# Targeted Cross-Section Calculations for Plasma Simulations

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

- Running a plasma simulation;
- Identifying species significantly influencing the densities of specified target species;
- Calculating missing cross-sections.

#### 2.1. Plasma Simulation

#### 2.2. Identifying Key Species

#### 2.3. Cross-Section Calculations

## 3. Example SF${}_{6}$/O${}_{2}$ Plasma

- Electron collision processes for F were taken from [31];
- Neutral–Neutral reactions, specifically the creation of SOF${}_{x}$ species, were taken from [7];
- Ion–Ion recombination and charge exchange, both symmetric and asymmetric, were included for all possible combinations with generic rate coefficients;
- Electron collision ionization and dissociation for SOF${}_{x}$ were included with estimated rate coefficients in analogy to SF${}_{x}$, e.g., SF${}_{5}$ rate coefficients were used for SOF${}_{4}$. We assumed that the neutral dissociation process splits one F and the ionization produces the SOF${}_{x}^{+}$ ion. One exception is SOF${}_{4}$ which produces SOF${}_{3}^{+}$ + F on ionization.

- Power: 500 W;
- Pressure: 10 Pa;
- Radius: 10 cm;
- Height: 10 cm;
- Total flow: 100 sccm;
- Relative oxygen flow: 10–90%.

- The calculated ionization cross-section is significantly larger than the estimated ones, by about a factor of 4 throughout the entire energy range up to a 1000 eV. However, the threshold energy is also larger, 15.19 eV compared to 11.8 eV for the estimated cross-sections. As a result, the ionization rate coefficient for the calculated cross-section is smaller for low electron temperatures and larger for high electron temperatures. The rate coefficients differ by about a factor of 2 at most.
- While the calculated dissociation cross-section shows significantly smaller values over a large range of energies, it also has a lower threshold energy; concerning the rate coefficients, the larger values of the estimated cross-section has a larger influence than the higher threshold energy. Hence, the estimated rate coefficient is significantly larger than the precisely calculated one over the majority of the investigated electron temperature range.
- The analysis of the neutral dissociation also showed that a breakup into SOF${}_{2}$ + 2F is more likely than into SOF${}_{3}$ + F (see the explanation above). Hence, this dissociation reaction was also changed with regard to the reaction products.

- improve the accuracy of our plasma simulation by calculating precise cross-sections which were formerly missing and had to be estimated;
- save time by ruling out species for which precise cross-section calculations will unlikely improve the simulation significantly.

## 4. Conclusions

- Run a plasma simulation such as a global model with a chemistry set containing estimates for missing cross-sections.
- Use the results of the plasma simulation in a species ranking algorithm. This identifies the species with missing cross-sections who potentially influence the densities of target species such as major etchants.
- Calculate precise cross-sections for high-ranking species and substitute these for the estimated ones.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Species ranking with regard to the productions of electrons and F for a relative oxygen flow of 50%. Cross-sections for all SOF${}_{x}$ species are estimated.

**Figure 4.**Electron and F density for a variation of the relative oxygen flow. The graphs compare the respective densities between the chemistry set with precisely calculated and the set with estimated cross-sections for the dissociation and ionization of SOF${}_{4}$.

**Figure 5.**Energy per electron–ion pair for SOF${}_{4}$ as a function of electron temperature derived from calculated and estimated cross-sections.

**Figure 6.**Electron energy per electron–ion pair weighted by their respective relative densities for SOF${}_{4}$, F, and O as a function of the relative oxygen flows.

**Figure 7.**Species ranking with regard to the productions of electrons and F for a relative oxygen flow of 50%. Cross-sections for SOF${}_{4}$ are precisely calculated.

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Mohr, S.; Tudorovskaya, M.; Hanicinec, M.; Tennyson, J. Targeted Cross-Section Calculations for Plasma Simulations. *Atoms* **2021**, *9*, 85.
https://doi.org/10.3390/atoms9040085

**AMA Style**

Mohr S, Tudorovskaya M, Hanicinec M, Tennyson J. Targeted Cross-Section Calculations for Plasma Simulations. *Atoms*. 2021; 9(4):85.
https://doi.org/10.3390/atoms9040085

**Chicago/Turabian Style**

Mohr, Sebastian, Maria Tudorovskaya, Martin Hanicinec, and Jonathan Tennyson. 2021. "Targeted Cross-Section Calculations for Plasma Simulations" *Atoms* 9, no. 4: 85.
https://doi.org/10.3390/atoms9040085