Facile fabrication of wafer-scale, micro-spacing and high-aspect-ratio silicon microwire arrays

Jimu Yan ab, Shaolong Wu*ab, Xiongfei Zhaiab, Xiang Gaoab and Xiaofeng Li*ab
aCollege of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China. E-mail: shaolong_wu@suda.edu.cn; xfli@suda.edu.cn
bKey Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province & Key Lab of Modern Optical Technologies of Education Ministry of China, Soochow University, Suzhou 215006, China

Received 28th July 2016 , Accepted 5th September 2016

First published on 6th September 2016


Abstract

One dimensional micro-/nanostructures of semiconductors are attracting increasing attention due to their large specific surface area, their high aspect ratio and their unique physical/chemical properties. However, fabrication of large-area, micro-spacing and high-aspect-ratio silicon microwire arrays (SiMWAs) is still challenging relative to the usual SiMWAs with submicro-/nanoscale spacing or a small aspect ratio. In this work, we combined metal assisted chemical etching (MACE) and photolithography to successfully prepare wafer-scale (2.5 × 2.5 cm2) SiMWAs, with 4 μm spacing which are over 70 μm long with unconventional etching conditions. We find that the key factor for achieving the desired SiMWAs is to keep the metal catalyst uniformly and steadily migrating into the substrate during the MACE for as long as possible. The most effective route to realizing this delicate control is a combination of a bilayered metal catalyst and low etching temperature (instead of the conventional room or higher temperature). Furthermore, metal migrations into the substrate are systematically investigated and the corresponding processes are categorized as various models. This work promotes an understanding of the MACE mechanisms and supplies guidelines for etching Si with a large lateral dimension.


Introduction

Recently, semiconductor-based micro-/nanowire arrays have drawn great interest because of their unique morphological, physical and chemical characteristics, attracting considerable research in the fields of the optoelectronics,1–7 photocatalysis,8,9 thermoelectrics10–12 and sensors.13–15 Among the vast number of potential applications, photon conversion is especially interesting owing to the outstanding light-trapping performance and the separation of the photon-incident and carrier-collection directions. However, the undesirable carrier combination in the micro-/nanostructured device is demonstrated to be much larger than that in the bulk/planar counterpart, limiting or even degrading the system performance.1,2 Besides surface passivation or modification, tailoring the sizes/geometries of the micro-/nanostructures has proved to be effective in suppressing carrier recombination.2–4,11,12,16–18 For Si micro-/nanowire radial-junction devices, the radius should be comparable to the minority diffusion length (usually in microns);19 otherwise, if the wires are too small they may be completely depleted (leading to a small built-in electric field and substantial bulk recombination) and if they are too large they may show poor light trapping and lots of useless (un-extracted) carriers. Therefore, Si microwire arrays (SiMWAs) are expected to be promising for high-performance optoelectronic devices because they provide a good compromise between the effects of doping concentration, wire radius and corresponding optical/electronic properties.1,4–7,19

