On calculating scanning capacitance microscopy data for a dopant profile in semiconductors

One scanning probe method that holds great promise for two-dimensional dopant profiling in semiconductors is scanning capacitance microscopy (SCM). The usual way to calculate the SCM signal (the derivative capacitance) for a dopant profile is to use the finite-element method to solve Poisson's equation. The domain region includes the doped semiconductor substrate region, the oxide layer region, and the air region bordering the metallic probe-tip of the SCM. The bias between the probe and sample is determined by using Dirichlet boundary conditions along the boundary of the probe-tip and the grounded plane deep within the doped semiconductor substrate. Managing this mesh-object (domain geometry and boundary conditions) requires sophisticated software. To help simplify these calculations, it is proposed that the SCM signal be found by using the areal capacitance distribution of the oxide and the air near the probe-tip 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. For this configuration, it is found that linear finite-elements can give inaccurate results, and that cubic finite-elements can give accurate results. To help speed the calculations, it is proposed that a reasonably coarse mesh may be used, as well. The method is applied to a. model one-dimensional 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.