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 $\bm j(\bm r_\|)$. Predicting the local value of $\bm j(\bm r_\|)$ and the corresponding porosity $p(\bm r_\|)$ and etching rate $v(\bm r_\|)$ is desirable for their systematic design. We develop a simple analytical model for the calculation of $\bm j(\bm r_\|)$ within a prism shaped cell. Graded single layer PS samples were synthesized and their local calibration curves $p$ vs $\bm j$ and $v$ vs $\bm j$ 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.