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Consider avalanche breakdown in a silicon $p^{+}n$ junction. The $n$-region is uniformly doped with a donor density $N_{D}.$ Assume that breakdown occurs when the magnitude of the electric field at any point in the device becomes equal to the critical field $E_{crit}.$ Assume $E_{crit}$ to be independent of $N_{D}$. If the built-in voltage of the $p^{+}n$ junction is smaller than the breakdown voltage, $V_{BR}$, the relationship between $V_{BR}$ and $N_{D}$ is given by

  1. $V_{BR}\times \sqrt{N_{D}} = \text{constant}$
  2. $N_{D}\times \sqrt{V_{BR}}= \text{constant}$
  3. $N_{D}\times {V_{BR}}= \text{constant}$
  4. $N_{D}/{V_{BR}}= \text{constant}$
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