2 where n i is the ion density, v i the ion velocity, the potential, e the elementary charge and M the mass of the ions. Finally, Poisson's equation c

Transcription

1 Numerical Modelling of the Plasma Source Ion Implantation Process in 2D Linda Stals Michael Paulus y Ulrich Rude Bernd Rauschenbach y 1 Introduction The application of ion beams for alternating the properties of materials is well known. It became widely adopted by the semiconductor industry. High uency ion implantation is also a highly developed tool for modifying the structure and properties of materials, ceramics and polymers. Current research interests have expanded from the friction and wear studies to include other areas such as corrosion, oxidation, fatigue, conductivity, superconductivity, optical applications and basic metallurgical studies. In comparison with alternative surface modication technologies, ion implantation has a number of advantages. But, the ion implantation requires more or less complicated manipulations of the target when homogeneous ion beam treatment of non-planar workpieces are desired. Recently, plasma immersion ion implantation (PII) has initially been developed for material modication by ion implantation for circumventing the line-of-sight restrictions of beam-line ion implantation. In PII the target is immersed in a weakly ionised plasma and biased with a high negative voltage. In the time scale of the inverse electron plasma frequency,! pe?1, electrons near the surface are repelled due to their negative charge, leaving behind an uniform ion sheath. In the time scale of the inverse ion plasma frequency,!?1 pi, the positive charged ions are accelerated through the sheath towards the target from all sides and are implanted. Consequently the ion density in the sheath drops causing the sheath-plasma edge to recede and uncover enough ions to shield the target potential. As a result the sheath expands. Predictions of the concentration of ions implanted in the target requires an understanding of the dynamics in the implantation process. The structure of the plasma sheath and its temporal evolution is a very important part of this understanding as it determines the trajectories and therefore the implantation current, angle and energy of the ions striking the surface. This process is modelled by a time-dependent uid simulation [18]. The ion motion is assumed to be collisionless, which is an appropriate assumption for the low pressures achieved in PII. Furthermore thermal motion is neglected because the applied voltage is much greater than the plasma electron temperature. The movement of the ions is hence given by the equation of continuity and + r (n iv i ) = 0 + (v ir) v i =? e r; (2) M Institut fur Mathematik, Universitat Augsburg, D Augsburg, Germany y Institut fur Physik, Universitat Augsburg, D Augsburg, Germany 1

2 2 where n i is the ion density, v i the ion velocity, the potential, e the elementary charge and M the mass of the ions. Finally, Poisson's equation connects the potential to the electron and ion densities n e and n i r 2 =? e 0 (n i? n e ) ; (3) where 0 is the vacuum permittivity. The electron density n e is assumed to obey the Boltzmann relation e n e = n 0 exp ; (4) k B T e where n 0 is the uniform initial density in the plasma, k B the Boltzmann constant and T e the electron temperature. The assumption is held to be valid because the characteristic time scale for the implantation (order 10?6 sec.) is long compared with! pe?1 (order 10?9 sec.). Therefore electron inertia eects can be neglected and the electrons obey the Boltzmann approximation. This improvement allows the simulation to follow only the ion dynamics, but at the cost of Poisson's equation becoming highly nonlinear in the potential. This non-linearity makes equation (3) dicult to solve numerically. When (4) is substituted into (3) a term of the form n 0 e= 0 arises, which is typically of the order of In this paper we propose two dierent algorithms for the solution of equation (3); Newton-Multigrid method and the FAS scheme. Straightforward implementations of both of these methods either diverged or converged very slowly, depending upon the grid size and the shape of the target. We shall describe the modications which were necessary to overcome these diculties and compare the performance between the two methods. As described above, the potential within the sheath grows rapidly and furthermore the shape of the sheath changes over time. To model such behaviour numerically adaptive renement must be used. We also present a method of renement which `captures' the sheath and lets us change the shape of the grid to follow the movement of the sheath. Our data structure is based upon the node edge data structure. It is a very exible data structure and can be used to rene the grid and solve the potential equation in parallel computations. We plan to presents results evaluating the parallel eciency in a later paper. This paper mainly focuses on the solution of the potential equation, discussions about the ion equation, equation (1), can be found in the thesis currently being written by Paulus [13]. We start the paper by describing the data structure and our method of renement. It then compares the Newton-Multigrid method with the FAS scheme and gives some example results from our numerical simulation. Within this report we assume that the reader is familiar with the multigrid method. A description of this method can be found in numerous references such as [2, 3, 5, 6, 8]. 2 Dimensionless Potential Equation During the numerical simulation it is more convenient to work with a dimensionless version of equation (3), see [15]. Using the width of the planar ion matrix sheath d im = p (?2 0 t )=(en 0 ) [4], the bias on the target t and the plasma density n 0 the new variables are dened by: R = R d im or r = d im r

