Simulation of the laser-material interaction of ultrashort pulse laser processing of silicon nitride workpieces and the key factors in the ablation process

Laser ablation is highly popular for industrial applications. However, there are various research gaps in this field, especially for ultrashort pulse laser ablation of difficult-to-cut materials like ceramics. Controlling the laser ablation is performed by laser parameters. The laser parameters, which are related to the wavelength and laser power intensity, have influences on the laser-material interaction mechanism and the portion of transferred laser energy to the material. Because of the high sensitivity of ceramic materials against the thermal shocks and the considerable energy transmitted by the laser to the material, studying the effects of different laser types with different pulse duration on the thermal shocks and surface integrity is an interest of many researchers [1].

In an identical wavelength, a decrease of the pulse time leads to an improvement of the quality of the ablated surface. Oosterbeek et al. [2] investigated the influence of nano- and femtosecond lasers on alumina material with pulse times of 80 ns and 110 fs, respectively. They illustrated that material ablation without thermal degradation is possible when using the femtosecond laser. More molten material and thermal damages on the workpiece were observed with increasing pulse time from several hundred femtoseconds to several nanoseconds.

In addition to the pulse time, the amount of pulse energy affects the material ablation. Herrmann et al. [3] compared in an analytical work the ablation mechanisms of picosecond and femtosecond laser pulses in a range of 5 ps–10 fs with a maximum pulse energy of 1.0 × 10⁻¹⁵ J and a focus of 25 Å on a 100 Å × 100 Å × 50 Å silicon sample. However, the ablation strongly depends on the pulse length; the main material removal takes place on a time scale of a few picoseconds. While the damage initiation depends on the pulse energy, the beginning of the material removal shows a strong dependence on the pulse duration.

In the study of Jandeleit et al. [4], the laser ablations of Si₃N₄, SiC, and WC were compared. They utilised pico- and nanosecond lasers with laser power of 3–9 W. They investigated the material removal threshold as a limit, above which the material is removed. However, only the thermal effect without material removal is presented below this threshold. Accordingly, a pulse time reduction from 10 ns to 40 ps leads to a considerable increase (up to 20 times) in the material removal threshold. However, by increasing the pulse time beyond the mentioned threshold, the Material Removal Rate (MRR) was reduced up to 200%. For a single pulse, the average laser power intensity is introduced as below to simultaneously investigate the influence of pulse time and pulse energy [5]: where E_{l − pulse}is the laser pulse energy (J), t_pulse is the pulse duration (s), and A_foc is the laser irradiation area in the focus point (cm²).

According to Eq. (1), the average laser power intensity has a linear relation with the laser pulse energy and is reversely proportional to the pulse time and laser-irradiated area. Karnakis [6] studied the influence of a single laser pulse Nd:YAG with a wavelength of 355 nm, a maximum pulse duration of 44 ns, and maximum pulse energy of 0.5 mJ on a single crystal of silicon. Here, it was investigated how an increase of laser power intensity up to 20 × 10⁹ W cm⁻² affects the ablation depth and material removal mechanism. It was concluded that the removal mechanism is based on evaporation and melting. The ablation depth increased exponentially with laser power intensity, while the crater diameter varied linearly. As a result of growing laser power intensity, the crater shape changed from V-shape to a parabolic shape.

In ultrashort laser ablation, the material heating does not follow Fourier equations. However, the temperature of the surface increases with increasing irradiation time and the laser power density up to a maximum value (T_max). The T_max may exceed a phase transformation threshold within the pulse time (t_pulse). Short and ultrashort pulsed lasers have pulse times in the range of 100 fs < t_pulse < 100 ns. The laser pulses in the mentioned range usually induce laser power picks in orders of several mega- to terawatts [7]. A portion of laser energy is delivered on the surface within the thermal relaxation time and is localised in a skinny layer of material without any thermal diffusion, which causes electron excitation and electron heating [8]. In the case of ultrashort pulse lasers, the laser energy of a single pulse, which was radiated in less than 10 ps, leads to a material breakdown through a thin layer on the surface with ionisation of atoms, which converts a normal transparent material into absorbing plasma [9].

Ren et al. [10] modelled the interaction between an ultrafast laser and a gold layer using the electron warming model. In this model, the laser energy is first transferred to the free electrons and increases the electron temperature. The peak temperature of the free electrons, which was stimulated by the electron warming model, reached 40,000 K in about 100 fs. In contrast, the lattice temperature increased in a period of more than 5 ps. The lattice peak temperatures grew more extensive than the normal melting and the boiling points inside the film, and melting occurs from the surface exposed to the laser, followed by evaporation. At t = 20 ps, the liquid portion of the gold film was superheated up to 3880 K (the normal boiling point is 3127 K), and the solid part was superheated to 3840 K (the normal melting point is 1336 K).

