Fabrication of high-aspect-ratio SiO2 nanopillars by Si thermal oxidation for metalenses in the visible region

We propose a fabrication method of metalenses in the visible region with high-aspect-ratio SiO2 nanopillars by thermal oxidation of Si nanopillars. We first evaluated the expansion of the nanopillars in width due to thermal oxidation, which affects the phase shift on metalenses. Next, considering expansion due to thermal oxidation and processing errors, a metalens pattern was fabricated, and the pillar width distribution was measured. The highest aspect ratio was 8.7. Finally, the focusing of the fabricated reflective metalens was confirmed, which indicates that the proposed method can fabricate metalenses in the visible region with SiO2 nanopillars including transmissive metalens.


Introduction
As the demand for mobile devices such as smartphones grows, there is a need for compact, high-performance camera modules. 1) In addition, along with the progress of image sensors with higher pixel counts and narrower pitches, lens configurations have also become more complex. In order to meet the demands for higher functionality, the number of lenses has increased, reaching more than a dozen, but as a result, it is difficult to reduce the overall height of the camera module.
Conventional bulk lenses cannot be made thinner because the surface shape controls the refraction of light. Fresnel lenses are thinner than bulk lenses because they are concentrically serrated, 2) but there are problems such as the deterioration of the imaging efficiency due to diffraction, making them unsuitable for camera applications except for specially designed ones. 3) Conversely, as a lens that positively utilizes the effect of diffraction, there is a diffractive lens represented by a Fresnel zone plate. [4][5][6] However, due to disadvantages such as an image blur caused by the 0th-order diffracted light, its use is limited to being incorporated as part of the conventional lens system. In recent years, attention has been paid to a method called a lens-less camera, in which a seemingly blurred image is obtained through a mask pattern and processed arithmetically to restore the image of the object. 7) An extremely thin camera can be realized because only a mask pattern is placed directly above the image instead of lenses for image formation. It also has the advantage of being able to produce an image that is in focusing at any distance through computational post-processing. Although it is a promising technology, it is not as good as a lens-based camera for current imaging applications due to issues such as the computational time required for image restoration and the limited image resolution.
Research on metalens is also progressing as a method to solve the above problems. [8][9][10][11][12][13][14][15][16][17][18][19][20][21] Metalenses are the ultimate thin lenses that control the refraction of light by designing the in-plane distribution of a single-layer subwavelength structure. Thanks to the subwavelength structure, no diffracted light occurs, and full imaging efficiency can be theoretically achieved. Metalenses are one of the structures called metasurfaces and have been proposed with various functions, wavelength bands, and materials. 22) In particular, in order to make metalenses that function in the visible light region, it is necessary to make nanostructures of transparent materials such as SiO 2 . For example, Park et al. achieved transparent metalens by directly etching a SiO 2 substrate to make nanopillars. 23) However, the collapse of nanopillars with an aspect ratio of more than 6.6 has been reported, suggesting that it is difficult to directly etch a SiO 2 substrate to fabricate high-aspect-ratio SiO 2 nanopillars. Using materials with a higher refractive index than SiO 2 , such as TiO 2 , [24][25][26][27][28][29][30][31][32][33][34] SiN, 35,36) GaN [37][38][39][40][41] and ZnS, 42) has the advantage of lowering the required aspect ratio of nanopillars and making them easier to fabricate. However, the reflection loss at the interface with air or the substrate materials increases due to Fresnel reflection. There are also reports using Si, [43][44][45][46][47][48][49] which is an opaque material in the visible, and metals that exhibit a plasmon response, [50][51][52] but the absorption loss due to the material is fatal.
Although Si is an opaque material in the visible range, it is known as a material capable of forming nanopillars with a high aspect ratio. 53) In addition, Si is also known as a material that becomes SiO 2 when thermally oxidized. 54) Therefore, there is a possibility that metalens made of SiO 2 nanopillars with an aspect ratio of more than 6.6 can be fabricated by making Si nanopillars and thermally oxidizing them.