Various methods have been developed to prepare silicon microwires, such as chemical vapor deposition (CVD),20,21 molecular beam epitaxy (MBE),22 electron beam lithography (EBL),23 deep reactive ion etching (DRIE)2,4,6,9,24 and metal-assisted chemical etching (MACE).5,7,25–27 However, only CVD, DRIE and MACE have been widely employed to obtain large-area and uniformly-aligned SiMWAs. N. S. Lewis et al. employed a CVD approach along with photolithograph technology to produce vertically-aligned SiMWAs over a large area (>1 cm2) with a length of ∼100 μm, which are the longest SiMWAs in publications to date.20,21 Since the SiMWAs are grown on a substrate through a bottom-up method, the adhesion between the wires and the substrate is not firm, and carrier transport is blocked to a certain degree by the wire-substrate interface barrier. DRIE was also successfully used to prepare SiMWAs, but the surface defect density is relatively high and the length is relatively short (due to the large width-to-depth ratio of the etched gaps). Moreover, both the CVD and DRIE methods need specific and complicated equipment, and the fabrication processes are usually complex and expensive. In contrast, MACE is a simple and cheap method for fabricating wafer-scale Si micro-/nanostructures. For Si nanowires with diameters of several tens or hundreds of nm prepared by MACE,28–32 the aspect ratio may be very large (or the length can be up to several tens of or over one hundred μm), but the internal spacing is also nanoscale, which leads to a very large specific surface area, substantial surface recombination for optoelectronic applications and poor wettability for coating an external layer to construct the hybrid architecture.32 In contract, desired Si microwires with micro-spacing have some specific advantages (such as smooth surfaces, microscale spacing, and relatively small specific surface area), which are beneficial for hybrid optoelectronic devices.2,6 K. Seo et al. fabricated variously-spaced (i.e., 1–5 μm) SiMWAs via the MACE method with an Au catalyst, but the largest length obtained was only ∼23 μm.5 Together with arranged polystyrene spheres as a template, MACE has been widely applied to fabricate diameter/spacing-controllable Si micro-/nanowires, but the internal spacing between the adjacent wires is usually in the nanoscale.1,18,33–36 Subsequent research into SiMWA hybrid devices showed that the externally coated layer (like an organic hole-transporting poly) cannot immerge into the bottom of the above-mentioned SiMWAs, meaning that a good conformal heterojunction cannot be formed, although the wire length is less than 10 μm.18 Therefore, achieving both large lateral and longitudinal dimensions in the MACE fabrication of a Si substrate is significant, and the large-area, micro-spacing, high-aspect-ratio SiMWAs can extend their applications to hybrid devices and microelectromechanical systems (MEMS) where large pattern features (both in width and depth) should be engraved.

Note that most previous work has been focused on producing high-aspect-ratio wires: that is, micrometer-long nanowires, tens of nanometers in diameter and internal spacing.17,33,34,37,38 In this work, MACE and photolithography are used to successfully fabricate wafer-scale (2.5 × 2.5 cm2) SiMWAs, with micro-spacing (4 μm) and a high-aspect-ratio (>18). The key to the desired SiMWAs is keeping the migration of metal catalyst into the Si substrate uniform and stable over a long time, which is realized by using a bilayered metal catalyst and nonconventional etching temperature. Potential models related to the various migrations of metal catalyst into the substrate are discussed, which deepens the understanding of the MACE mechanism and provides guidance for preparing various-size Si micro-/nanostructures.

Experimental details

An N-type Si (100) wafer (0.9–1.1 Ω cm) with a thickness of 520 ± 25 μm is cut into 2.5 × 2.5 cm2 pieces, from which the SiMWAs are fabricated. Before coating the photoresist, the Si substrates were ultrasonically cleaned in acetone, ethanol, and deionized water for 20 min in sequence. The main processes of the fabrication are shown in Fig. 1. The patterns of the photomask were transferred to the substrates through photolithography technology, as shown by the SEM images of Fig. 1b. To remove the photoresist residues of the exposed regions and to improve the side-wall slope of the pattern features, oxygen-plasma treatment was conducted in a plasma tripping machine (M4L, PVA TEPLA) with a power of 200 W for 10 min. Then, Ti and Au films were deposited in order on the photoresist-pattern-coated substrates by an electron beam evaporator (Ei-5z, ULVAC) with a deposition rate of 0.3 Å s−1 and 0.5 Å s−1, respectively. The lift-off treatment was performed by immersion into acetone and mildly shaking for 3 min, and then the patterned Au/Ti film was obtained, as shown in the SEM image of Fig. 1d. Finally, MACE of the metal-coated substrate was executed in an aqueous solution of HF and H2O2. Different concentrations of H2O2, etching temperature and etching duration were used. After the MACE process, the Au/Ti films were removed by using aqua regia solution and then cleaning in acetone and deionized water. The morphologies of the patterns of the photoresist, metal film and as-etched Si were characterized by field-emission scanning electron microscopy (FE-SEM, Hitachi SU8010).
image file: c6ra19104e-f1.tif
Fig. 1 Schematic of the fabrication of SiMWAs. (a) Photoresist-coated Si substrate for UV radiation; (b) patterned photoresist on Si substrate; (c) deposition of the Au/Ti films on the dephotoresist patterns; (d) patterns of Au/Ti films on Si substrate; (e) as-etched Si after MACE.

