#
Low-Energy Electron Scattering from c-C_{4}F_{8}

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{4}F

_{8}were investigated at low energies by using the R-matrix method. The static exchange (SE), static exchange with polarization (SEP), and close-coupling (CC) models of the R-matrix method were used for the calculation of the scattering cross-section. The shape resonance was detected with all the models at around 3~4 eV, and a Feshbach resonance was detected with the SEP model at 7.73 eV, in good agreement with the previous theoretical calculation. The resonance detected was also associated with the experimental dissociative electron attachment of c-C

_{4}F

_{8}, which displayed the resonances at the same energy range. The cross-sections calculated are important for plasma modeling and applications.

## 1. Introduction

_{4}F

_{8}is widely used in thin-film etching processes, such as Si, SiO

_{2}, HfO

_{2}, Si

_{3}N

_{4}, and SiO

_{2}-Si

_{3}N

_{4}-SiO

_{2}stacks [2,3,4,5,6]. It is diluted with various gases rather than a pure c-C

_{4}F

_{8}molecule or is often used with SF

_{6}in a multiple-step deep-Si etching process [2]. However, since various ions and radicals are generated in c-C

_{4}F

_{8}mixed plasmas, it is challenging to understand c-C

_{4}F

_{8}plasmas. Therefore, many research groups have conducted research to analyze c-C

_{4}F

_{8}plasmas and optimize the process. Li et al. analyzed experimentally the effect of mixing additional gases to c-C

_{4}F

_{8}inductively coupled plasmas (ICP) on the oxide etch rate [3]. Experiments were conducted by mixing Ar, He, and Ne in c-C

_{4}F

_{8}plasmas, and the results showed that the highest ion current density was obtained when Ar was mixed. Moreover, the etch rate of SiO

_{2}depends on the type of noble gas added to c-C

_{4}F

_{8}. Hua et al. investigated the effect of N

_{2}dilution on the Si

_{3}N

_{4}and SiC etch rate in c-C

_{4}F

_{8}and c-C

_{4}F

_{8}/Ar discharges [4]. The results showed that a change in the steady-state fluorocarbon film thickness caused by the addition of N

_{2}to the c-C

_{4}F

_{8}/Ar gas mixture has a negligible effect on Si

_{3}N

_{4}. Takahashi et al. conducted HfO

_{2}etching experiments in ICP discharge in which Ar was mixed with CF

_{4}or c-C

_{4}F

_{8}, and they found that the etch rate depends on various external and plasma parameters [6].

_{4}F

_{8}/O

_{2}CCP discharges [8]. Although many reactions are considered in simulations, several reaction rate coefficients are still assumed, and some electron collision reactions were neglected. Above all, the collision cross-sections are essential input data for plasma simulations, and accurate data feeds directly into the accuracy of simulations. Electron impact ionization, dissociation, attachment cross-sections, and momentum transfer and excitation collision cross-sections are essential to obtain more accurate simulation results [9]. Due to the importance of fluorocarbons in plasma applications, there has been a surge of studies for fluorocarbons electron collision cross-sections. Recently, we investigated electron collision studies with C

_{2}F

_{2}[10], C

_{3}F

_{4}[11], and C

_{4}F

_{6}[12] isomers, for which there were very little data in the literature, highlighting the importance of the need for more investigation. The importance of investigating fluorocarbons and other feedstock gases for replacing currently used higher global-warming-potential gases has been highlighted in a recent review [13]. For c-C

_{4}F

_{8}, experimental data of total and ionization cross-sections are available, but clearly, there is a lack of detailed theoretical investigation for this important target.

_{4}F

_{8}. Jelisavcic et al. [15] measured the absolute cross-sections for elastic scattering of electrons from c-C

_{4}F

_{8}in the energy range of 1.5–100 eV and over the scattering angles of 10°–130°. The most recent recommended data for c-C

_{4}F

_{8}was provided by Yoon et al. [16] for the integral elastic (Q

_{el}), momentum transfer (MTCS), and differential (DCS) cross-sections. Measurements for total cross-sections (TCS) were made for this gas by Makochekanwa et al. [17], Sanabia et al. [18], and Nishimura et al. [19]. Winstead and Mckoy [20] calculated the Q

_{el}, DCS, and MTCS using the Schwinger multichannel (SMC) using a limited CC and SE approximation. In this article, we concentrate on the low-energy Q

_{el}, MTCS, DCS, and excitation cross-sections (Q

_{exc}) using the SE, SEP, and CC approximation using the R-matrix method.

