Photonic hyperuniform networks by silicon double inversion of polymer templates

Hyperuniform disordered networks belong to a peculiar class of structured materials predicted to possess partial and complete photonic bandgaps for relatively moderate refractive index contrasts. The practical realization of such photonic designer materials is challenging however, as it requires control over a multi-step fabcrication process on optical length scales. Here we report the direct-laser writing of hyperuniform polymeric templates followed by a silicon double inversion procedure leading to high quality network structures made of polycrystalline silicon. We observe a pronounced gap in the shortwave infrared centered at a wavelength of $\lambda_{\text{Gap}}\simeq $ 2.5 $\mu$m, in nearly quantitative agreement with numerical simulations. In the experiments the typical structural length scale of the seed pattern can be varied between 2 $\mu$m and 1.54 $\mu$m leading to a blue-shift of the gap accompanied by an increase of the silicon volume filling fraction.


INTRODUCTION
Photonic crystal structures have drawn a lot of attention over the last two decades but the routine design of full bandgap materials in three dimensions for optical wavelengths has proven elusive [1][2][3][4][5][6][7]. Recently, disordered and isotropic photonic materials have been suggested as an alternative [8][9][10][11][12][13]. A hyperuniform structure combined with short range order and an open network architecture is widely considered to be a strong candidate for an optimized photonic material design [10,11,14]. Moreover the isotropic structure should offer additional advantages such as the possibility to incorporate waveguides with arbitrary bending angles [15][16][17]. Numerical calculations in two and three dimensions suggest the presence of a full photonic bandgap in the near infrared if the material is made out of silicon with an refractive index n 3.6 [10,11]. Previous experimental studies of hyperuniform photonic materials have reported partial and full bandgaps for 2D hyperuniform network structures in the microwave regime [15,18]. Recently our group has reported on the fabrication of silicon hyperuniform materials that show a pseudo gap in shortwave infrared [13]. The material however also contained substantial amounts of titania (TiO 2 , n 2.4) somewhat lowering the refractive index of the material. The gold standard to achieve silicon photonic bandgap materials is the silicon double-inversion method, a rather complex multi step process to transfer polymer templates into silicon replica [19,20]. Despite its complexity it has been successfully applied to periodic structures [20][21][22]. Full photonic bandgaps have been reported for silicon woodpile photonic crystals in the shortwave infrared [20] and the near-infrared at telecom wavelengths [23]. However, the application of this approach to open hyperuniform network structures is even more complicated. In particular retaining the mechanical stability of the network is a challenging task considering the harsh conditions when removing the sacrificial material components after each processing step. Here we report the successful realization of silicon double inversion approach for hyperuniform network structures designed by direct-laser writing into a polymer photo resist [20,22,24].

Direct laser writing (DLW)
Polymeric templates on the mesoscale are then fabricated using a commercially available direct-laser writing (DLW) system (Photonic Professional GT, Nanoscribe GmbH, Germany) in Dip-In configuration. [25,26] The structures are written on infrared transparent CaF 2 substrates (Crystan, UK) by dipping an oil-immersion objective (63x, NA=1.4) inside a liquid negative-tone photoresist (IP-DIP, Nanoscribe GmbH, Germany). One should note that the writing process is started in a virtual depth of approximately 0.5µ m inside the glass substrate to guarantee a continuous laser writing process along the axial direction which is necessary to ensure the adhesion of the polymer template to the substrate. The actual height of the structures is thus reduced by 0.5 µ m as well. The highest resolution is achieved by tuning the laser power close to the photopolymerization threshold of the photoresist. The structures are tempered at 600 • C for ¿8 hours in order to transform the amorphous silicon into its polycrystalline phase. This procedure reduces the refractive index slightly but also leads to a lower residual absorption coefficient [6]. Consecutive plasma etching results in a further reduction of the silicon overlayer until it was observed to brake off revealing the bare network structure.

RESULTS AND DISCUSSION
We first fabricate polymeric templates with direct-laser writing (DLW) into a liquid negative-tone photoresist as reported previously in [25,26]. As a seed pattern, we use the center positions of a jammed assembly of spheres [27] which is then numerically converted into a three-dimensional hyperuniform (HU) network structure by following the procedure described in reference [25]. The average distance between the points is set by the diameter  (Figure 1(b)). The rods possess an elliptical cross section oriented in plane [26] with a length of about 210 nm and 580 nm along the short and long axis, respectively. These values corresponding to a mean rod diameter of ¡D¿≈ √ 250 · 580 nm= 350 nm and a corresponding silicon volume fraction of φ ≈ 0.13. The compositional analysis of the final structure by Energy Dispersive Spectroscopy (Table S-1) indicates the presence of pure silicon.
The optical spectra of the silicon hyperuniform materials are recorded using a Fourier trans- gas mixture. Next, the sample is tempered at 600 • C for ¿ 8 hours in order to transform the amorphous Si into its poly-crystalline phase.
measuring at oblique or normal incidence. This is an important observation since disordered materials are isotropic nature and therefore the photonic features are expected to be nearly angular independent [10]. The dip extends from λ ≈ 2.2-3 µm while at the same time no strong specular reflectance is observed for the corresponding wavelengths. This means that the light that is not transmitted is diffusely reflected over the whole hemisphere. Residual oscillations in some of the reflectance spectra can be attributed to Fabry-Perrot interference effects.
Next we study the evolution of the optical transmission spectra as a function of the typical structure length scale of the seed pattern a. To this end a discrete set of network structures is designed and fabricated where the parameter a is reduced from a= 2 µm down to a= 1.54 µm (Figure 3 (b,c)). Correspondingly the central gap wavelength shifts from λ Gap ≈ index is increased as discussed in more detail in [13]. Numerical calculations on hyperuniform disordered network structures have shown that a complete photonic bandgap appears for filling fractions of φ=0.15-0.4 and for refractive indices n≥ 3 [11]. The optimal filling fraction is predicted to be in the range φ=0.15-0.25. Indeed, it is in this region that we observe experimentally the most pronounced gaps. For a = 1.82 µm we find experimentally λ Gap 2.6 µm and we estimate a silicon filling fraction of φ = 0.13. For comparison, in the numerical study a central gap wavelength for the silicon network (n=3.6,φ = 0.15) is found at λ Gap 1.6 · a = 2.9 µm [11] which almost quantiatively matches our experimental result. The slight 10% difference can be attributed to the slightly lower filling fraction and to shrinking effects during the fabrication process. Similar shrinking effects have been reported previously due to the thermal decomposition of the polymer and the high temperature silicon chemical vapor deposition in the fabrication of silicon woodpile photonic crystals [29].
It would be desirable to reduce the structural length scales even further in order to bring the photonic band gap closer to telecommunication wavelengths. As we have shown here this can only be realized if the cross section of the laser writing 'pen' can be reduced substantially. Otherwise the filling fraction will increase beyond φ = 0.25 and the photonic properties are expected to weaken as shown in [11]. Indeed, such a higher resolution can be achieved for example by employing a direct-laser writing scheme based on a 405 nm wavelength diode laser for the fabrication of polymeric templates as reported in [30]. We believe the structural length scale a could be reduced to below a =1 µm, thereby shifting the gap to the near-infrared wavelengths around λ Gap ≈ 1.3-1.5 µm.