In this study, we propose a fabrication method of metalens in which SiO 2 nanopillars were formed from Si nanopillars by thermal oxidation. When transforming from Si to SiO 2 , it is accompanied by volumetric expansion, so it is necessary to take it into account in the pattern design. We formed nanopillars on the device Si layer of a silicon-on-insulator (SOI) substrate and performed thermal oxidation. First, we fabricated nanopillar arrays with different widths to confirm the width change associated with thermal oxidation, and then fabricated nanopillar patterns that function as metalens. Since the handle Si layer of an SOI substrate acts as an Si-mirror, the metalens fabricated in this study is a reflective metalens. Using a microscope-based experimental setup, we evaluated the light focusing properties of the fabricated reflective metalens and demonstrated that the metalens can be fabricated by the proposed method, which indicates that it is also possible to fabricate transmissive metalens as well by using wafers with Si on a transparent substrate.

Design
When designing a metalens, it is necessary to know what the phase shift is exhibited by the subwavelength-sized SiO 2 nanopillars. We first investigated the relationship between the pillar width and the phase shift using an electromagnetic field simulation software based on the rigorous coupled-wave analysis (RCWA) method (DiffractMOD, Synopsys, Inc.). Figure 1(a) shows the unit cell of the periodic structure used in the simulation model. A pillar was defined as a square column made of SiO 2 with a side width of w and a height of h on a substrate made of SiO 2 . The refractive index of SiO 2 used in the simulation was shown in the prior literature. 55) The period between the adjacent pillars was p. The light source with a wavelength of λ was placed inside the substrate in order to ignore optical interference caused by the presence of the substrate. The phase of transmitted light was calculated when the pillar width was changed. In order to compare structures with different periods, the ratio of width w to period p was defined as fill factor FF = w/p. The simulation was performed by changing the fill factor from 0.475 to 0.85 by 0.001. Figure 1(b) shows the calculation results of the relationship between the fill factor and the phase and transmittance when the pillar height was 2.5 μm, the period was 400 nm or 260 nm, and the wavelength was 532 nm. In the case of the period of 400 nm, if the phase was 0 when the fill factor was 0.85, that is, the pillar width was 340 nm, the phase was −2π when the fill factor was 0.487, corresponding to the pillar width of about 195 nm. In other words, by varying the pillar width from 195 to 340 nm, a phase shift in the range of 2π was achieved. A metalens can be realized by fabricating pillars with different widths and causing a phase shift distribution equal to that of the desired lens. At a peak with a fill factor of 0.716, likely due to diffraction, the minimum transmission was about 88%.
A phase distribution that should be satisfied by a lens that collects light of a specific wavelength to one point is expressed by the following equation 8) Here, r is the distance from the center of metalens, and the phase at the center is set to 0. Based on the phase and pillar width shown in Fig. 1(b), the phase in Eq. (1) can be replaced with the pillar width. Thus, a relationship is obtained that defines the pillar width with respect to the distance from the center, which determines the pattern of nanopillars on the metalens. Note that when using the effective refractive index distribution, n eff (r), the phase shift is described as follows 8) Here, n O is the effective refractive index at the center. From Eqs. (1) and (2), the following equation is obtained Here, mod(x, y) is a modulo function that returns z that satisfies x = my + z and 0 ⩽ z < y where m is an integer. By using Eq. (3), it is also possible to give the effective refractive index distribution ranging from n O to n O − λ/h, that is, the metalens pattern, using the pillar height and focal length as parameters.
