Viridiana Aca-López

Enrique Quiroga-González

Estela Gómez-Barojas

Jolanta Światowska

J Alberto Luna-López

EFFECTS OF THE DOPING LEVEL IN THE PRODUCTION OF SILICON NANOWALLS BY METAL ASSISTED CHEMICAL ETCHING

Keywords: Si etching, metal assisted chemical etching, nanowalls, Ag chemical deposition

Si nanowalls have been prepared out of Si substrates in a wide range of resistivities, by Metal Assisted Chemical Etching (MACE). Si (100) boron doped wafers with three resistivities 15-25 Ω·cm, 0.01-0.02 Ω·cm and 0.0005-0.0007 Ω·cm were used in this work. MACE was carried out in two steps: first, silver particles were deposited; and second, the etching solutions were optimized by varying the HF and H2O2 concentrations. This is the first time that nanowalls are obtained out of substrates of resistivity smaller than 0.001 Ω·cm. The relation of sizes and characteristics of the walls with the doping level of the Si substrates has been emphasized.

Introduction

Semiconductor nanostructures have proven to be good candidates for the development of low dimensional devices. They have gained great interest in the scientific community due to the improvements in operation and manufacturing of some optoelectronic devices [START_REF] Peng | Silicon nanowires for photovoltaic solar energy conversion[END_REF], biosensors [START_REF] Cui | Nanowire Nanosensors for Highly Sensitive and Selective Detection of Biological and Chemical Species[END_REF], energy storage devices [START_REF] Quiroga-González | How to Make Optimized Arrays of Si Wires Suitable as Superior Anode for Li-Ion Batteries[END_REF], and also their use as sacrificial layers in micromachining [START_REF] Bell | Porous silicon as a sacrificial material[END_REF][START_REF] Pérez-Díaz | Silicon microstructures through the production of silicon anowires by metal-assisted chemical etching used as sacrificial material[END_REF]. Si nanostructures have been obtained mainly by two fabrication techniques. First, a top-down approach based on silicon wafer processing by means of high resolution lithography and dry or wet etching [START_REF] Choi | Fabricaction of Sub-10-nm Silicon Nanowire Arrays by Size Reduction Lithography[END_REF][START_REF] Chang | An Alternative Process for Silicon Nanowire Fabrication with SLP and Wet Etching System[END_REF][START_REF] Juhasz | Controlled Fabrication of Silicon Nanowires by Electron Beam Lithography and Electrochemical Size Reduction[END_REF][START_REF] Liu | Selflimiting oxidation for fabricating sub5 nm silicon nanowires[END_REF], and the conventional lithography techniques that offer great flexibility and precision in device positioning, but the fabrication of nanowires with high-aspect ratio has to be improved. Second, a bottom-up approach based on the vapor-liquid-solid deposition (VLS) growth by means of chemical-vapor deposition with SiH4 or SiCl4 as precursor and Au nanoparticles as catalysts [START_REF] Westwater | Growth of silicon nanowires via gold/silane vapor-liquid-solid reaction[END_REF][START_REF] Zhang | Synthesis of thin Si whiskers (nanowires) using SiCl4[END_REF][START_REF] Gao | Selective Growth of Si Nanowire Arrays via Galvanic Displacement Processes in Water-in-Oil Microemulsions[END_REF]. Both of these techniques are expensive. An alternative top-down approach is the electrochemical etching of Si in solutions containing hydrofluoric acid (HF), which allows the fabrication of silicon structures like luminescent porous silicon and silicon nanowires [START_REF] Zhang | Nondegrading Photoluminescence in Porous Silicon[END_REF][START_REF] Peng | Fabrication of Single-Crystalline Silicon Nanowires by Scratching a Silicon Surface with Catalytic Metal Particles[END_REF]. Taking this method as base, an even simpler method was developed, called metal assisted chemical etching (MACE). This method is a relatively simple and effective top-down technique to obtain nanowalls and nanowires of Si. It uses a noble metal such as Ag, Au or Pt as catalyst to etch Si. MACE can be performed either in one or two successive steps. The first step consists on the deposition of a noble metal on the Si substrate and the second step on the immersion of Si wafer into the etching solution composed of HF and hydrogen peroxide (H2O2) as the usual oxidant agent. Other oxidant reactants can be: iron (III) nitrate (Fe(NO3)3) [START_REF] Peng | Metal-Particle-Indiced, Highly Localized Site-Specific Etching of Si and Formation of Single-Crystalline Si Nanowires in Aqueous Fluoride Solution[END_REF] or potassium permanganate (KMnO4) [START_REF] Bai | Template-free fabrication of silicon micropillar/nanowire composite structure by one-step etching[END_REF].

Different techniques can be used to deposit the noble metals on semiconductors to enable the MACE process, such as: sputtering [START_REF] Backes | Temperature Dependent Pore Formation in Metal Assisted Chemical Etching of Silicon[END_REF], spin-coating [START_REF] Harada | Catalytic Amplification of the Soft Lithographic Patterning of Si. Nanoelectrochemical Orthogonal Fotoluminescent Porous Si Pixel Arrays[END_REF] and the electroless deposition in HF solutions [START_REF] Shimizu | AgNO3-Dependent Morphological Change of Si Nanostructures Prepared by Single-Step Metal Assisted Etching Method[END_REF][START_REF] Gorostiza | Simultaneous platinum deposition and formation of a photoluminesccent porous silicon layer[END_REF]. In particular, the electroless process is relatively simple, of low cost since it does not require special equipment and it can be performed at classical chemical laboratories at room conditions. However, the effect of the etching solution composition and their relation to the properties of Si substrate (e.g. conductivity) is not well understood. In many investigations, Si nanostructures have been fabricated by MACE using Ag as a catalyst. However, the deposition of Ag particles on highly doped Si substrates in the most cases has been carried out with vacuum techniques such as sputtering or thermal evaporation [START_REF] Backes | Impact of doping level on the metal assisted chemical etching of p-type silicon[END_REF][START_REF] Geyer | Influence of the doping level on the porosity of silicon nanowires prepared by metal-assisted chemical etching[END_REF], since in the case of the electroless deposition, the process time should be adjusted according to the doping concentration to achieve a uniform film of particles [START_REF] Chiappini | Biodegradable Porous Silicon Barcode Nanowires with Defined Geometry[END_REF]. On the other hand, the concentration of the Ag precursor solution should also be optimized [START_REF] Nichkalo | Role of Ag-catalyst morphology and molarity of AgNo3 on the size control of Si nanowires produced by metal-assisted chemical etching[END_REF].