In the study of Chen et al. [11], which was about the applicability of the electron warming model for a gold layer, the temperature of the workpiece surface increased up to 8000 K. The mentioned temperature was achieved after 1.0 ps. It is enough for material sublimation without melting. Moreover, it was reported that the increasing electron temperature in 0.3 ps after laser radiation results in a considerable acceleration of free electrons faster than the sound speed, and a repulsion pressure of electrons reach 1.24 GPa in the depth of 0.3 μm. The pressure caused by electrons and the thermal expansion of the material results in the workpiece destruction. Accordingly, so long as the laser power intensity (I_L) value was less than 10⁸ W cm⁻³, the irradiated material is molten and then evaporated. If I_L is in the range of 10⁸ to 10¹⁰ W cm⁻³, the substance is partially sublimated without melting. With higher I_L, ionisation is a dominant mechanism of material ablation.

Brenk and Rethfeld [12] studied the electron warming model for silicon oxides. Here, the workpiece was an insulator without the free electrons in the conduction band. Multiphoton ionisation (MPI) was introduced as a crucial mechanism to generate free electrons with material ionisation. At the beginning of MPI, different critical parameters, such as the critical density of free electrons and lattice temperature, were determined. They showed that the laser absorption increases by increasing the density of electrons beyond a threshold that intensifies laser energy’s influence on the workpiece.

In insulators like structural ceramics, since there are no free electrons for direct excitation, if the incident photons provide enough energy to shift an electron from the valence into the conduction band, the electrons can detach from its atom [7]. No direct absorption can theoretically take place if the incoming photon energy level is lower than the bandgap. However, the density of low-energy photons becomes high enough, and they can simultaneously provide the required energy for a jump from a valance band to a higher conduction band. The excited electron by several photons participates in transferring the laser energy into the other valance electrons. The excitation of electrons by several low-energy photons is introduced as multiphoton absorption (MPA). The ionisation, which is attributed to the MPA, is the MPI. Ultrashort pulse lasers with frequencies lower than the UV range usually excite electrons with a multiphoton absorption mechanism, as shown in Fig. 1. This phenomenon leads to localised ionisation of material and transforms the material into plasma [9].

The generated conduction electrons by MPI absorb the laser energy during a pulse laser radiation, while the lattice remains cold. Therefore, the heat conduction should be described by two heat flow equations, for electrons and lattice. Here, the Drude model (or free electron model) provides the free electrons’ basic formulation in the material response to electromagnetic waves of a specific angular frequency. Accordingly, the free electrons’ heat capacity is linearly dependent on the electron temperature up to the fermi temperature [13].

In a MPA, transferring the energy to the electron depends on two factors: the energy level of the external field and the dissipated energy during collisions of the phonons with the atoms. The ratio of the energy lost by the electron in a collision is of the order of the ratio of the electron’s mass to the mass of the ion [14]. In a MPI mechanism, a valence electron absorbs enough photons to excite to a higher conduction band through the simultaneous absorption of several photons by an electron. Moreover, the conduction band’s excited electron can absorb several laser energy and act as a photon [15].

Sugioka et al. [16] summarised various investigations concerning the laser micro-fabrication allocated to the different interaction mechanisms between laser and material. In the ultrashort pulse lasers (USPL), I_L may vary from 10 ⁸ to 10¹⁴ W cm⁻², contributing to the MPI mechanism, which is the considerably dominated material removal. Here, electron heating begins in a radiation time of fewer than 100 fs. In radiation times higher than 10 ps, thermal energy is transferred from free electrons to the atoms. Bäuerle [5] gathered the results of several studies in a graph, as shown in Fig. 2. It indicates the effect of I_L on laser-material interaction for different metals, polymers, and dielectrics. In a range of I_L = 10 ⁸ – 10¹⁰ W cm⁻², the small amount of the material is removed by melting and evaporation while the sublimation is dominant. In contrast, the material removal occurs through ionisation of the atoms without melting and evaporation for laser intensities more than 10¹⁰–10¹² W cm⁻².

In the previous study, Soltani et al. [17] focused on a simulation of a laser ablation process on a silicon nitride workpiece by an ultrashort pulse laser, which predicted the depth of laser ablation. Despite a good correlation between the simulation and experimental results, there is a lack of finding the effects of different experimental parameters on the laser power intensity and the absorbed energy. In addition, the precise temperatures of the lattice and free electrons during laser processing were not calculated.

The laser-material interaction mechanism has an influence on the temperatures, which is affected by the laser parameters. It should be analysed how the experimental parameters affect the laser-material interaction mechanism. The main criterion of the ablation in different laser-material interaction mechanisms should also be studied in more details. Also, the effects of the laser parameters on the material removal rate were not considered in the previous work of Soltani et al. [17].

The goal of the current work is to develop a reliable simulation method to calculate the workpiece temperature, find the laser-material interaction mechanism during laser processing, and predict ablation depth. Following the previous work [17], it is aimed to investigate the interaction between an USPL and ceramic alloy Si₃N₄ (SL200-BG-Ceramtec Co), which is a gas-pressure-sintered Si₃N₄ (SL200-BG from Ceram Tec GmbH) with Y₂O₃ (~3 wt.%) and Al₂O₃ (~3 wt.%) as sintering aid additives [18]. The experimental tests were carried out by a USPL with a wavelength of λ = 1032 nm.

