Calculate the maximum space charge width for a given semiconductor doping concentration.
Consider silicon at T=300 \mathrm{~K} doped to N_{a}=10^{16} \mathrm{~cm}^{-3}. The intrinsic carrier concentration is n_{i}=1.5 \times 10^{10} \mathrm{~cm}^{-3}.
From Equation (10.4), we have
\phi_{f p}=V_{t} \ln \left(\frac{N_{a}}{n_{i}}\right) (10.4)
\phi_{f p}=V_{t} \ln \left(\frac{N_{a}}{n_{i}}\right)=(0.0259) \ln \left(\frac{10^{16}}{1.5 \times 10^{10}}\right)=0.3473 \mathrm{~V}
Then the maximum space charge width is
x_{d T}=\left[\frac{4 \epsilon_{s} \phi_{f p}}{e N_{a}}\right]^{1 / 2}=\left[\frac{4(11.7)\left(8.85 \times 10^{-14}\right)(0.3473)}{\left(1.6 \times 10^{-19}\right)\left(10^{16}\right)}\right]^{1 / 2}
or
x_{d T} \cong 0.30 \times 10^{-4} \mathrm{~cm}=0.30 \mu \mathrm{m}
Comment
The maximum induced space charge width is on the same order of magnitude as pn junction space charge widths.