For the MACE process, the concentrations of HF and H2O2 must be modified depending on the resistivity of the Si substrate, to produce Si nanostructures [START_REF] Chartier | Metal-assisted chemical etching of silicon in HF-H2O2[END_REF][START_REF] Liu | Fabrication and photocatalytic properties of silicon nanowires by metal-assisted chemical etching: effect of H2O2 concentration[END_REF]. In particular, a highly doped Si is more complicated to etch, since a high porosification level of Si occurs all over the Si sample regardless of the position of the catalyst particles [START_REF] Han | Metal-assisted chemical etching of silicon and nanotechnology applications[END_REF]. This occurs due to the high availability of electronic holes (from the etching solution or from Si substrate), which are necessary for the etching process. The etching profiles and rates are determined by two competing processes: etching catalyzed by the metal particles, and porosification caused by the high availability of electronic holes in highly doped Si. There are a limited number of papers reporting on the production of Si structures by MACE using highly doped substrates. Additionally, this "high doping" is limited to resistivities of 0.003 Ω·cm or higher [START_REF] Mcsweeney | Metal-assisted chemical etching of silicon and the behavior of nanoscale maaterials as Li-ion battery anodes[END_REF][START_REF] Vinzons | Unraveling the morphological evolution and etching kinetics of porous silicon nanowires during metal-assisted chemical etching[END_REF]. Particularly, Si nanowalls produced by MACE without the need of photolithography are relatively scarce, and they have been produced mainly using substrates of resistivities higher than 1 Ω·cm [START_REF] Duran | Fabrication of ultrahigh aspect ratio silicon nanostructures using selfassembled gold metal-assisted chemical etching[END_REF][START_REF] Yilbas | Surface characteristics of silicon nanowires/nanowalls subjected to octadecyltrichlorosilane deposition and noctadecane coating[END_REF].

To the knowledge of the authors, there are no works on MACE explaining the mechanism of chemical deposition of Ag as a function of the doping level, neither how it affects the etching process. Shimizu et al. [START_REF] Shimizu | AgNO3-Dependent Morphological Change of Si Nanostructures Prepared by Single-Step Metal Assisted Etching Method[END_REF] reported the formation of Si structures with morphologies "nanowire", "porous wall" and "polished" out of p-type (100) Si substrates with resistivities of 1000, 10 and 0.01 Ω·cm. They used a single-step MACE using Ag metal as the catalyst. They concluded that regardless of the resistivity of Si substrate, by increasing the AgNO3 concentration in the etching solution, the surface morphology of etched Si changes from nanowire to porous wall, and finally, polished. Yousong et al. [START_REF] Liu | Fabrication and photocatalytic properties of silicon nanowires by metal-assisted chemical etching: effect of H2O2 concentration[END_REF] used Si wafers (100) (with not detailed resistivity) to fabricate Si nanowires by MACE using Ag as catalyst. They stated that by controlling the H2O2 concentration in the etching solution, SiNWs can be obtained. Zhang et al. [START_REF] Zhang | Preparation of Large-Area Uniform Silicon Nanowires Arrays through Metal-Assisted Chemical Etching[END_REF] fabricated nanowires arrays on p-type Si substrates with both (100) and (111) orientation and different resistivity: 7-13 Ω•cm and 0.003-0.005 Ω•cm, by the 2 step MACE method using Ag nanoparticles as catalyst. In the second step they used a room-temperature aqueous solution of hydrofluoric acid and hydrogen peroxide. For every kind of silicon wafer, the effect of etching conditions, such as components of etchant, temperature and time, were systematically investigated. They found that the silicon nanowires arrays possess the same type and same doping level of the starting wafer and that Si substrates with both orientations presented the (100) crystallographic direction. However, the effect of the Ag particles deposition on Si substrates with different resistivity was not considered. This work deals with the production of Si nanowalls by MACE, using Si wafers in a wide range of resistivities: 15-25 Ω·cm, 0.01-0.02 Ω·cm and 0.0005-0.0007 Ω·cm. This is the first time that nanowalls are obtained out of substrates of resistivity smaller than 0.001 Ω·cm. The mechanisms of chemical deposition of Ag particles and MACE using the different substrates are explained. Special attention is paid at the effect of the doping concentration. A correlation between doping level, Ag deposition and MACE is presented here.

Experimental

The procedure to produce Si nanowalls consists of two steps: first, a deposition of dendritic silver particles (a), leaving elongated interstices in between, then the second, the MACE process (b), optimizing the etching solutions for each resistivity of Si used. Fig. 1 shows a schematic illustration of the deposition of the particles (a), and of the further etching process (b). As it can be observed, the interstices between Ag particles become the Si nanowalls after the etching process. Around the positions of the particles are located, trenches are produced. Single side polished p-type (boron doped) (100) Si wafers with resistivities of 15-25 Ω·cm, 0.01-0.02 Ω·cm and 0.0005-0.0007 Ω·cm, were used for the experiments. Prior to the production of nanowalls, the Si substrates were cleaned with ethanol (≥ 99.8 % purity, Sigma-Aldrich) and acetone (≥ 99.9 % purity, Sigma-Aldrich) for 10 min in ultrasonic bath, and they were rinsed with ultra-pure (UP) water of 18.2 MΩ·cm afterwards. Then, the samples were immersed in a solution of HF (48 %, EMSURE) 10 % v/v for 5 min, and they were rinsed with UP water afterwards.

