# Roughness Evolution and Charging in Plasma-Based Surface Engineering of Polymeric Substrates: The Effects of Ion Reflection and Secondary Electron Emission

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{+}ions.

## 2. The Modeling Framework

#### 2.1. The Surface Charging Module

#### 2.1.1. Particle Trajectory Model

#### 2.1.2. Secondary Electron-Electron Emission Model

_{e}, equal to δ + η, which is commonly defined as the number of emitted electrons per incident (primary) electron. According to this definition, the yield includes three categories of emitted electrons [33]: (a) elastically reflected primary electrons, (b) inelastically reflected primary electrons, and (c) true secondary electrons. δ is the secondary electron emission yield including (c) and η is the backscattering coefficient including (a) and (b). All coefficients, σ

_{e}, δ, and η, may depend on the energy and the angle of incidence of the primary electrons, as well as on the substrate material. In the following, a model for σ

_{e}, δ, and η is described. It is based on available information in the literature for PMMA and other polymers in the energy range, which is of interest in plasma etching.

_{2}trench in view of SEEE [29,30].

_{e}at low energies for PMMA. However, there are analytical expressions describing δ and η for the whole energy spectrum in the case of PMMA such as the Lin and Yoy law [37] for δ. Yu et al. [38] also proposed an analytical expression to describe δ and used an analytical equation derived by Burke [39] for η. Regarding the computational studies, Dapor et al. [40] developed a Monte Carlo model for the emission of secondary electrons from PMMA. They calculated δ in the energy domain ranging from a few keV down to a few tens of eV [40]. Dapor [41] also calculated the total electron yield σ

_{e}as a function of the primary electron kinetic energy varying from 0 to 1500 eV. σ

_{e}from the latter work is adopted in this work (Figure 2), as it is the only describing σ

_{e}in the energy range of interest (0–50 eV).

_{0}is the energy of the primary electrons. Equation (1) was also utilized by Yu et al. [38] for PMMA. Generally, it expresses η in polymers consisting of H, C, N, and O as a function of E

_{0}(eV). It should be mentioned that, in the energy range of interest (0–50 eV), we assume that η represents only elastically-reflected electrons. This simplification is prompted by Monte Carlo calculations for Teflon demonstrating that only elastically reflected electrons contribute to η for energy lower than 50 eV [42].

_{e}for 0 to 16 eV, something that is not realistic. A compromise is to consider that n below 16 eV is equal to σ

_{e}. Thus, δ, i.e., σ

_{e}− η, is considered equal to zero for energy lower than 16 eV (Figure 2). It should be noted that the value of 16 eV is not far from the value of 12.6 eV, i.e., the average energy required to produce one secondary electron for PMMA [36]. It is also not far from the value of 10 eV, the general threshold for the secondary electron emission process [43]. For energy greater than 16 eV, δ is calculated as the difference of σ

_{e}and η (Figure 2).

#### 2.1.3. Ion Reflection Model

_{2}substrates. Radmilovic-Radjenovic et al. [50] also considered specular reflection for SiO

_{2}substrates. Wang and Kushner [53] considered both specular (at high energies) and diffusive (at low energies) reflection for SiO

_{2}substrates. In all of the previous works [47,48,49,50,51,52,53] it was considered that the incident ions deposited their charge and were reflected as hot neutral species.

^{+}on a PMMA surface, specular and elastic reflection of Ar

^{+}is considered, although Ar

^{+}may be implanted or may lose energy at the collision. Additionally, it is considered that ions drop their charge at the spot of the impact and are reflected as hot neutral species [47,53].

**n**is the unit normal vector on the surface and

**d**is the unit vector on the direction of the incident ion, the direction of a specularly-reflected ion is given by vector

**r**, i.e.,:

**r**=

**d**− 2 (

**d·n**)

**n**

#### 2.1.4. Surface Charge Density Model

#### 2.1.5. Charging Potential Model

#### 2.2. Surface Etching Model

^{+}ions impinging on a PMMA surface, the etching yield (the etching rate is the product of the ion flux with the etching yield) is expressed by [26]:

_{+}is the ion energy and E

_{th}is the threshold ion energy for PMMA sputtering. E

_{th}is regarded equal to 4 eV and A equal to 0.1 monomers/(ion·eV

^{0.5}) [26]. θ in Equation (4) is the angle of incident ions. The angle dependence being manifested by f(θ) is depicted in Figure 3. It is typical for cases of physical sputtering [55,56] and is approximated by a simple polynomial function, the form of which can be found in [26].

