© The Author(s), published by EDP Sciences, 2025

1 Introduction

Reactive-ion-etching (RIE) is the process that enables the fabrication of nanometer-sized features with a high degree of selectivity and anisotropy [1, 2]. This process relies on a plasma that contains chemically reactive radicals and ions generated from a neutral gas with the help of oscillating electric and/or magnetic fields (e.g., ICP). These reactive species chemically interact with the sample, forming volatile byproducts that are subsequently evacuated from the reaction chamber [3].

In the region between plasma and solid surface (sample), called the plasma sheath, plasma neutrality breaks due to the different mobility of electrons and ions [4]. This imbalance gives rise to a potential drop across the sheath, accelerating positive ions toward the substrate. Applying an additional electric field oscillating at radio-frequency (RF) to the substrate electrode further increases ion energy, which often reaches hundreds of electron volts (eV) [1].

Simulation of the plasma sheath region is crucial for understanding ion dynamics and predicting etch rates. Analytical models [5–7] provide approximate solutions for two limiting cases: the low-frequency regime, in which ions respond to the instantaneous electric field, and the high-frequency regime, when ions respond to the average sheath potential [8]. Solutions for an intermediate regime can be obtained using numerical models based on fluid approximation [9, 10], and more accurate but computationally more expensive kinetic models [11, 12].

While the effect of plasma parameters on etching rates has been extensively studied on Si wafer with thin oxide coatings [13–15]. Much less is known about how the thickness of a substrate affects the discharge. Kawata et al. [16] has observed a decrease in the etching rate on thicker samples and linked it to the sample’s impedance. This is particularly relevant in applications involving optical components for highpower laser systems, such as pulse compression gratings [17] or polarization control gratings [18, 19], which can be several millimeters thick.

In this study, we investigate how substrate thickness impacts plasma etching. We use fluid-based simulations combined with an equivalent electric circuit model to compute the sheath potential and the ion energy distribution function (IED) at the substrate surface. These results are validated through etching experiments on samples of varying thickness.

2 Etching experiments

We performed a series of experiments using fused silica samples with thicknesses of 2, 5, and 10 mm and radius r = 20 mm. Etching was done in a CHF3/Ar/O₂ inductively coupled plasma (ICP) discharge using SI 500 (PTSA 400 ICP) from SENTECH Instruments GmbH, Berlin. A schematic of the etching chamber is shown in Figure 1a. A current applied to a spiral coil generates a time-varying magnetic field, which in turn induces the electric field that sustains plasma. The inductive source power (ICP power) controls plasma density, while the substrate electrode power (RF power) independently controls sample biasing [20, 21].

Figure 1 Simplified model of the ICP reactor and EEC used to model the sheath region.

The discharge conditions – gas composition, pressure, RF power, DC bias, and ICP power – are summarized in Table 1. Before the etching, samples were patterned with grating lines by laser interference lithography in the photoresist (Az^® MIR 701, Microchemicals GmbH). Etching was performed for 2 min. Final etching depths were measured with an atomic-force microscope (AFM) (Dimension 3100d, Veeco).

Figure 2 shows the etching rates as a function of the ICP power for all etched samples. The 2 mm sample exhibits a steady increase in etching rate with increasing ICP power. At 1000 W ICP power, DC bias drops (Table 1), which leads to the saturation of the etching rate. In contrast, samples with thickness 5 and 10 mm show a different trend. Etching rates increase to the maximum at ICP power values 500 W and 300 W, respectively, and then decline at higher power levels.

Figure 2 Etching rates for vs ICP power for samples with different thicknesses.

For the h = 10 mm sample, the etching completely stops at 1 kW ICP power. EDS analysis on this sample showed an elevated presence of carbon together with a high fluorine content (20%), suggesting the presence of a C_xF_y film deposited on the sample. Oehrlein et al. [22] reported that during SiO₂ etching, surface passivation by CF_x radicals dominates the process below a threshold value of the voltage bias.

