Si with 10¹⁶ As: carriers, conductivity, Fermi shift

Arsenic is a group V donor, so the ingot is n-type.

Data used (Si, 300 K): $n_i = 1.45\times10^{10}\ \text{cm}^{-3}$, $kT = 8.617\times10^{-5}\times300 = 0.02585\ \text{eV}$, $\mu_e = 1350$, $\mu_h = 450\ \text{cm}^2\text{V}^{-1}\text{s}^{-1}$, $e = 1.6\times10^{-19}\ \text{C}$.

Carrier concentrations

All donors are ionised at 27°C and $N_d = 10^{16} \gg n_i$:

$$\begin{aligned} n &\approx N_d = 10^{16}\ \text{cm}^{-3}\\ p &= \frac{n_i^2}{n} = \frac{(1.45\times10^{10})^2}{10^{16}} = 2.1\times10^{4}\ \text{cm}^{-3} \end{aligned}$$

Conductivity

$$\begin{aligned} \sigma &= e(n\mu_e + p\mu_h)\\ &= 1.6\times10^{-19}(10^{16}\times1350 + 2.1\times10^{4}\times450)\\ &\approx 1.6\times10^{-19}\times1.35\times10^{19}\\ &= 2.16\ \text{S cm}^{-1}\ (= 216\ \text{S m}^{-1}) \end{aligned}$$

Resistivity $\rho = 1/\sigma = 0.463\ \Omega$ cm. (If the doped-Si mobility from Kasap's graph, $\mu_e \approx 1200$, is used, $\sigma \approx 1.92$ S cm⁻¹.)

Fermi level shift

$$\begin{aligned} E_{Fn}-E_{Fi} &= kT\ln\frac{n}{n_i} = 0.02585\ln\frac{10^{16}}{1.45\times10^{10}}\\ &= 0.02585\times13.44 = 0.348\ \text{eV} \end{aligned}$$

Answer: $n = 10^{16}\ \text{cm}^{-3}$, $p = 2.1\times10^{4}\ \text{cm}^{-3}$, $\sigma \approx 2.16$ S cm⁻¹, and the Fermi level lies 0.348 eV above the intrinsic Fermi level.

 Ec ------------------------
 EFn ....................... (0.348 eV above EFi)
 EFi - - - - - - - - - - - -
 Ev ------------------------