Thickness Uniformity and Mass Efficiency in Sputtering

Thickness uniformity of sputtered films is also of great practical importance. Model calculations exist which determine thickness distributions in the substrate plane below the sputter cathodes not only for static but also for moving substrates 

P = angle between the normal of surface at the observed point and the tie line to the source Q 

Above a rectangular sputter cathode, the total rate at an observed point in the substrate plane is obtained by integration over all elementary sources, that is the erosion zone of the cathode  

In the case of a long planar magnetron, there exist two parallel erosion zones of the broadness b and the length 1. At first the rate in the surface plane for one emission zone is calculated according to (92) and finally by superposition completed with the 

As the next step, the inner integral of (93) is calculated and this intermediate result has now to be integrated according to X. Edge effects on the cathode are not considered. After Integration the final result is  

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