In the next chapters, we show that the thickness of a dielectric sample directly affects the sheath potential and, consequently, the ion energy at the surface. If the sample is too thick, it behaves like a floating surface due to the sample’s low capacitance. This results in suppressed etching because of the insufficient ion energy.

3 Simulations

We simulate a one-dimensional plasma-sheath region spanning from the sample to the sheath-plasma boundary. We adopt a fluid model combined with an Equivalent Electric Circuit model (EEC) and take plasma bulk parameters from the device’s datasheet. In this chapter, we provide the model description and governing equations.

3.1 Fluid model

We consider a low-pressure, non-magnetized, charge neutral plasma in contact with an RF-biased electrode in the presence of a dielectric substrate. A positively charged plasma sheath region with thickness s is formed between the plasma and the substrate. The following assumptions are made to simplify the model:

• Ions are considered cold, and their thermal motion is neglected, since the ion thermal velocity is not only much smaller than the electron thermal velocity, but also much smaller than the ion velocity due to the applied voltage bias;

• Only positively charged ions are considered. Negative ions generally do not enter the plasma sheath due to the positive sheath potential;

• No collisions or ionization processes are assumed to occur within the sheath. In high-density, low-pressure discharges, the mean free paths of ions and neutrals are typically larger than the sheath thickness;

• Electrons are in thermal equilibrium with a constant temperature T_e;

• The electron and ion densities in the plasma bulk are time- and space-invariant.

The potential distribution V (x) within the sheath is governed by the Poisson equation:$$\frac{{\partial }^{2}V\left(x\right)}{\partial x^2}=-\frac{e}{{\epsilon }_0}\left(n_i\left(x\right)-n_e\left(x\right)\right),$$(1)where e is the electronic charge, ε₀ is the permittivity of free space, and n_i and n_e are ion and electron densities respectively.

Electron density follows the Boltzmann distribution:$$n_e\left(x,t\right)=n_p\exp \left(\frac{\mathrm{eV}(x,t)}{k_{\mathrm{B}}{T}_e}\right),$$(2)where n_p is the plasma density at plasma-sheath boundary, and k_B is the Boltzmann constant.

Ion dynamics is described by the ion momentum and flux conservation equations:$$\frac{\partial u_i(x,t)}{\partial t}+u_i\left(x,t\right)\frac{\partial u_i\left(x,t\right)}{\partial x}=-\frac{e}{m_i}\frac{\partial V\left(x,t\right)}{\partial x},$$(3) $$\frac{\partial n_i(x,t)}{\partial t}+\frac{\partial \left(n_i\left(x,t\right)u_i\left(x,t\right)\right)}{\partial x}=0,$$(4)where m_i is ion mass, and u_i is the ion velocity.

At the sheath-plasma boundary, ions enter the sheath with Bohm velocity [23]:$$u_i\left(x=s\right)= u_{\mathrm{B}}={(k_{\mathrm{B}}{T}_e/m_i)}^{1/2}.$$(5)

We assume that the plasma density at the sheath-plasma boundary is n_p = 0.61n₀, where n₀ is the bulk plasma density [1]. The potential at the sheath-plasma interface is set to zero, while the potential at the substrate V(0, t) = V_sub(t) is obtained from the EEC model, described in the next section.

3.2 Equivalent electric circuit model (EEC)

In the EEC, the plasma-sheath model is represented as a parallel combination of: an ion current source, a time dependent capacitor for the plasma sheath, and a diode, representing the flow of electrons. A sample is simulated as a capacitor and connected in series to the sheath. It was pointed out by Yu et al. [24] that in the general case, the substrate size is smaller than the size of the electrode table. A plasma sheath is formed over the open part of the table in addition to the plasma sheath over the sample. We incorporate this extra sheath in our EEC in parallel to the sheath over the substrate. An RF voltage source is connected to sheaths through the blocking capacitor as shown in Figure 1b. The electrode is covered by a Si carrier wafer with a relative permittivity ε_Si = 11.7. The sample (fused silica ${\epsilon }_{\mathrm{Si}{\mathrm{O}}_2}=4.0$) is placed on top of the carrier wafer.

