As a microfabrication engineer, you etch a deep vertical groove of $1m$ in $Si.$

The $Si$ surface is covered by $S{i}_3{N}_4$ which serves as an oxidation mask, see left

figure. The structure is then oxidized in steam at $1100C,$ see right figure.

a$.$ What is the width of the $Si{O}_2$ region $($denoted $x)$ after the groove is

completely filled with oxide? To solve this problem, you will need to

consider how much $Si$ is consumed per mole of $Si{O}_2,$ as well as the

densities of $Si$ and $Si{O}_2.$ Express your answer for $x$ in $m.$

b$.$ Using Deal$-$Grove analysis, calculate the amount of time the above

oxidation process will take. Assume that the coefficients $B$ and $\frac{B}{A}$ have

Arrhenius$-$type temperature dependencies, i$.$e$.,$ Cexp$(-\frac{E}{k}T),$ where

experimentally$-$determined values of $C$ and $E$ are presented below.

Table $8\mathrm{.}1$ Deal$-$Grove coefficients, B and B$/$A$.($From $[2]).$