Extraction of semiconductor dopant profiles from spreading resistance data: An inverse problem

Abstract

The traditional method of solving, on a layer-by-layer basis, the inverse problem of extracting resistivity vaues from spreading resistance measurements is found to produce wildly oscillatory, physically unacceptable resistivity profiles in the case of p-type silicon structures, where a resistivity-dependent probe contact radius is used in conjunction with the probe calibration data. These oscillations are manifestations of the fact that the inverse problem has non-unique solutions; they occur because the problem is inherently ill-posed. The well-known Tikhonov regularisation technique, which converts the present set of highly non-linear integral equations to an equivalent variational problem, is applied to stabilise the solution. Tests are performed on a variety of simulated profiles, and they reveal the existence of an optimum value for the regularisation parameter that is to be used with a second difference expression for the stabiliser of the cost function. When applied to measured spreading resistance data, the technique is found to produce results of reconstruction that are stable and physically reasonable.