Electrostatic Interpretation of Phase Separation Induced by Femtosecond Laser Light in Glass

Abstract

Numerous studies on the effect of the femtosecond laser pulses in oxide glasses have been achieved over the last two decades, and several specific effects pointed out. Some of them are classical with respect to a laser treatment, such as thermally related effects, and are widely taken into account for applications. Other effects are directly induced by light, caused by its intricated spatiotemporal structure and associated properties: ponderomotive and polarization effects or coherence within the focal volume. These effects enable the development of forces that can lead to orientation effects. Among the specific resulting transformations from the light-induced effects in glass, the formation of so-called nanogratings was first pointed out in 2003 in silica glass. From this date, asymmetric organization into parallel nanoplanes, perpendicular to the laser polarization, have been found in many vitreous and crystalline compounds. While it is accepted that they arise from the same origin, i.e., a plasma organization that is eventually imprinted inside the material, uncertainties remain on the formation process itself. Indeed, since it exists several categories of nanogratings based on the final structuring (nanoporous phase separation, crystallization, and nanocracks), it can be expected that several processes are at the roots of such spectacular organization. This paper describes an approach based on electrochemical potential modified by an electronic excitation. The electric field induced during this process is first calculated, with a maximum of ~4500 kV/µm and a distribution confined within the lamella period. The maximal chemical potential variation is thus calculated, in the studied conditions, to be in the kJ/mol range, corresponding to a glass-to-crystal phase transition energy release. The kinetics aspect of species mobility is subsequently described, strengthening the proposed approach.

1. Introduction

One of the most spectacular effects of the interaction of tightly focused femtosecond (fs) laser pulses inside or at the surface of materials is the production of a special nanostructure called nanogratings (NG) or “Type II” modification regime. One example, inscribed in silica (SiO₂) glass, is provided in Figure 1. Inside the focal volume (here approximately 3 µm and 25 µm in width and length, respectively), the observed contrast takes the shape of dashes or lamellae, which are parallel to each other and perpendicular to the laser polarization orientation. After the discovery of these structures inside silica glass in 2003 [1], the substructure was found to be similar to an assembly of disks being a few µm in diameter [2].

This exemplar of NGs is the conventional view, since most of the work has been carried out in silica glass. But they have also been observed and reported in many other solid compounds. Overall, NGs can be sorted into several categories. The most popular, as mentioned above, is the one primarily revealed in SiO₂ with an arrangement of porous nanoplanes containing spherical nanopores of 10–20 nm diameter in size [1]. It is produced by oxide decomposition due to local temperature increase [3], sensitive to the laser polarization, and giving rise to form birefringence with a large retardance amplitude as a resulting property. This type of NGs induced in volume and by femtosecond (fs) laser were observed in several glasses or crystals, including doped silica with F, P, Ge, Cl, OH [4], GeO₂ glass [5,6,7], SiO₂-GeO₂ glasses [8,9], TiO₂-SiO₂ glasses [10,11], TeO₂ single crystal [12], sapphire [13], Al₂O₃-Dy₂O₃ binary glass [14], lithium niobium silicate [15], or titanium silicate glasses (ULE, Corning) [11] and even in multicomponent aluminoborosilicate glasses (Borofloat 33, Schott) [11]. In β-Ga₂O₃ [16,17] and in the Ba₂O₃-GeO₂-Ga₂O₃ glass family (BGG) doped with Na, K, or La, there is no laser irradiation conditions pointing out NG formation, although doping with Ta, Gd, or Zn showed otherwise [18,19]. This is an example to highlight that a small addition of dopants can significantly impact the dynamics of NGs formation.

To continue the above list, NGs were obtained in alumino-silicate families [19,20,21], i.e., alumino-silicate glasses including B₂O₃, Na₂O, and/or CaO that can vary in a large proportion. NGs are also found in crystals with small forbidden gaps such as Si, GaP, 4H-SiC, GaN, or GaS [22]. In recent work, the possibility to inscribe NGs was shown in sodium borosilicate glass (SBS) with the molar composition of 68% SiO₂, 27% B₂O₃, 4% Na₂O, and 1% Al₂O₃ [23]. This composition lies in the range of metastable liquation, which makes it possible to form a phase-separated structure in it by means of thermal treatment. Microregions with polarization-sensitive birefringence are formed when typ. ≥10⁵ fs laser pulses are deposited inside the glass. The laser pulse energy range enabling the formation of nanogratings and the phase shift obtained in the birefringent regions increase with the increasing number of writing pulses, and the formation of nanogratings is accompanied by the migration of sodium cations outside the laser-modified region.

