Regression procedure for determining the dopant profile in semiconductors from scanning capacitance microscopy data

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Abstract
A regression procedure has been developed to correlate scanning capacitance microscope (SCM) data with dopant concentration in three dimensions. The inverse problem (calculation of the dopant profile from SCM data) is formulated in two dimensions as a regularized nonlinear least-squares optimization problem. For each iteration of the regression procedure, Poisson’s equation is numerically solved within the quasistatic approximation. For a given type model ion-implanted dopant profile, two cases are considered; the background doping is either the same or the opposite type as that ion-implanted. Due to the long-range nature of the interactions in the sample, the regression is done using two spatial meshes: a coarse mesh and a dense mesh. The coarse mesh stepsize is of the order of the probe-tip size. The dense mesh stepsize is a fraction of the coarse mesh stepsize. The regression starts and proceeds with the coarse mesh until the spatial wavelength of the error or noise in the estimated dopant density profile is of the order of the coarse mesh stepsize. The regression then proceeds in like manner with the dense mesh. Regularization and filtering are found to be important to the convergence of the regression procedure.

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