3 3 u = t Note that u is positive since t is negative. Equation (3) thus becomes (r ) 2 ni u = 2? exp u e t : n 0 k B T e Dropping the astrix we get ni u = 2? exp(?u) ; (5) n 0 where =?(e t )=(k B T e ). Values of are of the order of Figure 1 gives a schematic view of a PII experiment. The bulk plasma region (non-shaded region) is the region which is modelled in the numerical simulation. We take the `outer boundary' to be the boundary between the bulk and quasineutral plasma. The boundary values are ( 0 on the outer boundary; u = g(t) on the surface of the target: After the voltage is applied to the target it takes a while for the sheath to fully develop. incorporate this feature into our model we set 8 >< 1 for t s = 0; g(t) = t=t s for t t s ; t s 6= 0; >: 1 for t > t s ; t s 6= 0: (6) To for some given start-up time t s. The values of u must vary continuously between 0 and g(t) throughout the domain, therefore, if t > t s we expect the right hand side term in equation (5) to take values between 2n (when u! 1) and 2n? 1 (when u! 0), but with a sharp transition region due to the large values of. 3 Finite Element Grids The discretisation method we use is the nite element method with linear basis functions on triangular grids. By using triangular meshes we can easily discretise irregularly shaped domains that are of particular interest for industrial applications in the plasma ion immersion technology. Our way of storing the nite element grid is based upon a node-edge data structure similar to the one described by Rude [14]. Within this data structure the node table stores the geometrical information while the edge table stores the topological information. We do not explicitly store the triangles but rather use a loop which iterates through the nodes and edges to nd the triangles. The general idea is easily shown by an example. Suppose the grid given in gure 2 is the mth level grid in the multigrid structure, then it can be stored in the following node and edge tables: Node N m = f1(0.0, 0.0), 2(3.0, -3.0), 3(5.0, 0.0), 4(3.0, 3.0), 5(7.0, 3.0), 6(5.0, 5.0)g Edge E m = f(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (3, 1), (3, 4), (3, 5), (4, 1), (4, 3), (4, 5), (4, 6), (5, 3), (5, 4), (5, 6), (6, 4), (6, 5)g

4 4 Figure 1: Example schematic view of PII simulation when t > t s. In the results presented in this paper we set c = 6. To nd the triangles within the data structure let B(E; N i ) = fn j : (N i ; N j ) 2 Eg and use the following algorithm; Algorithm 1 Find Triangle(N m ; E m ) f for N i 2 N m for N j 2 B(E m ; N i ) for N k 2 B(E m ; N i ) if (N j ; N k ) 2 E m g then fn i ; N j ; N k g is a triangle One other important table used in the node-edge data structure is the connection table which stores the algebraic information. For example, in Newton's method the connection table is used to store the Jacobian matrix. Two other matrices stored within the connection table are the interpolation, Im?1, m and restriction, Rm m?1, matrices. For example in gure 3 there is a restriction connection between nodes N a and N b and an interpolation connection between nodes N b and N a. We use linear interpolation and dene the restriction matrix to be the transpose of the interpolation matrix. This data structure is very exible (it can also be used to store rectangular and tetrahedral grids) and thus allows us to handle the complex data dependencies dened by the multigrid algorithm when working in parallel. A detailed description of this data structure s given in [16, 17]. In