Additionally, nanogratings inscription was demonstrated in nanoporous silica (sol-gel, Vycor) [24] and in binary sodium silicate glasses xNa₂O-(100−x)SiO₂ with x = 5 and 15. It revealed the effect of nano-periodical chemical differentiation inside NGs in the x = 15 glass composition, with the Na concentrated near the nanoplanes [25,26]. Very recently, Wang et al. [27] showed the formation of NGs in sodium germanate (SG) glass with a 5Na₂O-95GeO₂ molar composition. Unlike in sodium silicate glasses, an increasing Na₂O content to 10 mol% was reported to prevent NG formation in these compositions.

In Ref. [28], the authors also investigated the formation of NGs in SG glasses, but for Na₂O contents ranging from 3 to 22 mol.%. They revealed aspects of their inscription drastically different from those in fused silica, although their structure is shown to have much in common. This is another exemplar that the glass composition has a critical effect on the NG writing conditions. Moreover, a minimal number of pulses required to induce form birefringence grows exponentially with the Na₂O content. The formation of NGs in 22Na₂O·78GeO₂ glass is accompanied with precipitation of Na₂Ge₄O₉ crystals inside and around them. For completeness, it must be pointed out that a form of birefringent nanostructure, appearing at a low pulse energy regime, has been recently observed by Sakakura et al. [29] (labeled Type “X”, as opposed to “Type II” for NGs). It is a homogeneous production of asymmetric nanopores, oblate in the direction parallel to the laser polarization. One possible explanation is the destabilization of the oxide [30] or nanocavitation [31,32] by multiphoton ionization on lattice irregularities. The effect of nanopores flattening is attributed to near field enhancement with linear polarization [33]. For larger pulse energies (i.e., larger intensities), the induced plasma is observed to reorganize in arrays as predicted for higher energies [34,35] and drag the nanopores into nanoplanes.

A second type on NGs is found in oxide crystals and semiconductors (TeO₂ [12], Al₂O₃ [13], and Si [22]), where mostly arrangements of cracks, perpendicular to laser polarization, reveal the nanogratings. A third kind of NG is found in quartz crystal by periodic amorphization likely due to local fluctuations of thermal effect induced by the electron plasma structure [36]. Finally, a fourth kind of NG appears to be a partially crystalline nano-periodic structures. They were observed, for example, in 33Li₂O-33Nb₂O₅-34SiO₂ glasses (LNS glass precipitating LiNbO₃, [15]), in (33Li₂O-33Nb₂O₃-34-xSiO₂)-xB₂O₃ glasses (LNSB non-congruent glasses) [37], in 65Al₂O₃-35Dy₂O₃ glass (congruent glass composition for Dy₃Al₅O₁₂) [14], and in 20Na₂O-80GeO₂ glasses (congruent glass composition for Na₂Ge₄O₉ [28]). Obviously, the periodicity arises from an interplay between multiphoton ionization and the glass constituents.

In the case of 20Na₂O-80GeO₂ glasses, the crystallization occurs mainly around the NGs and in the experimental conditions of the study. However, it is not the case for 65Al₂O₃-35Dy₂O₃ glasses, for which crystallization is inside the NG region, but the experimental resolution cannot bring up information on its distribution. As for LNS, the only noncongruent glass studied, the NGs are produced through an allotropic phase separation (SiO₂ lamellas and LiNbO₃ in between) [15]. A thorough study of this irradiated glass in Ref. [38] demonstrated that the spatial periodicity of the phase separation is influenced by the induced plasma. Indeed, it is much larger than the spontaneous one, i.e., simply induced by a temperature elevation. However, for NGs composed of nanopores or cracks such as in silica, there is no periodic chemical migration. Moreover, in all models describing the mechanisms on NG formation, there is no feature that can explain a phase separation, because the material is always considered homogeneous; just some defects can play a role in the mechanism by seeding a hot spot of nanoplasma [33] and triggering a local thermal effect. The distribution difference between positive and negative charges is not taken into account and cannot explain the influence of light on the phase separation [39]. For that purpose, in this paper, we consider the charge distribution induced upon light irradiation, analyzing more deeply compared to our previous publication and showing semi-quantitatively that the quasi-free electron plasma arrangement may interact with the phase separation, opening a way to its engineering.

