1. Introduction

Modern silicon-based photonics is based on functional nano-scale [1,2] and micro-scale [3] structures. In turn, femtosecond laser ablation has been successfully used for creation of broadband highly sensitive photodetectors [4,5], hierarchical superhydrophobic surfaces [6,7], and multifunctional structures based on such kind of structures. Generally, it was concluded that the surface nano- and microstructuring of different materials by laser pulses with ultrashort (femtosecond) duration has an advantage over longer pulses owing to reduction of the heat affected zone [8–14], allowing the formation of the hierarchical surface structures in scanning mode [15] with tailored morphology [16]. For relatively long (sub-ns and longer) laser pulse durations surface hydrodynamycal instability is widely accepted as the major mechanism of the microstructures formation [11,17,18]. At ultrashort laser pulse durations (much less than electron-phonon coupling time, i.e. shorter than ps-scale) this mechanism was proven to be responsible for nanostructures formation [19,20], because the molten depth is reduced down to submicron level [21] and becomes too shallow to support development of multi-micron scale instabilities. Fs-laser induced ablation takes place even in case of relatively shallow molten pool, causing in most cases removing of material layer with thickness of about 10 – 100 nm per pulse [21]. It means that each ultrashort laser pulse can precisely imprint subwavelength features near any nanostructure [22], yielding nonlinear optical feedback during multipulse irradiation of the surface. The nonlinear feedback was studied recently for periodical nanostructures [20,23] and quasiperiodical few-microns microstructures [20,24]. However, detailed study of nonlinear optical feedback during large conical microstructures formation is still missing.

In this work, we demonstrate and study nonlinear optical feedback during fs-laser microstructures formation on silicon. We show their growth from subwavelength to multi-micron sizes, covering wide range of possible silicon-based photonic designs. The analysis of influence of wavelength, fluence, exposure time, and air pressure allows us to build detailed physical picture of the fs-laser micropatterning.

2. Experimental methods

In our experiments, 744-nm, τ_IR ≈100-fs linearly-polarized laser pulses of a Ti:sapphire laser system (Avesta Project Ltd.), including regenerative and multi-pass amplifiers, are frequency-tripled to provide third-harmonic (248 nm) pulses with pulsewidth τ_UV ≈60 fs (FWHM). Maximum pulse energies at repetition rate 10 Hz are 5 mJ at λ = 744 nm (IR) and 0.5 mJ at λ = 248 nm (UV) in laser radial TEM₀₀ mode. The laser pulse energy is varied and monitored by means of an attenuator and a pyroelectric energy meter (OPHIR), respectively. The IR (UV) laser pulses are focused by a BK-7 lens with focal length 11 mm (a fused silica lens with focal length 215 mm) at the normal incidence into a spot with 1/e-level radius r_1/e≈150 μm (50 μm) on a dry polished surface of a 400-μm thick monocrystalline Si(100) slab, mounted in a motorized three-dimensional translation stage (Standa). Local laser fluence F is calculated as following: F(r) = F₀∙exp(-r²/ r²_1/e), where F₀ is peak laser fluence and r is radius from the laser beam center [25]. Laser microstructuring of the sample is carried out in air at two pressures (normal and 10⁻² Torr) via static accumulation of laser pulses: N = 20, 30, 1 × 10², 3 × 10² and 1 × 10³. The structured areas are characterized by means of a field-emission scanning electron microscope (FE-SEM, JEOL 7001F) in the relief mode with magnification up to 500 000 × .

3. Influence of different parameters on surface microstructure formation

3.1 Effects of dielectric permittivity dependence on laser fluence

Figure 1(a) represents SEM image of the silicon surface irradiated by N = 20 IR fs-laser pulses with F ≈1.8 J/cm², where the typical transition between different types of periodical surface structures is shown. Particularly, ripples with period about 0.6 μm are formed within area of 0.3 < F < 0.6 J/cm², while microripples with transversal orientation to ripples and period 2 – 3 μm correspond to fluence range 0.6 < F < 0.8 J/cm². Also, one can observe ripples at higher fluences, which period is equal to the laser wavelength 0.74 μm (Fig. 1(c)).