After the cleaning procedure, Ag particles were deposited by an electroless process by immersion of the Si substrates in an aqueous solution composed of 0.01 M of AgNO3 and HF (48%), in a volumetric proportion 25:0.5, for 30 s at 30 °C. The Si substrates were rinsed with UP water afterwards. The etching solution consisted of HF, H2O2 and H2O. The starting composition was 4:4:32 v/v/v. The composition of the etchant was optimized to be able to obtain nanowalls out of Si with different resistivities. The etching temperature was 30 °C, and the time was set to 30 min for all samples. After carrying out the MACE process, the samples were rinsed with UP water. All the samples were analyzed using a Zeiss Ultra55 microscope with high-resolution Schottky thermal field emission gun (Schottky SEM-FEG).

Results and discussion

3.1.Chemical deposition of Ag nanoparticles

The reaction mechanism of the deposition of Ag particles using AgNO3, Si and HF as precursors, can be described by the following equations ( 1) and ( 2) [START_REF] Pérez-Díaz | Silicon microstructures through the production of silicon anowires by metal-assisted chemical etching used as sacrificial material[END_REF]:

6HF + Si → H2SiF6 + 4e -+ 4H +
(1)

Ag + + e -→ Ag 0 (2)
HF etches Si (slow reaction) and produces electrons that are caught up by Ag + ions, which are transformed to Ag metallic particles. However, it is known that the dissolution of Si proceeds faster with the availability of electronic holes according to reaction (3) [START_REF] Quiroga-González | Fundamentals of Silicon Porosification via Electrochemical Etching[END_REF]:

6HF + Si + 4h + → H2SiF6 + 4H + (3)
Nevertheless, if there is a high availability of electronic holes in the substrate (e.g. when Si p-type is highly doped), the production of electrons, necessary for the electroless deposition of Ag, is limited. The overall dissolution can be understood as a combination of (1) and (3):

6HF + Si + xh + → H2SiF6 + ye -+ 4H + (4)
where x + y = 4.

If there is a high availability of electronic holes, the production of electrons tends to zero.

Fig. 2 shows SEM micrographs of the Ag particles deposited by the electroless method on p-type Si substrates of three resistivities. In all the samples the Ag particles coalesce, as needed for the production of nanowalls by MACE. Nevertheless, there are big differences in the size and spatial distribution of the Ag particles from sample to sample. Fig. 2a shows the Ag particles deposited on substrates of 0.0005-0.0007 Ωcm. The average size of the agglomerated particles is 213 nm. In (b) one can see a micrograph of the Ag particles deposited on substrates of 0.01-0.02 Ω cm. They have an average size of 68 nm. A micrograph of the Ag particles deposited on substrates of 15-25 Ωcm is shown in (c). The particles have an average size of 87 nm. Fig. 3a explains schematically how is the deposition mechanism for the sample with resistivity 0.0005-0.0007 Ω·cm. Particularly in this sample, the production of electrons for the reduction of Ag + is limited, because of the abundance of electronic holes, as indicated in equation ( 4). However, once the first Ag particles nucleate, they attract the electrons through an electric field of a space charge region (SCR) formed under the particles [START_REF] Quiroga-González | Light enhanced metal assisted chemical etching of silicon[END_REF]. The SCR is thin with strong field, because of the high doping density. Then, an over deposition of Ag on the first nucleated Ag particles has a much higher probability than a nucleation of new particles. This is why the distance between particles is large. Even though, the particles could start coalescing with the time (see Fig. 2a). In the case of the samples with resistivities of 15-25 and 0.01-0.02 Ω·cm, the deposition process (presented in Fig. 3b) is similar. Below the deposited particles, an SCR is formed with a similar shape as for the sample with low resistivity. However, it is wider and its field is lower. On the other hand, there is a higher availability of electrons for the deposition, given by the lower doping of Si wafer. Due to these conditions, after nucleating the first Ag particles, there is a similar probability to deposit on the exiting Ag particles than nucleating new particles. Thus, as observed in Fig. 2b and Fig. 2c, the distances between particles are smaller, and there is a larger distribution of sizes. The only difference that can be found when comparing the sample of 15-25 Ω·cm with the one of 0.01-0.02 Ω·cm, is that the particles for the former are larger (the deposition rate is faster at this resistivity). The reason is the higher availability of electrons for the reduction of Ag + , as indicated by equation ( 4). It is important to mention that in the three cases the particles agglomerate to form dendrites leaving interstices, as needed for the production of nanowalls by MACE.

3.2.Fabrication of Si nanowalls by MACE

Nanowalls from Si wafers of 15-25 Ω·cm

At first, the MACE process (with Ag catalyst) involves the formation of electronic holes at the sites of the catalyst particles. The chemical equation describing this process is as follows [START_REF] Li | Bubble-regulated silicon nanowire synthesis surfaces by metal-assisted chemical etching[END_REF]:

H2O2 + 2H + 𝐴𝑔 → 2H2O + 2h + (5)
The electronic holes speed up the dissolution of Si in the presence of HF. The overall dissolution reaction can be described by equation [START_REF] Bell | Porous silicon as a sacrificial material[END_REF]. According to this reaction, H + ions can be reduced and lead to H2 evolution, which can be described by equation ( 6) [START_REF] Li | Bubble-regulated silicon nanowire synthesis surfaces by metal-assisted chemical etching[END_REF]:

6HF + Si + nh + → H2SiF6 + nH + + (2-n/2)H2 (6)
where n depends strongly on the proportion of HF to H2O2 (etching to oxidation rate). It is important to mention that the etching rate and the evolution of H2 depend on the availability of electronic holes in the Si substrate, which is given by the doping level. More details on these dependencies will be discussed below. where a value of 30 µm/h has been reported for the production of SiNWs out of a Si wafer with resistivity of 15-25 Ω·cm [START_REF] Pérez-Díaz | Silicon microstructures through the production of silicon anowires by metal-assisted chemical etching used as sacrificial material[END_REF]. However, the morphology of the pores is not the appropriate to form nanowalls. Isolated pores are obtained instead of coalesced pores, in contrast to what was expected when looking at the morphology of the merged Ag particles used for etching. H2 bubbles generated during the etching of Si may evolve from around the Ag particles, considering that there is an excess of H2O2 that caused an additional porosification (and additional reaction mechanisms), not just at the pore tips. The bubbling during the etching process could lead to splitting the Ag particles and their aggregates. To support this theory, the amount of H2O2 was increased.