Ion and electron currents are governed by equations (6) and (7) respectively.$$I_i\left(x,t\right)=\mathrm{eA}{u}_i\left(x,t\right)n_i\left(x,t\right),$$(6) $$I_e\left(x,t\right)=\frac{\mathrm{eA}{u}_e\left(t\right)n_e(x,t)}{4},$$(7)where A is the area, such as for the sample path in EEC A = A_sub and for the path through the open part of the electrode A = A_d = A_electrode − A_sub. The electron mean velocity u_e follows equation (8).$$u_e={\left(\frac{8k_{\mathrm{B}}{T}_e}{m_e\pi }\right)}^{1/2}.$$(8)

The sheath region acts as a time-dependent capacitor with the corresponding displacement current [10] (Eq. (9)).$$I_d\left(t\right)=\frac{\mathrm{d}\left(C_s\right(t\left)V_{\mathrm{sub}}\right(t\left)\right)}{\mathrm{d}t}=C_s\left(t\right)\frac{\mathrm{d}{V}_{\mathrm{sub}}\left(t\right)}{\mathrm{d}t}+V_{\mathrm{sub}}\left(t\right)\frac{\mathrm{d}{C}_s\left(t\right)}{\mathrm{d}t},$$(9)where the sheath capacitance $C_s\left(t\right)$ defined by equation (10).

$$C_s\left(t\right)=\frac{{\epsilon }_{0}A}{s\left(t\right)},$$(10)with s(t) being a time-dependent sheath thickness.

The following differential-algebraic system of equations (DAE) is formed by the EEC:$$\begin{array}{ll}& \{\begin{array}{l}\frac{\mathrm{d}{V}_{\mathrm{sub}}^{\left[1\right]}}{\mathrm{d}t}(C_s^{\left[1\right]}+C_{\mathrm{eff}})-\frac{\mathrm{d}{V}_3}{\mathrm{d}t}{C}_{\mathrm{eff}}=I_i^{\left[1\right]}-I_e^{\left[1\right]}-V_{\mathrm{sub}}^{\left[1\right]}\frac{\mathrm{d}{C}_s^{\left[1\right]}}{\mathrm{d}t}\\ \frac{\mathrm{d}{V}_{\mathrm{sub}}^{\left[2\right]}}{\mathrm{d}t}(C_s^{\left[2\right]}+C_c^{\left[2\right]})-\frac{\mathrm{d}{V}_3}{\mathrm{d}t}{C}_c^{\left[2\right]}=I_i^{\left[2\right]}-I_e^{\left[2\right]}-V_{\mathrm{sub}}^{\left[2\right]}\frac{\mathrm{d}{C}_s^{\left[2\right]}}{\mathrm{d}t}\\ -\frac{\mathrm{d}{V}_{\mathrm{sub}}^{\left[1\right]}}{\mathrm{d}t}{C}_{\mathrm{eff}}-\frac{\mathrm{d}{V}_{\mathrm{sub}}^{\left[2\right]}}{\mathrm{d}t}{C}_c^{\left[2\right]}+\frac{\mathrm{d}{V}_3}{\mathrm{d}t}(C_{\mathrm{eff}}+C_c^{\left[2\right]}+C_b)=\frac{\mathrm{d}{V}_{\mathrm{RF}}}{\mathrm{d}t}{C}_b\end{array}\\ & \end{array},$$(11)where C_c is the capacitance of the Si carrier wafer. The upper script [1] represents the path through the sample, and [2] is for the path through the open electrode covered by the Si carrier wafer. A combination of the carrier wafer and the sample is represented as capacitors in series with the effective capacitance C_eff:$$C_{\mathrm{eff}}=\frac{C_{\mathrm{sub}}{C}_c^{\left[1\right]}}{C_{\mathrm{sub}}+C_c^{\left[1\right]}},$$(12)where C_sub is the substrate capacitance, $C_c^{\left[1\right]}$ is the capacitance of the carrier under the sample computed according to equations (13a)–(13c).