2. Background

2.1. NG Structure

For suitable laser parameters that are described in Ref. [40], the previously depicted micro-to-nano structuring for LNS glass can be observed using transverse electron microscopy (bright field). The results, taken from Raf. [38] and displayed in Figure 2a, highlight the observed nanoplanes (i.e., nanogratings) oriented perpendicular to the laser polarization. Additionally, the chemical analyses reported in Figure 2b show that, for a suitable laser energy, there is alternation of rather pure silica nanoplanes. Between them, there is an important concentration of LiNbO₃ for high pulse energies or a small one (nanocrystals) surrounded by the original glass otherwise. Moreover, it must be recalled here that there is a difference of nanostructure between the matter irradiated (large spatial period) and the matter surrounding this irradiated volume but nevertheless heated by thermal diffusion from the irradiated volume (small period of the phase separation). We pinpoint this transition from one to the other in Figure 2c. At this transition, the thickness of the white (crystalline) lamellae decreases as one moves out of the focus point (i.e., away from the light), and new lamellae start to appear between them (the periodicity decreases). The phase separation period decreases roughly from 100 nm to 10 nm. As the NG structuring appears in a large number of compounds and in several types of NG, we infer that the periodicity is imposed by the electron plasma structuring, in relation with some physical properties of the matter (still unknown to this date). The crystallization does not disturb the nanogratings organization. A direct deduction from this is that chemical separation is achieved before the crystallization.

2.2. About the Plasma Spatial Structuring

Plasma structuring is carried out by photoionization of the solid during several pulses [41], with the help of levels in the bandgap created by the previous pulses [42]. This explains how the quasi-periodicity can be maintained along the laser scan [43], because the defects are gradually concentrated in certain areas, those where the electric charge density is the strongest. Moreover, all it takes is a few pulses and the plasma self-structures as calculated in silica glass in Ref. [43]. Nevertheless, the electron mobility must be sufficient to produce self-organization, and the ion mobility not excessively large to avoid annihilation of this structuring at high temperatures.

It is worth considering how the very short time period of a laser pulse may trigger an effect on the matter. In fact, at a scanning speed of a few µm/s and with a repetition rate of 100 kHz, the pulse density is on the order of 10⁴ pulses/µm. Consequently, there is enough time to induce and maintain the fields, along with the induced potentials, which are imprinted in the solid. These fields will act in two ways: (i) thermodynamically by intervening on the constituent potentials and modifying the phase diagram and (ii) kinetically, by exerting forces on these constituents that can lead to structuring. A question that directly arises from this discussion is how does the plasma impose its structuring on the chemical separation?

3. Influence of the Electrostatic Field Induced by the Laser

In Ref. [43], the physical origin of the nanogratings in silica, constituted by nanopores arranged in quasi-periodic porous nanoplanes, is the cavitation of nanopores coupled to electromagnetic waves through temperature modulations perpendicular to the laser polarization induced by electron plasma energy transfer to the lattice. There is no chemical diffusion in this scenario but a destabilization of SiO₂ at the place of high electron density where the energy deposition is thus the largest. Consequently, in this paper, we investigate an alternative route to the observed chemical separation, where the plasma imprints its spatial structure on the chemical separation achievement through electrochemical potential, leading to the uniformity of the Gibbs-free energy.

At the ultrashort time scale, the space modulation of the light-excited electrons is recognized as being trapped into the glass lattice and re-excited by the following pulse, thus reinforcing the nanograting structure.

If the plasma structure is achieved during a first pulse of a few 0.1 ps, 10 to 100 pulses are necessary for building a nanograting in pure silica lattice, as shown in Ref. [2], for example. This mechanism plays a memory role in the plasma structure formation and thus may develop transformations on a much longer time scale than during a single pulse duration. We suggest that the mechanism here starts similarly. However, we notice that, if the starting excitation follows the light intensity profile, this is no longer the case after several pulses. This means that positive ions produced at the beginning and negative ions produced by the electron trapping do not have the same distribution, yielding to an electric field modulation along the laser polarization direction.