## 2. Theoretical Methodology

_{4}F

_{8}. QEC is new expert system that replaces Quantemol-N [26] and runs the upgraded R-matrix code. Quantemol-N has been successfully used for low-energy collision cross-section calculations for a variety of molecular targets [27,28,29,30,31]. Initial studies were performed with Quantemol-N, while the final results given below were all obtained using QEC.

^{2}configurations. These are multi-center quadratically integrable functions constructed by placing all the N + 1 electrons in the target molecular orbitals (MOs).

_{0}, and the energy dependence of the scattering electron is carried in this region, where all the required quantities, such as eigen phase sum and scattering cross-sections, are calculated.

_{0}until it is matched with asymptotic functions given by the Gailitis expansion [34]. After matching to the boundary conditions, the symmetric K-matrices are determined, and all the observables, such as cross-sections, are obtained using the K-matrix elements. Resonances, an essential part of the low-energy calculations, are identified and detected using the RESON [35] module by fitting them to the Briet–Wigner profile [36] to obtain the energies and widths. From the K-matrices, we can obtain the T-matrices as follows:

^{2}configuration in Equation (1). The third model (CC) is a more sophisticated approximation than SE and SEP, and in this case, one can include many electronic excited states into the calculation in the expansion of equation (1). Here, some electrons are frozen in the ground, and some are allowed to move freely in the active space, which helps to incorporate electronic states into calculations, leading to the much better description of the polarization effects, which gives us the more accurate cross-sections and resonance energies. Due to the inclusion of the excited states into the calculations, this approximation is well-suited for detecting Feshbach/core-excited resonances at low energies.

#### Target Models

_{4}F

_{8}. The equilibrium geometry of c-C

_{4}F

_{8}was obtained by fully optimizing the molecular structure and orbital parameters using DFT-ωB97X-D [39] hybrid functionals and Dunning’s [40] aug-cc-pVTZ basis set. The optimized structure of c-C

_{4}F

_{8}is given in Figure 1.

_{4}F

_{8}is a closed-shell target that belongs to a ${D}_{2d}$ point group symmetry. The calculations are performed in the ${C}_{2v}$ symmetry, which is the subgroup of ${D}_{2d}$. The ground-state Hartree–Fock electronic configuration of c-C

_{4}F

_{8}molecule is 1b

_{2}

^{2}, 1a

_{1}

^{2}, 1b

_{1}

^{2}, 2a

_{1}

^{2}, 2b

_{1}

^{2}, 3a

_{1}

^{2}, 4a

_{1}

^{2}, 2b

_{2}

^{2}, 5a

_{1}

^{2}, 3b

_{2}

^{2}, 3b

_{1}

^{2}, 6a

_{1}

^{2}, 7a

_{1}

^{2}, 4b

_{2}

^{2}, 4b

_{1}

^{2}, 8a

_{1}

^{2}, 9a

_{1}

^{2}, 5b

_{1}

^{2}, 5b

_{2}

^{2}, 10a

_{1}

^{2}, 11a

_{1}

^{2}, 6b

_{2}

^{2}, 6b

_{1}

^{2}, 12a

_{1}

^{2}, 13a

_{1}

^{2}, 1a

_{2}

^{2}, 14a

_{1}

^{2}, 7b

_{2}

^{2}, 7b

_{1}

^{2}, 8b

_{1}

^{2}, 8b

_{2}

^{2}, 15a

_{1}

^{2}, 2a

_{2}

^{2}, 9b

_{2}

^{2}, 9b

_{1}

^{2}, 16a

_{1}

^{2}, 3a

_{2}

^{2}, 10b

_{2}

^{2}, 10b

_{1}

^{2}, 17a

_{1}

^{2}, 4a

_{2}

^{2}, 18a

_{1}

^{2}, 11b

_{2}

^{2}, 11b

_{1}

^{2}, 19a

_{1}

^{2}, 12b

_{2}

^{2}, 12b

_{1}

^{2}, 5a

_{2}

^{2}in the ${C}_{2v}$ symmetry. For the scattering calculations, we used the complete active space–configuration interaction (CAS-CI) model to represent the target wavefunction with cc-pVTZ basis set and 15 target states in our calculations. Of 96 electrons, only 8 electrons are located in the active space composed of 19a

_{1}, 20a

_{1}, 21a

_{1}, 12b

_{1}, 13b

_{1}, 12b

_{2}, 13b

_{2}, and 5a

_{2}molecular orbitals in this CAS-CI model. The accuracy of the scattering data depends on the choice of the target wave function, and hence, careful and critical assessment of the target wavefunction is essential. The number of configuration-state functions (CSF) generated for the ground state is 900, and 225 channels are included in the present scattering calculation. To accommodate the target electrons’ charge cloud inside the inner region, the inner region radius of 10a

_{0}was sufficient and provided a stable calculation in the present case. Two virtual orbitals were included in the SE, SEP, and CC scattering calculations.