Results and discussion

The schematic of fabricating SiMWAs by the MACE method integrating photolithography is shown in Fig. 1. After ultraviolet radiation and development, micro-disk arrays of the photoresist with a diameter (period) of 4 μm (8 μm) are obtained on the Si substrate [see the scanning electron microscopy (SEM) images of Fig. 1b]. Next, a metal (Au, Ag or Au/Ti) layer is deposited, and a patterned metal film is achieved after lifting off the photoresist [see Fig. 1d]. Finally, MACE of the Si substrate is performed. Vertically-aligned SiMWAs can be obtained only if the metal migrations into the substrate are well controlled. However, for the MACE of Si with a microscale lateral dimension, metal migration can easily get out of control during the etching process. This is the reason why the lengths of the uniform SiMWAs with a microscale spacing were relatively short (≤30 μm) in previous reports.5,18 Therefore, the key procedure in the fabrication of ordered, micro-spacing and long SiMWAs is controlling the metal migration. We systematically investigated this topic and achieved successful control of metal migration for over 30 hours.

The widely accepted MACE process involves the oxidation of Si followed by dissolution, with the metal catalyst working as a charge transport mediator.1,35 In contrast, it has been argued whether silicon oxide is formed at the surface before the dissolution of Si and whether H2 is produced.39,40 Since gas bubbles are observed in our experiments with a HF concentration much higher than the H2O2 concentration, we confirm that H2 is generated and the hole-injected Si is directly dissolved in a divalent state, based on our previous studies.41,42 The MACE of Si can be regarded as a micro primary cell, and the cathodic reaction is expressed as:

 
H2O2 + 2H+ + 2M → 2H2O + 2M+ (1)

The anodic reaction is:

 
Si + 2M+ + 6F + 2H+ → 2M + SiF62− + H2 (2)
where M (M+) represents the as-deposited (hole-injected) metal. If the rate of hole injection into the metal is obviously higher than that into the Si, i.e., the cathodic reaction is more drastic than the anodic reaction, the metal catalyst will be partially dissolved in the solution, as indicated previously.35,43 As the MACE proceeds, the metal gradually migrates into the substrate, forming the micro/nanostructures via scratching the Si substrate.

Based on the above understanding of the MACE mechanism, we carry out the fabrication of wafer-scale and micro-spacing SiMWAs with the length as long as possible. The type of metal catalyst is considered first. For the MACE of Si with a nanoscale-lateral dimension, Ag is frequently employed due to its chemical activity and low cost.33,35,41 However, Ag is excluded for the case with the microscale lateral dimension because of its easy dissociation into the solution in the form of Ag+. What's more, isolated larger-size Ag nanoparticles (with sizes of tens of or even over one hundred nanometers) are formed by re-nucleation from the as-patterned Ag film consisting of high-density small-size Ag nanoparticles (with sizes of several nanometers), as observed in our experiments. Therefore, we choose Au as the catalyst.5,34,38 Since the adhesion between Au and Si is poor, an intermediate layer (such as Ti and Ag) is sometimes introduced.18,38 However, the differences between the cases with or without an intermediate layer have not been systematically investigated, especially for an etching gap with a large lateral size. Considering that the lateral dimension is in the microscale and the desired etching depth is over 50 μm, the influence of the intermediate layer (i.e., Ti) on the MACE for specific sizes is studied. Fig. 2 shows a comparison of the typical cross-sectional SEM images of the as-etched Si with single Au [Fig. 2a and c] or Au/Ti bilayered catalyst [Fig. 2b and d]. Under an etching temperature of 30 °C [Fig. 2a and b], disordered micro-/nanopores are produced at the bottom of the as-formed SiMWAs and the metal/Si interfaces are non-uniform, which can be attributed to the cleavage of the catalyst film. Some microwires are dumped due to the disordered etching at the bottom, which is obviously undesirable. In contrast, vertically-aligned SiMWAs with smooth surfaces and interfaces are achieved by lowering the etching temperature to 8 °C. Note that the lengths and diameters of the wires obtained for the sample without Ti are discrete, as shown in Fig. 2c. Different lengths are from the crispation and/or lifting off of the catalyst film during the etching process, and the abridged diameters at the upper region can be ascribed to the secondary MACE from the curled Au contacting with the as-etched SiMWAs. Note that, under the same etching conditions, the overall etching depths for the sample without Ti are longer than those of the sample with Ti. This suggests that the intermediate layer slows down the etching rate due to the poorer catalyst activity. But the positive effect is that the metal/Si interfaces and the as-formed SiMWAs are very uniform and smooth, as shown in Fig. 2d, implying that the etching duration can be further increased without introducing the disordered and secondary MACE. These results certify that the intermediate Ti can suppress the curling of the catalyst and improve the Au/Si compactness, which guarantees a uniform, stable and long-time MACE with a micro-spaced lateral dimension.


