As a first step analytical expressions are used for a direct approximation of the incident particle distributions. However, the accuracy of such an approach is limited. More detailed information about the incidence conditions for the particles can be obtained when the complete reactor system is included into the considerations. Analytical derivations as well as Monte Carlo simulations can be used to correlate the behavior of the impinging particles with the emission characteristics of the target and with the transport and collision processes in the reactor chamber. The thus obtained distributions represent the real conditions prevailing at the substrate surface much closer.
Fig. 6.2 outlines a sputter reactor and polar plots of the local emission characteristics at the target as well as the incoming distributions at the wafer surface. By means of a sputtering deposition process as a prototypical example the relation between emitted and incoming distributions will be discussed.
Starting point for the considerations is the experimentally determined target erosion profile. Semiconductor manufacturing routinely uses rotating magnet sputter systems, leading to circular racetrack grooves in the sputter target which means, that physical sputtering is not conformal across the sputter target but shows radially symmetric emission maxima with certain distance to the center of the target. The depth of the groove corresponds to the emission intensity, indicated in Fig. 6.2 by the variable size of the circles denoting the emitted distribution.
Moreover the angular distribution of the emitted particles is of significant
importance. For most of the commonly used semiconductor materials the emission
profile can be approximated with cosine 
functions of the emission angle measured with respect to the surface
normal. Still, the crystallographic structure of some metals like aluminium or
titanium leads to sputtering conditions, where the emission maximum is not
normal to the surface , which can be described with sub-cosine
functions  of the form
Once the emission characteristic is known, investigation about the path of the particles resulting from collisions on their way from the sputter target to the substrate is the next step. Due to the inherent stochastic nature of the collision processes they can be simulated best with Monte Carlo methods. Monte Carlo simulations trace the pathways of the particles including various models for the change in direction when particles collide. The models for the particle-particle interactions range from simple two-body-interactions to Lennard-Jones, Born-Mayer, or Abrahamson potentials . Additionally, Monte Carlo methods model the acceleration of charged particles in electric fields as used for target bias in order to increase the directionality of etching and deposition processes . The final results of comprehensive Monte Carlo simulations of particle transport in sputter reactors are angular and energy dependent distributions of the particles arriving at the wafer surface .
In  a fitting model was introduced as faster alternative to the
above mentioned MC methods. It determines the parameter of a
distribution function by comparing simulated and
experimentally obtained film profiles. We have decided for a similar
approximation model, but instead of fitting the final profiles the initially
implemented cosine and sub-cosine functions have been extended with expressions
which are able to fit the angular distributions resulting from Monte Carlo
simulations. As an example the exponential function
Three interesting aspects of the resulting flux distributions have to be mentioned. Firstly, there will be a certain amount of particles arriving at angles larger than which is the limiting angle for particle incidence if only direct transport from the target to the wafer is permitted. Since pressures in the region of 1mTorr do not allow the assumption of collisionless transport, it is obvious, that scattered particles account for this lateral fraction of particles. With increasing pressure the number of collisions per particle increases, the distribution broadens and the angle of maximum particle incidence increases. This is in excellent agreement with the MC results from . The MC results also reveal that the particles typically undergo 1 to 3 collisions until the arrive at the wafer.
Secondly, three-dimensional simulations obviously require three-dimensional
functions for the particle distributions. According to the circular racetrack
observed for the target erosion profiles it is reasonable to assume radially
symmetrical distribution functions which are independent on the polar angle
. The distribution function from (6.3) thus reads as
Finally, the MC results from Fig. 6.3 indicate that there is no particle incidence perpendicular to the wafer. This is understandable with the knowledge about the geometric configuration used for the Monte Carlo simulations. The center of the wafer is aligned with the center of the sputter target, from where almost no particles are emitted downwards to the wafer. It is therefore obvious, that the situation changes for peripheral positions on the wafer, located, e.g., directly below the emission maximum. At these positions a significant amount of particles with normal incidence will be observable. It is clear that the variations in the particle distributions as indicated by the polar plots of the emitted and impinging distributions in Fig. 6.2 strongly influence the local profile evolution. A method for the derivations of the non-uniformities in the particle distributions from the position on the wafer will be demonstrated in the next section.
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