$$C_{\mathrm{sub}}=\frac{{\epsilon }_{\mathrm{Si}{\mathrm{O}}_2}{\epsilon }_0{A}_{\mathrm{sub}}}{h_{\mathrm{sub}}},$$(13a) $$C_c^{\left[1\right]}=\frac{{\epsilon }_{\mathrm{Si}}{\epsilon }_0{A}_{\mathrm{sub}}}{h_c},$$(13b) $$C_c^{\left[2\right]}=\frac{{\epsilon }_{\mathrm{Si}}{\epsilon }_0{A}_c}{h_c},$$(13c)where h_sub and $h_c$ are thicknesses of the substrate and carrier wafer respectively.

The model is solved iteratively until the solution converges. At first, constant ion current $I_i(0,t)=\mathrm{eA}{u}_{\mathrm{B}}{n}_p$ and constant sheath $s\left(t\right)=I_{\mathrm{RF}}/e{n}_p{\omega }_{\mathrm{RF}}A$ are assumed and the DAE system (11) is solved. Then equations(2)–(4) are solved with the obtained sheath potential $V_{\mathrm{sub}}\left(t\right)$. After that, the DAE system (11) is solved with new values for ion and electron currents and sheath thickness.

Generally speaking, finding the sheath-plasma boundary is not trivial. Godyak [25] suggested that the electric field at the sheath-plasma boundary is non-zero and equals $E\left(x\right)=\frac{k_{\mathrm{B}}{T}_e}{e{\lambda }_D}$ with Debye length ${\lambda }_D$:$${\lambda }_D={\left(\frac{{\epsilon }_0{k}_{\mathrm{B}}{T}_e}{n_p{e}^2}\right)}^{1/2}.$$(14)

Riemannn [26] and Kaganovich [27] showed that the sheath boundary proposed by Godyak lies deep in the sheath. We define the sheath thickness s(t) as the position where the electric field reaches:$$E\left(x\right)=-\frac{\mathrm{d}V\left(x\right)}{\mathrm{d}x}=\frac{k_{\mathrm{B}}{T}_e}{10e{\lambda }_D}.$$(15)

We use MATLAB^® ode23t solver for the DAE system (11). PDEs (3), and (4) are solved by the method of lines with the finite difference in the space domain and MATLAB^® ode23 solver in the time domain. Equation (1) was solved with the MATLAB^® bvp4c solver at each time step.

4 Simulation results

We simulate a carbon-fluorine discharge, assuming the presence of only CF₃₊ ions. A uniform electron temperature k_B T_e = 3 eV was used (from the machine datasheet). Plasma density n₀ = aP_coil was modeled as a linear function of the ICP power P_coil, where a is the slope coefficient [28]. The maximum plasma density n₀ = 5 × 10¹⁷ m⁻³ was taken from the datasheet of the device and corresponds to the maximum P_coil = 1200 W. RF frequency of the biased electrode f_RF = 13.56 MHz and the driving voltage V_RF was adjusted for different P_coil to ensure that desired DC bias V_DC is reached in the absence of the sample.

As an example, we show results for a SiO₂ substrate with radius r = 50 mm (half of the diameter of the electrode) and thickness h = 2 mm. Figure 3a shows calculated waveforms for the voltage across the sheath. As expected, the waveform shape is close to the half-rectified sinusoid [29]. Nevertheless, the potential drop in the sheath over the substrate is much smaller than that over the open part of the electrode.

Figure 3 Results of simulations with sample height h = 2 mm and radius r = 50 mm. (a) Potential drop across plasma sheath, (b) and (c) current waveforms for the sample and the open part of the electrode respectively, (d) ion energy distribution (IED), (e) and (f) ion velocity and density waveforms. In (a), (d)–(f) the red-dashed line is for the open electrode path in EEC, and the blue line is for the sample path in EEC.