#### 2.3. Profile Evolution Module

## 3. Case Study

^{+}. The ion energy and angle distribution functions (IEADFs) for Ar

^{+}, as well as the electron energy and angle distribution functions (EEADFs) are the same as in [25,60]. The mean energy of ions and electrons are 90 and 4 eV, respectively. The ion angular distribution resembles a Gaussian and the electron angular distribution is isotropic. The ion flux is 1.86 × 10

^{20}m

^{−2}·s

^{−1}. The dielectric constant of PMMA is equal to 3. A substrate with a high (infinite) thickness is considered. The role of the substrate thickness on charging has been analyzed previously [24] and it has been found that although the thickness of the dielectric substrate affected the charging time, i.e., the time required for reaching a steady state charging potential, it did not affect the roughness evolution: the charging time was much shorter than the etching time for all values of thickness (0.1 μm to infinite thickness for surface roughness at the microscale).

## 4. Results and Discussion

## 5. Conclusions

^{+}ions on a PMMA surface, a simple model of non-interacting collisions was considered, i.e., specular reflection of the ions on the surface with no energy losses.

_{2}chemistry, which is widely used for the surface modification and roughening of polymeric substrates.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The modeling framework and the procedure of the computations. The coupling among modules of the framework, as well as the flow of data in the framework, is depicted.

**Figure 2.**The total electron yield, σ

_{e}, the secondary electron emission yield, δ, and the backscattering coefficient, η, being utilized in the SEEE model.

**Figure 3.**The function f(θ) of Equation (4) vs. the angle of ion incidence, θ. The maximum is at 75°.

**Figure 4.**The initial sinusoidal profile of the PMMA surface. The steady state potential, as well as some of the ion (in black) and electron (in red) trajectories, are also depicted. h is the height of the vacuum space, and b is the thickness of the PMMA layer. h is 2.2 μm, and b >> h.

**Figure 5.**Snapshots of the surface profile for different etching times (

**a**) without charging (multimedia view, please see Figure5a.avi), (

**b**) with charging (multimedia view, please see Figure5b.avi), and (

**c**) with charging and SEEE (multimedia view, please see Figure5c.avi), when the ion reflection is not taken into account. The profiles are cut from the middle of the first valley to the middle of the last one. The charging potential for the snapshots of Figure 5b,c is also depicted.

**Figure 6.**Snapshots of the surface profile for different etching times (

**a**) without charging (multimedia view, please see Figure6a.avi), (

**b**) with charging (multimedia view, please see Figure6b.avi), and (

**c**) with charging and SEEE (multimedia view, please see Figure6c.avi), when the ion reflection is taken into account. The profiles are cut from the middle of the first valley to the middle of the last one. The charging potential for snapshots of Figure 6b,c is also depicted.

**Figure 8.**The charging potential (average value of potential at the four valleys of the profile) vs. parameter m (Equation (5)). The arrows on the curves denote the time path during etching. Values above 500 s of etching for the curves corresponding to cases without ion reflection have been removed as the profiles are almost flat (cf. Figure 7). The small difference (~3 V) observed in the initial potential between the two cases of SEEE is an expected difference between two runs of a stochastic process.

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**MDPI and ACS Style**

Memos, G.; Lidorikis, E.; Kokkoris, G.
Roughness Evolution and Charging in Plasma-Based Surface Engineering of Polymeric Substrates: The Effects of Ion Reflection and Secondary Electron Emission. *Micromachines* **2018**, *9*, 415.
https://doi.org/10.3390/mi9080415

**AMA Style**

Memos G, Lidorikis E, Kokkoris G.
Roughness Evolution and Charging in Plasma-Based Surface Engineering of Polymeric Substrates: The Effects of Ion Reflection and Secondary Electron Emission. *Micromachines*. 2018; 9(8):415.
https://doi.org/10.3390/mi9080415

**Chicago/Turabian Style**

Memos, George, Elefterios Lidorikis, and George Kokkoris.
2018. "Roughness Evolution and Charging in Plasma-Based Surface Engineering of Polymeric Substrates: The Effects of Ion Reflection and Secondary Electron Emission" *Micromachines* 9, no. 8: 415.
https://doi.org/10.3390/mi9080415