Note that the period of 400 nm used in this design was not an optimal value in terms of transmittance but was chosen as an example of what could be fabricated in practice. In designing a metalens, it is desirable to satisfy the Nyquist criteria ( where NA is the numerical aperture of the designed metalens) and the condition for only the 0thorder diffracted light to exist ( refractive index of the substrate). [56][57][58] However, this design does not satisfy the latter; therefore, diffracted light other than the 0th-order occurs on the substrate side and the transmittance decreases. Because the wavelength and the refractive index of the substrate at that wavelength is 532 nm and 1.461, respectively, the period must be 364 nm or less in order to have only the 0th-order diffracted light. Furthermore, assuming that the lower limit of visible light is 400 nm and the refractive index of the substrate is about 1.5, a period of about 260 nm or less is required for the entire visible region. In order to achieve the same fill factors as the period of 400 nm ranging from 0.475 to 0.85, SiO 2 nanopillars must be formed with minimum widths of 124 nm. Assuming that the expansion coefficient in the width direction due to oxidation was 1.51 as described later in Sect. 3.1, Si nanopillars must be formed with widths of about 82 nm before oxidation. Because the height of the nanopillars in this design is 2.5 μm, the required aspect ratio is about 30. Si nanopillars with similar dimensions and aspect ratios have been reported. 59) Also, thermal oxidation of Si nanowires with dimensions of 100 nm or less have been considered possible. 60) Thus, the period of 400 nm used here does not represent the limit of the proposed method, and it would be possible to fabricate metalenses with narrower periods. As shown in Fig. 1(b), when the period is 260 nm, the transmittance is higher than that when the period of 400 nm while the range of phase is almost the same at the same fill factor; the minimum transmittance increased to about 96% from 88%. On the other hand, a minimum transmittance of 87% has been reported for TiO 2 nanopillars for metalens with The Japan Society of Applied Physics by IOP Publishing Ltd a design wavelength of 532 nm. 25) Although the design wavelength is the same, the transmittance of our SiO 2 nanopillars is higher. This is because the refractive index of TiO 2 (∼2.4) is higher than those of the SiO 2 substrate (∼1.5) and air, resulting in greater Fresnel reflection. Therefore, by using SiO 2 nanopillars, the increase of transmittance by about 9% can be expected from 87% to 96%.

Fabrication
A designed metalens pattern was fabricated on an SOI substrate. Figure 1(c) shows the fabrication processes. An SOI substrate with a handle Si layer of 525 μm, the buried SiO 2 layer of 4 μm, and the device Si layer of about 2.5 μm with a variation was used as a starting substrate. First, electron-beam (EB) resist (ZEP520A, Zeon Corporation) was deposited on a cleaned substrate and patterned by EB lithography (JBX-6300SK, JEOL Ltd.). The resist was diluted two-fold with anisole and spin-coated at a rotation speed of 4000 rpm for 60 s to obtain a film thickness of approximately 140 nm. The acceleration voltage was 100 kV, the beam current was 2 nA, and the dose was 200 μC cm −2 . Second, the device Si layer was etched by inductively coupled plasma reactive ion etching (ICP-RIE). The passivation process using C 4 F 8 gas and the etching process using SF 6 gas was repeated for 20 cycles using a silicon deep RIE system (MUC-21 ASE-SRE, SPP Technologies Co., Ltd.). Third, wet thermal oxidation was performed by a bubbling method in which O 2 gas passed through heated ultrapure water was introduced into the furnace. The flow rate was 1 l min −1 , the furnace temperature was 1100°C, and the processing time was 3 h. The temperature of the ultrapure water used in the bubbling method was 80°C. As a preliminary experiment, a Si substrate was oxidized, and a SiO 2 film of 450 nm was obtained on the Si substrate surface under the above conditions. The maximum width of the SiO 2 used in the metalens design was 340 nm. Thus, it was considered that the pillars would be fully oxidized if oxidation proceeded from the side surfaces of the pillars.

Fabrication results of nanopillar arrays
In order to investigate the expansion rate from Si pillars to SiO 2 pillars by thermal oxidation, arrays of Si pillars with uniform width were fabricated and thermally oxidized. Figure 2 shows the observation results of the fabricated pillars before and after thermal oxidation by scanning electron microscopy (SEM). Figures 2(a), 2(c) and 2(e) show the fabrication results of Si nanopillars with different widths, which were approximately 210 nm, 260 nm and 310 nm, respectively. Note that the designed and patterned resist features were square as shown in Fig. 2(a), but the top view of nanopillars fabricated after Si etching was closer to circular. This is probably because the corners of the resist were also etched and rounded during the Si etching. As shown in Fig. 2(e), some of the closely spaced Si pillars were connected to each other because the resist pattern was not well formed. Excluding the 310 nm pillars, which changed shape due to contact with neighboring pillars, the expansion rate was 1.43 times for the 210 nm pillars and 1.46 times for the 260 nm pillars. According to previous research, it is known that 44% of the oxide film produced by thermal oxidation is the original Si; 61) thus, when Si is completely oxidized to SiO 2 , the volume increases by 2.27 times. If there is no expansion in the height direction, the width should be 1.51 times. From the measurement results, it is thought that the expansion in the height direction is at most a few percent, and the expansion in the width direction is dominant.