Fig. 5 shows SEM micrographs of a Si 15-25 Ω·cm sample etched in a solution of HF:H2O2:H2O 2:4:34 v/v/v. (a) shows well isolated pores, even more isolated than when using the previous composition of the etchant (HF:H2O2:H2O 4:4:32 v/v/v). This confirms the theory of Ag particle dissociation, most probably due to the bubbling. This bubbling has been used in other works even for controlling the shape of the produced structures [START_REF] Li | Bubble-regulated silicon nanowire synthesis surfaces by metal-assisted chemical etching[END_REF]. The cross section image (Fig. 5 (b)) shows pores with conical shape. This shape is common when an excess of H2O2 is used [START_REF] Tsujino | Helical nanoholes bored in silicon by wet chemical using platinum nanoparticles as catalyst[END_REF]. The average depth of the pores is 4.0 ± 0.2 µm, indicating an etching rate of 0.12 µm/min. S1 of supplementary information). This solution was selected as the optimal etchant to prepare nanowalls with the lowest porosity. Fig. 6 (a) shows a micrograph of the surface with the optimized morphology which consists of nanowalls surrounded by trenches (seen as black regions). Fig. 6 (b) shows the nanowalls in cross section. The height of the nanowalls is 11.0 ± 0.7 µm, thus the etching rate is 0.37 µm/min. The optimized etchant contained less H2O2 than the solutions used for the two previous samples, minimizing bubbling and porosification of the walls. This etching solution was used as a starting etchant for the experiments for Si substrates with resistivities of 0.01-0.02 and 0.0005-0.0007 Ω·cm. To minimize the bubbling problem and the porosification of the walls, the amount of H2O2 was greatly reduced. The optimized solution was HF:H2O2:H2O 4:0.2:35.8 v/v/v. Some other tests were performed varying one at a time the amount of both HF and H2O2 (Table S2 of supplementary information), and etching solution formed by HF:H2O2:H2O 4:0.2:35.8 v/v/v was selected as optimal to prepare nanowalls with the lowest porosity. Fig. 8 shows SEM micrographs of the Si sample with resistivity of 0.01-0.02 Ω·cm etched in the optimized solution. Fig. 8 (a) shows a top view, where nanowalls can be observed. The structures tend to collapse, probably due to their small thicknesses and their significant height of around 13.0 ± 0.9 µm, as shown in a cross section image in Fig. 8 (b). The etching rate was of 0.47 µm/min. Taking a look at the HF:H2O2 ratio, it can be inferred that for etching samples of medium resistivity, the concentration of HF in the etchant must be dominant. Ω·cm). Some Ag agglomerates can be observed on the surface but no structures characteristic of etched Si. The inset at 250,000X shows some surface modifications indicating a negligible etching, most probable due to the high surface porosification originating from the availability of electronic holes. The etching seems to occur everywhere and not only at the sites of Ag particles, because of the high availability of electronic holes. The porosification should have been so dense, that HF has dissolved the porous Si. The etching solution was optimized to obtain nanowalls. Before, several tests were carried out, using different compositions of the etchant (see Table S3 of the supplementary information), until an optimal solution was found: HF:H2O2:H2O 20:0. Comparing the composition of the etchant used for this sample with the one for the sample of 0.01-0.02 Ω·cm, it is evident that much more HF was needed. The small amount of H2O2 is necessary to form a small gradient of electronic holes at the sites of presence of Ag particles. An excess of HF is needed to etch away without limitation of the etching rate. Due to the larger availability of electronic holes at the pore tips, the etching rate is higher at these sites and Si nanowalls can be formed. However, the walls are short, and may be porous. The presence of pores can be deduced from the charging effect observed during the SEM measurements. Table 1 shows a summary of the optimized compositions of the etchants. It can be seen that for MACE, to move from a high to a medium resistivity, it is necessary to reduce the amount of oxidant (H2O2). This is mandatory due to the high availability of electronic holes in the semiconductor. It is just important to create a small difference in the concentration of holes, to inject them right under the catalyst (Ag) particles to increase the etching probability at those positions. For the sample of the lowest resistivity, in addition, it is necessary to increase the amount of HF (so that the etching reaction rate is not the limiting factor).

Conclusions

Dendritic growth of Ag particles has been accomplished chemically, on Si substrates of high, medium and low resistivities. The growth depends of the electric field of the space charge region form below the first nucleated particles, which is dependent of the doping level. For low resistivity Si samples, the electric field is much higher than for medium and high resistivity Si, generating bigger Ag particles with a large separation between them, while in the other ones, the distances among particles are smaller.

For the first time nanowalls were obtained out of substrates of resistivity smaller than 0.001 Ω·cm. The resistivity of the Si substrates and concentrations of HF and H2O2 in etching solution affect the formation of nanowalls. For substrates with high resistivity (15-25 Ω·cm), it was observed that a high concentration of H2O2 causes additional lateral porosification. This effect is more evident in samples with a high concentration of electronic holes (samples with low resistivities), so it is necessary to reduce considerably the H2O2 concentration. On the other hand, the optimization of etching solution allowed that the bubbling of H2 do not displace the dendritic Ag particles. Additionally, the HF had to be increased for the sample of lowest resistivity, so that the etching reaction rate is not the limiting factor. Considering all these factors, the etching process led to formation of nanowalls.

Fig. 1 .