 

Fig. 1 (a) SEM image of a part of irradiated silicon surface by N = 20 IR fs-laser pulses at F ≈1.8 J/cm². White arrow shows the values of incident fluence F and corresponding densities N_eh of generated electron-hole plasma (adopted from [32]). White double-headed arrow with latter e in the picture (a) indicates direction of fs-laser polarization. Spatial Fourier spectra of area within (b) red and (c) blue frames in picture (a), and its whole spectrum (d).

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Such transition between different structures occurs at fluences higher than the melting threshold F_IR,melt ≈0.23 J/cm² [21], the spallative ablation F_IR,spall ≈0.4 J/cm² [21] and near the fragmentation ablation F_IR,frag ≈0.7 J/cm² [21]. Indeed, there is dramatic variation of dielectric permittivity at such fluences owing to dense electron-hole plasma generation, band gap renormalization, and screening of ions Coulomb potential [26]. One of the most important effects for fs-laser surface structuring is switching a sign of a real part of dielectric permittivity at some fluence, when surface plasmon polaritons (SPPs) excitation and inhomogeneous laser energy deposition becomes possible [27,28]. The sign switching in silicon usually occurs during the pulsewidth [23,28], resulting in enhanced laser energy deposition at some moment. Therefore, according to such time-dependence of ε, it is more correctly to divide fs-laser pulse onto two parts: the first part changes dielectric permittivity, and the second part participates in inhomogeneous energy deposition on rough surface (SPPs excitation, diffraction, etc.).

The SPPs excitation is known to provide ripples formation with wavevector parallel to vector of laser polarization (type (i) in Fig. 1(a)) owing to their interference with the incident fs-laser pulse [27–30]. The period of such ripples depends on laser fluence, because their wavenumber depends on dielectric permittivity ε of the photoexcited material [31] as k_SPP = 2π/Λ = k₀(ε/(ε + 1)), where k₀ is the wavenumber of light in air. Concerning the silicon photoexcitation by IR fs-laser pulse, SPPs excitation becomes possible at F(t > τ/2) > 0.3 J/cm², where Re(ε) < −1 [32]. Thus, the ripples can be formed, if local fluence in maxima of SPPs-laser interference picture exceeds the spallation ablation threshold F_IR,spall ≈0.4 J/cm² [21]. It means that the threshold of ripples formation lies in the fluence range 0.3 – 0.4 J/cm², being in a good agreement with previous studies [25,33]. This range corresponds to electron-hole plasma density around 0.5∙10²² cm⁻³, according to previous measurements [32].

Under specific conditions, when −1 < Re(ε) < 1, it is possible to excite specific surface electromagnetic (EM) modes, which correspond to formation of another periodical surface structures (microripples) with wavevector parallel to the laser polarization and period by 2–4 times larger than the laser wavelength [30]. The microripples were observed at IR fs-laser irradiation at fluences higher than threshold of ripples formation [33]. Indeed, at fluence of IR fs-laser pulse F(t = τ/2 – τ) > 0.65 J/cm² value of Re(ε) appears in range between −1 and 1 [29]. In our experiments the ring of the microripples corresponds to fluence range F ≈0.7 – 0.9 J/cm² and their period is about (2 ± 0.5) μm ≈3λ (type (ii) in Fig. 1). This range corresponds to electron-hole plasma density around 10²² cm⁻³, according to previous measurements [32].

At fluences F > 0.9 J/cm² the complex dielectric permittivity consists of Re(ε) << −1 and Im(ε) >> 1 [32], yielding k_SPP ≈k₀, which provides relatively small SPPs fields within surface layer, resulting in less pronounsed ripples formation with near-wavelength period (Fig. 1(a),1(c)). In opposite, the strongest SPPs fields correspond to Re(ε) ≤ −1 and Im(ε) < 1, i.e. where ripples (i) and microripples (ii) are overlapped.

The results of this Section show the essential role of fluence-dependent optical response on the type and position of periodical surface structures. Further, we will show, why it is also important for the microcones formation.

3.2 Effect of diffraction by single photoexcited microcone

The microconical surface structures were observed at much higher fluences F ≈1.8 J/cm² (Fig. 2). In opposite to the (micro)ripples, they appear randomly. Further, we will show that they are strongly affected by laser fluence, the number of laser shots and ambient gas pressure, but in this Section, we discuss the influence of diffraction on their growth. Indeed, one can clearly see a ring around each microcone (Fig. 2) caused by enhanced local ablation due to diffraction pattern formation.

