WDS' Proceedings of Contributed Papers, Part II, 5 9,. ISBN 978-8-778-85-9 MATFYZPRESS The Computational Simulation of the Positive Ion Propagation to Uneven Substrates V. Hrubý and R. Hrach Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. The low-temperature plasma is commonly used to treat the surfaces of a wide range of substrates. The deposition of layers on the surfaces of conductive substrates may be achieved by various ways. In comparison with neutral atom deposition, the use of ions has got considerable advantages. The energy of ions and the flux to the substrate can be controlled by the voltage bias and the pressure. in this contribution, the propagation of ions from low-temperature plasma to substrates with uneven surfaces is studied by means of computational simulations. Instead of simulating the whole system of discharge, sputtering and deposition, this study is devoted to the transport of ions to a complex substrate, which demands the fully three-dimensional approach, independently of the source of plasma. The computational simulations are based on a hybrid model which consists of a fluid and a particle part trying to take advantage from both of them. The fluid part reaches the global results in a short time, whereas the particle part gives detailed results, such as local energy distributions and transport parameters. The hybrid model achieves the solution by coupling these two parts iteratively. Introduction The propagation of charged particles from the plasma to the surface of some object influences the results of the treatment. In the case of the deposition of layers on surfaces of conductive substrates, parameters of the layer depend on the flux and energy of deposited particles. The main advantage of the deposition of ions is that their trajectory and energy may be controlled by the electric and magnetic field and the pressure of the neutral background gas [e.g. Rossnagel et al., 994]. The distribution of the ion flux at the surface of the substrate is especially significant when the geometry of the surface is rather complex, e.g. with protrusions, trenches etc. The energy distribution is of high importance when the deposition is accompanied by a chemical reaction, e.g. the deposition of TiN thin layers. Generally, the computational simulation can advance our knowledge of the process of ion deposition. Many computational studies of the whole deposition system have been published, but they are usually restricted to simple geometries of the substrate [e.g. Kolev et al., 9]. The propagation of ions from plasma to the substrate can be studied separately from the ion source to reduce computational costs. Still, most studies are restricted to D or simplified D [Macak et al., a and b]. In our contribution, a fully D model is used to analyze the ion flux to a substrate with a matrix of hemispherical protrusions. To simplify the computation, argon ions are used as an general example of heavy positive ions. Model The hybrid model is based on a combination of a fluid model with a non self-consistent particle model [Hruby et al., ]. Its aim is to take advantage from both parts. The fluid part reaches the solution with reasonable requirements on the computer memory and time, while the particle part provides the detailed information about energy and flux distributions. The two parts are coupled iteratively. In each step, the stationary solution of the fluid part supplies the particle part with the spatial distribution of electric potential and consequently, the particle part yields corrected transport coefficients. 5
Fluid part The transport equations for electrons () and positive ions () together with the Poisson s equation () form the set of partial differential equations solved in the fluid part of the hybrid model n e t + (µ en e ϕ D e n e ) =, () n i t + ( µ in i ϕ D i n i ) =, () ϕ = e ε (n i n e ). () The transport coefficients, i.e. the mobilities µ k and the diffusion coefficients D k, are provided by the particle part. Following boundary conditions are used on the surface of the substrate Ion flux Γ i n = µ i n i ϕ n + 4 n ic i Electron flux Γ e n = µ e n e ϕ n + 4 n ec e Bias ϕ = U s, where c e and c i are local mean chaotic velocities and n is the normal to the surface. The expression for electron flux is valid, only if Γ e n is positive. Otherwise the condition n e = is used. The conditions n e = n i = n and ϕ = are applied to the boundary adjoining the bulk plasma. The zero flux and zero charge conditions are used on side walls of the domain. The set of PDEs is solved by the finite element method using COMSOL Multiphysics. Particle part The motion of charged particles is computed in the particle part. It is determined by the electric potential resulting from the fluid part and by collisions with neutral particles of the background gas. The Verlet s algorithm [Verlet, 967] is used as the propagator. The collisions are treated by the null-collision method [Skullerud, 968] with energy dependent collisional cross-sections summarized in [Bogaerts et al., 996]. In order to obtain the local transport coefficients, the chaotic velocity c k, square of chaotic velocity c k and collisional frequency ν k are sampled