other, showing a tilt around the vertical (X) and horizontal (Y) axes.
Using a double-tilt sample holder, we determined the rotation angles of the three nanoantennas with respect to the  (X) and [11̅0] (Y) directions by comparing their electron diffraction maps (Figure 4D, the one of pillar 2) to the one of the GaAs substrate. It appears that the AlGaAs pillars have lost their epitaxial relationship with the GaAs wafer due to the presence of the AlOx amorphous layer. The measured rotations around the X axis with respect to the substrate are θ≈5.5°, 2.9°, and 0.5° for nanoantennas 1, 2, and 3, respectively, whereas the rotations around the Y axis are less significant.
3 Modeling and discussion
To gain further insight in our experimental results, we performed finite-element numerical simulations with COMSOL Multiphysics. As the oxidation of the Al-rich layer induces a substantial volume shrinkage of 10%–13%, which in principle might lead to a significant strain in the proximity of the AlOx , , we dedicated a first part of our numerical analysis to the evaluation of the mechanical strain distribution in the nanoantennas and its possible effect on SHG. AlxGa1−xAs was considered as an anisotropic material with the following mechanical properties: poisson’s coefficient υ=0.31+0.1x and elasticity tensor components C11=(118.8+1.4x) GPa, C12=(53.8+3.2x) GPa, and C44=(59.4–0.5x) GPa . As AlOx is amorphous and not stoichiometric, its exact mechanical properties are unknown. As an approximation, we used γ-Al2O3 parameters (Young modulus: E=400 GPa, υ=0.21) . We have considered an elastic strain of 13% in the oxide layer along the direction perpendicular to the substrate plane. As shown in Figure 5, the strain in the nanodisks seems to be extremely low (<10−3) compared to the values employed for SHG enhancement in silicon ,  and is mainly located in the immediate vicinity of the AlOx/AlGaAs interface. From these numerical results, along with TEM observations, we can assert that no significant deformation of the χ(2) tensor occurs.
Finally, we performed fully vectorial frequency-domain finite-element calculations to assess the impact of the crystalline rotation on the tilt of the SH polar plots from single nanoantennas. We model this stress-induced rotation of the principal axes by defining an angle θ between the  crystalline axis and the linearly polarized electric field of the pump impinging at normal incidence on a stress-free nanoantenna, as shown in Figure Figure 6 (inset). For definition, θ is thus completely equivalent to the tilt angle around the X axis, measured with electron diffraction (see 4D). We inferred its impact on the angle φ between the prevailing SH linear polarization and the [11̅0] axis by calculating polarization plots of the emitted SH shown in Figure 6. From these results, we can conclude that the values of θ consistent with electron diffraction measurements (2–5°) give rise to a range of values for φ (5–13°) that are in good agreement with the SHG measurements.
We have assessed the limitations of the AlOx substrate on the SHG in AlGaAs-on-insulator monolithic nanoantennas. We found in both simulations and TEM observations that the mechanical stress induced during the oxidation procedure does not extend to the whole nanoantenna, leaving the nonlinear susceptibility tensor basically unaltered. On the contrary, electron diffraction showed a rotation in the (001) plane, in agreement with both measured and simulated SH radiation diagrams.
We would like to warmly thank P. Biagioni, D. Neshev, and A. Zayats for fruitful discussions and S. Suffit for help in fabrication.
Funding: Erasmus Mundus NANOPHI (2013 5659/002-001), NanoSpectroscopy COST Action MP1302, CARIPLO Foundation SHAPES Project (2013-0736), and “Double Culture” Ph.D. Program of Sorbonne Paris Cité (V.F. Gili). We acknowledge the financial support from the SEAM Labex (“Programme d’investissements d’avenir” of the French Government), within the PANAMA Project.
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