 

Fig. 2 SEM image of microcones formed on a silicon surface after exposure by N = 20 IR fs-laser pulses at local fluence F ≈1.8 J/cm².

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To understand the role of diffraction, we analyze intensity distribution on a silicon surface with a single microcone in two-dimensional case. Such kind of problem has been solved analytically elsewhere [34]. However, to deal with real geometrical parameters and dielectric permittivity, we use the finite-elements method (Comsol Multiphysics) of numerical solving the wave equation for electric field E (∇ × ∇ × E + (k₀)²εE = 0, where ∇ is the vector of partial derivative operators) at given boundary conditions. Two-dimensional computational area with nonreflecting boundaries has length 100 μm and height 20 μm. In Fig. 3(a) the decimal logarithm of the ratio of resulted E and incoming E₀ electric fields is shown. Arrows with inscribed values of intensity enhancement coefficient ξ point to the most intensive maxima of interference between incoming and diffractive fields.

 

Fig. 3 (a) Calculated distribution of local electric field E enhancement relatively to incident E₀ electric field at λ = 744 nm near a microcone with height H, diameter D, and curvature radius r. Blue arrows with inscribed values of intensity enhancement coefficient ξ point to the most intensive maxima of the interference pattern. Black arrows show polarization (e) and wavevector (k) of incident plane wave. Dependencies of the intensity enhancement coefficient ξ on the microcone height H at different aspect ratios both for (b) λ = 248 nm and (с) λ = 744 nm.

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Under EM intensity enhancement coefficient ξ on surface (along the line z = ˗ 0) we imply ξ = (|E|²/|E_sm|²)_{z =} ˗ ₀, where E (E_sm) is the electric field in case of presence (absence) of microcones on surface in the computing area, and the line z = 0 corresponds to border between silicon and vacuum (ε = 1). Due to generation of dense (N_eh = 10²¹-10²² cm⁻³) electron-hole plasma near the ablation threshold for IR fs-laser pulse, dielectric permittivity of photoexcited silicon strongly depends on laser fluence. For our calculations at λ = 744 nm and F ≈1.5 J/cm², we take the dielectric permittivity of photoexcited silicon as ε(744 nm) = 1.1 + i10.6, taking into account such effects as Drude metallization, plasma screening of ions Coulomb potential and band gap renormalization [32]. Concerning the UV fs-laser photoexcitation, these effects are not so sufficient owing to decay of Drude part in ε with laser frequency as ω⁻² (i.e. approximately by one order of magnitude in comparison with IR radiation). Moreover, band gap renormalization is weak due to relatively small deviation of ε at λ > 248 nm. Therefore, we use the data for silicon at normal conditions ε(248 nm) = - 7.39 + i11.15 [35].

The results of the modeling show that, interference maxima are formed near single microcone, and enhancement in the first maximum (Fig. 3(a)) increases with growth of the microcone height H and diameter D at fixed aspect ratios AR = H/D. Moreover, the enhancement decreases with increasing of the aspect ratio. It means that the intensity enhancement can demonstrate nonmonotonic character in case of microcone growth without aspect ratio conservation. Such tendencies are relevant for both wavelengths (Fig. 3(b),3(c)), but ξ is higher for shorter wavelength at same geometrical parameters of microcone owing to more effective diffraction [34]. The radius of curvature is taken as r = D/6. It should be noted that such single isolated microcones can be formed only on early stage of their growth, and their height in such state usually does not exceed 5 – 7 μm. Our calculations show that for the early stage of microcone growth the following enhancements can be achieved: ξ < 2.7 at λ = 248 nm (Fig. 3(b)) and ξ < 2 at λ = 744 nm (Fig. 3(c)). In next sections, we will analyze next stages of microcones growth, where they become larger and affect neighbor microcones via diffraction.

3.3 Effect of the number of laser pulses

Threshold of silicon surface nanostructuring at UV fs-laser irradiation is known to be approximately by 4 times smaller than for IR pulses [33]. Therefore, the evolution of microcones on silicon surface is studied in range of exposure parameters of N = 30 – 10³ fs-laser pulses both at λ = 744 nm and 248 nm with peak fluences 1.5 J/cm² < F₀ < 4 J/cm² and 0.1 J/cm² < F₀ < 0.5 J/cm², respectively.