 1 Fig. 1. Schematic illustration of the processes to fabricate Si nanowalls: (a) deposition of Ag dendritic particles leaving elongated interstices; (b) MACE process leading to formation of trenches located at the sites of particles, leaving nanowalls in between.

Fig. 2 .

 2 Fig. 2. SEM micrographs of Ag particles deposited by electroless on p-type Si substrates with resistivity a) 0.0005-0.0007 Ω·cm, (b) 0.01-0.02 Ω·cm, and (c) 15-25 Ω·cm using 0.01 M fresh solution of AgNO3 and HF for 30 s.

Fig. 3 .

 3 Fig. 3. Schematic illustration of the deposition mechanism of Ag particles: (a) for the sample of 0.0005-0.0007 Ω·cm; (b) for the samples of 15-25 and 0.01-0.02 Ω·cm. The dotted lines indicate the sizes of the Ag particles (filled ovals) at longer deposition times.

Fig. 4 shows

 4 Fig.4shows SEM micrographs of the Si sample with resistivity of 15-25 Ω·cm, etched in the solution of HF:H2O2:H2O 4:4:32 v/v/v. The micrograph of (a) presents the surface morphology of the etched sample, where black circular regions denote pores. The micrograph (b) is a cross sectional view, where vertical pores are evident. The depth of the pores is of 39 ± 1.5 µm, indicating an etching rate of 1.3 µm/min. The etching rate is good compared with another work where a value of 30 µm/h has been reported for the production of SiNWs out of a Si wafer with resistivity of 15-25 Ω·cm[START_REF] Pérez-Díaz | Silicon microstructures through the production of silicon anowires by metal-assisted chemical etching used as sacrificial material[END_REF]. However, the morphology of the pores is not the appropriate to form nanowalls. Isolated pores are obtained instead of coalesced pores, in contrast to what was expected when looking at the morphology of the merged Ag particles used for etching. H2 bubbles generated during the etching of Si may evolve from around the Ag particles, considering that there is an

Fig. 4 .

 4 Fig. 4. SEM images of Si wafer with resistivity of 15-25 Ω·cm etched in a solution of HF:H2O2:H2O 4:4:32 v/v/v: (a) surface morphology and (b) cross section view.

Fig. 5 .

 5 Fig. 5. SEM images of Si wafer with resistivity of 15-25 Ω·cm etched in a solution of HF:H2O2:H2O 2:4:34 v/v/v: (a) surface morphology and (b) cross section view.

Fig. 6

 6 Fig. 6 shows SEM micrographs of Si 15-25 Ω·cm sample etched in an optimized etching solution (HF:H2O2:H2O, 4:1:35 v/v/v) to obtain Si nanowalls. In the way to reach the optimized etchant some other tests were performed varying the amount of either HF or H2O2 (TableS1of supplementary information). This solution was selected as the optimal etchant to prepare nanowalls with the lowest porosity. Fig.6(a) shows a micrograph of the surface with the optimized morphology which consists of nanowalls surrounded by trenches (seen as black regions). Fig.6

Fig. 6 . 3 . 2 . 2 .

 6322 Fig. 6. SEM images of Si wafer with resistivity of 15-25 Ω·cm etched in a solution of HF:H2O2:H2O 4:1:35 v/v/v: (a) surface morphology and (b) cross section view.

Fig. 7 .

 7 Fig. 7. SEM images of Si wafer with resistivity of 0.01-0.02 Ω·cm etched in a solution of HF:H2O2:H2O 4:1:35 v/v/v: (a) surface morphology and (b) cross section view.

Fig. 8 .

 8 Fig. 8. SEM images of Si wafer with resistivity of 0.01-0.02 Ω·cm etched in a solution of HF:H2O2:H2O 4:0.2:35.8 v/v/v: (a) surface morphology and (b) cross section view.

3. 2 . 3 .

 23 Fig. 9 shows a SEM micrograph of Si 0.0005-0.0007 Ω·cm etched in a solution of HF:H2O2:H2O 4:1:35 v/v/v (the optimized etchant to obtain nanowalls out of Si wafers with resistivity of 15-25

Fig. 9 .

 9 Fig. 9. SEM micrograph of the Si surface of a sample with resistivity of 0.0005-0.0007 Ω·cm etched in a solution HF:H2O2:H2O 4:1:35 v/v/v. The inset is a magnification at 250,000X.

  2:19.8 v/v/v. The SEM micrograph presented in Fig. 10 (a) shows a dendritic surface morphology of white sections (Si nanowalls) surrounded by dark regions (trenches). Fig. 10 (b) shows a cross section view of the same sample, indicating the formation of nanowalls with a height of 3.0 ± 0.1 µm and the etching rate 0.10 µm/min.

Fig. 10 .

 10 Fig. 10. SEM images of Si with resistivity of 0.0005-0.0007 Ω·cm etched in a solution HF:H2O2:H2O 20:0.2:19.8 v/v/v: (a) surface morphology and (b) cross section view.

Table 1 .

 1 Summary of the optimal etching solutions used to produce nanowalls out of Si with the three resistivities studied in this work.

		Volume proportion
	Si Resistivity	HF	H2O2 H2O	Height	Etching rate
	(Ω·cm)				(µm)	(µm/min)
	15-25	4	1	35	11.0 ± 0.7	0.3-0.4
	0.01-0.02	4	0.2	35.8	13.0 ± 0.9	0.4-0.5
	0.0005-0.0007	20	0.2	19.8	3.0 ± 0.1	0.08-0.10

Acknowledgements

The authors acknowledge the support of BUAP with the project VIEP2019-100523072 and the support of CONACyT with the project CB-2014-01-243407 and through the Mexican Energy Storage Network. The author V. Aca-López thanks CONACyT for a graduate student's scholarship.