The excited electrons have a large mobility and an excited lifetime large enough, whereas the positive counterparts are fixed to the lattice under the defect forms such as self-trapped holes or Oxygen Deficient Centers [42].

Based on the above discussion and to calculate the electric fields, we assume that the excited charges on a period of NG (Λ) are collected on a circular plate of infinitely small thickness. This is exemplified in Figure 3.

The electric field E_Neg, assumed to represent the electron accumulation into nano-disks can be calculated along the z axis, as represented in Figure 3. The electric field developed by a circular plate infinitely thin and the radius R (set as 1.5 μm, i.e., the beam radius) yields to the following classical expression:

$$E_{Neg}\left(z\right)=\frac{\sigma }{2{\epsilon }_0{\epsilon }_r}\left[sign\left(z\right)-\frac{z/R}{\sqrt{1+{\left(z/R\right)}^2}}\right]$$

(1)

where σ is the surface charge density (σ = −Λρ, with Λ the spatial period of the NG around 200 nm and ρ the volume charge density of excited electrons), ε₀, and ε_r (≈4, see Appendix A) correspond to the vacuum and relative permittivity values, respectively.

The electric field of the positive counterpart, in turn, can be obtained assuming an homogeneous distribution on the period Λ. The expression is the following:

$$E_{Pos}\left(z\right)=\frac{\rho }{2{\epsilon }_0{\epsilon }_r}{{\displaystyle \int }}_{-\mathsf{\Lambda }/2}^{\mathsf{\Lambda }/2}\left[sign\left(z-za\right)-\frac{\left(z-za\right)/R}{\sqrt{1+{\left(\left(z-za\right)/R\right)}^2}}\right]dza$$

(2)

By setting the variable $x=\frac{z-za}{R}$, the above expression can be integrated into the following form:

$$\mathrm{that} \mathrm{is} \mathrm{read} \mathrm{as} E_{Pos}=\frac{\rho R}{2{\epsilon }_0{\epsilon }_r}\left(f\left(\frac{z-\frac{\mathsf{\Lambda }}{2}}{R}\right)-f\left(\frac{z+\frac{\mathsf{\Lambda }}{2}}{R}\right)\right), \mathrm{with} f\left(x\right)=\sqrt{1-x^2}-x·sign\left(x\right).$$

(3)

The sign(x) takes a value of −1, 0, or 1 when x is respectively negative, equal to zero, or positive.

Finally, to obtain the shape of the total electric field contributions, one simply needs to perform the sum of the two contributions (positive E_Pos + negative E_Neg). The results are displayed in Figure 4a, with individual contributions of E_Pos and E_Neg, and the sum of the two contributions. Moreover, a close up of the total electric field is provided in Figure 4b. As can be seen, especially from Figure 4b, the total electric field is confined in the period (from −100 to 100 nm). The amplitude (assuming 10¹⁹/cm³ electron excited [44,45,46]) reaches ±4500 V/μm, which is a considerable field (around 1% of the characteristic atomic electric field strength 5.1∙10⁵ V/μm [47]) that is probably partially rapidly screened by charged defects movement, even if the viscosity is sluggish. However, the plasma structure is recovered by the next pulse. On the other hand, the potential (assuming it is null at the infinity) reaches −200 V on the negative plate. Therefore, for a NG, the variation of the electric field and the potential along the axis of the NG appear as in Figure 4c. It is worth pointing out that the thickness of the negatively charged plate can be considered, instead of having an infinitely thin plate. This can be done by taking the expression of E_Neg similar to the one provided by E_pos but by substituting Λ = 200 nm by 20 nm (i.e., typical for a lamella) and by multiplying the constant term in front of the integral by –(Λ/e). This yields to a decrease of only ~10% of both the electric field and the potential values. Additionally, if the size of the lamellae is increased (i.e., R in Equation (1)), the value of the maximum and the shape of the total field does not change.