5 5 6 v AA 4 v A 5 Av v??? T TT v v 2 Figure 2: Example grid stored in the node-edge data structure. In our simulations for the square target we take c = 6. particular, [16] shows how the dependencies within the data structure may be exploited to build a parallel program which can adaptively rene the grids, solve the system of equations and balance the load. All of the work is done on the processors, we only use the host machine to dene the initial coarse grid. 4 Calculations with the Exponential Term Physically it is clear that the solution of the boundary value problem (5) with boundary conditions (6) can only take values between 0 and g(t). Numerically, however, slight oscillations may occur in the transition region between u 0 and u > 0 leading to slightly negative values of u. In our experiments the main reason for such oscillations were numerical errors when assembling the load vector with numerical integration. Fortunately, a simple modication was enough to solve the problem. Instead of assembling the mass matrix by integration we simply used a point-based evaluation of u resulting in an eect similar to lumping of the mass matrix. Negative results also cause problems in the numerical calculations. Due to the large value of, the evaluation of exp(?u) can result in overow even if u is only slightly negative. Lumping the mass matrix removed the problems with the oscillations, but the Newton-Multigrid and FAS schemes may still produce negative values during their iteration process, irrespective of whether the nal result is non-negative, so traps had to be put into the code to stop the values of?u becoming too large. 5 Renement To help clarify the discussion within this section we shall rstly give an example result. Figure 4 shows the solution of equation (5) with = 50000, t = t s = 0 ( n i = n 0 ). The potential along the inner boundary, which represents the target material is u = 1 while the potential along the outer boundary is u = 0. The area where the potential rises rapidly is the sheath. To rene the grids we use the newest node bisection method. In this method the triangles are split along the edges which sit opposite the newest nodes. These edges are called the base edges. See gure 5 for an example of non-adaptive renement. In the case of adaptive renement our approach is a modication of the method of conforming triangles described by Mitchell [9, 10, 11]. Here we dene an interface base edge to be a base edge which sits between two dierent levels of renement, as shown in gure 6. The interface-base

6 6 N a N b N c Figure 3: Example set of inter-grid connections. The interpolation matrix, I m m?1, and the restriction matrix, Rm m?1, are stored in the connection table. Figure 4: Solution of the dimensionless potential equation (5) with a square target, = 50000, t = t s = 0 and using 6 levels of adaptive renement.

7 7 B h B??? B h a) @@? x h h x?@ x h h??? @@? h h x h x h h c) Figure 5: Example renement using newest node bisection. Figure a) shows the initial grid. We assume that the centre node is our initial `newest node'. The corresponding bases edges are marked by a B. Figure b) shows the result after one renement sweep. Figure c) shows the nal grid. edges are used to control the order of renement by ensuring that the neighbouring coarse triangles are split before the ne triangles. It is possible that the neighbouring coarse triangle also has an interface-base edge, in which case we would have to split its neighbouring coarse triangle rst (etc.). For example, in gure 6 c) if we wanted to split I 11 we would have to split I 8 rst, but before we can split I 8 we must split B 4. The advantage of this approach is that the angles within the triangles remain bound away from 0 and and it therefore avoids long thin triangles. Mitchell gives a proof of this @@ B 3 B a) B 1@@?? 8 B I 11? 8 I B c) Figure 6: Example of adaptive renement. In gure a) I 1 and I 2 are two interface-base edges while B 3 and B 4 are two base edges. Not all of the base and interface-base edges have been marked to help reduce the clutter. The base edge B 3 must be split before the interface-base edge I 1. When B 3 is split the interface-base edge I 1 is updated to a base edge B 1 as shown in b). The edge B 1 can now be split to give the nal grid shown in c). 5.1 Error Indicator Roughly speaking, the program picks which triangles to rene by looking at how well the coarse grid approximation solves the problem on the ne grid. This error indicator is well suited to the multigrid framework. Lets rstly dene the error indicator in the context of some linear system of equations A m u m = f m ; (7) where A m and f m are the stiness matrix and load vector dened on the grid level m (i.e. the grid