The applied model is a finite-difference numerical model. The effects of the angle of incidence and the Gaussian factors on the absorbed energy of the laser were considered. The two temperatures model (TTM) was applied to calculate the heat conductions in electrons and the lattice. The Drude model provided the basic formulation of energy transferring to the free electrons and their heat capacity. Here, the energy of the external fields and energy dissipation by the collision was considered. It was evaluated how the valence electrons absorb the incident energy in all contributed components. The absorbed energy by the valence electrons can initiate the ionisation and MPI mechanism.

Therefore, the chemical composition’s influence on the ablation mechanism and temperature was studied, and the minimum laser power intensity to remove material without thermal damage was computed. The ions and free electron densities were calculated to simulate the ablation of the material.

Finally, the precise temperature values of the lattice and free electrons were calculated considering the laser parameters. The effects of the laser parameters on the laser-material interaction mechanism and the main criterion of the ablation in different laser-material interaction mechanisms were determined. The sublimation temperature was computed as the ablation criterion in the thermal ablation mechanism. The critical density of ions and free electrons as the ablation criterion in the cold ablation mechanism was also calculated. Besides, the material removal rate affected by the laser parameters was analysed. The simulation data of the ablation depth and material removal rate were validated by comparison to the experimental results. The critical laser power intensity as a threshold of the cold and the thermal ablation mechanism was calculated in the simulation.

The experiments were conducted to investigate the experimental effects of a USPL on Si₃N₄ alloy (Ceram Tec - SL200 BG) as a high-performance ceramic material. Due to excellent chemical stability and mechanical strength, SL200-BG is utilised to produce welding components, gas nozzles, and forming tools. However, due to the high values of hardness and brittleness of the workpiece compared with the metals, a conventional machining process is still a challenge with dramatic tool wear and numerous surface defects. Therefore, the laser ablation is considered a non-contact machining process without any tool wear.

The laser process parameters and the applied alloy properties are provided in Tables 1 and 2, respectively.

As mentioned above, the value of I_L influences the laser-material interaction mechanism. The amount of ablated material is highly dependent on the ratio of laser energy to the irradiated area, so-called total input energy density. An increase in total input energy density leads to a deeper ablation irrespective of the total laser radiation time. Equation (2) provides the calculation of laser input energy density (E_{L − input}) [21]: where E_{L − total} is the total laser energy (J), w_L is the width of laser ablation (mm), t_L is the radiation time (s), and v_L is the laser scan speed (mm s⁻¹).

The laser input energy density has a very similar definition to laser fluency. However, a laser-ablated structure on a workpiece, for instance, a dash line structure, has several ablated elements with specific gaps. Here, the laser input energy density is applicable to connect the average delivered energy to the ablation depth and the gaps between them. It means that, despite the same laser parameters, the laser input energy density will decrease with an increase in the ablated elements’ gaps. Therefore, the effect of the ablated structure geometry is considered in the laser input energy density.

For a laser at a constant power and radiation area, the E_{L − input} is a function of v_L. Figure 3 illustrates the calculated E_{L − input} according to Eq. (2) at various v_L, which are in the range of practical experiments. The w_L, according to the experimental measurements, is approximately equal to the laser spot diameter. The E_{L − total} and the t_L values were calculated for laser cuts with a length (L_L) of 10 mm: where E_{L − total} is the total laser energy (J), t_L is the radiation time (s), v_L is the laser scan speed (mm s⁻¹), P_L is the laser power (W), and L_L is the length of the laser cut (mm)

Accordingly, the E_{L − input} has lower sensitivity at v_L > 20 mm s⁻¹ compared to v_L = 0–20 mm s⁻¹. Here, a growth of v_L from 0 to 20 mm s⁻¹ causes a dramatic drop of the E_{L − input} from higher than 300 J mm⁻² to less than 50 J mm⁻². The current study, laser scan speed, ranging 2–100 mm s⁻¹, results in a laser input energy density of 6.25–312.5 J mm⁻². Furthermore, the influence of E_{L − input} on the ablation depth (a_e − L) and the width of thermal damages was examined.

According to Fig. 4, increasing laser input energy density from E_{L − input} = 6.25 to 62.5 J mm⁻² led to deeper laser ablations. No effect of thermal damage can be seen for the ablations with E_{L − input} = 6.25 J mm⁻², whereas increasing the E_{L − input} to 62.5 J mm⁻² resulted in thermal cracks, as shown in Fig. 4b. The majority of thermal cracks were initiated from the bottom of the laser ablation where it should have the maximum thermal gradient and temperature enhancement. The growth of cracks adjacent to the laser ablations toward the surface of the workpiece is a representation of a thermal tension field in the vicinity of the ablated zone. Therefore, it is expected that the laser energy on the wall of the laser cut results in a localised rise of temperature and thermal expansion. The wall surfaces absorb a noticeable portion of laser energy in deep laser cuts through several reflections of laser light. The laser beam reaches the bottom of the laser ablation, which has about > 100 μm depth, through several reflections. Therefore, the surface roughness, recast layer, trapped plasma cloud, and the increase of angle of incidence (AoI) affect the absorption and diffraction of the laser energy on the laser cut. Hence, the practical I_L is reduced noticeably, which is a crucial factor determining the laser-material interaction mechanism.

