pn junction: potential, widths, field, Fermi levels

Data used: $n_i = 1.45\times10^{10}\ \text{cm}^{-3}$, $kT/e = 0.02585$ V at 300 K, $\varepsilon = \varepsilon_0\varepsilon_r = 8.85\times10^{-14}\times11.9 = 1.053\times10^{-12}\ \text{F cm}^{-1}$, $e = 1.6\times10^{-19}$ C. $N_a = 10^{16}\ \text{cm}^{-3}$ (p side), $N_d = 10^{17}\ \text{cm}^{-3}$ (n side).

i) Built-in potential

$$\begin{aligned} V_0 &= \frac{kT}{e}\ln\frac{N_aN_d}{n_i^2} = 0.02585\ln\frac{10^{16}\times10^{17}}{(1.45\times10^{10})^2}\\ &= 0.02585\ln(4.756\times10^{12}) = 0.02585\times29.19\\ &= 0.755\ \text{V} \end{aligned}$$

ii) Width of depletion layer

$$\begin{aligned} W_0 &= \sqrt{\frac{2\varepsilon(N_a+N_d)V_0}{eN_aN_d}}\\ &= \sqrt{\frac{2\times1.053\times10^{-12}\times1.1\times10^{17}\times0.755}{1.6\times10^{-19}\times10^{33}}}\\ &= 3.31\times10^{-5}\ \text{cm} = 0.331\ \mu\text{m} \end{aligned}$$

iii) Maximum electric field (at the junction)

$$\begin{aligned} \mathcal{E}_0 &= \frac{eN_dW_n}{\varepsilon} = \frac{2V_0}{W_0}\\ &= \frac{2\times0.755}{3.31\times10^{-5}} = 4.57\times10^{4}\ \text{V cm}^{-1} \end{aligned}$$

iv) Width on n and p sides

From $N_aW_p = N_dW_n$:

$$\begin{aligned} W_n &= W_0\frac{N_a}{N_a+N_d} = 3.31\times10^{-5}\times\frac{1}{11} = 3.01\times10^{-6}\ \text{cm}\\ W_p &= W_0\frac{N_d}{N_a+N_d} = 3.31\times10^{-5}\times\frac{10}{11} = 3.01\times10^{-5}\ \text{cm} \end{aligned}$$

The depletion region lies mostly (91%) in the lightly doped p side.

v) Fermi level on n and p sides

$$\begin{aligned} E_{Fn}-E_{Fi} &= kT\ln\frac{N_d}{n_i} = 0.02585\ln\frac{10^{17}}{1.45\times10^{10}} = 0.407\ \text{eV}\\ E_{Fi}-E_{Fp} &= kT\ln\frac{N_a}{n_i} = 0.02585\ln\frac{10^{16}}{1.45\times10^{10}} = 0.348\ \text{eV} \end{aligned}$$

Check: $0.407 + 0.348 = 0.755$ eV $= eV_0$. ✔

 p side                     n side
 Ec ______
          \________________ Ec
 EF -------------------------- EF
 EFi_____ (0.348 above EF)
          \_______ EFi (0.407 below EF)
 Ev ______
          \________________ Ev