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Steady-state photocarrier collection in silicon imaging devices

Solid-state imagers lose resolution when photocarriers generated in one imaging site diffuse to a nearby site where they are collected. These processes are modeled by solving the steady-state diffusion equation for minority carriers. A source term represents the absorption of photons and the generation of photocarriers, and a linear term represents the loss of photocarriers by recombination. This is equivalent to studying the Helmholtz equation with an inhomogeneous term. The problem is simplified when the light source has symmetry. A line source or a cylindrically symmetric source leads to a two-dimensional problem. The approach of Seib, Crowell, and Labuda allows a solution by quadrature if the further assumption of a smooth top boundary is made. We calculate the integrated normal flux over each imaging site to see how many carriers diffuse from under the illuminated site to another site. We compare our predicted line- and point-spread functions to those measured on imagers and find reasonable agreement. This allows us to extract minority-carrier diffusion lengths. Further calculations show how the diffusion of carriers depends on the photon wavelength and the pixel size. We generalize Seib's approach and apply it to a solid-state imager covered with color filters. This allows us to see the extent of color mixing due to carrier diffusion. We also discuss a finite-difference solution of the diffusion equation that employs the method of conjugate gradients. This approach is useful for problems where the top boundary is not smooth.