Figures 3b and 3c illustrate current waveforms for the sample path, and for the electrode covered by the carrier wafer, respectively. During the negative half-cycle of the RF signal, the electron current is effectively zero, and ions are accelerated towards the sample with a delay due to their inertia. The plasma sheath responds to the current imbalance by increasing the displacement current. While I_RF approaches zero, the displacement current becomes negative to balance the ion current since the potential is too high and the electron current is still zero. During the positive part of the RF cycle, electrons react immediately to the positive current imbalance, and the electron current rises instantly. This results in sheath collapse, and the displacement current approaches zero. Numerical calculations cause noise in the displacement current due to strong oscillations of the $V_w\left(t\right)\frac{\mathrm{d}{C}_s\left(t\right)}{\mathrm{d}t}$ part in equation (9) during the plasma sheath collapse.

Solution of the ion conservation equations allows us to obtain the instantaneous ion energy as:$$E_{\mathrm{ion}}\left(t\right)=\frac{m_i{u}_i^2\left(t\right)}{2},$$(16)

and compute ion energy distribution (IED) (Fig. 3d). The IED for the substrate is shifted towards lower energies with a narrower peak separation $\Delta E_{\mathrm{ion}}^{\mathrm{sub}}$ = 93 eV and $\Delta E_{\mathrm{ion}}^{\mathrm{table}}$ e = 143 eV.

This difference is explained by the reduced sample capacitance, which limits charge accumulation and reduces the displacement current required to balance the sheath. This results in a smaller potential drop and a weaker electric field, reducing ion acceleration. Correspondingly, ion velocity and density near the substrate are shown in Figures 3e, 3f. The decreased ion velocity is compensated by increased ion density to maintain current continuity.

4.1 Sheath behavior without substrate

We begin by analyzing the simplest case of an electrode without a substrate. The RF voltage amplitude VRF was adjusted such that maximum sheath potential remained constant at V_sub = 200 V for each simulated ICP power value. Maximum values for sheath potential, ion energies, and peak separation distance Δ IEDs as functions of ICP power are presented in Figure 4a.

Figure 4 (a) Voltage potential, max ion energy and Δ IED over electrode without a sample. (b) Ion current without a sample and (c) ion energy distribution without a sample.

Despite the constant sheath potential, the maximum ion energy increases with rising ICP power, accompanied by a broadening of the IED. This behavior is consistent with previous findings [8] and can be attributed to the increase in plasma density, and consequently higher plasma ion frequency ${\omega }_{\mathrm{ion}}=(n_o{e}^2/({ϵ}_0{m}_{\mathrm{ion}}){)}^{1/2}$. Ions respond stronger to the time-varying electric field, causing oscillations of the ion current (Fig. 4b). Broad and bimodal IED is the result at higher plasma ion frequency as can be seen in Figure 4c.

4.2 Effect of sample geometry

In this section, we show how sample geometry affects the plasma sheath. We start by considering the case when the sample diameter equals the diameter of the electrode. Figures 5a, 5b illustrate the impact of the sample capacitance on the sheath potential and IED at ICP power of 1000 W and 300 W, respectively. The sample capacitance was varied by changing the thickness of the fused silica samples, and the utmost right point corresponds to the 500 $\mathrm{\mu }$m thick Si wafer.

Figure 5 Voltage potential, max ion energy and ΔIED (a) vs substrate capacitance at 1000 WICP (b) vs substrate capacitance at 300 W ICP.

All the values (sheath potential, ion energy, and ΔIED) plummet with the decreased capacitance (increased thickness) and both the potential and the maximum ion energy exhibit a power law dependence on it, such as y = ax ^b + c, where y is ion energy or sheath potential, x represents the sample capacitance, and a, b, and c are fitting coefficients. This effect is more pronounced at higher ICP power.

Figure 6 shows how the maximum ion energy changes for samples with different thicknesses. It is interesting that while for Si wafer (C_sub = 7 nF) the ion energy rises with ICP power, this trend is reversed for SiO₂ sample with thickness h > 2 mm (C_sub < 0.56 nF).

Figure 6 Maximum ion energy vs ICP power.