8 8 level we are rening). Each base and interface-base edge is assigned an error indicator which is equal to a weighted residual calculated at the midpoint. That is, if v m is the current approximation to the system given in equation (7) and if the node N d is the midpoint of the edge (N i ; N j ) then the error indicator, e m, assigned to that edge is, where q e m N d = r m+1 N d A m+1 N d ;N d ; (8) r m+1 = f m+1? A m+1 I m+1 m v m : (9) A m+1 and f m+1 are the stiness matrix and load vector that we would get if the edge was split at node N d to form a new set of triangles. Note that we do not construct the whole stiness matrix (or load vector), we only calculate the row of the stiness matrix which corresponds to node N d. Im m+1 interpolates the coarse grid approximation to the ne grid. In the examples given in this report we use linear interpolation. This error indicator is similar to the error indicator described by Mitchell ([9, 10, 11]) and to the one used by Rude ([14]). To handle the exponential term in equation (5) we modied equation (9) to give: r m+1 N d = f m+1 N d? (A m+1 Im m+1 v m ni ) Nd? 2w m+1 N d exp(?v m+1 N n d ) : 0 where the weight w m+1 N d is the sum of the area of the triangles of the form fn d ; N k1 ; N k2 g, k 1 < k 2. The value assigned to the midpoint is the average of the two endpoint, that is v m+1 N d = (vn m i +vn m j )=2. Here we take A m to be the stiness matrix corresponding to the nite element discretisation of the Laplacian, and f m is the load vector derived from the right hand side equation 2n i =n 0. Finally, the error indicator for our non-linear problem, e m, is q e m N d = r m+1 N d A m+1 N d ;N d : (10) Figures 7 and 8 show two example renements results for = 50000, t = t s = 0. The initial coarse grids are given in the appendix in gures 15 and 16. Around the target equation (5) is roughly equal to u = 2 and is therefore linear in that region. So our method of renement is working like we would expect it to for, say, an L-shaped target. Around the outer boundary, u 0, and equation (5) looks like u = 0. In this region the solution does not change much so there is no need for extra renement. At the bottom of the sheath the error indicator is picking up the transition between these two regions. Figures 9 and 10 compare the results for non-adaptive and adaptive renement applied to the square shaped target. Practical and theoretical results imply that the bottom of the sheath should be shaped like an oval. Figures 9 and 10 show that we do not see this eect until we have at least 4 levels of non-adaptive renement and 6 levels of adaptive renement. We argue that the results for 2, 4, 6 and 8 level of adaptive renement are comparable to those for 2, 3, 4 and 5 levels of non-adaptive renement, but require far fewer nodes. Five levels of non-adaptive renement produces nodes on the nest grid, but 8 levels of adaptive renement only gives 2024 nodes on the nest grid.

9 9 Figure 7: Resulting grid after six levels of adaptive rene when using a square target ( = 50000, t = t s = 0). Figure 8: Resulting grid after six levels of adaptive rene when using a C-shaped target ( = 50000, t = t s = 0).

10 Figure 9: Solution of the dimensionless potential equation (5) with a square target ( = 50000, t = t s = 0) using non-adaptive renement. The gures going from left to right and top to bottom show the results after 2, 3, 4 and 5 levels of non-adaptive renement. The nest grid contains 432, 1632, 6332 and nodes respectively. The projections drawn on the plane are the contour plots of u. These have been included as they more clearly show the shape of the sheath. 10

11 Figure 10: Solution of the dimensionless potential equation (5) with a square target ( = 50000, t = t s = 0) using adaptive renement. The gures going from left to right and top to bottom show the results after 2, 4, 6 and 8 levels of adaptive renement. The nest grid contains 120, 222, 556 and 2024 nodes respectively. The projections drawn on the plane are the contour plots of u. These have been included as they more clearly show the shape of the sheath. 11

12 12 6 Time Stepping To handle the time dependence shown in the ion movement equation (1) we used a `feed-back system'. Namely, the potential equation was solved at one time step in order to nd the new position of the ions, which was in turn used to form the new potential equation, etc. To solve the ion equation (1) a particle method is used as described in [7]. Particles of nite size, such as the ions, are moved by dv i dt = F M : (11) The force F at an ion's position is given by the potential : F = ee =?er; (12) where E is the electric eld. The charge density at the nodes is then obtained by weighting the nal distribution of the particles to the grid. This denes the new right-hand-side of equation (3) for the next time step. Figure 11 shows how the sheath grows when 0 < t t s = 0:2. Figure 12 shows an example calculation based upon the L-shaped target. In this case t > t s so the boundary condition around the target is g(t) = 1. The gures show how the sheath slowly spreads out and how that the bottom of the sheath is shaped like an oval. Since the sheath evolves over time, it is necessary to add new triangles at each time step. However, when working with the multigrid algorithm this is not as straightforward as it may rst appear. It is not sucient to just rene the ne grid, the coarse grids must also be taken into account, if they do not share some of the properties of the ne grid then the coarse grid correction step will be ineective. Therefore in our renement routine we move through all of the grid levels splitting those triangles which have a high error indicator. Note that if an edges is split on a coarse grid then it must also be split on the ne grid (if it is dened there) otherwise the grids will not remain nested. We do not explicitly remove triangles from the grids, but rather rely on the coarse grids as a means of implicitly removing triangles. Before the renement routine is called, we step down through the grid levels and copy the grid from level m-1 into level m (1 m). This gives a good starting point for the renement and is far cheaper then moving through the grid levels and actually deleting triangles. 7 Parallel Implementation The exibility of the node-edge data structure means that it is well suited to the parallel environment. We have used our code to solve linear elliptic partial dierential equations [16, 17] and much of the ground work carries over to the non-linear case. In this paper we do not give eciency results, they will be given in another paper which focuses on the parallel concepts in more detail, but to verify our claims that these methods work in parallel we show an example grid in gure Newton's Method Theoretical studies of Newton's method shows that it has super linear convergence when the initial guess is close to the solution. For small values of we found that it is sucient to interpolate the