image file: c6ra19104e-f2.tif
Fig. 2 Typical cross-sectional SEM images of the as-etched Si produced with a catalyst of 40 nm-thick Au [(a) and (c)] or 40/5 nm-thick Au/Ti [(b) and (d)]. The etching temperatures are 30 °C [(a) and (b)] and 8 °C [(c) and (d)], respectively. The etching duration and concentration of H2O2 are 12 h and 0.2 M, respectively. Scale bar is 20 μm in all figures.

The normal route to lengthening the SiMWAs is straightforwardly increasing the etching duration and/or the etching rate (including using higher etchant concentration and higher temperature). Etching duration is a key and easily controlled factor, through which the etching depth can be adjusted. However, the obtained largest length is unsatisfactory due to the disintegration of the catalyst film if the etching duration keeps increasing, as testified in Fig. S1. The etchant concentration and temperature have a positive correlation with chemical reaction rate. Usually, the etching depth in the MACE of Si with the same duration is larger under a higher etchant concentration and a higher temperature. Since the used HF concentration is usually very high (>4 mol L−1), the etching rate is determined by the hole-injection rate from H2O2 to Si (i.e., H2O2 concentration). Fig. S2 shows that the etching depth obviously increases with increasing H2O2 concentration from 0.2 to 0.6 mol L−1, while the case of 0.6 mol L−1 H2O2 shows disordered etching, that is, the duration of the uniform and vertical etching is less than 2 h. MACEs under various temperatures (see Fig. S3) also demonstrate that the etched depth is larger under a higher temperature, while vertically-aligned, uniform and surface-smooth SiMWAs can only be achieved for the case at 8 °C. In contrast, the surfaces of the as-formed microwires for the 50 °C temperature case are mesoporous and look like a “sponge”, which can be attributed to a combination of the excess holes diffusing from the Si underneath the metal to the wall of the as-etched wires and the metal dissolution/re-nucleation at high temperature.39,41

Through Fig. S1–S3 we find that the uniform and stable MACE cannot be sustained for a long duration (e.g. > 10 h) at the conventionally used temperatures (20–50 °C). Increasing the temperature will decrease the sustainable time of the uniform and stable MACE, meaning that it is difficult to obtain a largest length of the desired SiMWAs exceeding 30 μm. Only the case with 8 °C temperature shows relatively long-duration desired MACE and SiMWAs over 30 μm long. Note that all the terminations of the desired MACE come from the disintegration of the patterned metal film into isolated nanoflakes or nanoparticles, which inspires us to increase the thickness of catalyst film and to use a low temperature. Fig. 3 compares the typical morphology of the as-etched Si with different thicknesses of Au film (with 5 nm Ti). For the case of 30 nm-thick Au, lots of nanoneedles sprout out of the metal/Si interfaces at the center positions of the as-formed gap after 6 h etching [see Fig. 3a]. As the etching duration increases, the nanoneedles gradually grow up to be nanocones; meanwhile the microwires are lengthened [see Fig. 3b]. When the Au thickness is 40 nm, the uniform and stable MACE without a byproduct of nanostructures can be kept for a much longer duration, with lengths up to 38 μm after 22 h etching. Further increasing the Au thickness to 50 nm, the overall uniformity and etching rates are obviously degraded. After 18 h etching, the etched depths in some regions are much larger than those in other regions, and non-uniform diameters are obtained, which is similar to the case with a single Au film of 30 nm. The differences in the etching depths and diameters come from the curling of the thick metal film, which may be induced by the larger internal stress for the thicker film and the substantial differences in the etching rates at various sites (i.e., the etching rates in the regions close to the edges of the metal film openings are much higher than in the middle regions directly beneath the metal film). The MACE almost cannot happen when the Au thickness is over 60 nm, because hole injection into the Si substrate and etchant diffusion into the metal/Si interfaces are very difficult. If the Au thickness is 20 nm or less, a dense and continuous metal film cannot be obtained, resulting in the random nanostructures observed by H.-D. Um et al.5 Therefore, the optimized thickness of Au is around 40 nm for this specific SiMWAs, because a thinner metal film is easily split during the long-duration MACE and a thicker metal film substantially reduces the etching rate.