Zakariya Ghalmane

Chantal Cherifi

Hocine Cherifi

Mohammed El Hassouni

A Community-Aware Backbone Extractor for Weighted Networks

Introduction

Network science provides effective tools to model and analyze complex systems. However, the increasing size of real-world networks becomes a major obstacle in order to understand their structure and topological features. Therefore, reducing the original network into a smaller one while preserving its topological features is an important issue. Extracting the so-called backbone of a network is a very challenging problem that is generally solved either by coarse-graining or filter-based methods. Coarse-graining methods reduce the network size by grouping similar nodes, while filter-based methods map the network by discarding nodes or edges based on a statistical property. In this work [START_REF] Ghalmane | Extracting Backbones in Weighted Modular Complex Networks[END_REF], we propose and investigate a filter-based method exploiting the overlapping community structure in order to extract the backbone in weighted networks. In the proposed method, called "overlapping nodes ego backbone", the backbone is formed with the set of overlapping nodes and their neighbors. It is inspired from the results of a previous study [START_REF] Ghalmane | Exploring Hubs and Overlapping Nodes Interactions in Modular Complex Networks[END_REF] showing that overlapping nodes and hubs are generally neighbors in real-world networks. Once the sub network made of the overlapping nodes and their neighbors have been extracted, links with low weights are removed from the network as long as the biggest connected component is preserved.

Experimental evaluation has been performed with real-world weighted networks originating from various domains (social, co-appearance, collaboration, biological, and technological) and different sizes. Extensive Comparisons with the popular disparity filter method [START_REF] Serrano | Extracting the multiscale backbone of complex weighted networks[END_REF] demonstrate the greater ability of the proposed method to uncover the most relevant parts of the network.

Methods

The"overlapping nodes ego backbone" consists of the set of overlapping nodes and their neighbors. The algorithm used to extract the backbone is detailed below:

Step 1: Form the overlapping nodes ego sub-network. It is made of the union of the set of overlapping nodes and the set of their first neighbors. It is obtained by removing all nodes that do not belong to one of the two sets.

Step 2: Remove the edges with low weights from the overlapping ego sub-network. To do so, edges are sorted in decreasing order according to their weights. Then, edges with low weights are removed as long as the sub-network largest component is not split into two components. This ensures that the backbone is formed with a single component.

Step 3: Control the size of the overlapping ego backbone with a parameter s. This parameter allows to preserve only the top-ranked nodes of this backbone. To this end, all the nodes of the overlapping nodes ego backbone are sorted in decreasing order according to the weighted degree centrality [START_REF] Opsahl | Node centrality in weighted networks: Generalizing degree and shortest paths[END_REF]. Finally, according to the value of this threshold parameter, the nodes with low degrees are removed from the network.

Results

Table 1. N is the network size. A n represents the estimated values the proportion of common nodes of two backbones. < β > is the average node betweenness. < w > is the average link weight. OE stands for the overlapping nodes ego backbone, while DF stands for the disparity filter backbone. A set of experiments is performed to compare the overlapping nodes ego backbone with the disparity filter which is recognized as one of the most effective alternative method. Only two experiments are reported here. The SLPA algorithm is used to uncover the overlapping community structure of various real-world networks. Note that in all the experiments, the size of the backbone is limited to 30% of the size of the original network by setting the parameter s to 0.3. Accordingly, the parameter α is tuned in order to obtain a disparity filter backbone of the same size.

Network N A n (%) < β > < w > OE-
At first, the proportion of common nodes extracted by the two methods is computed. It is defined as the fraction of the size of the intersection between the two sets divided by their size. Table 1 reports the proportion of common nodes computed between the proposed backbone and the disparity filter. Overall, the overlap is more or less pronounced. The main difference lies in the fact that the disparity filter concentrates on links while the proposed method is based on nodes in its extraction process. This can be observed in Les Miserables network reported in Figure 1. One can notice that the disparity filter backbone misses some very important nodes such as "Marius" and "Cosette" that are among the main characters in the Victor Hugo's novel. This example is a good illustration that the proposed approach preserves almost all high-connectivity nodes and essential connections.

In the second part of the experiments, the performance of the two backbone extraction methods is compared by measuring the average betweenness. information can pass through the nodes of the backbone. The values of the average betweenness of the proposed backbone are higher than the ones computed in the disparity filter backbone. This implies that the nodes extracted by the overlapping nodes ego backbone act as a better information gateway of the original network as compared to the disparity filter backbone. Its performance is higher than the disparity filter backbone for all the tested empirical networks. Finally, the average link weight between both backbone extractors is compared. A higher value of this measure demonstrates that the picked links are quite relevant. Experimental results reported in Table 1 show that the backbones extracted by the proposed method have a higher average link weight as compared to the disparity filter backbones. Thus, some very relevant connections are missed in the disparity filter backbone. All these experiments confirm the ability of the overlapping nodes ego backbone to preserve nodes playing a major role in the network.

Fig. 1 .

 1 Fig. 1. The backbone extraction of different methods for Les Miserables network. (a) overlapping nodes ego backbone, (b) disparity filter backbone. Nodes are highlighted in different colors according to the community they belong to. Nodes with the same color belong to the same community while those in gray represent the overlapping nodes. The size of the nodes is proportional to their weighted degree, while the size of links is proportional to their weights.

  Table 1 reports the results for all networks under test. The average betweenness indicates how much


LAREMA Jean-Pierre Magnot
email: jean-pierr.magnot@ac-clermont.fr

Inconsistency in pairwise comparisons as an Abelian Yang-Mills theory

Keywords: Pairwise comparisons, Abelian Yang-Mills theory, inconsitency

We precise the correspondence between (classical) pairwise comparisons and some inconsistency indicators with the Abelian Yang-Mills theory in quantum physics. This work has already been performed in a more general framework by the author, who feels the need of a more explicit description of this correspondence in the classical framework used in decision theory. This communication is addressed to nonmathematician/physicists, and intends to provide a more comprehensive description of already published results.