13 Figure 11: Solution of the dimensionless potential equation (5) with a C-shaped target ( = 50000, 0 t 1, t s = 0:2) using adaptive renement. The gures going from left to right and top to bottom shows the results after 5, 10, 15 and 20 time steps. There were 100 time steps in total. Two level of whole grid followed by six levels of adaptive renement were used. 13

14 Figure 12: Solution of the dimensionless potential equation (5) with a L-shaped target ( = 50000, 0 t 1, t s = 0:2) using adaptive renement. The gures going from left to right and top to bottom show the results after 30, 50, 70 and 90 time steps. There were 100 time steps in total. Six levels of adaptive renement were used. The original coarse grid is shown in gure

15 15 Figure 13: Resulting grid after six levels of adaptive rene when using a star-shaped target ( = 50000, t = t s = 0), calculated on four processors. coarse grid solution up to the ne grid and use that as an initial guess. However, for the larger value of which we are interested in this approach did not always work and Newton's method sometimes diverged. We tried using a continuation type of method by slowly increasing the values of, but found that the step size had to be too small for this to be a viable solution. The approach which did work was to use a damped Newton's method. But this too had its diculties. If the damping factor is too large the method diverges, if it is too small it takes many iterations to converge. Therefore, we used an algorithm similar to the one given in section 8.7 of [12] (which cites [1] as the original reference) to automatically calculate the damping factor. Table 1 looks at the solution of the dimensionless equation 5 with = 50000, 0 < t 1 and g(t) = t. Notice that there are examples where the damping factor is of the order of 10?2. If we do not use such small values the method will diverge, it does not just slowly converge. However, these small values are usually only needed for the rst few iterations. Table 2 shows the time for an example calculation carried out on the digital workstation. At each time step, we iterated over Newton's method until the maximum residual was less then 10?6. It is interesting to note that the time does not appear to be directly related to the grid size, but rather depends on whether interpolating the coarse grid solution gave a good enough initial guess. If it did, then we did not need to use the damping factor, if did not then a small damping factor is required and the number of iterations consequently increases

16 16 t Damp Factor t Damp Factor , 1.00, , 1.00, 1.00, 1.00, 1.00, , 1.00, 1.00, 1.00, 1.00, , 1.00, 1.00, 1.00, 1.00, , 1.00, 1.00, 1.00, 1.00, , 1.00, 1.00, 1.00, 1.00, , 0.23, 1.00, 1.00, 1.00, , 1.00, 1.00, 1.00, 1.00, , 0.34, 0.39, 0.13, 1.00, , 1.00, 1.00, 1.00, 1.00, , 1.00, 1.00, 1.00, 1.00, , 1.00, 1.00, 1.00, 1.00, , 1.00, 1.00, 1.00, 1.00, , 1.00, 1.00, 1.00, 1.00, , 1.00, 1.00, 1.00, 1.00, , 1.00, 1.00, 1.00, 1.00, , 1.00, 1.00, 1.00, 1.00, , 0.01, 0.71, 1.00, 1.00, , 1.00, 1.00, 1.00, 1.00, , 0.06, 0.06, 0.18, 0.05, 0.78 Table 1: Table showing the damping factor for the rst six iterations of the Newton-Multigrid method ( = 50000, 0 < t 1 and g(t) = t). The method converged within three iterations for t = 0.0. step No. Nodes Total Newton V-cycle Table 2: Table showing time (sec.) required to solve equation 5 on the L-shaped target when using adaptive renement and Newton's method ( = 50000, 0 < t 1 t s = 0:2). The Total time is the total time required to solve the equation at each time step. During adaptive renement the equation is solved on the coarse grids to help calculate the error indicator, therefore the time given for Newton's method (and the cle) is the total time over all of the grid levels. To complete the calculations 100 time steps are required, we have only shown the rst 21 steps here. The calculations were carried out on the Digital Personal Work Station 500au using the Alpha 500 Mhz processor.