_{4}F

_{8}to be −946.85 Hartree. The triplet and singlet excited-states thresholds are 8.80 and 9.04 eV, which compares well with the 8.52 and 9.13 eV triplet and singlet excited-states thresholds computed using the single-excitation configuration-interactions (SECI) calculations [20]. At the energy minimum of the ground state, the vertical excitation energies to the six lowest-lying electronic excited singlets and triplets are provided in Table 1.

## 3. Results and Discussion

^{2}A

_{1},

^{2}E, and

^{2}A

_{2}of the c-C

_{4}F

_{8}system using the SE, SEP, and CC models in the C

_{2v}point group symmetry. The eigenphase diagram is important for the study of resonances at low-energy regimes. In Figure 2, the scattering state

^{2}A

_{1}shows a hump at around 3~4 eV with all three models. As expected, the SE model detected the resonance at slightly higher energy than the other two models due to the exclusion of the polarization and correlation effects in its calculations. The SEP model detected a shape resonance at 3.12 eV and a Feshbach resonance at 7.73 eV, as indicated by a hump at the same energies in the eigenphase sum due to the

^{2}A

_{1}and

^{2}E scattering states. The position of the resonance and their corresponding widths for c-C

_{4}F

_{8}below 10 eV are presented in Table 2 along with the resonance data of Winstead and Mckoy [20] and the experimental dissociative electron attachment (DEA) thresholds [41,42,43,44], which can be associated with resonance at different positions.

_{el}and MTCS from the R-matrix method’s SE, SEP, and CC models. The results obtained with different models are compared with the experimental, theoretical, and recommended datasets. In Figure 3a,c, the Q

_{el}and MTCS are compared that are calculated using different approximations. All the models detect the presence of a shape resonance at around 3~4 eV, with the SEP model predicting the resonance at lower energy compared to the other two models, as seen for both Q

_{el}and MTCS. The 7.73 eV resonance is detected in the SEP model and is also supported by the eigenphase diagram. The shape resonance is detected at 4.21, 3.12, and 3.37 eV, with the corresponding widths of 1.01, 0.95, and 0.92 eV in the SE, SEP, and CC models, respectively.

^{2}A

_{1}state of the C

_{2V}symmetry is associated with the

^{2}B

_{2}resonance of Winsted and Mckoy [20] in the D

_{2d}symmetry. The Feshbach resonance detected at 7.73 eV due to

^{2}E state with the SEP model in the present calculations can be associated with the 8.1 eV resonance of Winstead and Mckoy due to the

^{2}E state in the D

_{2d}symmetry. The present shape resonance at 3.12, 3.37, and 4.21 eV due to SEP, CC, and SE models could also be associated with the 3.75 eV observed dissociative attachment maximum to c-C

_{4}F

_{8}of Lifshitz and Grajower [41]. The present Feshbach resonance at 7.73 eV could be associated with the experimental dissociative electron attachment to c-C

_{4}F

_{8}at 8.0, 8.2, 8.5~8.8, and 7.9 eV due to Lifshitz and Grajower [41], Bibby and Carter [42], Harland and Thynne [43], and Sauers et al. [44].

_{el}and MTCS results are compared in Figure 3b,d with other available datasets for elastic and total cross-sections. The present and the data of Winstead and Mckoy [20] for Q

_{el}show a large disagreement with the experimental data [18,19] and recommended dataset of Christophorou [14] and Yoon et al. [16] at low energies below 8 eV, after which they follow the experimental and recommended data. The shape resonance detected in the previous and the present calculations is missing in the experimental data [18,19] for total cross-section. Since the target is quite large and complex, the effects of correlation and polarization are not sufficiently well-modelled, which may be a cause of the discrepancy between the experiment and the theoretical calculations at low energies. The MTCS follows the calculation of Winstead and Mckoy [20] but is in less good agreement with the recommended data of Yoon et al. [16].

_{exc}from the ground state of c-C

_{4}F

_{8}to the six low-lying excited states. The vertical excitation threshold of the first excited state (

^{3}A

_{2}) is around 8.8 eV. The triplet states contributes maximum to the Q

_{exc}, and for

^{3}E excited state, a maximum cross-section is found approximately at 15 eV.