As can be seen in Fig. 4, increasing laser input energy density from E_{L − input} = 6.25 J mm⁻² to E_{L − input} = 62.5 J mm⁻² led to a deeper laser cut. No effect of thermal damage can be seen for the ablations with E_{L − input}= 6.25 J mm⁻², whereas increasing the laser input energy density to 62.5 J mm⁻² resulted in thermally induced cracks (Fig. 7b). The majority of thermal cracks were initiated from the bottom of the laser cut. It means that the maximum thermal gradient and temperature enhancement are at the bottom of the laser cut. Growing cracks adjacent to the laser cut toward the workpiece’s surface represent an increase in thermal tension or shear stresses in the laser cut’s vicinity. Therefore, the laser energy on the laser cut wall results in a localised rise of temperature and thermal expansion. The wall surfaces absorb a noticeable portion of laser energy in deep laser cuts through several laser light reflections. The laser beam reaches the bottom of the laser cut through the reflections, which is about > 100 μm deeper than the laser focal point. Therefore, the surface roughness, recast layer, trapped plasma cloud, and the increase of the AoI affect the absorption and diffraction of the laser energy on the laser cut. Hence, the practical I_L is reduced noticeably, which is a crucial factor determining the laser-material interaction mechanism.

A plasma selective etch process was applied on a workpiece to investigate the physical changes in the laser cut’s vicinity. Figure 5 shows the etched surface of the workpiece. As illustrated in Fig. 8, the heat-affected zone (HAZ) and microstructural alternation can clearly be observed with the E_{L − input} = 312.5 J mm⁻². Moreover, a recast layer inside of the laser cut is detectable. The microstructural alternation occurs because of the elimination of silicon oxide, located between crystal grains [22, 23]. The presence of HAZ and thermal damages are indicators of high heat generated by laser beam energy.

According to the previous investigations, the recast layer on the laser cut wall is SiO₂, which is formed due to the chemical reaction of silicon ions or molten material with Oxygen as described in Eq. (5) and (6) [24]:

The generated energy in this reaction is negligible compared to the laser energy and cannot be considered an influential factor in microstructural alternation and thermal damage.

After etching the cross-sections with H₃PO₄, the maximum width of thermal damages was measured on three different cross-sections for every laser cut, as shown in Fig. 6a. Figure 6b indicates how the HAZ changes with laser input energy density, E_{L − input}. Accordingly, in E_{L − input} < 40 J mm⁻², the material has no thermal damages, whereas, for E_{L − input} > 60 J mm⁻², considerable thermal damages took place (see also Figs. 4 and 6). The range of laser input energy density, E_{L − input}, between 40 and 60 J mm⁻², is introduced as a transient area between ablation without and with thermal damages. Here, the laser energy is delivered to the workpiece's surface within the laser penetration depth. The laser penetration depth is about L = 19 nm, and it is assumed that 100% of the laser energy is absorbed through the laser penetration depth.

Therefore, it should be analysed how the applied laser parameters affect laser power intensity, the laser input energy density, the ablation geometry, irradiated surface temperature, and the mechanism of the laser-material interaction mechanism. An analytical model should be provided to predict the ablation depth, the effect of different parameters on the laser power intensity and absorbed energy, the exact temperature of free electrons and atoms during laser processing, and the mechanism of laser-material interaction.

In the applied theoretical model, appropriate thermal conduction equations should be utilised to calculate the temperatures. Then, it should be possible to predict the depth of material ablation and temperature changes during the laser irradiation and the effect of laser parameters on it. Also, the reasons for the onset of thermal cracks and their growth parallel to the laser cutting wall can be studied by monitoring temperature gradients on cutting edges.

The developed simulation is based on the 2D analysis of laser-material interaction to calculate the depth of material ablation and the workpiece temperature in laser treatment of a silicon nitride sample. In this regard, the laser power intensity I_L, which is delivered to the surface, is calculated by the following equation [7]: where I_L is the laser power intensity (W cm⁻²), I_pk is the maximum laser power intensity (W cm⁻²), G_space is the Gaussian function for spatial laser beam shape, G_time is the Gaussian function for temporal laser beam shape, and G_mod is the Gaussian function for beam motion.

