Here,
N(
r) represents the number of particles in an annular disc of radii
r and
r
+

Δ
r drawn with a particle at the center and
ρ the number of particles per square nanometer. The pair distribution function corresponding to the SEM image displayed in Figure

A is shown in Figure

B as black solid line. The peak positions of the array coincide with the predicted peak positions for a perfect hexagonal lattice that are depicted in grey. From the curve we can extract the distance between the nearest particles as well as its variation, and we can obtain a rough estimate on the quality of the order by considering the visibility of the various peaks. Briefly, a higher degree of order or larger domain sizes result in a slower decay of the visibility of the peaks. The mean interparticle distance was slightly influenced by the surface properties of the Si substrate. Either the hydrophilic Si surfaces resulting from piranha treatment or the hydrophobic, H-terminated Si surfaces obtained by immersion in 4.8

M HF solution were used for the fabrication of highly ordered nanoparticle arrays. The interparticle distance of the nanoparticle arrays decreased from 111

±

13

nm for the hydrophilic Si to 106

±

12

nm for the hydrophobic Si. In addition the silicon surface properties had a strong impact on the subsequent plasma treatment step. The polymer matrix of the gold nanoparticles prepared on piranha-treated Si was completely removed in W10 (90% argon + 10% hydrogen) plasma after 45

min. In contrast, polymer residues were still present on HF-cleaned Si substrates after W10 (90% argon + 10% hydrogen) plasma treatment for 90

min. Consequently, these samples were not further processed. In order to investigate metal-assisted etching on Si substrates immersed in HF prior to spin-coating, another technique, referred to as flame annealing, was chosen for the removal of the polystyrene. Here, Si substrates decorated with polystyrene-coated gold nanoparticles were pulled through a propane/butane flame. The diameter of the gold nanoparticles was not significantly influenced by thermal treatment and was determined from analyzing SEM images to be 55

±

9

nm. The key parameter of the fabricated gold nanoparticle arrays are presented in Table

, and representative SEM images are shown in the
Additional file 1. In summary for metal-assisted etching, three differently prepared gold nanoparticle arrays on Si were used which vary in the cleaning procedure of the Si substrates prior to spin-coating and/or in the thermal treatment method employed for the removal of the polymer matrix of the gold nanoparticles: (1) piranha treatment

+

plasma treatment, (2) piranha treatment

+

flame annealing, and (3) HF immersion

+

flame annealing. The first etching experiments were performed using gold nanoparticle arrays which were deposited on piranha-cleaned substrates and subsequently plasma-treated. The samples were immersed in a solution containing 4.8

M HF and 0.4

M H
2O
2 for different periods. SEM images of the resulting porous silicon layers are displayed in Figure

. After 1

min etching time, most gold nanoparticles sank into the silicon (Figure

A). However, the silicon etching rate was not reproducible in this case. Even on the same silicon substrate, gold nanoparticles could be found on top of the surface and burrowed in pores. An etching time of 10

min led to the formation of straight pores with a diameter of approximately 60

nm, reflecting the gold nanoparticle diameter (Figure

B). In addition the pore density seemed to be correlated to the nanoparticle density before etching. The etching rate was determined to be 400

nm/min. Cross-sectional SEM images also show the generation of microporous silicon in between the larger pores. After etching for 60

min the pore openings are larger than the gold nanoparticle diameter indicating the dissolution of porous silicon pore walls close to the surface (Figure

C). Figure

displays SEM images of porous silicon samples prepared by metal-assisted etching using different fabrication techniques for the generation of the gold nanoparticle mask. All samples were etched for 10

min. In Figure

A a porous silicon sample is shown whose gold nanoparticle mask was deposited on piranha-cleaned silicon. In this case the polymer matrix was removed by plasma treatment. Obviously, the particle density and accordingly the pore density decreased significantly upon etching. Polymer matrix removal by flame annealing improved the pattern transfer as can been seen in Figure

B,C. The ordered gold nanoparticle pattern was almost perfectly transferred into the silicon substrate (Figure

B,C). This observation is supported by the appearance of rings in the fast Fourier transform (FFT) of the SEM images which show porous silicon originating from flame-annealed Si substrates (see
Additional file 2). The surface properties of the Si substrate which result from piranha (hydrophilic) or HF treatment (hydrophobic) do not have a significant influence on the pattern transfer. However, we would like to emphasize that the gold nanoparticles did not sink homogenously into the silicon. Another major obstacle in the etching experiments is the formation of gas bubbles which interfere with the fabrication of homogenous porous silicon layers. Nevertheless, our results suggest that the fabrication method for the gold nanoparticle etching mask has an influence on the formation of porous silicon using metal-assisted etching.