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\begin{enumerate}
\item Growth of a TiSi intermetallic layer

  \begin{center}
    $\ $\pdfximage{tisi.png}\pdfrefximage\pdflastximage$\ $
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  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 AlTi 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:
  \begin{itemize}
  \item Diffusion of titanium through the TiSi layer with diffusivity $D$.
  \item Reaction of silicon at Si-TiSi interface with reaction rate coefficient
    $k$.
  \end{itemize}
  You may assume the following:
  \begin{itemize}
  \item The TiSi-Ti interface is at equilibrium.
  \item The concentration profile across the intermetallic layer is linear
    ({\i.e.} pseudo-steady-state diffusion).
  \item 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.)
  \item The (simplified) phase diagram looks something like:
  \end{itemize}
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    $\ $\pdfximage{tisiphase.png}\pdfrefximage\pdflastximage$\ $
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  \begin{enumerate}
  \item Write the mass transfer Biot number which compares the limiting
    reaction and diffusion mechanisms.  Which mechanism limits the growth rate
    if this number is large?

  \item Sketch the concentration of titanium $C_{\rm Ti}$ 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''?

  \item Sketch the relationship between TiSi layer thickness $Y$ and time $t$,
    giving the exponents of both short- and long-time behavior ({\em i.e.} the
    $n$ in $Y\propto t^n$).
  \end{enumerate}
\end{enumerate}
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