Surface and Coatings Technology 156 (2002)

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1 Surface and Coatings Technology 156 (2002) Two-dimensional particle-in-cell plasma immersion ion implantation simulation of gearywindmill geometry in cylindrical co-ordinates along the (r u) plane D.T.K. Kwok, R.K.Y. Fu, P.K. Chu* Department of Physics and Materials Science, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong SAR, PR China Abstract Plasma immersion ion implantation (PIII) into gearywindmill structures is simulated by the particle-in-cell (PIC) method in cylindrical co-ordinates. In cylindrical co-ordinates, the gearywindmill geometry becomes a periodic regular structure. An equal number of particles are placed inside the cell with the same angular and radial distance. The ion density represented by each particle is obtained and varies according to the radial distance of the particle. PIII simulation of a rectangular trench is carried out and compared with the cylindrical gearywindmill. The evolution and distribution of the potential and ion density contour lines are the same for these two geometries. The incident doses of a gear (windmill) will be larger than that of the trench. It is due to the space compression in cylindrical co-ordinates along the decreasing radial direction. The incident doses will thus be underestimated when treating a gear tooth using a rectangular trench Elsevier Science B.V. All rights reserved. Keywords: Plasma processing and deposition; Ion implantation; Numerical simulation; Particle-in-cell simulation 1. Introduction Plasma immersion ion implantation (PIII) circumvents the line-of-sight restrictions of conventional beamline ion implantation w1 5x and is therefore an excellent method to treat large and irregular industrial targets like ball bearings w6 8x, inner surface of bores w9,10x, and gears. The gear crack propagation trajectories in a moving tooth w11x, influence of tooth friction in gear dynamic w12x, load distribution w13x, gear distortion, changes of residual stresses, and hardness during quenching w14x have been numerically investigated by the finite element method (FEM) in a rectangular coordinate system. A gear can be treated as a modified windmill geometry. A gearywindmill structure can be treated as an irregular geometry in rectangular coordinates. However, in cylindrical co-ordinates, it is a regular periodic structure. In this work, we numerically investigate plasma immersion ion implantation of a single tooth sector of a gearywindmill by the particlein-cell (PIC) method in two-dimensional cylindrical co- *Corresponding author. Tel.: q ; fax: q address: paul.chu@cityu.edu.hk (P.K. Chu). ordinates along the (r, u) plane w15x. The incident doses along the outer gear circumference, sidewall, and bottom circumference are compared with the simulated results of the PIII process of the trench. The structure of a trench is similar to a sector of a gear except that it is presented in rectangular co-ordinates. PIII of a trench has been numerically investigated by PIC w16x. The dimension settings of the trench are made comparable to the gearywindmill. The potential and density contour lines are derived at different voltage pulse durations. The incident doses along the top, side, and bottom circumference are also generated at different pulse durations. Our results show that the incident doses into a gearywindmill are greater than that into the trench. It is due to the space compression in cylindrical co-ordinates along the decreasing radial direction. 2. Numerical simulation The simulated region of a gear (windmill) tooth is depicted in Fig. 1a. The base and outer radii r1 and r2 are equal to 0.5 and 0.7 m. Therefore, the sidewall is of height 2.0 m; that is, the distance between the r boundary and the base line iss1.5 m. The thickness of the tooth is presented in degrees of the sector, and that /02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S Ž