17 17 1 Square Target C-Shaped Target L-Shaped Target 0.8 Convergence Rate step Figure 14: Convergence rate for the FAS scheme when solving equation 5 ( = 50000, 0 < t 1 t s = 0:2). 9 Full Approximation Scheme The FAS method, or Full Approximation Scheme, is a generalisation of the standard multigrid method which is designed to handle non-linear equations. In the FAS method, as with any multigrid method, it is essential to use an appropriate smoother. If the smoother does not eectively remove the high frequency components of the error then the coarse grid correction does not work. We tried several dierent methods including a non-linear SOR method where the weight was automatically calculated to ensure that the iteration converged. However a point-newton method worked the best and it was the simplest to implement. Figure 14 shows the convergence rate for the FAS scheme. For the square and L shaped targets the scheme gives good convergence results. Clearly the method had some diculty with the C- shaped target, but Newton's method also had diculties with this target. For example, it required damping factors of the order of 10?4 and for some time steps it did not converge within 50 iterations. Table 3 shows the time for an example calculation carried out on the digital workstation. The FAS-scheme was repeatedly applied until the maximum residual was less then 10?6. When comparing the results with those given in table 2 we see that the FAS-scheme outperforms the Newton-Multigrid method. References [1] G. Berman. Minimization by Successive Approximation. SIAM J. Numer. Anal., 3:123{133, [2] A. Brandt. Mulit-level adaptive solutions to boundary-value problems. Math Comput., 31(138):333{390, April 1977.

18 18 t No. Nodes Total FAS point-newton Table 3: Table showing time (sec.) required to solve 5 on the L-shaped target when using adaptive renement and the FAS-scheme ( = 50000, 0 < t 1 t s = 0:2). The Total time is the total time required to solve the equation at each time step. During adaptive renement the equation is solved on the coarse grids to help calculate the error indicator, therefore the time given for the FAS scheme is the total time over all of the grid levels. To complete the calculations 100 time steps are required, we have only shown the rst 21 steps here. The calculations were carried out on the Digital Personal Work Station 500au using the Alpha 500 Mhz processor. [3] W. Briggs. A multigrid tutorial. SIAM, [4] J. R. Conrad. Sheath thickness and potential proles of ion-matrix sheaths for cylindrical and spherical electrodes. J. Appl. Phys., 62, 777 (1987). [5] W. Hackbusch. Multigrid Methods and Applications. Springer Verlag, Berlin, [6] W. Hackbusch and U. Trottenberg, editors. Lecture Notes In Mathematics, 960. Springer- Verlag, [7] R. W. Hockney and J. W. Eastwood. Computer Simulation Using Particles. Institute of Physics Publishing, Bristol and Philadelphia, [8] S. McCormick. Multigrid Methods. SIAM Frontiers In Applied Mathematics, [9] W. F. Mitchell. Unied multilevel adaptive nite element methods for elliptic problems Ph.D. thesis, Technical report UIUCDCS-R , Department of Computer Science, University of Illinois, Urbana, IL, 1988 [10] W. F. Mitchell. A comparison of adaptive renement techniques for elliptic problems. ACM Trans. Math. Software, 15(4):326{347, December [11] W. F. Mitchell. Optimal multilevel iterative methods for adaptive grids. SIAM J. Sci. Stat. Comput, 13(1):146{167, January [12] J. Ortega and W. Rheinboldt. Iterative Solution of Nonlinear Equations in Several Variables. Adademic Press, 1970.

19 19 [13] M. Paulus. Modellierung und Simulation der Prozesse bei der Plasma-Immersions- Implantation. Diplomarbeit, Institut f"ur Physik, Universit"at Augsburg, D Augsburg, Germany [14] U. Rude. Mathematical and computational techniques for multilevel adaptive methods. SIAM, [15] T. E. Sheridan and M. J. Alport. Ion-matrix sheath around a square bar. J. Vac. Sci. Technol. B 12, 897 (1994) [16] L. Stals. Parallel Multigrid on Unstructured Grids Using Adaptive Finite Element Methods. PhD thesis, Department of Mathematics, Australian National University, Canberra, 0200, Australia [17] L. Stals. Adaptive Multigrid in Parallel. In R. May and A. Easton, editors Computational Techniques and Applications:CTAC95, Melbourne, Australia. Pages World Scientic, [18] M. M. Widner, I. Alexe, W. D. Jones, K. E. Lonngren. Phys. Fluids 13, 2532 (1970). 10 Appendix Figure 15: Initial coarse grid for square target.

20 20 Figure 16: Initial coarse grid for C-shaped target. Figure 17: Initial coarse grid for L-shaped target.