At both wavelengths, an average volume of the microcones increases with increasing of fluence and the number of absorbed laser shots (Fig. 4, 5(a)). The average volume V is determined as V = $V={\left({\displaystyle {\sum }_{i=1}^n{V}_i/n}\right)}_{F=F0}$, where V_i = (π/12)·(D_i)²·H_i is the volume of each microcone, D and H is the diameter and height of each microcone, n is the number of taken microcones (usually in range n ≈6 – 10) in the central area where fluence is almost constant (0.9F₀ ≤ F ≤ F₀). The main difference between the microcones geometry dependences on N for IR and UV fs-laser pulses is their aspect ratio (AR) behavior at the early stage of their evolution (Fig. 5(b)). In the case of IR pulses, AR demonstrates continuous growth, while under the UV irradiation AR decreases firstly and then becomes similar to the IR values. It is worth noting, that at the initial stage the values of the average microcones volume for IR and UV are comparable, but AR of UV-microcones is by several times larger than that of the IR-microcones. Concerning the dynamics of the microcones growth (H, D and V), it is much faster at the early stage (N < 100), but further it is saturated or even decreased (Fig. 4 and 5(a),5(c)). Also, there is similar dependence of typical scale (period Λ) between the microcones on the number of laser shots (Fig. 5(d)), indicating that the period Λ for UV fs-pulses by 2 times smaller in comparison with IR fs-pulses. However, with increasing of the laser shots number this difference is diminished.

 

Fig. 4 SEM images of microcones on silicon surface fabricated in air at normal pressure by IR fs-laser pulses with F ≈1.8 J/cm² and (a) N = 30, (b) N = 100, (c) N = 300, (d) N = 1000, and by UV fs-laser pulses with F ≈0.44 J/cm² and (e) N = 30, (f) N = 100, (g) N = 300, (h) N = 1000. The scale bar in the picture (a) is relevant to all images. Angle of view is 45 degrees.

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Fig. 5 (a) Experimental dependence of microcone average volume on the number of laser pulses N at atmospheric pressure for IR (λ = 744 nm) fs-laser pulses (red line) and for UV (λ = 248 nm) fs-laser pulses (blue line). (b) Experimental dependence of aspect ratio AR = H/D on the number of laser pulses N for IR (red line) and UV (blue line) irradiation at atmospheric pressure. (c) Experimental dependences of height H (triangles) and diameter (squares) of microcones on the number of laser shots at λ = 248 nm (blue) and λ = 744 nm (red). (d) Experimental dependence of mean period of microcones on the number of laser shots at λ = 248 nm (blue) and λ = 744 nm (red). For all pictures, IR and UV fs-laser pulses have local fluences around F ≈1.8 ± 0.2 J/cm² and F ≈0.44 ± 0.05 J/cm², respectively.

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3.4 Effect of ambient gas pressure

Besides the laser parameters, the microcones properties strongly depend on type and pressure of ambient medium (gas or liquid). For instance, chemically active gases (SF₆, HF, etc.) affect silicon microcones’ shape, gas pressure usually affects size of the microcones [15], and the use of liquids leads to formation of high-aspect ratio submicron spikes [18]. We also reveal strong influence of ambient air pressure on the microcones parameters in the experiments on microcones formation in vacuum (10⁻² Torr) both under IR and UV fs-laser urradiation. In Fig. 6, the comparison of microcones formation under UV fs-laser pulses at normal pressure and vacuum is shown. At normal pressure and N = 30 (Fig. 6(a)), the growth of the microcones is much more prominent (the same tendency for IR fs-pulses). However, at the next stages (N > 100), the microcones in vacuum become larger than corresponding microcones in air (Fig. 6(b)-6(e)).

 

Fig. 6 SEM images of microcones in the crater center after the irradiation by UV fs-laser pulses with F ≈0.22 J/cm² and (a) N = 30, (b) N = 100 at normal pressure, and in vacuum at (с) N = 30, (d) N = 100. (e) Dependence of average volume on the number of UV fs-laser pulses for F ≈0.44 J/cm² at normal pressure (black triangles) or at 10⁻² Torr (open triangles), and for F ≈0.22 J/cm² at normal pressure (black circles) or at 10⁻² Torr (open circles).