2 98 D.T.K. Kwok et al. / Surface and Coatings Technology 156 (2002) region is filled with cylindrical cells divided equally along the r and u direction. In this paper, drs0.02 m and dus18. However, the cell at a longer r has a greater area since the circumference is larger. One-hundred particles are equally placed in each cell. The spaces between the adjacent particles are dry10s0.002 and duy10s0.18. The particle density is equal to Fig. 1. (a) Simulation region of a sector of the gearywindmill geometry in cylindrical co-ordinates. (b) Simulation region of a trench in rectangular co-ordinates. The dimension of the trench is made comparable to that of the gearywindmill. of the tooth and base sectors is equal to 188. The entire simulated region is within a sector of rs2.0 mand us 368. However, the area covered by the gear is not used. The base surface is set at the center of the simulated region partitioning the tooth into two halves. The left and right boundaries are periodical; that is, particles crossing those boundaries will come back to the simulated region fromthe opposite boundary. The simulated du dr dr n o= xpccrq F ycry F 10=360 DD 20 G D 20 G du dr sn o= xp= =r 10= BB E2 B E2E where no is the ion density and r is the particle position along the r direction. Therefore, the particle at a greater r position will represent more ions. A total of ions are used in the simulation. The simulation settings of the trench are made comparable to the gear as depicted in Fig. 1b. The base line is located at ys0. The maximum y value is 1.5 m; that is, the distance between the y boundary and base lines 1.5. The sidewall is of height 0.2 m. The circumference of the top surface of the gear is not equal to its base line, although the degrees are the same. The circumference of the top surfaces0.22 mand the base lines 0.16 m. In the trench settings, we have compromised and equalized the length of the top surfacesbaselines 0.18 mgiving a total length along the x simulated region to be We will discuss the effects on the incident doses by compromising the top and baseline surfaces later in this article. The size of the dividing cell is dxs 0.01 and dys0.02 m. The baseline of the trench is made the center of the simulated region and the top surface is partitioned into two halves. The left and right boundaries are periodical. One-hundred particles are placed in each cell giving a total of particles. Each particle B E dx dy represents the same amount of ions as Cn o= = F. D G The electron temperature is equal to 2 ev and follows Boltzmann s distribution. The ions are assumed to be cold and collisionless at low pressure. The plasma 14 y3 density is set as 1=10 m. The plasma consists of singly charged nitrogen ions. They will only be driven by the electric field created by the high negative voltage pulses applied to both targets, i.e. trench and gear. The applied voltage is y20 kv. The potential of the simulated region is governed by Poisson s equation. The potential at each node, and therefore Poisson s equation, is estimated by the finite difference method w17x. The nodes within the simulated region will iterate until the potential of each node has relaxed and converged to y4 within a relative error of 1=10. The particle move- ment is handled by Newton s equations. After the particle position has been updated at the time step, the F G

3 D.T.K. Kwok et al. / Surface and Coatings Technology 156 (2002) Fig. 2. (a) Potential contour lines at 1, 5, 10 and 20 ms of the sector of the gearywindmill. The applied voltage is y20 kv. (b) Potential contour lines at 1, 5, 10 and 20 ms of the trench. The applied voltage is y20 kv. ion density of each node is obtained by weighing the particle density to its adjacent four corners w16x. The renewed ion s density is fitted back into Poisson s equation and the potential will be ready for the next time step. The whole process is repeated until the end of the voltage pulse. The voltage pulse is 20 ms with a zero rise time. The time step is automatically calculated dx as, where v is the maximum velocity which can v=10 2 be calculated from, 1y2mv sapv=q, m is the ion mass, Apv is the absolute applied voltage, and q is the ion charge. Fig. 3. (a) Density contour lines at 1, 5, 10 and 20 ms of the 14 y3 gearywindmill. The plasma density is 1=10 m. (b) Density contour lines at 1, 5, 10 and 20 ms of the trench. The plasma density is 14 y3 1=10 m. 3. Numerical results The potential contour lines of the sector of the gear (windmill) and trench at pulse durations of 1, 5, 10 and 20 ms are plotted in Fig. 2a,b. At 1 ms, the potential contours have bent inwards following the concave structure of both targets. As the ions are being removed and land onto the target surfaces, the ion sheath as well as potential lines evolve into a smooth circumference around the gear geometry and straight line in the trench co-ordinate.

4 100 D.T.K. Kwok et al. / Surface and Coatings Technology 156 (2002) The ion density contour lines at pulse durations of 1, 5, 10 and 20 ms are plotted in Fig. 3a,b. After the first few microseconds, the ions inside the volcano area of both structures are attracted and removed. The ions leaving the ion sheath will be accelerated by the strong electric field established between the ion sheath and target surfaces. The amount of ions is too small to fill up the empty spaces, and therefore, at 20 ms, the ion density near the surfaces of the targets is too small to be reflected in the contour plot showing a zero value in 13 y3 those areas. The 2=10 mion density contour line at 0.6 and 0.7 mfromthe base line is at 10 and 20 ms in both figures. It shows that the propagating speed of the ion sheath is similar in both geometries. The incident doses of the left-handed top surface, left-handed sidewall and bottomsurface of both geometries are plotted in Figs The incident doses are calculated by accumulating the incident particle densities divided by the cell width. The cell width for the sidewall is 0.02 m. In the trench geometry, the cell width of the top and bottom surfaces is 0.01 m. However, as mentioned before, in the gear geometry, the circumference of half of the top surfaces0.11 mand the base lines 0.16 m. Therefore, the cell width of the top surface is mand of the bottombase line is m. The cell bears the same distance of In principle, the incident dose of the top surface of a gear should be less than that on the top surface of a trench since it is divided by a larger length. On the opposite, the incident dose of the bottomsurface should be larger than that of the base surface of a trench since it is divided by a smaller value. However, we will shortly see that they are not the dominating factor when comparing doses between these geometries. The incident dose of the top surface always exceeds that of the sidewall and bottomsurfaces. It is because the same number of projected ions is shared by the sidewall and bottomsurfaces. There is an increase of the incident doses at the corner at the beginning of the implantation process due to the focusing of the electric field near the corner. The small increase becomes the background at a longer implantation time. The ions Fig. 4. Incident doses of the left top surfaces of the trench and gearywindmill at 1, 5, 10 and 20 ms. The incident doses of the gearywindmill are greater than those of the trench at longer implantation time. Fig. 5. Incident doses of the left side wall surfaces of the trench and gearywindmill at 1, 5, 10 and 20 ms. The incident doses of the gearywindmill are greater than those of the trench at longer implantation time.