The laser beam should be modelled in the first stage, considering that the laser power intensity varies over the whole laser radius and during each pulse time. In this context, the Gaussian distribution model is applied to define the laser power intensity in time and space with functions G_space (Gaussian function for the spatial laser beam shape) and G_time (Gaussian function for the temporal laser beam shape), respectively. Moreover, the laser beam’s motion should be considered by G_mod (Gaussian function for the beam motion), affecting the laser power intensity. Therefore, for a single pulse, Eq. (7) is reformulated without G_mod function, and with respect to Eq. (1), the energy of a pulse (E_{l − pulse}) can be extracted from the Gaussian factors in Eq. (7) as:

where \( {E}_{\mathsf{l}-\mathsf{pulse}} \) is the pulse energy (J), r₀is the radius of the laser beam (cm), t_p is the pulse time (s), I_pk is the maximum pulse intemsity (W cm⁻²), r is the radial component of the position vector (cm), θ is the angular component of the position vector (rad), and r₀ and t_p are respectively beam radius and laser pulse time, as specified by the manufacturer in the applied laser’s technical specifications.

G_space means the Gaussian distribution that is as a function of the laser beam radius r₀ and the distance from the laser beam centre, r [25]:

Figure 7 illustrates how the Gaussian distribution obtained from Eq. (9) changes as one moves away from the beam’s centre. As illustrated in Fig. 2, the Gaussian function for the spatial laser beam shape (G_space) varies from 0 to 1 and increases as one approaches the beam centre, which is followed by an increase in laser power intensity. As can be seen in Fig. 7, the G_space value reaches 10% of the maximum value at the edge of the laser beam. Therefore, in Eq. (9), the amount of laser power intensity at the edge of the laser beam reaches 10% of the intensity in the centre of the laser spot.

Since the applied laser in the current study is a Q-Switch pulse laser, in addition to the G_space, the temporal laser beam energy is varied according to a Gaussian function of laser radiation time (G_time). According to Paschotta [26], the temporal intensity of the laser increases up to a maximum value at t = t_max . The G_space determines how the pulse energy is delivered to the workpiece at each laser radiation t concerning t_max and the pulse time (t_p) [27]:

The values of G_space throughout the radiation times during a single laser pulse is illustrated in Fig. 8. The applied laser is a Q-Switch pulse laser. In the utilised laser, the pulse energy vs. time is a Gaussian graph. The pulse duration was estimated considering the full-width half-maximum (FWHM) method. Here, pulse time is the with of the laser radiation with the Gaussian values greater than 0.5. Accordingly, the G_space changes from 0.5 to 1 over the pulse duration, which results in a variation of the I_L up to 50%.

By solving Eq. (8), the maximum laser power intensity (I_pk) can be determined that is approximately 2.5 times of average laser power intensity (I_{L − avg}). The I_pk at the laser centre point is reachable at t = 0.5 t_p.

The influence of the laser scan speed on the pulse overlapping and laser power intensity considered by Gaussian function for the motion of the laser beam (G_mod) is formulated in Eq. (11) [7]: where G_mod is the Gaussian function for the motion of the laser beam, r₀ is the laser beam radius (mm), v_L is the laser scan speed (mm s⁻¹), and t is time (s).

Figure 9 shows how the G_mod changes with the radiation time of the laser at different v_L. Here, according to Eq. (11), the variation of G_mod versus the radiation time of the laser is illustrated in several v_L in the range of 1–100 mm s⁻¹. A more intense change in G_mod versus time occurs at a higher laser scan speed. In Eq. (11), the edge of the scanning laser beam reaches a certain location at the initial time t = 0. Since the value of the intensity of the laser power at the edge of the laser beam is 10% of the intensity at the centre of the laser spot, the value of the G_mod at t = 0 becomes nonzero. The G_mod value reaches its maximum when the laser beam travels a radius of the laser beam over a certain point. Therefore, as the laser scanning speed increases, the G_mod value is maximised in a shorter time. When the laser beam travels a laser beam diameter on the surface, the G_mod returns to the value as t = 0.

The laser’s AoI affects the radiation area, leading to variation of the laser power intensity. As the laser-cut deepens, not only does the shape of the V-shape laser-cut change but the angle of the laser beam on the surface also grows (Fig. 10). Accordingly, during the process, unlike the beginning of the laser radiation, the laser beam is not perpendicular to the ablated surface. The effect of changing the AoI of the laser could be introduced by a coefficient (C_AoI), based on the geometry of the laser-cut. It is mathematically calculated as below in the current study: where C_AoI is the coefficient of AoI, AoI₀ is the initial AoI (a_e–L = 0 μm) (rad), a_e − L is the depth of the laser cut (μm), and w_L is the width of laser ablation (μm).

Eventually, the laser power intensity, I_L, corresponding the angle of the incident at each time increment can be calculated as follows:

After calculation of the I_L the amount of the volumetric energy which is absorbed by the surface is calculated by Eq. (14) [16]: where E_abs is the volumetric absorbed laser energy by the material (J cm⁻³), R is the reflection coefficient, L is the maximum depth of laser diffusion (cm), and z is the subsurface depth through the material (cm).