From the above results, such a large electric field has to be taken into account in the physicochemical reaction of phase separation processes. Here, there are two aspects of interest in the present observations. The first is the involvement of the electric field (E) in the phase separation (thermodynamic aspect). The second aspect is the migration of species in an electric field (kinetics aspect).

3.1. Thermodynamical Aspects

We have to analyze how E acts on the phase separation. We can show that phase separation is guided by the gain in generalized free energy (quoted $\tilde{G}$) that is the sum:

$$\tilde{G}\left(E\right)={\displaystyle \sum }_i{n}_i{\tilde{\mu }}_i\left(E\right)$$

(4)

where n_i are the number of particles i, and ${\tilde{\mu }}_i$ corresponds to the generalized electrochemical potential expression (see Ref. [48] or [49]). By considering the problem as an electrostatic one, we can consider the following expression (please refer to the Appendix A for additional details):

$${\tilde{\mu }}_i={\mu }_i+z_iℱ\varphi +\frac{1}{2}{\epsilon }_0{E}^2\frac{\partial \left[\left({\epsilon }_{r,g}-1\right)V\right]}{\partial n_i}.$$

(5)

In the above expression, ${\mu }_i$ is the chemical potential, $z_i$ is the charge of the species i, F the faraday constant, ϕ the electric potential, ε_r,g the total relative permittivity of the glass, E the electric field with $E=-\nabla \varphi$, and V the total volume of the system. It is demonstrated in the Appendix A that this expression can be rewritten under the following form:

$${\tilde{\mu }}_i={\mu }_i+z_iℱ\varphi +\frac{1}{2}{\epsilon }_0{E}^2\left({\epsilon }_{r,i}-1\right){\mathsf{\Omega }}_i,$$

(6)

where ${\epsilon }_{r,g}={\displaystyle \sum }_i{\epsilon }_{r,i}{\phi }_i$ with ${\epsilon }_{r,i}$ is an effective permittivity, ${\mathsf{\Omega }}_i$ is the partial molar volume of the i constituent, and ${\phi }_i$ is the volume fraction.

Here, we will consider that the glass is mainly covalent and composed of SiO₂ and LiNbO₃ group species and compare the electrostatic term of the energy variation with the chemical energy freed in the phase separation, which is of the order of a few kJ/mol [50] starting from the supercooled liquid or from the frozen glass.

Considering the example of this work, i.e., the formation of nanogratings in LNS glass, we must therefore consider the change of the energy balance, $\mathsf{\Delta }\tilde{G}\left(E\right)$, in the following reaction:

LNS _(glass) → 1/3 SiO_{2 (glass)} + 2/3 LiNbO_{3 (crystal)}

The variation of the molar free energy gain in the process of crystallization, from the view of a solid-state transformation, is:

$$\begin{array}{c}\mathsf{\Delta }\left(\mathsf{\Delta }\tilde{G}\left(E\right)\right)=\frac{1}{2}{\epsilon }_0{E}^2[1/3\left(({\epsilon }_{r,pure Si{O}_2}-1){\mathsf{\Omega }}_{pure Si{O}_2}-({\epsilon }_{r,Si{O}_2 in glass}-1){\mathsf{\Omega }}_{Si{O}_{2 in glass}}\right)+\\ 2/3\left(\left({\epsilon }_{r,pure LiNb{O}_3}-1){\mathsf{\Omega }}_{pure LiNb{O}_3}-({\epsilon }_{r,LiNb{O}_3 in glass}-1\right){\mathsf{\Omega }}_{LiNb{O}_{3 in glass}}\right)].\end{array}$$

(7)

In this work, ${\epsilon }_{r,pure Si{O}_2}=$ 2.12 at 633 nm [51], and ${\epsilon }_{r,pure LiNb{O}_3}=$ 5.31 [52], which corresponds to an averaged value knowing that LiNbO₃ is a uniaxial crystal.

For the first member of the physicochemical reaction, relative to the glass, the permittivities are calculated in the Appendix A, considering as a starting point the linear additivity of the refractive index with respect to the volume fraction of the glass constituents [53]. We calculated ${\epsilon }_{r,Si{O}_2 in glass}=1,91 \mathrm{and} {\epsilon }_{r,LiNb{O}_3 in glass}=4.50$ and found that both values are lowered with respect to the ones in the pure solids. Consequently, the term $\mathsf{\Delta }\left(\mathsf{\Delta }\tilde{G}\left(E\right)\right)$, from above, is positive. We point out that the refractive index values were taken at 633 nm in order to calculate ${\epsilon }_{r,LiNb{O}_3 in glass}$ from preexisting data. However, the results and the conclusions are not expected to change when considering the situation at 1030 nm, i.e., at the laser wavelength.