image file: c6ra19104e-f3.tif
Fig. 3 Typical cross-sectional SEM images of the as-etched Si produced with various-thickness Au/Ti films. The metal thicknesses are 30/5 nm [(a) and (b)], 40/5 nm (c) and 50/5 nm (d), respectively. The etching durations corresponding to (a)–(d) are 6, 18, 22 and 18 h, respectively. The etching temperature is 8 °C. The H2O2 (HF) concentration is 0.2 mol L−1 (8 mol L−1). Scale bar is 10 μm in all figures.

The above experiments and analysis indicate that the optimization of etching temperature and catalyst is the essential guarantee for uniform and long-duration MACE. So in the following, we try to fabricate the desired SiMWAs as long as possible with an Au/Ti (40/5 nm) bilayered catalyst and low-temperature etching (i.e., ∼8 °C). Fig. 4 shows that the vertically-aligned and surface-smooth SiMWAs show increasing wire length with an etching duration beginning at 30 h. Increasing the etching duration further to 36 h, non-uniform roots of SiMWAs are observed, meaning that the uniform length of the SiMWAs is smaller than the largest etched depth limited by the fracture of the metal film. Our study indicates that the maximum length of the high-quality SiMWAs is ∼52 μm via properly increasing the etching duration under the conditions of a low H2O2 concentration and a low temperature.


image file: c6ra19104e-f4.tif
Fig. 4 Typical cross-sectional SEM images of the as-etched Si produced under low-temperature etching (8 °C). The etching duration of (a–d) is 12 h, 20 h, 28 h and 36 h, respectively. The etchant concentration of H2O2 and thickness of Au/Ti are 0.2 mol L−1 and 40/5 nm, respectively. Scale bar is 20 μm in all figures.

Next, we speculate and demonstrate that a moderate increment of H2O2 concentration may further improve the largest realizable wire length. Fig. 5 shows that the length of the desired SiMWAs can be up to ∼73 μm (∼35 μm) when the H2O2 concentration is increased to 0.8 mol L−1 (0.4 mol L−1). A comparison of the size features of the SiMWs prepared by various methods is shown in Table 1, which shows that the obtained length of the high-quality and wafer-scale SiMWAs with a micro-spacing herein is the largest obtained by the MACE method to the best of our knowledge, which means that the length of SiMWAs prepared by MACE can be comparable to those from other methods.9,20,21,24,44,45 With an H2O2 concentration greater than 1.0 mol L−1, as shown in Fig. 5c and d, serious curling and non-uniform lifting off of the metal film are present, with the result that the lengths of the as-formed microwires are changing substantially in different regions in the same sample. What's more, severe secondary MACE leads the as-formed microwires to shrink in diameter [see Fig. 5c] or even be cut off [see Fig. 5d]. This can be ascribed to: (1) the Au/Ti catalyst is quickly oxidized (while hole injection into Si is much slower) under a high H2O2 concentration, and (2) the obvious dissolution of Au/Ti gives rise to poor adhesion. It is anticipated that micro-spacing SiMWAs over 100 μm long may be achieved by simultaneously optimizing the etching temperature and H2O2 concentration.


image file: c6ra19104e-f5.tif
Fig. 5 Typical cross-sectional SEM images of the as-etched Si produced under low-temperature etching (8 °C) with increasing H2O2 concentration [i.e., 0.4 mol L−1 for (a), 0.8 mol L−1 for (b), 1.2 mol L−1 for (c) and 1.6 mol L−1 for (d)]. The thickness of Au/Ti and etching duration are 40/5 nm and 12 h, respectively. Scale bar is 20 μm in all figures.
Table 1 Comparison of the size characteristics of the SiMWAs prepared by different methods
Length [μm] Spacing [μm] Diameter [μm] Method Ref.
72 4 4 MACE This work
23 1–5 2 MACE 5
20 5 5 MACE 25
12 8 4 MACE 26
30 50 50 MACE 27
9 4 3 DRIE 2
50 7 4 DRIE 9
45 1.86–3.1 7–10 DRIE 24
75 1 1 Electrochemical etching 44
60 5 2 Plasma etching 45
100 5.4 1.6 CVD 20
100 4 3 CVD 21