Fresnel’s refraction formulas calculate the reflected coefficient (R) at the sample surface [5]. The amount of reflected energy depends on the frequency of the incident energy, the angle of incidence, the quality of the sample surface, and the material properties. Here, the sample’s refractivity index, which is equal to (n − ik), has a real part and an imaginary part, k. n varies slowly with temperature and k is related to the absorption coefficient, α = 4πk/λ, where λ is the wavelength. The density of the lattice dislocation, the free electron temperature, and free electron density increase with an enhancement of temperature. Hence, the absorption coefficient grows with a temperature elevation. The α is related to T by an exponential function, α = α₀ exp(T/T₀). Here, T₀ changes with the wavelength [16, 30]. Additionally, the evaporation and heterogeneous melting of the target leads to the formation of a plasma plume and molten droplets, which attenuate the radiated energy intensity [31]. Not only, the reflection coefficient values, R, depend on the material properties, the laser wavelength, and surface quality, but the maximum depth of laser diffusion (L) changes with them.

The refractivity and absorption coefficients also depend on the microstructure of the material. Silicon nitride contains a layer of an intergranular amorphous phase, which mainly holds a higher concentration of impurities and free electrons than Si₃N₄ grains. Due to the high concentration of impurities relative to the grains, this amorphous layer has a principal effect on the reflection and absorption of radiated energy. In β-Si₃N₄, the grains are elongated rod-shaped, the α-Si₃N₄ grains have oval shapes. The concentration of the intergranular amorphous phase is the lowest in the direction of the grain axis. Therefore, the amount of energy absorbed in this direction is the least [28]. As the wavelength of incident energy grows from the UV to the near IR, the samples with β− and α-Si₃N₄ show a monotonic decrease in the absorption coefficient as well as reflectivity [29].

In the steady state, Si₃N₄ is chemically stable up to a temperature of about 2170 K [28]. With a single pulse of the laser, since the temperature locally changes within 10 ps, the reactions, which directly depend on the diffusion of atoms, such as phase transformation or chemical oxidation, do not launch. However, during the ablation with several laser pulses, the chemical oxidation affected the reflectivity as well as melting and plasma formation.

The absorbed energy is initially allocated for heating free electrons, which results in the generation of photons. An atom ionises if a photon, which collides with the atom, has enough energy to excite a valence electron to the conduction energy layer. The photon energy (E_photon) can be calculated as follows: where E_photon is the photon energy (J), h is the Planck constant (6.626 × 10⁻³⁴ Js), λ is the laser wavelength (m), and c is the speed of light (m s⁻¹).

As indicated in Eq.(15), the photon energy is inversely proportional to the laser wavelength. In other words, atomic ionisation occurs when the laser wavelength (λ) is shorter than a threshold value. However, if the photon density is high enough, atomic ionisation due to MPI can occur even when the laser wavelength is longer than the threshold value.

In the case of activation of the MPI mechanism, some part of energy leads to the generation of free electrons [12]. Free electrons are supplied by elements in the material. To provide free electrons, the energy needed to jump the valence electrons to the conduction band must be provided. Therefore, according to the density of free electrons, the amount of volumetric energy required to generate free electrons, which is supplied by laser radiation on the surface, must be calculated: where N_free is the density of free electrons (cm⁻³), E_g is the energy for generating 1 mol of the free electrons (J mol⁻¹), and N_A is the Avogadro constant (6.02 × 10²³ 1 mol⁻¹).

E_g is the required energy to jump 1 mol of valence electrons to the conduction band. If the material includes several chemical compounds, the corresponding breaking energy should also be considered for the weakest chemical bonds. In this study, the material is a ceramic alloy with a base material of silicon nitride. The additives are ~3 wt% Al₂O₃ and ~3wt% Y₂O₃ (SL200-BG). Therefore, it should also be taken into account the bandgap energy of the valence electrons and the corresponding energy of chemical bonds of Al–O, Si–O, Y–O, and Si–N. Consequently, It is expected that the component with a minimum required energy to create 1 mol free electrons should be ionised before the others.

On the other side, an amorphous oxynitride phase is mainly concentrated in the grain boundaries between the β-Si₃N₄ grains. Thus, in the amorphous phase, metallic impurities more easily diffuse into the grain boundaries than inside the grains [22]. According to Kasap’s study [32], the electrical breakdown is initiated by structural imperfections, metallic impurity concentration, and local tension fields. Lombardo et al. [33] studied the electrical breakdown on Si components. As concluded, in an insulator, the metallic or semiconductor components as well as defected zones in the lattice in the first step are locally ionised and provided the free electrons for electrical conductivity. The free electrons or valence electrons, which have no contribution in a chemical bond, are concentrated in the defected zones. Therefore, those electrons cause an electrical breakdown in lower potentials rather than a flawless zone. Besides, the tensile stresses facilitate the breakage of chemical bonds. Accordingly, in the current laser process, the electron warming and MPI mechanism should locally be initiated in the defect zones at the grain boundaries. It is expected that owing to the laser radiation, the destruction and ionisation of the material propagate through the β-Si₃N₄ grains succeeding in the grain boundaries. According to the material specifications, the grain sizes are in the range of 1–5 μm, and the grain boundary width is about 1–200 nm. It means that the total volumetric concentration of the impurities and additive components in grain boundaries is < 10 vol.%. So, the free electron density locally increases at material imperfections such as grain boundaries, and through the diffusion of free electrons from the grain boundary through the grains, the ionisation inside the grain also takes place.