INTRODUCTION

We have recently linked explicitely inconsistency in pairwise comparisons (PC) with Yang-Mills theories, showing a possible (non-Abelian) generalization of the classical framework [START_REF] Magnot | A mathematical bridge between discretized gauge theories in quantum physics and approximate reasoning in pairwise comparisons[END_REF], [START_REF] Magnot | On mathematical structures on pairwise comparisons matrices with coefficients in a group arising from quantum gravity[END_REF]. After exchanges and comments, it appears that our works, are ill-received due to the very general framework considered, envolving mild mathematical notions which are not familiar to specialists of operations research. In this communication, we intend to propose a travel guide to understand in which way the (non generalized, classical) PCs fit with an Abelian Yang-Mills theory. We finish by an open problem, from the viewpoint of decision theory, which seems important to us. The same problem can be addressed in physics, namely in Electro-Magnetic fields encoded by a Yang-Mills action functional, but this version will be developped elsewhere in the future.

More precisely, we present a pedestrian approach of the announced correspondence the following way.

• We first recall basics on inconsistency in pairwise comaprisons. Our approach here is partial by choice. Indeed, there exists many non-equivalent ways to measure the inconsistency of a pairwise comparisons matrix. We concentrate our attention on one of them, Koczkodaj(s inconsistency indicator, which realizes the most obvious link with Yang-Mills theory. • After that, we briefly discuss the consitencization of a pairwise comparisons matrix, mostly along the lines of a recent paper [START_REF] Koczkodaj | On Orthogonal Projections on the Space of Consistent Pairwise Comparisons Matrices Fundamenta Informaticae[END_REF]. We address a brief critique of this method. Along our explanations, we need to describe the correspondence between multiplicative and additive pairwise comparisons matrices.

• Then, the link between multiplicative and additive pairwise comparisons matrices is analyzed from the viewpoint of Lie theory, which directly leads to the gauge theoric framework of Yang-Mills fields. The notions of holonomy and discretized connection are outlined. This section also intends to be an introduction to [START_REF] Magnot | On mathematical structures on pairwise comparisons matrices with coefficients in a group arising from quantum gravity[END_REF] for the reader who is not specialist of gauge theories. • We conclude by focusing on the notion of inconsistency indicator, which apears to us as central (while most people working in decision theory seem to not care about the choice of the inconsistency indicator they use). Here, we precise that, to our opinion and considering the goal of a consistencization procedure, the prefered method should be a gradient method, which is not the most employed method in the classical litterature for decision making with pairwise comparisons. If so, the choice of the inconsistency indicator appears as the main feature in a modelization procedure. These non consensual considerations should be explored in future studies.

I. PAIRWISE COMPARISONS MATRICES WITH

COEFFICIENTS IN R * + A paiswise comparisons matrix (a i,j ) is a n×n matrix with coefficients in R * + = {x ∈ R | x > 0} such that ∀i, j, a j,i = a -1 i,j .
It is easy to explain the inconsistency in pairwise comparisons when we consider cycles of three comparisons, called triad and represented here as (x, y, z), which do not have the "morphism of groupoid" property such as x * z = y, which reads as xz = y in the multiplicative group R * + . Evidently, the inconsistency in a triad (x, y, z) ∈ R 3 is somehow (not linearly) proportional to y -xz. In the linear space, the inconsistency is measured by the "approximate flatness" of the triangle. The triad is consistent if the triangle is flat. For example, (1, 2, 1) and (10, 101, 10) have the difference y -xz = 1 but the inconsistency in the first triad is unacceptable. It is acceptable in the second triad. In order to measure inconsistency, one usually considers coefficients a i,j with values in an abelian group G, with at least 3 indexes i, j, k. The use of "inconsistency" has a meaning of a measure of inconsistency in this study; not the concept itself. The approach to inconsistency (originated in [START_REF] Koczkodaj | A new definition of consistency of pairwise comparisons[END_REF] and generalized in [?]) can be reduced to a simple observation:

• search all triads (which generate all 3 by 3 PC sub matrices) and locate the worst triad with a so-called inconsistency indicator (ii), • ii of the worst triad becomes ii of the entire PC matrix.

Expressing it a bit more formally in terms of triads (the upper triangle of a PC sub matrix 3 × 3), we have:

Kii(x, y, z) = 1 -min y xz , xz y . (1)
According to [START_REF] Koczkodaj | Axiomatization of Inconsistency Indicators for Pairwise Comparisons[END_REF], it is equivalent to:

ii(x, y, z) = 1 -e -|ln( y xz )|
The expression | ln( y xz )| is the distance of the triad T from 0. When this distance increases, the ii(x, y, z) also increases. It is important to notice here that this definition allows us to localize the inconsistency in the matrix PC and it is of a considerable importance for most applications.

Another possible definition of the inconsistency indicator can also be defined as:

Kii n (A) = max 1≤i<j≤n 1 -e -ln a ij a i,i+1 a i+1,i+2 ...a j-1,j (2)
The first Koczkodaj's indicator Kii 3 allows us not only to find the localization of the worst inconsistency but to reduce the inconsistency by a step-by-step process which is crucial for practical applications. The second Koczkodaj's indicator Kii n is useful when the global inconsistency indicator is needed for acceptance or rejection of the PC matrix. Following [START_REF] Magnot | On mathematical structures on pairwise comparisons matrices with coefficients in a group arising from quantum gravity[END_REF], these [0; 1]-valued inconsistency indicators (notice that the terminilogy fixed in [START_REF] Koczkodaj | On normalization of inconsistency indicators in pairwise comparisons[END_REF] imposes to such maps to be normalized, i.e. bounded by 0 and 1) can be gathered in an "inconsistency map" with values in formal series, by assuming that each Kii n furnishes the coefficient of the n-th monomial.

II. FROM MULTIPLICATION TO ADDITION: THE BASIC VIEWPOINT FOR MAKING CONSISTENT A PC MATRIX

This is well-known that the map ln is a morphism of groups from the multiplicative group R * + to the additive group R, that is, ∀(x, y) ∈ (R * + ) 2 , ln(xy) = ln(x) + ln(y).