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3.5 Effect of local laser fluence

Concerning the very beginning stage of the microcones formation, several main types of their precursors are revealed. In Fig. 2 and 4(a), the scenario for the microcones formation on random roughness, corresponding to high-fluence regime at both wavelengths, is shown. On such relatively undisturbed surface one can see diffractive rings around the individual microcones The microcones are observed not only in the centers of irradiated area, but also at its periphery (Fig. 7), where local fluence lies in range 0.4 < F < 0.8 J/cm² at IR and 0.03 < F < 0.1 J/cm² at UV. Particularly, the microcones are mostly formed within the region of the periodical structures (ripples and micro-ripples) formation and near the edge of the spallative crater (Fig. 7). Their evolution with increasing of the number of shots is very similar to the central microcones one.

 

Fig. 7 SEM images of a periphery area irradiated by (a) N = 30, (b) 10², (c) 3·10², and (d) 10³ IR fs-laser pulses with peak fluence F₀ ≈1.8 J/cm² in air.

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Since two separated fluence-depended sets of microcones are formed within one area of surface irradiated by a Gaussian beam, the microcones at relatively low fluences (at the crater periphery) will be named by Low-Fluence Microcones (LFMs), while the microcones formed at higher fluences (in the center of irradiated area) are High-Fluence Microcones (HFMs).

The LFMs formation is also observed in vacuum (Fig. 8) at pressure 10⁻² Torr and, generally, they have similar character of evolution with that for normal pressure. More differences one can find between LFMs and HFMs. Walls of LFMs are covered by “fluffy” nanoroughness, while HFMs have smooth walls. AR of HFMs at the stage of intensive growth is much higher that of LFMs, where maximum AR value is about 1 (i.e. AR ≤ 1). Maximum growth rate of LFMs occurs around the border between ripples and microripples (Fig. 7 and 8).

 

Fig. 8 SEM images of periphery area irradiated by (a,b) N = 30, (c,d) N = 10², (e,f) N = 3·10², (g,h) N = 10³ IR fs-laser pulses at peak fluence F₀ ≈2.5 J/cm² in vacuum (10⁻² Torr).

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4. Discussion

4.1 Homogeneous versus heterogeneous nucleation of the microcones

Existence of two types of the microcones (HFMs and LFMs) cannot be explained only by hydrodynamical phenomenological explanations of the microcones growth, which were used for relatively long laser pulses [17,36]. Indeed, development of multi-micron hydrodynamic instabilities in a thick enough molten layer corresponds to rather high fs-laser fluence, which is typical for HFMs formation conditions only. In our experiments, HFMs formation on the silicon surface under N = 30 IR fs-laser pulses occurs at F > 1.5 J/cm², which correlates with previous studies [25]. This is significantly higher than the fragmentation ablation threshold of silicon for this wavelength F_frag(744 nm) ≈0.7 J/cm² [21], and corresponds to the molten pool with h_melt ≈150 nm [19]. Moreover, at F > 1.5 J/cm² the rapid expansion (v > 2 km/s) of fragmentation ablation plume (supercritical fluid), exerting recoil pressure with magnitude of about 20 GPa [21]. The corresponding artificial acceleration g ≈v²/Λ in the developing of hydrodynamic instability is about g ≈2·10¹² cm/s² at v ≈3 km/s and Λ ≈5 μm (see geometry in Fig. 2), being in range of values g = 10⁹ – 10¹³ cm/s², which is appropriate for giving raise of the instability development and growth of micro-scale structures [17,36]. We suppose, that such unstable initial surface behavior, assisted by diffraction on the spontaneously appeared microcones, causes rapid increase of local intensity and ablation enhancement near each microcone (Fig. 3), while SPPs are not sufficient at Im(ε) >> 0. Therefore, at rather high fluences, on smooth surface the microcones nucleation is homogeneous, i.e. the microcones are formed immediately from smooth surface.

Formation of LFMs under IR laser irradiation was observed at fluences about F ~0.4 – 0.7 J/cm², resulting in good correlation with our previous estimation of silicon microstructuring threshold F_th(N ≈30) ≈0.4 J/cm² at the same wavelength [33]. Under irradiation by UV (248 nm) fs-laser pulses, such microcones are observed at fluences high than F_th(N = 30) ≈0.18 J/cm² [33], being in good agreement with our measurements.

