Targeted Cross-Section Calculations for Plasma Simulations
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/O Plasma
- Electron collision processes for F were taken from ;
- Neutral–Neutral reactions, specifically the creation of SOF species, were taken from ;
- 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 were included with estimated rate coefficients in analogy to SF, e.g., SF rate coefficients were used for SOF. We assumed that the neutral dissociation process splits one F and the ionization produces the SOF ion. One exception is SOF which produces SOF + 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 + 2F is more likely than into SOF + 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.
- 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.
Data Availability Statement
Conflicts of Interest
- Mašín, Z.; Benda, J.; Gorfinkiel, J.D.; Harvey, A.G.; Tennyson, J. UKRmol+: A suite for modelling of electronic processes in molecules interacting with electrons, positrons and photons using the R-matrix method. Comput. Phys. Commun. 2020, 249, 107092. [Google Scholar] [CrossRef][Green Version]
- Cooper, B.; Tudorovskaya, M.; Mohr, S.; O’Hare, A.; Hanicinec, M.; Dzarasova, A.; Gorfinkiel, J.; Benda, J.; Mašín, Z.; Al-Refaie, A.; et al. Quantemol Electron Collision: An expert system for performing UKRmol+ electron molecule collision calculations. Atoms 2019, 7, 97. [Google Scholar] [CrossRef][Green Version]
- Lu, T.; Law, C.K. A directed relation graph method for mechanism reduction. Proc. Combust. Inst. 2005, 30, 1333–1341. [Google Scholar] [CrossRef]
- Lu, T.; Law, C.K. Linear time reduction of large kinetic mechanisms with directed relation graph: n-Heptane and iso-octane. Combust. Flame 2006, 144, 24–36. [Google Scholar] [CrossRef]
- Lu, T.; Law, C.K. On the applicability of directed relation graphs to the reduction of reaction mechanisms. Combust. Flame 2006, 146, 472–483. [Google Scholar] [CrossRef]
- Hanicinec, M.; Mohr, S.; Tennyson, J. Fast species ranking for iterative species-oriented skeletal reduction of chemistry sets. Plasma Sources Sci. Technol. 2021, 29, 125024. [Google Scholar] [CrossRef]
- Ryan, K.R.; Plumb, I.C. A model for the etching of silicon in SF6/O2 plasmas. Plasma Chem. Plasma Process. 1990, 10, 207–229. [Google Scholar] [CrossRef]
- Bartha, J.W.; Greschner, J.; Puech, M.; Maquin, P. Low temperature etching of Si in high density plasma using SF6/O2. Microelectron. Eng. 1995, 27, 453–456. [Google Scholar] [CrossRef]
- Pateau, A.; Rhallabi, A.; Fernandez, M.C.; Boufnichel, M.; Roqueta, F. Modeling of inductively coupled plasma SF6/O2/Ar plasma discharge: Effect of O2 on the plasma kinetic properties. J. Vac. Sci. Technol. A 2014, 32, 021303. [Google Scholar] [CrossRef]
- Maruyama, T.; Narukage, T.; Onuki, R.; Fujiwara, N. High-aspect-ratio deep Si etching in SF6/O2 plasma. I. Characteristics of radical reactions with high-aspect-ratio patterns. J. Vac. Sci. Technol. B 2010, 28, 854–861. [Google Scholar] [CrossRef]
- Gomez, S.; Jun Belen, R.; Kiehlbauch, M.; Aydil, E.S. Etching of high aspect ratio structures in Si using SF6/O2 plasma. J. Vac. Sci. Technol. A 2004, 22, 606–615. [Google Scholar] [CrossRef]
- Aachboun, S.; Ranson, P. Deep anisotropic etching of silicon. J. Vac. Sci. Technol. A 1999, 17, 2270–2273. [Google Scholar] [CrossRef]
- Boufnichel, M.; Aachboun, S.; Grangeon, F.; Lefaucheux, P.; Ranson, P. Profile control of high aspect ratio trenches of silicon. I. Effect of process parameters on local bowing. J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. 2002, 20, 1508–1513. [Google Scholar] [CrossRef]