**Summary:**This work investigates electron collision study of plasma-relevant molecular c-C

_{4}F

_{8}target using the SE, SEP, and CC models. The present calculations have reproduced the previous theoretical results [20] calculated using a similar approximation. We could also confirm the presence of shape resonance at around 3~4 eV and Feshbach resonance at 7.73 eV, in accordance with the earlier calculation and the experimental dissociative attachment study to c-C

_{4}F

_{8}. The present study suggests that we could use similar models and approximations to study more complex targets, such as c-C

_{5}F

_{8}, c-C

_{6}F

_{8}, C

_{7}F

_{8}, C

_{7}F

_{14}, and c-C

_{10}F

_{8}, and test their validity for replacing the PFC gases with higher global-warming-potential ones, as highlighted in a recent review article [13]. It is quite clear that there is lack of studies for larger fluorocarbons, and even for a smaller targets, scarcity of data is seen, and hence, we hope this study can motivate others to investigate more on this subject. Moreover, the present data would find applications in low-temperature plasma modelling and simulation.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Equilibrium structure of neutral c-C

_{4}F

_{8}at ωB97X-D/aug-cc-pVTZ. The ring-puckering angle is 16.33°.

**Figure 2.**Eigen phase diagram for the doublet scattering states of e-c-C

_{4}F

_{8}scattering in C

_{2v}symmetry.

**Figure 3.**Elastic and momentum transfer cross-section of c-C

_{4}F

_{8}scattering; (

**a**) comparison among the different models (SE, SEP, and CC) for the elastic cross-section; (

**b**) comparison of the present elastic cross-section (SE and CC) with the available data in the literature; (

**c**) comparison among the different models (SE, SEP, and CC) for the MTCS; (

**d**) comparison of the present MTCS (SE and CC) with the available data in the literature.

**Figure 4.**Elastic DCS for the electron scattering of c-C

_{4}F

_{8}system for the energies of 1.5, 3, 5, 7, 8, and 10 eV.

**Figure 5.**Electronic excitation cross-section from the ground state to the six low-lying excited states of c-C

_{4}F

_{8}scattering.

**Table 1.**Electronic vertical excitation energy at the c-C

_{4}F

_{8}ground-state geometry. The results are compared with previous theoretical calculations [20] in eV.

State $\left({\mathit{C}}_{2\mathit{v}}\right)$ | Present | Theory [20] |
---|---|---|

^{1}A_{1} | 0 | |

^{3}A_{2} | 8.80 | 8.52 |

^{1}A_{2} | 9.04 | 9.13 |

^{3}A_{2} | 11.54 | |

^{3}E | 12.23 | |

^{1}A_{2} | 12.45 | |

^{1}E | 13.53 |

States $\left({\mathit{C}}_{2\mathit{v}}\right)$ | Present (SE) | Present (SEP) | Present (CC) | Winstead and Mckoy [20] | Experimental DEA Results | ||||
---|---|---|---|---|---|---|---|---|---|

Position | Width | Position | Width | Position | Width | Position | Width | Position | |

^{2}A_{1} | 4.21 | 1.01 | 3.12 | 0.95 | 3.37 | 0.92 | 3.0 | 0.33 | 3.75 [41] |

^{2}E | 7.73 | 1.07 | 8.1 | 1.2 | 8.0 [41], 8.2 [42], 8.5~8.8 [43], 7.9 [44] |

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## Share and Cite

**MDPI and ACS Style**

Gupta, D.; Choi, H.; Kwon, D.-C.; Su, H.; Song, M.-Y.; Yoon, J.-S.; Tennyson, J.
Low-Energy Electron Scattering from *c*-C_{4}F_{8}. *Atoms* **2022**, *10*, 63.
https://doi.org/10.3390/atoms10020063

**AMA Style**

Gupta D, Choi H, Kwon D-C, Su H, Song M-Y, Yoon J-S, Tennyson J.
Low-Energy Electron Scattering from *c*-C_{4}F_{8}. *Atoms*. 2022; 10(2):63.
https://doi.org/10.3390/atoms10020063

**Chicago/Turabian Style**

Gupta, Dhanoj, Heechol Choi, Deuk-Chul Kwon, He Su, Mi-Young Song, Jung-Sik Yoon, and Jonathan Tennyson.
2022. "Low-Energy Electron Scattering from *c*-C_{4}F_{8}" *Atoms* 10, no. 2: 63.
https://doi.org/10.3390/atoms10020063