Considering the phase separation reaction and the aforementioned positive term, a high field stabilizes the glass with respect to the zero-field scenario. Consequently, the phase separation should be more effective in the low-field regions.

Noting that the coefficient $\frac{1}{2}{\epsilon }_0{E}^2$ may reach ~9.10⁻² kJ/cm³ for the electric field previously calculated (4500 V/μm), and considering our value of electric field and knowing the partial molar volumes (see Appendix A) ${\mathsf{\Omega }}_{pure Si{O}_2}$ = 27.3 cm³/mol, ${\mathsf{\Omega }}_{Si{O}_{2 in glass}}$ = 25.5 cm³/mol, ${\mathsf{\Omega }}_{pure LiNb{O}_3}$ = 31.7 cm³/mol, ${\mathsf{\Omega }}_{LiNb{O}_3 in glass}$ = $34.4{ \mathrm{cm}}^3/\mathrm{mol}$, we obtain $\mathsf{\Delta }\left(\mathsf{\Delta }\tilde{G}\left(E\right)\right)=1.2 \mathrm{kJ}/\mathrm{mol}$. This variation of the Gibbs free energy change for the reaction is on the order of magnitude of the energy released for the phase transformation to occur.

3.2. Kinetics Aspects

A correlated question is which one from SiO₂ or LiNbO₃ initially migrates, as a chemical migration is observed in the process of phase separation?

On that point, we have to consider the principle that elemental migrations are conducted under forces introduced by the electrochemical potential gradient. This is well known in the thermodynamics of irreversible phenomena, i.e., flux parts ($L_{ij}\overrightarrow{\nabla }{\tilde{\mu }}_j \mathrm{in} \mathrm{the} \mathrm{expression} \mathrm{of} {\overrightarrow{J}}_i)$ related to each force evolves such that fluxes either balance or vanish:

$${\overrightarrow{J}}_i={{\displaystyle \sum }}_j{L}_{ij}\overrightarrow{\nabla }{\tilde{\mu }}_j,$$

(8)

where L_ij are the Onsager coefficients.

Forces are proportionnal to $\overrightarrow{\nabla }{\tilde{\mu }}_j$; therefore, they are proportionnal to $\overrightarrow{E}\overrightarrow{\nabla }\overrightarrow{E,}$ as we do not consider charged species (LiNbO₃ and SiO₂). In the case of ionic migration, the term arising from the potential gradient is usually compensated by Fick diffusion. In this case, a uniform electric field may give rise to inhomogeneities, although starting from a homogeneous material. It is worth noting that forces induced by dipole polarization in this case would be null. However, in our situation, the electric field strongly varies in space, and the force is large enough to drive the dipole species towards the zero field region, provided their mobilities are large enough. The largest effect will be for the species with the largest relative molar permittivity (i.e., LiNbO₃).

In addition, if we compare the two situations, one for which SiO₂ is in the high-field region and LiNbO₃ in the low-field region, with the opposite case, the former constituent distribution is about 7 kJ/mol lower than for the latter one.

Following this line of reasoning, LiNbO₃ would thus concentrate between the charged lamellae over silica that would preferentially remain at the high-field regions. The equilibrium of the fluxes is established by chemical potential gradient resulting from the induced chemical inhomogeneities (Fick’s law based on $\overrightarrow{\nabla }\mu$). When the low-field regions are sufficiently enriched enough in LiNbO₃ species, crystallization may occur. This explanation agrees with the experimental observations, as described in the above sections. A scheme of the proposed scenario is therefore shown in Figure 5.