5 D.T.K. Kwok et al. / Surface and Coatings Technology 156 (2002) indeed depend on the covered area of the ion sheath. In the trench geometry, the ion sheath area will more or less maintain the same amount of area and the ion flux will not exceed certain values. However, in the case of a gear geometry, as the ion sheath expands in the increasing r direction, the ion sheath area will uncover more and more ions. We can think of it as a focusing effect of ions froma large ion sheath area into a smaller target surface. Therefore, the incident dose difference in the gear geometry becomes more obvious at a larger pulse duration. 4. Conclusion Fig. 6. Incident doses of the bottomsurfaces of the trench and gearywindmill at 1, 5, 10 and 20 ms. The incidence doses of the gearywindmill are greater than those of the trench at longer implantation time. obtain enough momentum at later time and will equally distribute along the surfaces. The curvature of the bottomsurface also reaches a background level at longer pulse durations as shown in Fig. 6. The sidewall shows a distinct ladder distribution of incident doses with the highest dose at the top open area. The ladder distribution is retained at longer pulse durations reflecting that it is always harder for the ions to land on the bottomarea of the sidewall. The maximum dose of the sidewall is greater than that of the bottomsurface, but it is outnumbered at longer times of 10 and 20 ms. At a longer implantation time, the momentum of the ions is so high that it is hard for the electric field to pull themto the sidewall. It is observed that the incident doses of the gear geometry are higher than the trench co-ordinates. The difference becomes more obvious at larger pulse durations. The implantation ions originate from the uncovering process of the ion sheath because when expanding, the ion sheath uncovers more ions. The ion flux will We have simulated plasma immersion ion implantation into a gear (windmill) geometry by the particle-incell method. The simulation is carried out in cylindrical co-ordinates in which the gear (windmill) becomes a periodic regular structure. The angular and radial distance of each cell is the same and an equal number of particles is placed inside the cell. The ion density represented by each particle is different according to its radial distance. Plasma immersion ion implantation into a rectangular trench is simulated to compare with that into a cylindrical gear. It is found that the evolution and distribution of the potential and ion density contour lines are the same between these two geometries. The non-uniformincident doses along the top and bottom surfaces will reach background values at larger implantation time since the ions will gain larger momentum. The incident doses of the sidewall always show a ladder shape with a maximum at the top open area because it is harder for the ions to land on the bottomarea of the sidewall. In cylindrical co-ordinates (r, u), the circumference will increase as r increases. Therefore, the ion sheath will uncover more ions at longer implantation time (larger r). The extra ions will compress and focus into the smaller target surface area. The incident doses of a gear (windmill) will be larger than that of the trench. The incident doses will thus be underestimated when treating a gear s tooth by a rectangular trench. Acknowledgments The work was supported by City University of Hong Kong SRG References w1x J.R. Conrad, S. Baumann, R. Fleming, G.P. Meeker, J. Appl. Phys. 65 (1989) w2x A. Chen, J.T. Scheuer, C. Ritter, R.B. Alexander, J.R. Conrad, J. Appl. Phys. 70 (1991) w3x P.K. Chu, S. Qin, C. Chan, N.W. Cheung, L.A. Larson, Mater. Sci. Eng. Rep. 17 (6 7) (1996) 207. w4x S.Y. Wang, P.K. Chu, B.Y. Tang, X.C. Zeng, Y.B. Chen, X.F. Wang, Surf. Coat. Technol. 93 (1997) 309.

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