On calculating scanning capacitance microscopy data for a dopant profile in semiconductors
Jay F. Marchiando, Joseph Kopanski
While the calculating the dopant profile from SCM data (the inverse problem) can be formulated as a regularized nonlinear least-squares optimization problem, wherein Poisson equations are solved within the quasi-static approximation in each iteration of the regression procedure, the time-consuming part is calculating the SCM signal from the dopant profile (the forward problem). To speed the calculations, it is proposed that the SCM signal (the derivative capacitance) be found by solving three Poisson equations using the finite-element method, and the oxide capacitance areal distribution of the oxide and the air near the probe-tip be used to specify a natural boundary condition along the probed surface of the doped semiconductor sample, so that only the doped semiconductor substrate region needs to be meshed. While using a natural boundary condition here is less stable than meshing the regions of oxide and air near the probe-tip and using Dirichlet boundary conditions along the probe-tip boundary, it is found to be workable. Here, linear finite elements are found to give inaccurate results, and third-order finite-elements are found to give acceptable results. To further speed the calculations, it is proposed that a reasonably coarse mesh be used. The method is applied to a model one-dimensional (1D) ion-implanted dopant profile in a two-dimensional geometry, and the results of calculation for the forward problem are compared with that of using Dirichlet boundary conditions along the probe-tip.