Finally, we suggest that this process drives the LiNbO₃ migration that is extracted from the place of high electron density, that is, where the electric field is at its maximum. Reversely, under the effect of LiNbO₃ concentration gradient, SiO₂ migrates to high field region and pure silica lamellae are developed as it is observed in Figure 6a. This behavior agrees with the chemical reorganization observed in LNS [38]. The chemical composition of the remaining glass between lamellae is therefore impoverished in silica and, after a while, is ready for crystallization (Figure 6b). As shown in Ref. [38],the observed chemical separation in the irradiated region being influenced by the plasma spatial structuration is produced before any crystallization.

Then, crystallization texturing is induced by the peripheral solid—solid crystallization transformation for the laser parameters domain investigated here and as recalled earlier in this paper.

In such a model, an additional force (physical, not chemical), an electrostatic one, is developed from the self-organization of excited electron plasma. This force is super-imposed with the thermal force (thermodynamical phase separation) and leads to a quasi-period of the chemical separation appearing larger than those observed under simple thermal treatment.

Therefore, we scheme below the revised mechanism, and the associated timescales, integrating several aspects from previous authors:

• When the first pulse is deposited inside the material, over the several few fs, electrons are excited from the valence band to the conduction band through multiphoton ionization. Electron density in the conduction band increases until it saturates [56]. After this “equilibrium” is reached, the remaining pulse energy deposited will compensate the energy loss due to electron relaxation. At this time, the fixed positive charges remain in the valence band, since they are bound to the lattice, while negative charges are located in the conduction band. Their lifetime, in the conduction band, is typically in the 100–200 fs range at low electron density [56] and of the order of ps for a high one [57].
• On the other hand, the electron mobility can be considered of the order of 30 cm²/(V·s) [58]. This makes a maximum speed of a few 10⁷ m/s with an electric field of the order of kV/µm, i.e., an electron can move over ~100 nm during ~100 fs. A direct consequence is that electron plasma may reorganize or can be sensitive to the electromagnetic field during the rest of a sub-ps pulse, i.e., reorganized starting from a uniform electron density. According to the results and proposals from [33,35], electro-magnetic field concentration may appear in some random locations of the focal volume, including at point defects sites where the photoionization energy is weaker. Hot spots are formed under spherical nanoplasma [33].
• Following this line of reasoning, the spherical nanoplasma is becoming oblate due to field enhancement effect described in [59], with the work of [60].
• Then, long range interactions of the oblate nanoplasma are put into quasi-periodic gratings due to the fact that the interaction between nanoplasma are minimized perpendicularly to the light polarization (alignment into the nanoplane), whereas they are repulsed from each other along their short axis, i.e., along the laser polarization. At this stage, the plasma is roughly structured/organized, but this also corresponds to the end of the pulse duration. There are, therefore, now different distributions between positive and negative charges. As described above, the electrostatic force between them is compensated by negative–negative repulsion of nanoplasma, maintained by light concentration on the nanoplasma.
• When the pulse deposition is over, the plasma structuring should disappear if the excited electron lifetimes were long enough for developing the organization. As mentioned above, it is in the ps (or less) timescale, so on the order of a pulse duration. This means that during plasma reorganization, there was an equilibrium between excitation and trapping, which inscribes the plasma structure progressively, but temporarily. When light goes away, all electrons relax and are permanently trapped on some structural changes in the lattice (produced by the hot spot, local heating). Note that for a typical 100 kHz repetition rate, the second pulse would be deposited 10 µs later, which is a much longer timescale compared to the timescales presently discussed. After the pulse, both negative and positive charge distributions are “recorded”, i.e., trapped in the lattice which exhibits a low mobility. The thermal stability now depends on the lattice temperature and the associated viscosity.
• If the temperature decreases enough between pulses, such that the charge distribution is not destroyed, the negative charge carriers (fixed on structural defects) may act as sources to further seed a plasma during the subsequent pulses. From this view, the plasma structuring progresses towards a steady-state regime. This is the so called “memory effect”, introduced by [33]. From the aforementioned discussion, the lattice temperature-time dependence during the irradiation process must be discussed.
• For this purpose, we can say first that the lattice temperature increases just after the pulse vanished, within a few ps. We consider a very simple model, such as a spherical heat source deposited in volume (few µm³), andcorresponding to the laser heat deposition in the focal volume. The physicochemical parameters of the LNS glass are taken from [61] (page 86, and recalled below), and the temperature is defined by the formula without heat accumulation i.e., the energy deposited vanishes during the period of time between two pulses. This is valid for period of a few µs. The expression is:

$$T\left(r,z,t\right)=T_A+T_{00}·\frac{w^3}{{\left(w^2+4D_{th}·t\right)}^{3/2}}·exp\left[-\left(\frac{r^2}{w^2+4D_{th}·t}\right)\right],$$

(9)

$$\mathrm{With} T_{00}=\frac{A·E_p}{{\pi }^{\frac{3}{2}}·\rho ·C_p{w}^3}$$

(10)

The parameters, along with their definition, are provided in

Table 1. Parameters and typical values employed in LNS glass.

