Compact analytical expression of a multilayer system electric resistance measured at the upper surface

A compact expression for the resistance of a rectangular multilayer system, measured on the top surface by linear four points is given. This expression has no approximation when contacts approach the sample borders. The determinants method is avoided to calculate the potential distribution on the top surface. On the other hand, this solution, which is a double series, is arranged into two parts: the first called the “resistivities” part, decreases to zero exponentially when the layer thickness increases. The second called the “geometric” part, depends only on the sample width and length, contacts position and the top surface layer resistivity. The “geometric” part is a slowly converging series that the convergence depends strongly on contacts dimensions. It is transformed into a quickly converging series with terms decreasing exponentially to zero. The contacts width is taken into account as an additive correction factor, vanishing to zero for very small dimensions.

► A compact expression for the resistance of a multilayer system is found.
► This expression has no approximations when contacts approach the boarders.
► The determinants method is avoided to find the potential distribution on the top surface.
► Convergence is rapid and does not depend on contacts dimensions.