According to the study of Brenk et al. [12], the plasma frequency is a function of the density of free electrons (N_free): where f_p is the plasma frequency (rad s⁻¹), N_free is the free electron density (m⁻³), e is the electron charge (−1.6 × 10⁻¹⁹°C), m_e is the electron mass (9.11 × 10⁻³¹ kg), and ϵ₀ is the dielectric constant of vacuum (8.85 × 10⁻¹² F·m⁻¹).

When the laser frequency equals the plasma frequency, the density of free electrons reaches the critical value (N_crt) in which the material can absorb the maximum laser energy and electron warming and MPI mechanism are dominant. Based on laser specifications, this value, N_free, would be around 1.12 × 10⁻²¹ cm⁻³. Since the laser-free electron interaction is available locally in the grain boundaries, the N_free should reach the critical value at the grain boundary. The material ablation requires the repulsive force among free electrons which is higher than the mechanical strength of the material. The repulsive tension (σ) by free electrons can be calculated as follows [11]: where c_e0 is the initial electron-specific heat capacity (7 × 10⁻⁴ J mol⁻¹ K⁻²) [34], ρ is the electron density (mol cm⁻³), T_e is the electron temperature (K), and σ is the repulsive tension (N cm⁻²).

Early efforts in the simulation showed that the maximum free electron temperature should be in the range of 1000–10,000 K. Also, as the density of free electrons reaches 4.5 × 10²¹ cm⁻³, the calculated repulsive tension in Eq. (18) is enough to damage the amorphous silicon (with mechanical strength 7 MPa [35]) located in the grain boundaries. Therefore, the N_crt in this study equals to 4.5 × 10²¹ cm⁻³.

Otherwise, if the density of photons is not enough, and the laser wavelength is longer than the threshold value, the heating of the lattice structure takes place rather than activation of the MPI mechanism. In this case, electron-phonon coupling volumetric energy (E_eph) is transferred to the material as a result of a collision between heated free electrons and atomic structure. Equation (19) provides the mathematical calculation of E_eph [12]: where E_eph is the electron-phonon coupling volumetric energy (J cm⁻³), α_K is the electron-phonon coupling coefficient (J (cm³ K¹)⁻¹), T_e is the electron temperature (K), and T_l is the lattice temperature (K).

The E_eph is the amount of energy that is ultimately used to heat the lattice at the laser penetration depth. Equation (20) provides the calculation regarding increasing the lattice temperature [36]: where c_p is the material-specific heat capacity (J (g K)⁻¹), ρ is the material density (g cm⁻³), K is the material heat conductivity (W (cm K)⁻¹), and T_l is the lattice temperature (K).

Finally, the volumetric amount of energy leading to heating of the electrons (E_he) is obtained by subtraction of heating lattice structure, E_eph, and generation of free electrons (E_eGen) from the volumetric-absorbed laser energy by the material (E_abs) [7]:

Equation (22) provides the electron warming based on the volumetric energy leading to E_he [36]: where c_e0 is the initial electron-specific heat capacity (7 × 10⁻⁴ J mol⁻¹ K⁻²) [34], ρ_e is the electron density (mol cm⁻³), K_e is the electron heat conductivity (1.05 W (m K)⁻¹) [34], and T_e is the electron temperature (K).

The material ionisation with the maximum depth of laser diffusion (L) depends on the magnitude of the E_abs. The material ionisation causes local damages in the lattice and ablates the workpiece material. Hence, a new workpiece surface is generated. For each pulse, the volumetric absorbed laser energy reduces after material ionisation. The new value of the E_abs again leads to ionisation until the value of E_abs becomes less than the E_eGen. Due to the Gaussian distribution of the laser energy in the spot area, the V-form ablation occurs, which in turn causes the variation of the AoI. Hence, the I_L reduces with the ablation depth.

The laser ablation process should be divided into discrete time steps. In every time step, the practical laser power intensity and the absorbed energy of the laser were calculated according to Eqs. (13) and (14). The electron warming model calculated the absorbed energy, according to Eqs. (21) and (22). Also, Eqs. (16)–(18) were pplied to analyse the atomic ionisation, the repulsive stresses, and activation of the MPI mechanism. If the density of free electrons becomes higher than a specific threshold value, the MPI mechanism should be activated as described. The amount of energy that ensures the minimum density of free electrons in the absorption layer was the threshold value required to activate the MPI mechanism. Otherwise, the energy should only be used to heat up the electrons. The MPI mechanism causes the density of the free electrons to change. Therefore, at each time step with the MPI, the free electron density should be updated along with the temperature of the free electrons and other physical properties. Then Eqs. (19) and (20) was used to transfer the energy of the electrons to the atoms.