Additionally, periodic patterns were observed on the sides of the thermally oxidized pillars. This is thought to be due to scallops that occur during Si etching processes. Such a structure may affect the progress of thermal oxidation and the optical characteristics of nanopillars. One of the methods to reduce it is to add processes such as H 2 annealing techniques after the Si etching to smooth the sides of the Si pillars. [62][63][64]

Fabrication results of metalens patterns
The designed metalens pattern was fabricated in consideration of fabrication errors in resist patterning and etching, and expansion in thermal oxidation. The wavelength used in the design was 532 nm, and the focal length was 1000 μm. The thickness of the device layer assumed in the design was 2.5 μm, but unfortunately, step height metrology after Si etching revealed that the true thickness of the device Si layer of the SOI substrate was approximately 1.5 μm. Assuming that the distance from the center of the metalens is sufficiently smaller than the focal length, Eq. (3) can be approximated as follows Since the effective refractive index distribution, that is, the metalens pattern was designed with a pillar height of 2.5 μm, the focal length varied according to an actual pillar height of 1.5 μm. From Eq. (4), when the effective refractive index distribution is given, pillar height and focal length must be inversely proportional. Therefore, the actual focal length was estimated to become 1667 μm from the designed focal length of 1000 μm. Figure 3 shows the fabrication results of the metalens pattern. As shown in Fig. 3(a), concentric contrast was confirmed by SEM observation of the entire pattern. This is because pillars with different widths were formed depending on the distance from the center. Although the concentric contrast could be seen discretely, it was not intentionally formed and was likely a fabrication error dependent on the EB lithography conditions. Furthermore, when zooming in on the part with the largest contrast difference as shown in Fig. 3(b), the boundary between the thin pillars and the thick pillars was confirmed. This boundary was due to the wrapping of the phase shift realized in Eq. (1) in the range from 0 to 2π. Thus, all phase elements on the metalens were realized by repeating pillars with corresponding widths ranging from 195 to 340 nm. As can be seen in Fig. 3(b), our proposed method was able to fabricate pillars of all designed widths without collapsing. Figure 3(c) shows the results of measuring the width of the fabricated pillars with respect to the distance from the center of the metalens. The measured value and the design value were roughly the same, and the coefficient of determination (R 2 ) was 0.635. The minimum width of the successfully fabricated pillars was 173 nm. If the expansion in the height direction due to thermal oxidation was ignored, the pillar height was considered to be 1.5 μm, which was the same as the thickness of the device layer of the SOI substrate, and the aspect ratio of the fabricated SiO 2 pillar was about 8.7. This value is larger than the value of 6.6 shown in previous studies that directly etch SiO 2 , demonstrating the superiority of this study in thermally oxidizing Si nanopillars. Furthermore, since the verticality is excellent due to the high Si etching selectivity, it is considered that the pillar shape can be easily controlled.

Evaluation as a reflective metalens
In order to confirm that the fabricated sample functions as a metalens, an experiment was conducted to evaluate the light focusing. Since the sample fabricated in this study was a reflective metalens in which the handle Si layer of the SOI substrate functions as a mirror, the focusing of the reflected light was evaluated. Figure 4(a) shows the experimental setup. The optical system was based on a microscope (BX51, Evident Corporation). The microscope had a Köhler illumination system, and the light emitted from the light source was collimated and illuminated the sample as a reflective metalens. The magnification of the objective lens used for observation was 5 times. An aperture was inserted in the imaging plane in the middle of the optical path. The light reflected by the reflective metalens formed an image of the aperture near the focal point.