- Blauw, M.A.; van der Drift, E.; Marcos, G.; Rhallabi, A. Modeling of fluorine-based high-density plasma etching of anisotropic silicon trenches with oxygen sidewall passivation. J. Appl. Phys. 2003, 94, 6311–6318. [Google Scholar] [CrossRef]
- Anderson, H.M.; Merson, J.A.; Light, R.W. A Kinetic Model for Plasma Etching Silicon in a SF6/O2 RF Discharge. IEEE Trans. Plasma Sci. 1986, 14, 156–164. [Google Scholar] [CrossRef]
- Rauf, S.; Dauksher, W.J.; Clemens, S.B.; Smith, K.H. Model for a multiple-step deep Si etch process. J. Vac. Sci. Technol. A 2002, 20, 1177–1190. [Google Scholar] [CrossRef]
- Marcos, G.; Rhallabi, A.; Ranson, P. Topographic and kinetic effects of the SF6/O2 rate during a cryogenic etching process of silicon. J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. 2004, 22, 1912–1922. [Google Scholar] [CrossRef]
- Tennyson, J.; Rahimi, S.; Hill, C.; Tse, L.; Vibhakar, A.; Akello-Egwel, D.; Brown, D.B.; Dzarasova, A.; Hamilton, J.R.; Jaksch, D.; et al. QDB: A new database of plasma chemistries and reactions. Plasma Sources Sci. Technol. 2017, 26, 055014. [Google Scholar] [CrossRef][Green Version]
- Dijkstra, E.W. A note on two problems in connexion with graphs. Numer. Math. 1959, 1, 269–271. [Google Scholar] [CrossRef][Green Version]
- Tennyson, J. Electron-molecule collision calculations using the R-matrix method. Phys. Rep. 2010, 491, 29–76. [Google Scholar] [CrossRef]
- Kim, Y.K.; Rudd, M.E. Binary-encounter-dipole model for electron-impact ionization. Phys. Rev. A 1994, 50, 3945. [Google Scholar] [CrossRef][Green Version]
- Graves, V.; Cooper, B.; Tennyson, J. The efficient calculation of electron impact ionization cross sections with effective core potential. J. Chem. Phys. 2021, 154, 114104. [Google Scholar] [CrossRef]
- Werner, H.J.; Knowles, P.J.; Knizia, G.; Manby, F.R.; Schütz, M. Molpro: A general-purpose quantum chemistry program package. WIREs Comput. Mol. Sci. 2012, 2, 242–253. [Google Scholar] [CrossRef]
- Herron, J.T. Thermochemical Data on Gas Phase Compounds of Sulfur, Fluorine, Oxygen, and Hydrogen Related to Pyrolysis and Oxidation of Sulfur Hexafluoride. J. Phys. Chem. Ref. Data 1987, 16, 1. [Google Scholar] [CrossRef]
- Phelps, A.V. Tabulations of Collision Cross Sections and Calculated Transport and Reaction Coefficients for Electron Collisions with O2; Technical Report; University of Colorado: Boulder, CO, USA, 1985. [Google Scholar]
- Krishnakumar, E.; Srivastava, S.K. Cross-sections for electron impact ionization of O2. Int. J. Mass Spectrom. Ion Process. 1992, 113, 1–12. [Google Scholar] [CrossRef]
- Itikawa, Y.; Ichimura, A. Cross Sections for Collisions of Electrons and Photons with Atomic Oxygen. J. Phys. Chem. Ref. Data 1990, 19, 637. [Google Scholar] [CrossRef][Green Version]
- Christophorou, L.G.; Olthoff, J.K. Electron Interactions with SF6. J. Phys. Chem. Ref. Data 2000, 29, 267–330. [Google Scholar] [CrossRef][Green Version]
- Tarnovsky, V.; Deutsch, H.; Martus, K.E.; Becker, K. Electron impact ionization of the SF5 and SF3 free radicals. J. Chem. Phys. 1998, 109, 6596–6600. [Google Scholar] [CrossRef]
- Phelps, A.V.; Van Brunt, R.J. Electron-transport, ionization, attachment, and dissociation coefficients in SF6 and its mixtures. J. Appl. Phys. 1988, 64, 4269–4277. [Google Scholar] [CrossRef]
- Morgan, W.L.; (Kinema Research and Software). Personal Communication.
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
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
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/atoms9040085Chicago/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