Ramsauer–Townsend minimum in electron scattering from CF $$_4$$ 4 : modified effective range analysis

THE EUROPEAN PHYSICAL JOURNAL D Eur. Phys. J. D (2021)75:76 https://doi.org/10.1140/epjd/s10053-021-00083-x Regular Article - ATOMIC AND MOLECULAR COLLISIONS Ramsauer–Townsend minimum in electron scattering from CF4: modified effective range analysis Kamil Fedusa and Grzegorz P. Karwasz Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5/7, 87-100 Toruń, Poland Received 8 January 2021 / Accepted 13 February 2021 © The Author(s) 2021 Abstract. Elastic cross sections for electron scattering on tetrafluoromethane (CF4 ) from 0 up to 5 eV energy are analyzed using semi-analytical approach to the modified effective range theory (MERT). It is shown that energy and angular variations of differential, integral and momentum transfer cross sections can be parameterized accurately by six MERT coefficients up to the energy region of the resonant scattering. In particular, the model is used to determine the depth and the position of the Ramsauer–Townsend minimum as well as the s-wave scattering length. Moreover, we investigate the influence of the dipole polarizability value on the predictions of present model. To further validate our approach, the elastic data are combined with the Born-dipole cross sections for vibrational excitations as the input data for Monte Carlo simulation of electron swarm coefficients. 1 Introduction Carbon tetrafluoride (CF4 ) is used for etching Si and SiO2 from the beginning of nano-electronics era [1]. It is a great source of reactive species needed for plasma processing of materials. However, it could be also a reason of some undesirable effects such as enlarging the etched paths. This is due to the peculiarity of the cross sections for electron scattering: high thresholds for the ionization and relatively large contribution from the dissociation into unstable neutral and ionized fragments, see review papers on cross section for e− -CF4 collisions by Bonham [2], Christophorou et al. [3], Karwasz et al. [4] or Sakai [5]. Nevertheless, as recently showed [6], CF4 in appropriate mixtures with other gases (O2 , He, C4 F8 ) can be used for preparation of nanowires of SiGe and Si as thin as 20 nm in diameter. Obviously, to avoid laboratory try-and-error procedures, theoretical modeling of plasma is desirable. The very low-energy collisions (below 1 eV), apparently, seem to be less important in plasma processing than the inelastic processes starting at few eV such as dissociation and ionization. On the contrary, it is the Ramsauer–Townsend minimum of the integral elastic cross sections at 0.3 eV that defines the overall temperature of Ar-based discharges with practical applications in lighting [7]. Moreover, the elastic collisions are of primary importance in numerical modeling of plasma, particularly at low temperatures. In this paper we discuss the very low-energy electron elastic scattering on CF4 . Relatively, few papers explored this problem. The a first relative measurements of total cross sections by David Field’s group [8] (repeated later by Lunt et al. [9]) indicated the presence of Ramsauer–Townsend (RT) minimum somewhere at 0.15 eV. However, the evidence was not clear since the same energy regime corresponds to excitation thresholds for bending (symmetric ν2 at 0.054 eV and asymmetric ν4 at 0.078 eV) and stretching (symmetric ν1 at 0.112 eV and asymmetric ν3 at 0.157 eV) vibrational modes. Due to the high transition dipole moments of CF4 molecule the vibrational scattering channel may become the leading contribution to the total cross sections. Therefore, the separation of the two contributions: elastic one and vibrational one at low energies, is essential. Unfortunately, later absolute measurements of total cross sections were not carried out at such low energies: Gdańsk laboratory stopped at 0.4 eV [10], Jones [11] at 1 eV and Tokyo group at 1.5 eV [12]. Only the measurement of differential elastic cross sections (DCS) down to 0.3 eV by Mann and Linder [13] helped to confirm the presence of R-T minimum in integral elastic cross sections (IECS). This minimum was derived from experimental data using the modified effective range theory (MERT), as formulated in sixties of last century by O’Malley et al. [14]. MERT allows to extrapolate measured cross sections down to ultra-low energies which are inaccessible experimentally. Another evidence for the presence of R-T minimum in IECS was brought by the analysis of electron transport parameters measured in swarm experiments (i.e., a cloud of electrons adrift in external electric field through relatively dense CF4 gas) by Curtis et al. [15] and Hayashi [16]. However, swarm-derived IECS in both experiments e-mail: kamil@fizyka.umk.pl (corresponding author) 0123456789().: V,-vol 123 76 Page 2 of 9 do not agree on the energy position, the width and the depth of minimum. The theoretical research on the low-energy e− -CF4 collisions are even more scarce than experimental ones. Although there are several high-quality theoretical works for energies just above 1 eV [17–21] to the best of our knowledge, only Isaacs et al. [22] and Gianturco and Willner [23] computed scattering cross sections at much lower energies. Both models predict the presence of R-T minimum, but they do not agree, either with the experiments and with each other, on the exact position of this minimum. In the present work we re-analyze available experimental data of elastic cross sections for e− -CF4 collisions using a new approach to MERT as proposed by Idziaszek and Karwasz [24] and applied successfully for noble gases and some molecules (H2 , CH4 ) [25,26]. In the original MERT formulation by O’Malley et al. [14] the scattering phase shift of partial wave with given angular momentum is expanded into the energy series known as the effective range expansion. In contrast to the original approach, in the present work we take into account the exact contribution of the long-range polarization potential (∼ r−4 ) to the phase shifts, while the effective range approximation is applied exclusively to unknown short-range interaction. This allows to extend the applicability of MERT almost up to the threshold for electronic excitation, while the original MERT [14] (where contributions of both long and short-range parts of interaction are approximated) is valid only at ultralow energies (much below 1 eV) as proved by Buckman and Mitroy [27] for noble gases and Chang [28] for nonpolar molecules. In this work we show that a new approach to MERT can be used to describe elastic cross sections for e− -CF4 collisions up to such high energy as 5 eV, i.e., just below the threshold for resonant scattering. To validate our approach the MERT-derived momentum transfer cross section (MTCS) in the region of R-T minimum is combined with vibrational cross sections calculated using Born-dipole approximation as input data for METHES Monte Carlo collision code [29] in order to calculate electron drift coefficients. In (...truncated)