Analytical model for the current density in the electrochemical synthesis of porous silicon structures with a lateral gradient

Layered optical devices with a lateral gradient can be fabricated through electrochemical synthesis of porous silicon (PS) using a position dependent etching current density j(r∥). Predicting the local value of j(r∥) and the corresponding porosity p(r∥) and etching rate v(r∥) is desirable for their systematic design. We develop a simple analytical model for the calculation of j(r∥) within a prism shaped cell. Graded single layer PS samples were synthesized and their local calibration curves p vs j and v vs j were obtained from our model and their reflectance spectra. The agreement found between the calibration curves from different samples shows that from one sample we could obtain full calibration curves which may be used to predict, design, and fabricate more complex non-homogeneous multilayered devices with lateral gradients for manifold applications. [Display omitted] •A simple expression was found for computing the current density during the etching process of layered porous silicon systems with lateral gradients.•The expression is an exponentially fast converging series, each term of which is given by a simple analytical formula.•Full calibration curves for porosity and etching rate can be obtained from measurements on different positions along the surface of a single sample.•Useful for design and fabrication of multilayered PS GRIN devices with a lateral gradient of porosity and thicknesses..