Sheet Resistivity Measurements on Rectangular Surfaces - General Solution for Four Point Probe Conversion Factors

Sheet resistivity measurements, made with probes on a bounded surface, are converted to resistance per square by "conversion factors". These factors are a function of the geometry and relative dimensions of the parts. For rectangular slices, tables of such factors 1 , 2 are available for the special case of four equally spaced points symmetrically placed on a center line. These tables were computed by use of convergent infinite series, derived for insulated edge and doublesided sheets. A general closed form solution has been developed for rectangles with arbitrary point locations. The method is applicable to any other 2277 2 2 8 0 TIIE H E L L SYSTEM TECHNICAL JOURNAL, DECEMBER 1967 surface shape which, through a conformal transformation, can be converted into a semi-infinite plane. For a rectangle, the transformation is the complex Jacobian sine-amplitude function. New tables have been computed for four equally-spaced probe points in a line, but with the probe center at points distributed in a uniform pattern over the rectangular surface. With these tables and interpolation, determination of the sheet resistivity now can be made anywhere on the rectangular surface, for either insulated-edge or double-sided conduction. Of course, a new factor can be computed directly for any point not tabulated, rather than calculated by interpolation. (The new tables begin on page 2292.) The purpose of using the transformation is to change the boundary of the actual slice to the X-axis of a semi-infinite plane, and the points to equivalent locations in this semi-infinite sheet.