during the motion of particles. The transport coefficients are calculated as follows µ k = e m k ν k and D k = c k ν k (4) Further, the particles reaching the substrate are registered and the fluxes and energy distributions are calculated. Results and discussion The hybrid model was used to study the plasma interaction with an uneven substrate. The geometry of the substrate with four hemispherical protrusions is shown in Fig.. The computational domain is prism-shaped, the substrate being its base face. The height of the domain is h = mm. The boundary conditions simulate an infinite substrate with the matrix of protrusions. At first, the influence of pressure on fluxes and energies was studied. The substrate was biased to U s = V. The pressure range spanned from 4 Pa to Pa. The number density of ions in the plasma was 7 m, which roughly corresponds to ion deposition technologies [e.g. Rossnagel et al., 994]. 6
Figure. The geometry of the surface of the substrate exposed to the plasma. TotalfluxtosubstrateΓ[ 8 m s ] 7 6.5 5 4.5 p[pa] Relative flux Γrel Total flux to substrate Relative flux to hemispheres Relative flux to plane between hemispheres Figure. The pressure dependence of fluxes to the substrate. The pressure dependence of total ion flux to the substrate is shown in Fig.. The surface area used to calculate the total flux includes the surface of hemispheres. Further, the flux to hemispheres and to the flat part of the substrate in relation to the total flux is presented. The hemispheres are preferred at higher pressures, while the flux is more homogeneous at lower pressures. In Fig., the energy distributions of ions reaching the substrate are shown. The ions are getting thermalised at higher pressures due to collisions, while at lower pressures they retain more energy gained from the bias. The energy of ions collected by the hemispheres is higher than that of ions falling at the flat part of the substrate. This is caused by stronger electric field in the vicinity of hemispheres, where ions accelerate at shorter paths and so undergo less collisions. The influence of voltage bias was also considered. In Fig. 4 the results for bias range from V to V at p = Pa are shown. The flux increases together with the negative voltage bias. The distribution of flux to hemispheres and the flat part of the surface remains nearly unchanged over whole range of the bias. 7
. 4Pa.. 4 6 8... Pa 4 6 8 Plane Hemispheres 4Pa.8.6.4. 4 6 Pa 4 Figure. Energy distributions of ions reaching the substrate at hemispheres and at the flat part. TotalfluxtosubstrateΓ[ 8 m s ] 7.5 6 5 4.5 5 5 5 U s [V] Relative flux Γrel Total flux to substrate Relative flux to hemispheres Relative flux to plane between hemispheres Figure 4. The dependency of the ion flux on the voltage bias at pressure p = Pa. Conclusion In this contribution, the computational simulation was used to study the ion propagation to the substrate with hemispherical protrusions. The flux and energy distributions are influenced by the pressure and the voltage bias applied to the substrate. The flux to the hemispheres is generally higher than the flux to the flat part. The energy distribution of ions differs profoundly due to stronger electric field near the hemispheres. This work is a part of a wider study, which 8
aims to investigate the influence of more parameters, such as the size and distance of protrusions, the ionization degree and the plasma composition. The hybrid model allows to solve the D problems on a personal computer. One run of the model took approximately 6 h at Intel Core i7 98 at. GHz with OpenMP parallelization to threads. Acknowledgments. The work is a part of the research plan MSM684 financed by the Ministry of Education of the Czech Republic. The authors acknowledge the support of the Czech Science Foundation GA CR, the project /8/H57. References Bogaerts A., R. Gijbels, Two-dimensional model of a direct current glow discharge: Description of the electrons, argon ions, and fast argon atoms, Anal. Chem., 68, 96, 996. Hruby, V., R. Hrach, Three-Dimensional Hybrid Computer Modeling of Langmuir Probes of Finite Dimensions in Medium Pressure Plasmas, IEEE Trans. Plasma Sci., 8, 8,. Kolev, V., A. Bogaerts, Numerical study of the sputtering in a dc magnetron, J. Vac. Sci. Technol. A, 7, 8, 9. Macak, E. B., W. D. Munz, J. M. Rodenburg, Plasma-surface interaction at sharp edges and corners during ion-assisted physical vapor deposition. Part I: Edge-related effects and their influence on coating morphology and composition, J. Appl. Phys., 94, 89 86,. Macak, E. B., W. D. Munz, J. M. Rodenburg, Plasma-surface interaction at sharp edges and corners during ion-assisted physical vapor deposition. Part II: Enhancement of the edge-related effects at sharp corners J. Appl. Phys., 94, 87 844,. Rossnagel, S. M., J. Hopwood, Metal ion deposition from ionized magnetron sputtering discharge, J. Vac. Sci. Technol. B,, 449 45, 994. Skullerud, H. R., The stochastic computer simulation of ion motion in a gas subjected to a constant electric field, Brit. J. Appl. Phys.,, 567 568, 968. Verlet, L., Computer Experiments on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules, Phys. Rev., 59, 98, 967. 9