p-n Junction Doping and Depletion Width

Understanding the relationship between doping concentrations ($N_D$, $N_A$) and depletion widths ($x_n$, $x_p$) is crucial for semiconductor device physics.

Doping Relation from Charge Balance

Charge neutrality requires the total charge ionized on both sides of the depletion region to be equal:

$q N_D x_n = q N_A x_p$

This simplifies to the charge balance equation:

$N_D x_n = N_A x_p$

Depletion Width Dependence

The depletion width on the n-side ($x_n$) is approximately given by:

$x_n \approx \sqrt{\frac{2 \epsilon_s V_{bi}}{q} \frac{N_A}{N_D(N_D+N_A)}}$

This formula shows $x_n$ depends inversely on factors related to $N_D$ and $N_A$.

$N_D \gg N_A$ Approximation Analysis

When the donor concentration vastly exceeds the acceptor concentration ($N_D \gg N_A$), we approximate ($N_D + N_A \approx N_D$). The expression for $x_n$ simplifies significantly:

$x_n \approx \sqrt{\frac{2 \epsilon_s V_{bi}}{q} \frac{N_A}{N_D^2}} = \frac{1}{N_D} \sqrt{\frac{2 \epsilon_s V_{bi} N_A}{q}}$

This demonstrates the inverse proportionality between $x_n$ and $N_D$ under this common condition:

$x_n \propto \frac{1}{N_D}$

Proportionality $N_D$ vs $x_n$

Rearranging the inverse proportionality $x_n \propto \frac{1}{N_D}$ gives the relationship:

$N_D \propto \frac{1}{x_n}$

This relationship correctly describes the behavior under the approximation $N_D \gg N_A$.