Finally, Figure 7 shows that increasing the sample radius also increases the sheath potential and broadens the IED, due to a larger effective capacitance.

Figure 7 Voltage potential, max ion energy and ΔIED vs substrate’s radius at 1000 W ICP for h = 2 mm SiO2 sample.

These findings suggest that sample geometry significantly affects ion dynamics and must be considered in the design of the etching experiments.

5 Etching model and comparison with experimental values

The etching process in fluorocarbon plasma discharge is governed by the balance between the deposition of CF_x polymer layer and the material removal [3]. This process can be characterized by four dominant mechanisms: (1) physical sputtering by high-energy ions, (2) chemical etching by energetic ions, (3) etching by radicals, and (4) polymer deposition (passivation). The deposition of CF_x layer is primarily caused adsorption of CF_x radicals and assisted by the activation of the surface sites by low energetic CF_x⁺ ions [30]. This layer can both inhibit etching and act as a source of atomic fluorine depending on process conditions [31]. While a comprehensive list of reactions can be found elsewhere [30], we list here the most relevant ones.

Direct ion sputtering:$$\mathrm{Si}{\mathrm{O}}_2\left(\mathrm{s}\right)+{\mathrm{I}}^{+}⟶\mathrm{Si}{\mathrm{O}}_2.$$(17)

Polymer deposition by radicals:$$\mathrm{Si}{\mathrm{O}}_2\left(\mathrm{s}\right)+\mathrm{C}{\mathrm{F}}_x⟶\mathrm{Si}{\mathrm{O}}_2\mathrm{C}{\mathrm{F}}_x\left(\mathrm{s}\right).$$(18)

Ion enhanced etching:$$\mathrm{Si}{\mathrm{O}}_2\mathrm{C}{\mathrm{F}}_x\left(\mathrm{s}\right)+{\mathrm{I}}^{+}⟶\mathrm{Si}{\mathrm{F}}_x+\mathrm{C}{\mathrm{O}}_2.$$(19)

Etching by radicals through fluorination:$$\mathrm{Si}{\mathrm{F}}_3\left(\mathrm{s}\right)+\mathrm{F}⟶\mathrm{Si}{\mathrm{F}}_4.$$(20)

Surface activation and passivation by low-energy ions:$$\mathrm{Si}{\mathrm{O}}_2\mathrm{C}{\mathrm{F}}_{\mathrm{x}}\left(\mathrm{s}\right)+{\mathrm{I}}^{+}⟶\mathrm{Si}{\mathrm{O}}_2\mathrm{C}{\mathrm{F}}_{\mathrm{x}}\mathrm{*}\left(\mathrm{s}\right),$$(21) $$\mathrm{Si}{\mathrm{O}}_2\mathrm{C}{\mathrm{F}}_x\left(\mathrm{s}\right)+{\mathrm{CF}}_x^{+}⟶\mathrm{Si}{\mathrm{O}}_2{\mathrm{C}}_2{\mathrm{F}}_{2x}\left(\mathrm{s}\right).$$(22)

Here (s) denotes surface sites, I⁺ stands for ions, and * denotes activated surface. The minimum ion energy thresholds for etching are approximately 70 eV for sputtering by CF_x⁺ ions and 35 eV for ion-assisted chemical etching [30].

5.1 Comparison with experiment

Using parameters from Table 1, we calculated ion energy values for each ICP power and substrate thickness, keeping substrate radius r = 20 mm constant, similar to the etching experiments. Figure 8 shows how maximum ion energy changes as a function of the sample thickness for different ICP powers. A clear reduction in ion energy with increasing sample thickness is observed. For the 10 mm thick sample, ion energy at 500 and 1000 W ICP powers is below the chemical etching threshold, correlating with the observed decrease in the etching rate. In these cases, Ar⁺ ion sputtering may continue etching if ion energy exceeds the Ar⁺ sputtering threshold 20–45 eV [32]. At 1000 W ICP power, ion energy E_ion drops to 18.5 eV, and consequently, no etching was observed in the experiment.