There is significantly less probability of homogeneous microcones nucleation on smooth surface under LFMs formation conditions, i.e. at F < 0.7 J/cm². Indeed, at such fluences the molten pool has depth around h_melt ≈50 nm [21] and undergoes recoil pressure of 2 – 6 GPa by the spalled layer leaving the surface with a velocity around v ≈0.5 – 1.5 km/s [21]. From the estimation of corresponding artificial acceleration g_RT ~10¹⁰ – 10¹¹ cm/s², one can see that these values are much less than the value of g_RT for HFMs formation. On the other hand, excitation of SPPs dominates here, because real part of dielectric permittivity is close to the plasmon resonance, i.e. ε ≈-1 [31], in range F = 0.3 – 0.7 J/cm², resulting in strong local intensity enhancement [19,20,23,27,28]. Therefore, LFMs location along some annular line (threshold-like behavior), where ripples and microripples are crossed, can be attributed to optical properties of photoexcited silicon in this region. Relatively strong enhancements of EM field and ablation provide faster material accumulation and the microcones nucleation [37], resulting in their growth in the region between ripples and microripples along certain annular line. Additionally, the re-deposition of ablated material from the spot center takes place in this area. Since LFMs occur only on some surface features (periodical structures or spallative edge [37]), rather than smooth surface, this process should be considered as heterogeneous nucleation of the microcones. However, according to the experimental results, evolution of the both types of microcones on silicon demonstrates two main stages: nucleation and ripening. In the next sections, the origin of their growth will be discussed.

4.2. Diffraction: optical feedback of the microcones growth

Here we carry out systematic calculations for silicon, to reveal dependence of the enhancement factor ξ on different geometrical parameters of the microcones. The problem is solved numerically for two-dimensional case, where the modeled microcones have variable geometrical parameters: diameter D, height H, curvature radius of the microcone top r, and distance between tops of two neighbor microcones d (Fig. 9(a)). The chosen range of the parameters correlates with experimental ones taken from Fig. 4 and Fig. 5.

 

Fig. 9 (a) Calculated distribution of local electric field E enhancement relatively to incident E₀ electric field at λ = 744 nm near two microcones with height H, diameter D, curvature radius r, and separation distance Λ. Black arrows show polarization (e) and wavevextor (k) of incident plane wave. Dependences of maximum value of calculated intensity enhancement ξ between microcones separated by distance of Λ on their height H at fixed aspect ratios AR = 1 (blue), 2 (black) and 3 (red) for (b) Λ = 3.5 μm, λ = 248 nm and for (c) Λ = 7 μm, λ = 744 nm. Dash curves are given for comparison and correspond to the values of ξ for an individual cone depicted in Fig. 3.

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As was shown in Section 3, the enhancement factor ξ in the first diffraction maximum near a single microcone increases with increasing of height H at fixed aspect ratio AR, while ξ goes down with increasing of AR. Roughly, similar trends are observed for ξ between couple of identical microcones. However, at some distances Λ between tops of the microcones the enhancement becomes much higher in comparison with the case of a single microcone (Fig. 9). In Fig. 8(a), a typical picture of calculated intensity distribution near the couple of microcones is presented, where the diffraction maximum between the microcones yields high field enhancement.

We numerically investigate the dependences of ξ on H and λ at various AR, i.e. both H and D are varied. The distance between tops of the microcones pair is fixed for each wavelengths: Λ = 3.5 μm at λ = 248 nm and Λ = 7 μm at λ = 744 nm. It means that with increasing of H and AR (i.e. D) the flat surface section between two microcones decreases. The numerical analysis of diffraction by the couple of the microcones shows several main tendencies:

The comparison of microstructures evolution (rapid growth and saturation) and results of presented calculations shows qualitatively agreement between the experiment and theory, i.e. it explains the saturation in terms of decrease of the enhancement. However, the character of the microcones evolution with the number of absorbed laser pulses affected by specific of multipulse fs-laser ablation, which will be discussed in the next Section along with the process of ablative material redeposition on the microcones.