The experimental procedures were as follows. First, while observing with a camera, the observation image was focused on the surface of the metalens and set as the zero point of the stage position (z = 0). Next, the stage was lowered by 100 μm to obtain an observation image. The stage position where the image of the aperture was in focus and the observed image was brightest corresponds to the focal length of the metalens (z = f ). Figure 4(b) shows images observed by the camera while changing the stage position. When the stage position was 0 μm, the edge of the 100 μm square region where the metalens was formed was in focus. As the stage was lowered, the edge became blurred, while light was gathered near the center of the metalens. A diamond-shaped aperture was seen near the stage position of 1000 μm, indicating that the focal point of the metalens is roughly around this distance. As the stage position was further lowered, the shape of the aperture became blurred. In the above experiment, since white light was used, it can be said that the metalens functions in the visible light region. Although chromatic aberration still exists, it can be evaluated by using lasers with wavelengths corresponding to each color and corrected by devising the shape and arrangement of nanopillars. 31) For more detailed analysis, the brightness distribution of the obtained images was calculated using image analysis software (ImageJ, open source). The brightness of each pixel was the average value of the RGB values. Figure 4(c) shows the brightness distribution with respect to the distance from the center of the metalens and the stage position. Note that the stage position is equivalent to the distance from the metalens surface, since zero stage position represents the metalens surface. With increasing distance from the surface of the metalens, the brightness distribution becomes narrower, and the value at the center becomes larger. The value reaches a maximum at 1000 μm from the metalens surface, confirming that the focal length of this reflective metalens was approximately 1000 μm. A metalens focal length of 1667 μm estimated from the design and the true thickness of the device Si layer would hold if it were a transmission metalens. In the case of the reflective type, the focal length is halved to 833 μm because it is equivalent to two identical lenses with a focal length of 1667 μm lined up at a nearly negligible distance. One of the reasons for the difference between the expected value of 833 μm and the measured value of 1000 μm may be that the thickness of the device Si layer used in the design was not accurate resulting in a nonideal focusing. It is also possible that Si, which has a higher refractive index than SiO 2 , is insufficiently oxidized and remains in the center of the pillars, 60,65) increasing the effective refractive index of the pillars. If the pillar height is a constant in Eq. (4), the effective refractive index and focal length are inversely related. Therefore, if the focal length changes from 1000 to 833 μm, the effective refractive index is considered to be 1.2 times larger. Although it will be necessary to confirm that no Si remains in the center of the nanopillar in future works, complete thermal oxidation will be achieved by higher temperature thermal oxidation. From the above results, it was shown that part of the Si that constitutes the nanopillar changes to SiO 2 by thermal oxidation, and the metalens with a focal length close to the design was successfully fabricated. Additionally, the proposed method of fabricating metalenses is not thought to be limited to the reflective type. A transmissive metalens in the visible region with SiO 2 nanopillars will be fabricated by using Si on a transparent substrate.

Conclusions
In this study, we demonstrated a fabrication method of metalenses in the visible region made of SiO 2 nanopillars by thermal oxidation of Si nanopillars. First, Si nanopillar arrays with the same period of 400 nm and different widths were fabricated and thermally oxidized to observe the expansion of nanopillars. The pillars with a width of 210 nm and 260 nm expanded in the width direction by 1.43 times and 1.46 times, respectively, and the pillars with a width of 310 nm were in contact with the adjacent pillars. Since the expansion rate was close to 1.51 when Si completely changes to SiO 2 and expands only in the width direction, the expansion in the width direction was considered to be dominant. Next, the metalens pattern was fabricated in consideration of expansion due to thermal oxidation and processing error. The nanopillars with the designed width were successfully fabricated with almost no collapse, and the highest aspect ratio was 8.7. Then, the focal length of a reflective metalens with a diameter of 100 μm fabricated on an SOI substrate was evaluated. The light focusing at 1000 μm from the metalens surface was confirmed, which indicates that metalenses with SiO 2 nanopillars can be fabricated with the proposed method.