402 zyxwvutsrqponmlk IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 8, NO. zyxw 4, NOVEMBER 1995 Trajectory Split Method for Monte Carlo Simulation of Ion Implantation Walter Bohmayr, Alexander Burenkov, Jurgen Lorenz, Heiner Ryssel, zyxw Senior Member, IEEE, and Siegfried Selberherr, Fellow, IEEE Abstract-A new method for the acceleration of two- and three-dimensional Monte Carlo simulation of ion implantation into crystalline targets is presented. The trajectory split method ensures a much better statistical representation in regions with a dopant concentration several orders of magnitudes smaller than the maximum. As a result, the time required to perform a simu- lation with comparable statistical accuracy is drastically reduced. The advantages of the new approach have been confirmed by a thorough statistical analysis. I. INTRODUCTION HE Monte Carlo method is rapidly gaining acceptance T as a means for the simulation of ion implantation due to its capability of simulating channeling and damage accumu- lation phenomena in arbitrary multi-dimensional structures. A well-known disadvantage of the Monte Carlo approach is its considerable demand for computer resources to obtain results with satisfying statistical accuracy. The traditional Monte Carlo approach for crystalline targets is based on the calculation of a large number of “distinct” ion trajectories, i.e., each trajectory is usually followed from the ion starting point at the surface of the target up to the stopping point of the ion. Since the majority of ion trajectories ends at the most probable penetration depth inside the structure, the statistical representation of this target region is good. Regions with a dopant concentration several orders of magnitudes smaller than the maximum (in the following we call these areas “peripheral”) are normally represented by a much smaller number of ions (typically IO4 times lower than at the maximum). This results in an insufficient number of events at low concentration areas and leads to statistical noise that cannot be tolerated. Inspired by the striking message in [I] about the results of the rare event approach implemented in the UT-MARLOWE code [2] for one-dimensional structures, we developed the trajectory split method for the Monte Carlo simulation of ion Manuscript received February 20, 1995. This work has &en supported by PROMPT (JESSI project BTBB) and ADEQUAT (JESSI project BTI 1) and has been funded by the EU zyxwvutsrq as ESPRIT projects 8150 and 8002, respectively. W. Bohmayr and S. Selberherr are with the Institute for Microelectronics, Technical University of Vienna, Austria. A. Burenkov and J. Lorenz are with the Fraunhofer-Institut fur Integrierte Schaltungen Bauelementetechnologie Schottkystrasse, Erlangen, Germany. H. Ryssel is with the Fraunhofer-Institut fur Integrierte Schaltungen, Erlan- gen. Germany, and the Lehrstuhl fur Elektronische Bauelemente, Erlangen, Germany. IEEE Log Number 9414527. implantation. A similar method was first used in the work of Phillips and Price [3] to simulate hot electron transport. The algorithm drastically reduces the computational effort and is applicable for two- and three-dimensional simulations. 11. THE TRAJECTORY SPLIT METHOD The fundamental ideas of our new simulation approach are to locally increase the number of calculated ion trajectories in areas with large statistical uncertainty and to utilize the information we can derive from the flight-path of the ion up to a certain depth inside the target. For each ion, the local dopant concentration zyxw Clot is checked at certain points of the flight-path (checkpoints). In order to limit the computational overhead of this strategy the checkpoints are chosen according to the following conditions: First, the ion energy has to be less than 30% of its initial value, and second, there has to be a number of ion-target collisions between two successive checkpoints. For the second parameter we use 2% of the maximum number of collisions per ion trajectory. In later versions we will replace the 30% by an energy depending threshold value to take heed of low and high energy implants. At each checkpoint we relate Clot to the current maximum global concentration C, current by calculating the ratio Cloc/Cmax The result is compared with given relative concentration levels (we define ten levels at 0.3, 0.09, 0.027, . - ., 0.3”). A trujectory split point is defined at this checkpoint only if the current local concentration falls in an interval below the previous one. We store the position of the ion, its energy as well as the vector of velocity and use this data for virtual branches of ion trajectories starting at this split point. The shape of the ion trajectory in a crystalline material is determined by two different factors: (a) The impact point of the ion into the crystal lattice, and (b) the current positions of the target atoms. The distribution of the entrance points into the silicon crystal is determined by covering layers and masks and is homogeneous for exposed crystalline surfaces. The distribution of the entrance points is determined by standard procedures zyxw [5]. The current positions of the target atoms are determined by thermal vibrations which may have a pronounced influence on the channeling and dechanneling probabilities. We have assumed a three-dimensional Gaussian distribution for the displacement of the atoms centered at the ideal sites of silicon lattice [SI. The standard deviation of the thermal vibration is zyxwv 0.009 nm according to [7]. The momenta 08946507/95$04.00 zyxwvuts 0 1995 IEEE