root/trunk/matml/transport/problems/dopantdrive/dopantdrive.tex

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\begin{enumerate}
\item Drive-in diffusion of semiconductor dopant

  Ion implantation is used to create a phosphorous-rich (n-type) surface layer
  in one side of a silicon wafer 1 mm thick.  For the purposes of this problem,
  we'll assume the resulting phosphorous-rich layer has uniform phosphorous
  concentration 10$^{21}\rm\frac{atoms}{cm^3}$ from the surface to a depth of
  0.1$\mu$m.  The wafer begins with boron doping (p-type) at a uniform
  concentration 10$^{19}\rm\frac{atoms}{cm^3}$.

  At time $t=0$, this wafer is heated to a temperature 1200$^\circ$C, at which
  the diffusivity is $\rm2.49\times10^{-12}\frac{cm^2}{s}$ (which is orders of
  magnitude larger than it was before).  The phosphorous diffuses into the
  wafer, making a thicker n-type layer with lower concentration.  This step is
  called ``drive-in'' diffusion.

  \begin{enumerate}
  \item Sketch the concentration as a function of depth into the wafer,
    showing:
    \begin{enumerate}
    \item the uniform boron concentration $C_B$
    \item the inital phosphorous layer at concentration $C_{P0}$
    \item the time evolution of phosphorous concentration, at ``long'' times
      (the initial distribution isn't actually a uniform layer, so the erf-like
      short-time solution isn't very helpful).
    \end{enumerate}

  \item For how long can the 1 mm thick wafer be considered ``semi-infinite''?

  \item The silicon is n-type where the phosphorous concentration is greater
    than the boron concentration.  How thick is that n-type layer after 30
    mitutes (1800 seconds)?  After 90 minutes (5400 seconds)?
  \end{enumerate}
\end{enumerate}
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