Titanium vapor is deposited on silicon at a high enough temperature to produce a high$-$quality smooth

titanium film with few defects. At that high temperature, the titanium and silicon react, producing an

intermetallic titanium$-$silicon compound at the interface, which acts as a diffusion barrier preventing

aluminum interconnect wires deposited later from diffusing into the silicon and disappearing. $($A low$-$

diffusivity AITi layer also forms at the aluminum$-$titanium interface, but we'll focus on TiSi here.$)$

Suppose that layer is TiSi, that is$,50$mol$\%$ titanium, $50$mol$\%$ silicon. It is a contiguous layer, with

uniform thickness, and its growth is limited by one of two mechanisms:

Diffusion of titanium through the TiSi layer with diffusivity $D.$

Reaction of silicon at $Si-$TiSi interface with reaction rate coefficient $k.$

You may assume the following:

The TiSi$-$Ti interface is at equilibrium.

The concentration profile across the intermetallic layer is linear $(1.$e$.$ pseudo$-$steady$-$state diffu$-$

sion$).$

Ignore changes in the thickness of the Ti film over time. $($That is$,$ for simplicity, assume the whole

Ti film is deposited, then the intermetallic layer starts to grow.$)$

The $($simplified$)$ phase diagram looks something like:

$($a$)$ Write the mass transfer Biot number which compares the limiting reaction and diffusion mecha$-$

nisms. Which mechanism limits the growth rate if this number is large?

$($b$)$ Sketch the concentration of titanium $C_{Ti}$ in mol$\%$ as a function of distance from the top surface

$z$ for small and large values of the Biot number. Which corresponds to a "short time", and which

to a "long time"?

$($c$)$ Sketch the relationship between TiSi layer thickness $Y$ and time $t,$ giving the exponents of both

short$-$ and long$-$time behavior $($i$.$e$.$ the $n$ in $Yprop{t}^n).$