The finite-difference model and its numerical equation were used to calculate the thermal gradient and temperature values in the free electrons and atoms after laser energy absorption. In the model, the finite difference method was applied with Du Fort-Frankel formulation to calculate the temperature of electrons and the material: where c_p is the material-specific heat capacity, ρ is the material density, K is the material heat conductivity, \( {T}_{i\ j}^n \)is the temperature in node with row number i, column number j, and in time step n, ∆x is the width of every element on the workpiece surface, ∆y is the depth of every element through the workpiece material.

Each pulse includes the radiation time of t_p = 10 ps. Since the repetition frequency of the utilised laser is 400 kHz, the cooling time that is the rest time between two consecutive pulses would be 2.5 μs.

It is essential to select the appropriate time step ∆t in order to obtain the convergence for Eq. (23) [37]. If the material ablation for each node is expected, its corresponding c_p and K values should be rewritten so that the absorbed energy does not lead to the increasing temperature of this node and is entirely transferred to the nodes in underlying layers.

According to Fig. 11, the 2D simulation was carried out for the beam radius instead of the full beam. Therefore, the total time of the simulation is determined considering beam radius and velocity. The workpiece was discretised in elements with width and length of ∆x = 2 μm and ∆y = 20 μm, respectively, as shown in Fig. 11. The simulation was performed on the A-A cross-section parallel to the central laser beam axis. The data of the elements, including temperature values and the material properties, is stored in the corresponding matrixes. The absorbed energy of the pulse is calculated, considering the Gaussian and AoI factors, according to Eq. (13). The time increments for the radiation phase, and the cooling sections are not the same to maintain the stability of Eq. (23). Material state and its physical properties, like c_p and K change with temperature. Hence, at each time step, the absorbed energy of the laser and mechanism of laser-material interaction is determined. Then, the elements data should be updated, including the temperatures of lattice and electrons, thermal properties of the material and free electrons, the density of ions, and free electrons. Finally, regarding the volume of ablated material, the geometry of the model is refreshed. A significant portion of absorbed energy is dissipated through thermal conduction in the material in the cooling section. In this regard, thermal conduction follows classical equations like Fourier conduction equations.

As shown in Fig. 12, in every time increment, the updated properties of the material and the applied laser, the Gaussian equations, laser incident angle, and the practical laser power intensity were calculated. A comparison of the computed laser power intensity with a critical value has clarified the laser-material interaction mechanism. The ablation depths in the interacted nodes were calculated through the laser-material interaction mechanism. The necessary laser power intensity, a threshold to switch the laser-material interaction mechanism from a thermal interaction to MPI, should be calculated in the simulation. Besides, in the thermal laser-material interaction mechanism, the practical sublimation temperature should be determined. Hence, for two different ablation tests, the mentioned values should be obtained by comparing the depth of ablations in the simulations with the corresponding experimental values. The maximum depth of ablation in the simulation validates with the corresponding experimental results.

In this section, the ablation depth values, calculated by the simulation, were compared with the corresponding experimental results. Besides, the material removal rate, computed by the simulation, is also compared with the experimental values. The comparison showed a good agreement between the calculated values and the test results.

The next section describes the temperatures obtained from the simulation. The temperature values are compared with the HAZ in the practical results. The variables affecting temperature values and laser-material interaction mechanism can also be calculated in the simulation, such as laser absorption and power intensity during the laser process.

In the final part, determining factors such as the critical power intensity and the sublimation temperature were obtained by the simulation. The correct calculation of these values is crucial in the accurate and valid estimate of the cutting depth.

In the current investigation, the interaction mechanism between USPL with the wavelength of about 1000 nm and silicon nitride was studied. The minimum laser power intensity (I_{L − crt}) for material ionisation as well as the T_sep as the ablation temperature in thermal interaction were calculated.

The a_e − L in a mathematical model is found and compared with the practical results for E_{L − input} = 6.25, 20 and 62.5 J mm⁻². With appropriate values for I_{L − crt} and T_sep, the mathematical model precisely predicted the laser ablation depths for the given E_{L − input}. Next, the mathematical model was validated by the experiments with respect to laser ablation depth for different E_{L − input} = 6–12.5 J mm⁻². The model predicted the laser ablation depth with about 10% error in comparison to the corresponding experimental results. The amount of absorbed laser energy and the temperature of material ablation are other outputs of the mathematical model. The obtained results from this study are summarised below:

In the peripheral of the laser cut, since the intensity of the laser was considerably lower than in the beam centre, the temperature enhances. It results in a shallow HAZ area near the laser cuts on the surface with a depth of <5 μm.