Based on the above results, we conclude that the metal migration into the Si substrate for the micro-spacing and high-aspect-ratio SiMWAs is much more complicated than that for the case with a spacing of tens of nanometers. The differences and possible processes are described in various models, as summarized in Fig. 6. And the typical morphologies of the as-etched Si with residual metal catalyst corresponding to various models are shown in Fig. S4. For the conventional MACE of Si, the narrow gap can be easily and efficiently obtained: when the width of the metal catalyst is small, the Si underneath the metal nanoparticles can be uniformly oxidized and dissolved at a relatively high rate due to the efficient diffusion of the etchant (byproduct) into (out of) the metal/Si interfaces (see Fig. S4a), named Model 1. For etching with a lateral dimension of several micrometers, the employed metal catalyst is the patterned continuous film consisting of dense and stacked nanoparticles. The etchant (byproduct) can conveniently diffuse to (from) the metal/Si interfaces close to the openings of the patterned metal film (i.e., region i), while it is much more difficult for the middle region away from the openings (i.e., region ii). So as described in Model 2, the etching rate at region i is much larger than that at region ii, which is demonstrated by Fig. 2d and S4b. When the metal film has some small cracks or voids as deposited or after a period of time of MACE, the etchant and byproduct can diffuse via these cracks or voids, reducing the differences in the etching rates of the various regions (see Fig. S4c), as indicated by Model 3. However, when the sizes of the cracks or voids are larger, the Si underneath cannot be completely etched out (Model 4), producing some nanostructures, as shown in Fig. S4d. As the thickness of the metal film further decreases, the continuous metal will be disintegrated into isolated nanoparticles or nanoflakes, resulting in disordered etching [Model 5, as shown in Fig. S4e, Fig. 1a and b]. If the metal film is lifted off or bent from the original level, the secondary MACE of the as-formed structures or partial termination of MACE will happen (Model 6), which is certificated by Fig. 5d and Fig. S4f. To obtain micro-spacing and high-aspect-ratio SiMWAs, the key factor is controlling the patterned metal film migrating into the Si substrate as in Model 2 and stopping the MACE before Model 4 is activated. With MACE going on, cracks will inevitably be present, and a change in etching model will be present. When the optimized Model 2 gradually changes into Model 3, the etching process should be stopped in time; if not, the undesirable random nanostructures will be formed.


image file: c6ra19104e-f6.tif
Fig. 6 Mechanism models of metal migrations into the Si substrate during MACE.

Conclusions

Large-area, micro-spacing and high-quality SiMWAs with a uniform length over 70 μm are fabricated by MACE combined with photolithography under mild etching conditions. The key factor for implementing the MACE of Si with large lateral and longitudinal dimensions is controlling the metal catalyst migration to be uniform and stable for as long as possible. Our study shows that the corresponding optimized conditions involve low temperature, moderate H2O2 concentration, and an Au/Ti bilayered catalyst with an appropriate thickness. Moreover, the etching duration should also be controlled appropriately. Long-time etching most likely leads to the as-formed SiMWAs dumping (i.e., not vertically aligned) due to the disordered etchings at the metal/Si interfaces. Various potential kinds of metal catalyst migration into the Si substrate during MACE have been systematically investigated and categorized in the proposed model analysis. This work promotes an understanding of the MACE mechanisms and supplies guidelines for realizing large lateral-dimension etching and desired SiMWAs with specific sizes.

Acknowledgements

This work is supported by National Natural Science Foundation of China (61504088), Natural Science Foundation of Jiangsu Province of China (BK20140312, BK20141200), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (15KJD140004), the Youth 973 Program (2015CB932700), and Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.

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Footnotes

Electronic supplementary information (ESI) available: SEM images of typical morphology of the as-etched samples with different etching durations, H2O2 concentrations, and etching temperatures under conventional operations. See DOI: 10.1039/c6ra19104e
These two authors contributed equally to this work.

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