Parameters	Definitions	Value	Units
$A$	Fraction of the absorbed light	0.3	none
${\tau }_D$	$\mathrm{Heat} \mathrm{diffusion} \mathrm{time} {\tau }_D=\frac{w^2}{4D_{th}}$ (decrease by 1/e)	0.53	μs
$w$	Beam waist radius (at 1/e)	1.5	μm
$D_{Th}$	$\mathrm{Thermal} \mathrm{diffusivity} D_{th}=\frac{\kappa }{\rho ·C_p}$	1.06∙10−6	m2/s
$\kappa$	Thermal conductivity	2.65	W/(m∙K)
$E_p$	Pulse energy	0.5	µJ
$\rho$	Density	3830	kg/m3
$C_p$	Specific heat capacity	650	J/(kg∙K)
TA	Ambiante temperature	20	°C
Tm	Melting point	1260	°C
Tg	Glass transition temperature	580, Ref. [62]	°C

.

The reached maximal temperature T_A + T₀₀ (Equation (10)) remained during a fraction of diffusion time which is given in the table. It then decreases by 63% within 0.5 µs. We can understand that the maximum temperature does not overcome the boiling temperature as we have not detected any nanobubbles by TEM in our experiments.

Therefore, due to the temperatures reached, the glass viscosity will be small enough during the diffusion time to enable phase separation under the action of the electric field. One can estimate the migration length of the species for a time comparable to the diffusion time, using the well-known Stokes-Einstein equation D_v = k_BT/(3π·d·η). In this expression, k_B is the Boltzmann’s constant, T the temperature, D_v the diffusion coefficient, d the radius of the molecular species (calculated assuming a spherical volume and from a molar volume of 30 cm³/mol), and η the viscosity. Following this, a characteristic length for the mobility of the species can be determined from the classical diffusion length $l_c=2\sqrt{D_v·t}$ with t = τ_p. From this analysis, the diffusion length reached hundreds of nm. Furthermore, the presence of the electric field might increase this value.

Finally, the charge distribution decreases over the period, so is the electric field. In fact, it decreases at the same speed than SiO₂-LiNbO₃ phase separation is achieved. However, as opposed to the phase separation that progresses from one pulse to the other, the charge distribution is reinitiated by the subsequent pulse and thus drives the constituent separation. This separation is reinforced over a time corresponding to a few tens of pulses (order of magnitude), whereas the plasma structure evolves (in particular the spatial period of the nanogratings) in the same duration.

4. Conclusions

Among the different models proposed to describe the formation of nanogratings induced by femtosecond laser irradiation in glass, none of them consider the effect of charge redistribution due to high electronic mobility while the positive counterpart remains much lower (charged defects in the lattice). This paper takes into account the non-uniformity of excited electron density into the non-uniformity of electrochemical potential itself. From this view, we propose an explanation of how the formation of nanogratings are driven, especially in LNS glass family that crystallizes non congruently. This work also proposes an interpretation on how the phase separation distance is larger in the laser-irradiated areas relative to a purely thermally driven crystallization process. These new insights on the nature of the chemical phase separation could be applied to other phase transitions, for a better control of nanogratings formation especially in systems for which crystallization properties are key aspects. Following the idea that excited electron charge density fluctuation drives the phase separation, we can predict that the organization of NG of chemical type may be perturbed by an external electric field. Furthermore, the existence of large electric field produced by mobility difference of oppositely charged carriers by light excitation open to consider that other effects may be at play such as electrostriction or inverse piezoelectricity. This may trigger deformation fields and contribute to cavitation and other stress fields that should be detected in some materials.