4.3. Mechanism of the microcones growth: ablation in interference maxima

The microcones demonstrate two typical growth (“ripening”) stages: 1) in the early stage, they grow separately via process of material redeposition; 2) then, they additionally start to fuse together. According to our results and previous studies, such dynamics of the microcones growth under fs-laser irradiation shows universal character for different wavelengths and materials [37,38].

Another important parameter is pressure (or density) of ambient media (gas or liquid). In previous studies taller microcones formation on titanium at lower pressures in air was reported in multipulse regime [15]. Moreover, size of microcones on silicon and distances between them do not exceed the few-micron scale in liquid environment [18,39]. In our experiments we observe nontrivial dependence on gas pressure (Fig. 5(e)): at early stage (N < 100) the microcones growth is faster at higher pressure, but afterward (N > 100) the microcones average volume becomes larger at lower pressure. Such dependence points out to high importance of matter transport conditions in environment during the microcones formation.

To describe such influence of ambient medium pressure, the typical distance of ablation plume expansion from the surface should be taken into account. Main parameter in this problem is size of ablation “sources” – the interference maxima caused by diffraction on the microcones. Our calculations of intensity distribution near the microcones show, that the typical size of the maxima is approximately equal to the laser wavelength. At fluences higher than the ablation threshold, one can consider each interference maximum as a source of ablation micro-plume, which can be described in terms of point-blast model [37]. In this model, pressure P at the micro-plume spherical front can be expressed as P ≈P_s(R_s/R)³ [40], where P_s is pressure on surface in the source, R_s is the source radius, R is the distance between the plume front and the surface. Typical height of micro-plume propagation from surface H_mp is determined by condition of balance between pressure at ablation front and ambient media pressure P(R = H_mp) = P_a. Using this condition and expression for pressure P, one can roughly estimate typical height of micro-plume propagation:

Numerical estimations of H_mp and comparisons of its values at different pressures are carried out for IR fs-laser pulses (λ ≈744 nm), because the main ablation parameters for such conditions are well-studied [21]. The microcones, being formed at fluence F ≈1.8 J/cm² (Fig. 4), provide typical enhancement coefficient about ξ ≈2 – 3 (Fig. 8). Therefore, the fluence in the source is enhanced up to F < 5 J/cm², resulting in increase of pressure at the surface up to P_s < 50 GPa [19]. According to our numerical electromagnetic calculations, typical radius of the source is about R_s ≈0.3 μm (Fig. 7(a)). Thus, substituting the values to formula (1) gives micro-plume propagation heights at different pressures H_mp(P_a = 760 Torr) < 24 μm and H_mp(P_atm = 10⁻² Torr) < 600 μm.

The roughly estimated heights can give only qualitative insight, because at some moment micro-plume expansion cannot be described as free [41]. Additionally, the micro-plume interacts with the walls of the microcones at the initial stage of propagation, which makes the process of ablative mass transport even more complex. These two factors show that the presented simple approach overestimates the micro-ablation plume propagation. Indeed, the measured experimental maximum height of the microcones at normal pressure is about of H_exp ≈13 μm, which is less (but comparable) than our theoretical estimation H_mp ≈24 μm. Concerning the microcones formation at lower pressure, both maximum theoretical (H_mp ≈600 μm) and experimental (H_exp ≈50 μm) values are higher than the corresponding values at normal pressure. On flat surface the microcones growth (nucleation) should be easily initiated with shorter plume penetration lengths (i.e. at higher pressures) owing to higher probability of ablated material redeposition [42], correlating with the observed slower evolution of microcones at lower pressure and at the early stage.

Despite the discussed dependence on gas pressure cannot be applied to chemically active environments, this approach might be useful for chemically neutral liquids as well as for gases. For instance, the sub-micron scale microcones formation under liquid layer can be attributed to much higher density of liquid in comparison with gas, resulting in much shorter micro-plumes expansion. Additionally, decreasing of characteristic distances between the microcones under liquid layer may be caused by decrease of the laser wavelength in medium with higher refractive index.

5. Conclusion

Presented in this work experimental and theoretical study reveal following physical picture of the microcones formation on silicon surface:

To conclude, we have demonstrated that both micro- and nanoscale structures have similar principles of formation. We believe that our rigorous experimental and theoretical study paves the way to controllable and low-cost fabrication of advanced optoelectronics devices with multilevel patterning of functional elements.