Its inverse, the exponential map, satisfies also a similar property:

∀(x, y) ∈ R 2 , exp(x + y) = exp(x) exp(y).
Hence, PC matrices for multiplication can be transformed bi-univoqely, to PC matrices for addition. This correspondence is the key motivation for [START_REF] Koczkodaj | On Orthogonal Projections on the Space of Consistent Pairwise Comparisons Matrices Fundamenta Informaticae[END_REF] where an apparently natural way to link any PC matrix with a "prefered" consistent one is described. This impression of simplicity and clarity is obtained by the use of basic geometrical mathods, mostly the use of orthogonal vector spaces. However, we have to mention that this method my appear as very arbitrary in a multiplicative language. Indeed, one should prefer to make the inconsistency indicator, for example Kii 3 , as small as possible and with as few iterations as possible in the method.

III. MULTIPLICATION VERSUS ADDITION: THE VIEWPOINT OF LIE THEORY AND QUANTUM GRAVITY

In a Lie theoric viewpoint [START_REF] Kobayashi | Foundations of Differential Geometry I and II Wiley classics library[END_REF], R * + is called an Abelian Lie group, due to the smootness of the multiplication operation. Its tangent space at 1 is called its Lie algebra, and in this case the Lie algebra can be identified with R. The exponential map exp : R → R * + is here a (smooth) diffeomorphism. Now, following [START_REF] Magnot | A mathematical bridge between discretized gauge theories in quantum physics and approximate reasoning in pairwise comparisons[END_REF], [START_REF] Magnot | On mathematical structures on pairwise comparisons matrices with coefficients in a group arising from quantum gravity[END_REF], let n be the dimension of the PC matrices under consideration, and let us consider the n-simplex

∆ n = (x 0 , . . . , x n ) ∈ R n+1 | n i=0 x i = 1 ∧ (∀i ∈ {0, . . . .n}, x i ≥ 0)
as a graph, made of n(n-1) 2 edges linking n vertices, equipped with a pre-fixed numerotation. Then there is a one-to-one and onto correspondence between edges and the positions of the coefficients in the PC matrix. Hence, each PC matrix (a i,j ) assigns the coefficient a i,j to the (oriented) edge from the i-th vertex to the j-th vertex. These numbers are called holonomies in mathematics and physics, and the holonomy of a path is the product of the holonomies of its edges. Then one can read the formula (2) as the maximum the holonomy of a loop with n edges. Minimizing holonomies is required in a model in physics called Yang-Mills theory [START_REF] Rovelli | Covariant loop quantum gravity[END_REF], [START_REF] Sengupta | Gauge Theory in Two Dimensions: Topological, Geometric and Probabilistic Aspects[END_REF], [START_REF] Sengupta | Yang-Mills in Two Dimensions and Chern-Simons in Three[END_REF]. There exists various approaches, but the one which is more of interest for us is the quantum gravity approach [START_REF] Rovelli | Covariant loop quantum gravity[END_REF], where the seek of minimization of the distance between the holonomies and 1 is outlined. Let us only mention one apparent difference: the classical approaches of Abelian Yang-Mills theory very often consider the circle

U (1) = {z ∈ C | |z| = 1} instead of R *
+ as the suitable group for the coefficients (called structure group in the litterature). We have to precise that the presence of R * + simplifies technical issues.

In particular, it is possible to consider the (additive) PC matrix (ln(a ij )) without any restrictive additional assumption. This PC matrix, used in the "naive" approach [START_REF] Koczkodaj | On Orthogonal Projections on the Space of Consistent Pairwise Comparisons Matrices Fundamenta Informaticae[END_REF] represents the so called discretized connection along the lines of [START_REF] Whitney | Geometric Integration Theory[END_REF], which encodes (first order) infinitesimal holonomies. In this picture, a distance between the family i,j ln(a ij ) and 0; where the double indexes i, j run through the coefficients along a loop, needs to be minimized.

IV. OPEN PROBLEM: "WHICH DISTANCE" MEANS "HOW TO MINIMIZE INCONSISTENCY"

One of the most efficient ways to minimize a functionnal is the well-knwon gradient method. This method consists in determining in which direction (the gradient) the decay to the map is faster. The gradient is directly derived from the first derivative of the map with values in R. Other methods may work too, but if one requires the most efficient changes in the evaluations of inconsistency, with the less changes in the coefficients of the initial PC matrix, the best method uses the gradient.

Let us now turn back to Koczkodaj's inconsistency indicators, slighly reformulated:

Kii n (A) = 1 -exp -max 1≤i<j≤n ln a ij a i,i+1 a i+1,i+2 . . . a j-1,j .
The presence of the max operator refers to the intrinsic use of the norm ||.|| ∞ defined by

∀x = (x 1 , ..., x m ) ∈ R m , ||x|| ∞ = max{|x 1 |, ..., |x m |}.
This norm does not admit a derivative at each point of its domain. By the way, the "smooth" approach of [START_REF] Koczkodaj | On Orthogonal Projections on the Space of Consistent Pairwise Comparisons Matrices Fundamenta Informaticae[END_REF] proposes a way to minimize inconsistency, but this cannot the one with the less changes in the coefficients.

In order to recover now the quemtum gravity analog, one replaces max 1≤i<j≤n (...) by a quadratic mean, one recovers the so-called Yang-Mills action functional, which is smooth and on which the gradient method applies.

In such a context, our question is methodological and not mathematical. Minimization of the inconsistencies can be performed by many ways, and with one which has to be preferred, namely the gradient method. Even if for Koczkodaj's inconsistency indicator (for example) the non-differentiability of the formulas requires more technical attention, one may circumvent these difficulties via refined applications of the gradient method, which is largely generalized to many classes of not-so-regular functions. But before that, we have to open the question of what is meant by "minimizing inconsistency". This meaning is encoded in the inconsistency indicator under consideration (one may say that the inconsistency indicator is a modelization of the preferences of the users) and there exists many of them, very differents. This question is strengthened by the link with Yang-Mills in quantum gravity, where theoretical physicists prefer a mean of the holonomies to the maximal one.