Figure 8 Maximum ion energy as a function of the sample thickness for different ICP power values.

While the decrease in the ion energy with increasing ICP power does not necessarily lead to a decrease in the etching rate, the lower ion energy due to sample thickness will result in a reduced etching rate. In the next section, we show how the etching rate depends not only on the ion energy but also on the ion and neutral densities.

5.2 Etching rate model

Gottscho et al. [33] proposed a model to predict the etching rate based on plasma discharge parameters (Eq. (23)). It takes into account the incident ion flux Γ i, mean ion energy E_i, and etching effect due to the neutral flux Γ_N. S₀ in equation (23) stands for a radical sticking probability and v and k are volumes removed per neutral and ion respectively.$$\mathrm{ER}={\left(\frac{1}{\nu S_0{\mathrm{\Gamma }}_{\mathrm{N}}}+\frac{1}{k{\mathrm{\Gamma }}_{\mathrm{i}}{E}_i}\right)}^{-1}.$$(23)

However, this model does not account for etching due to direct sputtering, nor does it consider that a minimum ion energy E_th is needed for etching to occur. Additionally, the time dependence of the ion flux ${\mathrm{\Gamma }}_i\left(t\right)=n_i\left(t\right)u_i\left(t\right)$ and the ion energy E_i(t) adds to the complexity of the problem. Steinbruchel [34] showed that the etching yield is proportional to the square root 1/2

of the ion energy $Y\propto E_{\mathrm{ion}}^{1/2}$. To include those effects, we propose the modified model, given in equation (24).$$\mathrm{ER}={\left(\frac{1}{\alpha \mathrm{E}{\mathrm{D}}_{\mathrm{N}}}+\frac{1}{\beta \sqrt{\mathrm{E}{\mathrm{D}}_i^{\mathrm{chem}}}}\right)}^{-1}+\gamma \sqrt{\mathrm{E}{\mathrm{D}}_i^{\mathrm{sput}}},$$(24)

where ED_N and ED_i are energy densities (J/m³) of neutrals and ions respectively. The energy density of neutrals is estimated as$${\mathrm{ED}}_{\mathrm{N}}=n_N{E}_N,$$(25)

where E_N is the neutral energy and n_N is the density of neutrals estimated as [35]:$$n_N=\tau K{n}_e{n}_g,$$(26)

τ is the residence time, K is the rate constant obtained assuming Maxwellian electron energy distribution, and n_e and n_g are electron and feed gas densities, respectively.

Time-dependence was included by integrating over one RF period as shown in equation (27).$$\{\begin{array}{cc}\mathrm{E}{\mathrm{D}}_i=f\mathrm{RF}{\int }_0^T{n}_i(E_i-E_{\mathrm{th}})\mathrm{d}t,& E_i-E_{\mathrm{th}}>0\\ \mathrm{E}{\mathrm{D}}_i=0,& E_i-E_{\mathrm{th}}<0\end{array}$$(27)

The last term in equation (24) accounts for the etching due to sputtering. In our experimental discharge, Ar⁺ ions are the main source of the direct sputtering. In our simulations, we consider $E_{\mathrm{th}}^{\mathrm{chem}}=35\mathrm{ }\mathrm{eV}$ for chemical etching, and $E_{\mathrm{th}}^{\mathrm{sput}}=20 \mathrm{eV}$ for sputtering by Ar⁺ ions. While we have simulated our plasma conditions only for CF3⁺ ions, we assume that Ar⁺ ions acquire similar ion energy, since it was shown that in multi-ion plasma discharges, ions acquire velocities close to the system velocity [36].

Results of the linear fit with coefficients α = 0.9 (nm/min (J⁻¹ m³)^−1/2), β = 1466 (nm/min J⁻¹ m³), and γ = 89 (nm/min ((J⁻¹ m³)^−1/2) are shown in Figure 9. R² values of 0.97, 0.95, and 0.86 are obtained for sample with thickness h = 2, h = 5, and h = 10 mm respectively. These results indicate a good fit of the simulations to the experimental data, especially for identifying etch rate suppression due to low ion energy. In particular, the etch rate collapse observed for thicker samples at high ICP power is a direct consequence of the low ion energy due to low sample capacitance.

