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### 6.1.3.1 Adaption of Analytical Distribution Functions

Fig. 6.4 shows a typical arrangement of a magnetron sputter reactor with a 320mm target and a 200mm silicon wafer. The most important specifications for the reactor geometry are wafer diameter, target diameter, and target to wafer distance. In combination, these three parameters determine how the emission characteristic from the target (cf. Section 6.1.2) evolves to local flux distributions for different positions on the wafer.

Geometrically, the flux distribution at an off-center position can be formulated by shifting the origin of the analytical function used to describe the radially symmetric distribution of the incident particles for the center wafer position. The position on the wafer is determined by the polar angle , pointing from the position towards the center of the wafer and the azimuthal angle , given as angle between the positive, vertical z-direction and the direction to the center of the sputter target (cf. Fig. 6.4).

By shifting its origin, the distribution function is transformed to the local coordinates and . gives the polar angle in the plane normal to the direction pointing from the structure to the target center and the azimuthal angle with respect to this direction. The distribution function at the off-center position is now radially symmetric with respect to and can be written as

 (6.5)

Fig. 6.5 shows the three-dimensional representation of the exponential distribution function from (6.4). The left hand side shows the distribution for the center wafer position, which is radially symmetrical with respect to the z-axis. In the figure on the right hand side the tilt angle for the distribution is 15. For a target to wafer distance of 20cm, the 15 tilt angle represents a position shifted 55mm off the wafer center.

The shift in the flux distribution represents the change in the configuration of the position on the wafer with respect to the racetrack groove of the sputter target. The result is that the flux predominantly attacks those sidewalls of the feature, which are directed towards the center of the sputter target and thus exposed to the predominant direction of particle incidence. This effect is of significant importance, when the geometry of the considered structures is not radially symmetric. In this case the polar orientation plays an important role for the evolving profiles.

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