Figure 9 Experimental etching rates (solid lines) and result of the linear fit of equation (24) (dashed line) vs ICP power. Plots are for samples with thickness h = 2 mm (circles), h = 5 mm (squares), and h = 10 mm (triangles).

Conclusion

We extended a combined fluid simulation and equivalent electric circuit (EEC) model for plasma sheath simulation to include the effect of the sample. Our results demonstrate that a dielectric sample acts as a capacitor, significantly affecting the sheath voltage and the energy of ions reaching the surface of the sample. Simulations demonstrate that an increase in the thickness of the sample leads to lower sheath potential and, consequently, lower ion energy, resulting in lower etching rate. Etching experiments validated this behavior and showed a significant reduction in the etching rate for thicker samples. If the ion energy drops below the threshold value, the etching is dominated by the polymer deposition process, as for the 10 mm thick sample at 1000 W ICP power.

The proposed etching rate model, which accounts for chemical etching as well as sputtering, agrees well with experimental data. This validates the predictive capability of our simulation approach.

These findings highlight the importance of the sample size, especially thickness, in predicting plasma sheath behavior. Neglecting the effect of the sample may lead to significant overestimation of ion energies and etching rates, particularly for dielectric materials used in high-power optics.

Acknowledgments

We are grateful to Dr. Ing. Andreas Schulz, IGVP, University of Stuttgart, for a valuable discussion. We also wish to thank Thomas Schoder, ITO, University of Stuttgart, for his help with the etching equipment. Part of this work has been presented on the 126. Annual Conference of the DGaO 2025, 26.-30.5.2025, Stuttgart.

Funding

This work was funded by the German Federal Ministry for Economic Affairs and Energy by resolution of the German Federal Parliament, project KuHLGit. The etching equipment was funded by the German Science Foundation DFG, project 54555282.

Conflicts of interest

The authors declare that they have no competing interests to report.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Author contribution statement

Conceptualization, A. Savchenko and C. Pruss; Investigation and formal analysis, A. Savchenko; Funding acquisition, C. Pruss and S. Reichelt; Supervision, C. Pruss, A. Herkommer, and S. Reichelt; Writing – Original Draft Preparation, A. Savchenko; Methodology and Writing – review & editing, all authors. All authors discussed the results and contributed to the final manuscript.

References

All Tables

All Figures

	Figure 1 Simplified model of the ICP reactor and EEC used to model the sheath region.
In the text

	Figure 2 Etching rates for vs ICP power for samples with different thicknesses.
In the text

Figure 3 Results of simulations with sample height h = 2 mm and radius r = 50 mm. (a) Potential drop across plasma sheath, (b) and (c) current waveforms for the sample and the open part of the electrode respectively, (d) ion energy distribution (IED), (e) and (f) ion velocity and density waveforms. In (a), (d)–(f) the red-dashed line is for the open electrode path in EEC, and the blue line is for the sample path in EEC.

In the text

	Figure 4 (a) Voltage potential, max ion energy and Δ IED over electrode without a sample. (b) Ion current without a sample and (c) ion energy distribution without a sample.
In the text

	Figure 5 Voltage potential, max ion energy and ΔIED (a) vs substrate capacitance at 1000 WICP (b) vs substrate capacitance at 300 W ICP.
In the text

	Figure 6 Maximum ion energy vs ICP power.
In the text

	Figure 7 Voltage potential, max ion energy and ΔIED vs substrate’s radius at 1000 W ICP for h = 2 mm SiO2 sample.
In the text

	Figure 8 Maximum ion energy as a function of the sample thickness for different ICP power values.
In the text

	Figure 9 Experimental etching rates (solid lines) and result of the linear fit of equation (24) (dashed line) vs ICP power. Plots are for samples with thickness h = 2 mm (circles), h = 5 mm (squares), and h = 10 mm (triangles).
In the text